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How many significant figures are in the number 0.00208?
how many significant figures are in the number 0.00208?:-It used to be that the question, how many significant numbers, was asked only by those in mathematics.
Now, though, many people have become interested in answering this question too, as well as the way they can use them to make money through various methods. How many digits are there in decimals? How about in percentage terms?
how many significant figures are in the number 0.00208?
how many significant figures are in the number 0.00208?
Sig Figs
3
0.00208
Decimals
5
0.00208
Scientific Notation
2.08 × 10-3
E-Notation
2.08e-3
Words
zero point zero zero two zero eight
In simple terms, a decimal is a number (in the format of [pi, percent, seconds, etc.]) whose fraction is one or more digits? Let’s say, for instance, that we are computing the value of a percentage figure. We could either do it with decimals, which would yield four significant digits: the fraction itself, plus the entire fraction after rounding to the nearest whole number, plus one more digit, the leading zero or the denominator. The number, then, would be five digits.
This, then, can be written as the numerical value of the fraction (after rounding to the nearest whole number) and the leading zero is the number that tells us the decimal portion of one to the next decimal.
In other words, decimals can be thought of as a finite number of repeating measurements. And since the decimals, when added together, always add up to exactly what we are working with, it follows that there must be an infinite number of these measurements. But what if we were told that there are two ways to compute decimals? One method is known as the exact number method, while another is called the rounding to zero method.
Let’s assume for a moment that we know, for certain, that there are no less than three significant digits between the decimal point and the closest whole number to it. (It doesn’t matter if the next number is a fraction; it doesn’t matter if it’s not a number.)
We can also assume that any digit can be divided by any other digit, so long as all the possible divisions are made into multiples of ten. We also know, in fact, that any such division involving multiples of ten will yield a rounded fraction, which will always equal zero.
So we see that there are no trailing zeros. If there were, then the rounding to zero would be a problem because if the number were not significant, the rounding to zero would give rise to negative zeros instead of positive ones.
The second method of computing decimals is known as the exact-digit or exact numerator notation. It involves writing down the denominator in binary form, then completing the equation by adding the corresponding significant digits. For instance, if x is the numerator, then the expression representing the division by x is the exact-digitdivider. (The only real difficulty with this form of computation is that it’s only a tool that’s available to computers; a calculator would already have this type of computation.)
The third method of computing decimals is known as decimals using radii. This form of computation uses the fraction “00” through “9”. In this case, the fraction should be written down in binary form. Any digits not in binary form should be written in lower case, and the leading zeros should be considered as negative infinity (if they’re followed by a number other than one, which will be clearly shown on the graph).
The decimals are once again rounded to one hundredths of a degree, and decimals of ten are once again treated as zero, with all leading zeros as signifying the absolute value of their residue. (The exactness of these calculations will vary depending on the accuracy of the calculator used and the precision of the input number.)
The final method of computing decimals is known as the exact-digit method, which yields results which are mathematically similar to the first and second methods described. Here, when computing decimals, the digits which must be written down must always be in the right order. For example, if a number is written out, such as “1.2 trillion”, the decimals are written in order from left to right, starting with the smallest amount (the first digit, which is itself a number, of course, is always significant).
Any digits that are written after this need to be considered as being left out, since they will be written before or after the next digit. Once all the relevant digits are in place, this is the final result that is produced. And because this is always significant, the decimals of any amount are always significant digits.
If you were asked how many significant figures there are, you could use any one of these methods. If you were asked how many decimals of ten are in this example, your answer might contain all of them.
However, you would not have any way of knowing how many were written out without consulting a calculator. Therefore, it is recommended that you consult a calculator whenever you need to know or compute the number of digits.
Find here Sig Fig Calculator (Rounding)<|endoftext|>
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# Maximum Minimum Value of Quadratic Expression: Quadratic Equations Video Lecture - CSAT Preparation - UPSC
## CSAT Preparation
197 videos|151 docs|200 tests
## FAQs on Maximum Minimum Value of Quadratic Expression: Quadratic Equations Video Lecture - CSAT Preparation - UPSC
1. What is the maximum value of a quadratic expression?
Ans. The maximum value of a quadratic expression occurs when the coefficient of the quadratic term (i.e., the term with the variable raised to the power of 2) is negative. In this case, the maximum value is equal to the y-coordinate of the vertex of the parabola represented by the quadratic expression.
2. How can I find the minimum value of a quadratic expression?
Ans. To find the minimum value of a quadratic expression, determine the vertex of the parabola represented by the expression. The x-coordinate of the vertex can be found by using the formula -b/2a, where a and b are the coefficients of the quadratic term and the linear term, respectively. Then substitute the x-coordinate into the quadratic expression to find the corresponding y-coordinate, which represents the minimum value.
3. Can a quadratic expression have both a maximum and a minimum value?
Ans. No, a quadratic expression can have either a maximum or a minimum value, but not both. Whether the quadratic expression has a maximum or a minimum depends on the coefficient of the quadratic term. If it is positive, the expression has a minimum value, and if it is negative, the expression has a maximum value.
4. Is there a direct formula to find the maximum or minimum value of a quadratic expression?
Ans. Yes, there is a direct formula to find the maximum or minimum value of a quadratic expression. It is given by -D/4a, where D is the discriminant of the quadratic expression and a is the coefficient of the quadratic term. However, this formula can only be used when the quadratic expression is in standard form (ax^2 + bx + c = 0).
5. How can I determine whether a quadratic expression has a maximum or minimum value without graphing it?
Ans. To determine whether a quadratic expression has a maximum or minimum value without graphing it, you can look at the coefficient of the quadratic term. If it is positive, the expression opens upward and has a minimum value. If it is negative, the expression opens downward and has a maximum value.
## CSAT Preparation
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## Permutations and Combinations Quiz-6
Permutations and Combinations is one of the most important chapters of Algebra in the JEE syllabus and other engineering exams. For JEE Mains, it has 4% weightage and for JEE Advanced, it has 5% weightage..
Q1. The letters of word ‘ZENITH’ are written in all possible ways. If all these words are written in the order of a dictionary, then the rank of the word ‘ZENITH’ is
• 716
• 692
• 698
• 616
Solution
Q2.Let there be circles in a plane. The value of for which the number of radical centres is equal to the number of radical axes is (assume that all radical axes and radical centre exist and are different)
• 7
• 6
• 5
• None of these
Solution
For a radical centre, 3 circles are required. The total number or radical centres is nC3. The total number or radical axis is nC2. Now, nC2 = nC3 => n = 5
Q3. In a class tournament, all participants were to play different games with one another. Two players fell ill after having played three games each. If the total number of games played in the tournament is equal to 84, the total number of participants in the beginning was equal to
• 10
• 15
• 12
• 14
Solution
Q4. There are four letters and four directed envelopes. The number of ways in which all the letters can be put in the wrong envelope is
• 8
• 9
• 16
• None of these
Solution
There is concept of derangement. The required number is
4![1 - 1/1! + 1/2! - 1/3! + 1/4!] = 9
Q5.Ten IIT and 2 DCE students sit in a row. The number of ways in which exactly 3 IIT students sit between 2 DCE students is
• 10C3 X 2! X 3! X 8!
• 10! X 2! X 3! X 8!
• 5! X 2! X 9! X 8!
• None of these
Solution
Three IIT students who will be between the IIT students can be selected in 10C3 ways. Now, two DCE students having three IIT students between them can be arranged in 2! x 3! ways. Finally, a group of above five students and the remaining seven students together can be arranged in 8! ways. Hence, total number of ways is 10C3 x 2! x 3! x 8!
Q6. A variable name in certain computer language must be either an alphabet or an alphabet followed by a decimal digit. The total number of different variable names that can exist in that language is equal to
• 280
• 290
• 286
• 296
Solution
Total number of variables if only alphabet is used is 286. Total number of variables if alphabets and digits both are used is 26 x 10. Hence, the total number of variables is 26(1 + 10) = 286
Q7.A student is allowed to select at most n books from a collection of (2n + 1) books. If the total number of ways in which he can select at least one book is 63, then the value of n is
• 2
• 3
• 4
• 5
Solution
Q8.Let Tn denote the number of triangles, which can be formed using the vertices of a regular polygon of n sides. If Tn+1 - Tn = 21, then n equals
• 5
• 7
• 6
• 4
Solution
Q9.The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is
• 55
• 66
• 77
• 88
Solution
There are two possible cases
Case I Five 1’s, one 2’s, one 3’s
Number of numbers = 7!/5! = 42
Case II Four 1’s, three 2’s
Number of numbers = 7!/4!3! = 35
Total number of numbers 42 + 35 = 77
Q10.The number of ordered pairs of integers (x,y) satisfying the equation x2 + 6x + y2 = 4
• 2
• 8
• 6
• None of these
Solution
(x + 3)2 + y2 = 13
x + 3 = ±3 or x + 3 = ±3, y = ±2
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Difficulty Level: At Grade Created by: CK-12
Estimated18 minsto complete
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Progress
MEMORY METER
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#### Example 2
Round each value to the nearest tenth and then find the sum.
\begin{align*}4.56, 3.47, 2.04, 6.21\end{align*}
First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point. Look at each digit in the tenths place and then the digit to the right of that to decide if you will need to round up the digit in the tenths place or keep it the same.
• 4.56 rounds to 4.6
• 3.47 rounds to 3.5
• 2.04 rounds to 2.0
• 6.21 rounds to 6.2
Now, find the sum of the four rounded numbers. Start by arranging the numbers vertically so that the decimal points line up.
\begin{align*}4.6\\ 3.5\\ 2.0\\ \underline{+ 6.2}\end{align*}
Next, add from right to left. Start by adding 6, 5, 0, and 2 to make 13. Carry the 1 and add it with the 4, 3, 2, and 6 to make 16. Insert a decimal point into your answer directly under the decimal points in the original numbers.
\begin{align*}4.6\\ 3.5\\ 2.0\\ \underline{+ 6.2}\\ 16.3\end{align*}
The answer is \begin{align*}4.6+3.5+2.0+6.2=16.3\end{align*}.
#### Example 3
\begin{align*}4.56+1.2+37.89=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
First, arrange the numbers vertically so that the decimal points line up.
\begin{align*}& \quad \ 4.56\\ & \quad \ 1.2\\ & \underline{+ 37.89}\end{align*}
Next, add one zero after the 2 in the second number so that all numbers have two digits to the right of the decimal point.
\begin{align*}4.56\\ 1.20\\ \underline{+ 37.89}\end{align*}
Now, add from right to left. Start by adding 6, 0 and 9 to make 15. Carry the 1 and add it with the 5, 2, and 8 to make 16. Once again carry the 1 and add it with the 4, 1, and 7 to make 13. Carry the 1 one more time and add it with the 3 to make 4. Insert a decimal point into your answer directly under the decimal points in the original numbers.
\begin{align*}4.56\\ 1.20\\ \underline{+ 37.89}\\ 43.65\end{align*}
The answer is \begin{align*}4.56+1.20+37.89=43.65\end{align*}.
#### Example 4
\begin{align*}14.2+56.78+123.4=\underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
First, arrange the numbers vertically so that the decimal points line up.
\begin{align*}& \quad \ 14.2\\ & \quad \ 56.78\\ & \underline{+ \; 123.4 \;\;\;}\end{align*}
Next, add one zero after the 2 in the first number and one zero after the 4 in the third number so that all numbers have two digits to the right of the decimal point.
\begin{align*}14.20\\ 56.78\\ \underline{+ 123.40}\end{align*}
Now, add from right to left. Start by adding 0, 8, and 0 to make 8. Then add 2, 7, and 4 to make 13. Carry the 1 and add it with the 4, 6, and 3 to make 14. Once again carry the 1 and add it with the 1, 5, and 12 to make 19. Insert a decimal point into your answer directly under the decimal points in the original numbers.
\begin{align*}14.20\\ 56.78\\ \underline{+ 123.40}\\ 194.38\end{align*}
The answer is \begin{align*}14.20+56.78+123.40=194.38\end{align*}.
#### Example 5
\begin{align*}189.34+123.5+7.2=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
First, arrange the numbers vertically so that the decimal points line up.
\begin{align*}& \ \ \ 189.34\\ & \ \ \ 123.5\\ & \underline{\;\; + \, \, 7.2 \;\;}\end{align*}
Next, add one zero after the 5 in the second number and one zero after the 2 in the third number so that all numbers have two digits to the right of the decimal point.
\begin{align*}189.34\\ 123.50\\ \underline{\;\; + \ \ 7.20}\end{align*}
Now, add from right to left. Start by adding 4, 0, and 0 to make 4. Then add 3, 5, and 2 to make 10. Carry the 1 and add it with the 9, 3, and 7 to make 20. Carry the 2 and add it with the 8 and 2 to make 12. Carry the 1 and add it with the 1 and 1 to make 3. Insert a decimal point into your answer directly under the decimal points in the original numbers.
\begin{align*}189.34\\ 123.50\\ \underline{\;\; + \ \ 7.20}\\ 320.04\end{align*}
The answer is \begin{align*}189.34+123.50+7.20=320.04\end{align*}.
### Review
Find each sum.
1. \begin{align*}29.451+711.36\end{align*}
2. \begin{align*}0.956+0.9885\end{align*}
3. \begin{align*}13.69+42.23\end{align*}
4. \begin{align*}58.744+12.06+8.8\end{align*}
5. \begin{align*}67.90+1.2+3.456\end{align*}
Round to the nearest tenth, then find the sum.
1. \begin{align*}88.95+23.451\end{align*}
2. \begin{align*}690.026+831.163\end{align*}
3. \begin{align*}3.27+7.818\end{align*}
4. \begin{align*}0.99+0.909+0.048\end{align*}
5. \begin{align*}5.67+3.45+1.23\end{align*}
Round to the nearest hundredth, then find the sum.
1. \begin{align*}511.51109+99.0986\end{align*}
2. \begin{align*}0.744+7.005+0.7071\end{align*}
3. \begin{align*}32.368+303.409\end{align*}
4. \begin{align*}1.0262+1.0242\end{align*}
5. \begin{align*}1.309+3.4590\end{align*}
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
TermDefinition
Decimal In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).
Estimate To estimate is to find an approximate answer that is reasonable or makes sense given the problem.
Rounding Rounding is reducing the number of non-zero digits in a number while keeping the overall value of the number similar.
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# How do you solve 9x^2-12x-14=0 by completing the square?
Aug 13, 2016
$x = \frac{2}{3} - \sqrt{2}$ or $x = \frac{2}{3} + \sqrt{2}$
#### Explanation:
In $9 {x}^{2} - 12 x - 14 = 0$, while $9 {x}^{2} = {\left(3 x\right)}^{2}$, to complete the square, recall the identity ${\left(x \pm a\right)}^{2} = {x}^{2} \pm 2 a x + {a}^{2}$.
As -12x=-2×(3x)×2, we need to add ${2}^{2}$ to make it complete square.
Hence, $9 {x}^{2} - 12 x - 14 = 0$ can be written as ((3x)^2-2×(3x)×2+2^2)-4-14=0 or
${\left(3 x - 2\right)}^{2} - 18 = 0$, which is equivalent to
${\left(3 x - 2\right)}^{2} - {\left(\sqrt{18}\right)}^{2} = 0$
and using identity ${\left(a - b\right)}^{2} = \left(a + b\right) \left(a - b\right)$ we can write the equation as
$\left(3 x - 2 + \sqrt{18}\right) \left(3 x - 2 - \sqrt{18}\right) = 0$
i.e. either $3 x - 2 + \sqrt{18} = 0$ or $3 x - 2 - \sqrt{18} = 0$.
Now as $\sqrt{18} = 3 \sqrt{2}$
either $x = \frac{2}{3} - \sqrt{2}$ or $x = \frac{2}{3} + \sqrt{2}$<|endoftext|>
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In this short chapter, Ishmael reports that the crew of the Pequod comes near to another whaling vessel, called The Albatross, which has been bleached white by the sun, and on which the crew appears to be starving, half-mad, and clothed in “animal skins.” Ahab attempts to pull up next to The Albatross and speak to the captain of that vessel, asking if the man has seen the White Whale; but the captain cannot hear Ahab over the high winds that whip about the two boats. Ahab notices that the school of fish that had been swimming with the Pequod immediately darted off to follow the Albatross when the ships passed, and Ahab assumes that this is because the fish are afraid of the Pequod’s mission to sail around the world in search of Moby Dick. The two boats part, with Ahab yelling to the other captain, who still cannot hear him, that the Pequod is bound for the Pacific Ocean.
The first of the novel’s many “gams,” or conversations between ships. Gams are extremely important in the novel, as they were on the open seas during the 1800s. For ships had no other means of communicating with land—it was the exchange of letters from outbound to inbound ships, and vice versa, that allowed for occasional, if imperfect, information exchange between sailors and their families and friends. The crew of the Albatross, unfortunately, seems to have been wrecked by its time at sea, but some of the other ships the Pequod encounters seem positively bubbly despite years at sea. This shows the vast range of human experience possible aboard whaling vessels.<|endoftext|>
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Sending signals through fiber optic cable is reliable and fast, but because of internal absorption and other effects, they will lose photons—which is a problem when the number of photons being sent is small. This is of particular concern in quantum networks, which typically involve a small number of entangled photons. Direct transmission through free space (vacuum or air) experiences less photon loss, but it's very difficult to align a distant receiver perfectly with the transmitter so that photons arrive at their destination.
A group in China has made significant progress toward solving that problem, via a high accuracy pointing and tracking system. Using this method, Juan Yin and colleagues performed quantum teleportation (copying of a quantum state) using multiple entangled photons through open air between two stations 97 kilometers apart across a lake. Additionally, they demonstrated entanglement between two receivers separated by 101.8km, transmitted by a station on an island roughly halfway between them.
Though the authors do not make this clear in the paper, their method is currently limited to nighttime communication. Nevertheless, their results achieved larger distances for multi-photon teleportation and three-point entanglement than before, and the tracking system used may even enable ground-to-satellite quantum communication—at least if it happens at night.
Quantum communication requires transmitting an arbitrary quantum state between two points, similar to how ordinary communication sends bits (voice or other data) across distances. However, a quantum state is a small amount of information, typically carried by a single photon, so many methods used in ordinary communication are out of the question (including broadcasting).
In fiber optic quantum networks, photon loss is large over significant distances, requiring the use of quantum repeaters. Point-to-point free-space transmission—either open-air or through the vacuum of space—is better, though larger distances allow the beam of photons to disperse. Atmospheric turbulence also contributes to photon loss in the air, with the losses increasing the farther the signal must travel.
One of the biggest challenges in point-to-point communication, however, is target acquisition by the transmitter and/or receiver. If the ground shifts slightly due to settling or tectonic activity, or atmospheric turbulence makes the receiver appear to move, the laser transmitting the signal can miss its target entirely. With few photons to spare in quantum communication, real-time tracking and acquisition is necessary. The researchers solved this problem using beacon lasers, bright beams that carry no information, but can be used to aim both transmitter and receiver, and wide-angle cameras.
As usual in quantum entanglement experiments, the group created entangled photons by stimulating a crystal with ultraviolet light. This produces a pair of photons with the same wavelength, but opposite (and unknown) polarization values. These entangled photons were subsequently sent to detectors, where their polarization quantum states were measured and compared. In the first experiment, one photon was sent 97km across Qinghai Lake (using a telescope to focus the beam), while the second was analyzed locally. Using these photons, the researchers copied the quantum state from the laboratory to the far station, achieving quantum teleportation over a much larger distance than previously obtained.
However, quantum communication sometimes also requires coordination between two distant receivers, so the researchers set up the transmitter on an island in the lake. The receivers were 51.2 and 52.2 km from the photon source respectively, on opposite shores of Qinghai lake, forming a triangle with the transmitter. The distance between the receivers—101.8km—was far enough to create a 3 microsecond delay between measurements of the photon polarization.
Given this setup, there was no possible way for the two receiving stations to communicate. Yet the photons they registered were correlated, indicating entanglement was maintained.
These experiments provide not only a proof of principle for free-space quantum communication, but also a means to test the foundations of quantum theory over larger distances than before. With very large detector separation, quantum entanglement experiments can help differentiate between standard and alternative interpretations of the quantum theory.
Though the long-distance aspect is promising, the fact that they set up on the shores of a lake (where no intervening obstacles exist) and that the experiment could only be performed successfully at night indicate its limitations. Author Yuao Chen told Ars via e-mail that they are working on solving the problem for daytime communication, but since the signal consists of single photons, it's not clear how this will work—the number of received photons fluctuated with the position of the Moon, so noise appeared to be a significant problem for them. Point-to-point communication will need to solve that problem as well before satellite-to-ground quantum networks are practical.<|endoftext|>
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The Magna Carta is displayed in Salisbury, England February 27, 2015. REUTERS/Kieran Doherty
Magna Carta. Lawyers and judges continue to name drop it in court, but the document has been largely neutered in terms of any modern-day legal application.
So why does a medieval piece of parchment still matter more than 800 years later?
“All legal documents that flowed subsequent to 1215 derived their inspiration from the Magna Carta,” says Moin Yahya, vice dean of the law school at the University of Alberta.
He adds its impact borders on religious in that “all of modern-day legal apparatus stems spiritually from the Magna Carta.”
Historians credit Magna Carta — Great Charter in Latin — with being the first legal document to assert that all citizens are bound by the rule of law. In the original thirteenth-century context, this meant that the king, or monarchy, was not above the law.
Back then King John of England was in danger of being overthrown by a bunch of rebel barons, who were appalled by his inhumane treatment of his subjects. A bloody confrontation was narrowly avoided when the king signed the initial Magna draft, limiting the monarchy’s power and protecting the barons from illegal imprisonment — habeus corpus — and excessive taxation.
It also established the concept of a trial by jury of one’s peers to settle disputes between the barons and the Crown. And attempted to stop the king from solidifying his power by banning the practice of forcing wealthy widowers to marry his confidants and give up their estates.
This would later become a key step in the battle for women’s rights.
“Magna Carta has been cited in Canadian court cases but there are rulings demonstrating that the literal text of the Great Charter is not enforceable in Canada,” concedes Toronto-based historian Dr. Carolyn Harris, the author of Magna Carta and its Gifts to Canada.
While direct links may not exist, sections of Canada’s Constitution are steeped in Magna.
“When Canada's constitution was repatriated in 1982, the Canadian Charter of Rights of Freedoms included legal protections with precedents in Magna Carta such as mobility rights and fundamental justice,” says Harris.
Some of the Great Charter’s key principles, include:
- Nobody is above the law: the basis of equal justice at all levels of society.
- Habeas corpus: freedom from unlawful detention without cause or evidence.
- Trial by jury: rules to settle disputes between barons and the Crown established trial by a jury of one’s peers.
“Everybody can identify why we have it in the first place: it’s meant to be a check on the power of the state,” says Yahya. “It reminds people that people can make a change; whether it was the barons back in 1215 or we the people today.”
Magna Carta Canada
Magna Carta: an introduction<|endoftext|>
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# Mathematics
COMP 361/5611 – Elementary Numerical Methods Assignment 3 – Due Sunday, October 12, 2014
Problem 1. (20%) Show how to use Newton’s method to compute the cube root of 5. Numerically carry out the first 10 iterations of Newton’s method, using x(0) = 1 . Analytically determine the fixed points of the Newton iteration and determine whether they are attracting or repelling. If a fixed point is attracting then determine analytically if the convergence is linear or quadratic. Draw the “x(k+1) versus x(k)
diagram”, again taking x(0) = 1, and draw enough iterations in the diagram, so that the long time behavior is clearly visible. For which values of x(0) will Newton’s method converge?
Problem 2. (20%) Also use the Chord method to compute the cube root of 5. Numerically carry out the first 10 iterations of the Chord method, using x(0) = 1 . Analytically determine the fixed points of the Chord iteration and determine whether they are attracting or repelling. If a fixed point is attracting then determine analytically if the convergence is linear or quadratic. If the convergence is linear then determine analytically the rate of convergence. Draw the “x(k+1) versus x(k)
diagram”, as in the Lecture Notes, again taking x(0) = 1, and draw enough iterations in the diagram, so that the long time behavior is clearly visible. (If done by hand then make sure that your diagram is sufficiently accurate, for otherwise the graphical results may be misleading.)
Do the same computations and analysis for the Chord Method when x(0) = 0.1.
More generally, analytically determine all values of x(0) for which the Chord method will converge to the cube root of 5.
Problem 3. (20%) Consider the discrete logistic equation, discussed in the Lecture Notes, and given by
x(k+1) = cx(k)(1− x(k)), k = 0, 1, 2, 3, · · · .
For each of the following values of c, determine analytically the fixed points and whether they are attracting or repelling: c = 0.70, c = 1.00, c = 1.80, c = 2.00, c = 3.30, c = 3.50, c = 3.97. (You need only consider “physically meaningful” fixed points, namely those that lie in the interval [0, 1].) If a fixed point is attracting then determine analytically if the convergence is linear or quadratic. If the convergence is linear then analytically determine the rate of convergence. For each case include a statement that describes the behavior of the iterations, as also shown in the Lecture Notes.
1
Problem 4. (20%) Consider the example of solving a system of nonlinear equations by Newton’s method, as given on Pages 95-97 of the Lecture notes. Write a program to carry out this iteration, using Gauss elimination to solve the 2 by 2 linear systems that arise. Use each of the following 16 initial data sets for the Newton iteration:
(x (0) 1 , x
(0) 2 ) = (i, j) , i = 0, 1, 2, 3 , j = 0, 1, 2, 3 .
Present and discuss your numerical results in a concise manner.
Problem 5. (20%) The multiplicity of the zero x∗ is the least integer m such that f (k)(x∗) = 0 for 0 ≤ k < m, but f (m)(x∗) 6= 0. Show analytically that in the case of a zero of multiplicity m, the modified Newton’s method
xn+1 = xn −m f(xn)
f ′(xn)<|endoftext|>
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Foundation piles are groups of cylindrical or flat sections of wood, steel or concrete that are driven into soil to form part of a foundation. They are used when the soil near the surface is too weak to support the weight of the structure or building. Piles transfer this structural load to deeper soil or rock that is better able to carry the weight. Piles are classified depending on how they carry the structural load.
Typically timber piles are tree trunks that are trimmed of branches and bark and that have no bends, large knots, splits or decay and are uniform from end to end. Timber piles are the cheapest types of foundation piles but have a lower load bearing capacity than steel or concrete and may not be suitable for areas where they are exposed to seasonal elements.
Concrete piles may be of two types. Pre-cast concrete piles are pre-made using reinforced or pre-stressed concrete. They have a high load-bearing capacity but longer piles are difficult to transport and they're difficult to cut to a specific length.
Cast-in-place concrete piles are made by driving a cylindrical shell into the ground to the desired depth and then filling the shell with liquid concrete. The shell doesn't contribute to load bearing capacity but provides a hole in order to make the concrete pile. These are preferred, as builders can form any length and depth and it isn't necessary to transport them. However, conditions must be favorable as the concrete must be able to harden in the form.
Steel piles come in various shapes and sizes but are usually rolled steel sections of seamless pipes that can be welded to lengths of up to 70 meters. Steel piles are treated before being placed as corrosion may be an issue. They are typically embedded into the soil with open ends and can be placed into hard surfaces such as boulders or rock beds.
Composite piles are made of one material in the bottom section and another in the top section. The materials used depend on the weight of the structural load. A composite pile made of wood and concrete, for example, is used to support loads of 20 to 30 tons. One made of steel and concrete is used to support loads up to 50 tons.
Sheet piles are specially shaped interlocking piles made of steel, wood or formed concrete that are used to make a continuous wall that resists horizontal pressures from earth or water loads.
Driven vertically into the ground, bearing piles are used for the direct support of vertical loads, which they distribute through fairly soft soils that aren't able to support concentrated loads.
End-bearing piles are driven through very soft soil, and they come to an end in hard, somewhat impenetrable material such as rock or dense gravel. Their carrying capacity comes from the resistance of the bottom layer of the pile.
A friction pile is a pile that is driven into soil that is relatively uniform, but the tip of the pile is not settled into a hard layer. The weight of the structure is transferred downward and laterally by the friction between the pile and the surrounding soil.
Combination End-Bearing and Friction Piles.
A pile might pass through soft soil that gives some frictional resistance, but then go into a denser layer that has a better load-bearing capacity. In this case, these piles carry loads by a combination of end-bearing and friction.
Piles that are driven into the ground at an angle with vertical piles are called batter piles. When the foundation material fails to stop sideways movement of the vertical piles, batter piles will help bear the load. Batter piles are also used if the soil is compressible, in order to spread vertical loads over a larger area. Batter piles can be used alone (placed in opposite directions) as well.
- The Best Soils for Agriculture
- How Much Rebar to Put in Concrete
- Types of Masonry Work
- Distance Between Construction Joints in Retaining Wall
- Horizontal Vs. Vertical Foundation Cracks
- Elastic Modulus in Soils
- Add Fiber to Concrete
- Prepare Soil for a Concrete Slab
- Build Concrete Pads
- What Types of Bases Should a Gazebo Be Placed On?
- Parts of a Foundation Construction
- What Is Ovangkol Wood?<|endoftext|>
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One of our citizen scientist landholders from the Tocal ‘Who’s living on my land?’ survey thought she had Koalas on her property and had a tree with Koala scratching. So, she put her camera up at the base of the tree and crossed her fingers for a Koala to walk on by!
She was in luck!
Koalas (Phascolarctos cinereus) are one of Australia’s most iconic species. This tree-dwelling marsupial inhabits a range of eucalyptus-dominated environments spanning the length of the Australian continent from the far northern tropical rainforests of Queensland right through to the semi-arid communities of South Australia.
Koalas occur naturally in four states – Queensland, New South Wales, Victoria and South Australia – but the health and status of koala populations differ substantially across the continent. Victoria and South Australia have large and thriving koala populations, unlike Queensland and New South Wales where koala populations are in decline and are listed as vulnerable under the Environment Protection and Biodiversity Conservation Act 1999
Threats to Koalas
The main threat to Koala population survival is habitat loss and land clearing. This habitat loss decreases the area of suitable habitat available for koalas, and decreases the connectivity between quality habitat. When valuable feed and shelter trees are lost to land clearing; koalas have to travel further on the ground between trees. Koalas are most vulnerable on the ground, where they are susceptible to predator attack, particularly from dogs, and car collisions. Man-made and natural structures (e.g. roads, urban landuse, large rivers) can act as barriers to dispersal limiting connectivity. Climate change is another significant threat to Koalas, as it may accelerate extreme weather events like bushfires and heat waves. Some Koala populations have a high incidence of chlamydia which can lead to infertility. In the Hunter Region, groundwater extraction has also been recorded as a threat to koala populations as the activity affects some of their primary feed trees.
Koalas in the Hunter Region
The dramatic land clearing in the Lower Hunter Region since European settlement (e.g. approx. 75% reduction in koala habitat in Port Stephens LGA) has been attributed to the decline, and presumed local extinction, of koala populations in the Lower Hunter region.
While the koala population at Port Stephens is well researched and population estimates can be calculated, poor data from the other LGAs in the Lower Hunter has led to significant information gaps about the presence and prevalence of koalas in these areas. In Port Stephens, the koala presence increases with amount of habitat available and decreases with increasing density of roads. Consequently, a 2013 report Lower Hunter Koala Study called for the following conservation actions:
- ‘Conserving patches of greater than 100 ha with koala feed and roosting tree species; and
- ‘seeking to incorporate connective between patches for koalas’
The home ranges of koala within the study region is not consistent, with koalas having relatively small home ranges (10 ha) in high quality habitat areas like Port Stephens, but requiring much larger home ranges (approx. 80 ha) in low quality habitat areas. The availability of feed trees is related to the quality of habitat available for koalas, however not all feed trees are used equality, with the leaf chemistry of individual trees being an important determinant of their use. In general, highly fertile soils produce feed trees which are used more. However, these highly fertile areas are also prioritised for agricultural use, particularly on the valley floors of the Hunter region. Interestingly, recent studies have concluded that feed trees are not the only type of important vegetation for koalas. During extreme temperatures, they also rely on other species of ‘cool’ trees to provide shelter from the heat. In Port Stephens, koalas were reported using tree species like Angophora costata, Eucalyptus signata and Corymbia gummifera as shelter trees.
The Lower Hunter Koala Study classified the study area into four classes based on their suitability as koala habitat. 1) Lower koala habitat value; 2) Moderate koala habitat value; 3) High koala habitat value; and 4) Very high koala habitat value. The site in our study where the koala was recorded (Figure 5) was on the boundary between Lower and Moderate koala habitat. This demonstrates that even in areas which have been classified as Lower or Moderate koala habitat, efforts to conserve or rehabilitate areas can have positive outcomes for koalas.
It is important for landholders to conserve and rehabilitate areas on their land which could be good habitat for koalas. Although the current reserve system officially protects some areas of Very high and High koala habitat value land, over 75% of the land classified under these two categories in the Lower Hunter Region are on private land, or land not maintained for conservation purposes.
Hunter Region koala information from: Eco Logical Australia 2013. Lower Hunter Koala Study. Prepared for the Dept Sustainability, Environment, Water, Population and Communities.<|endoftext|>
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# math115
posted by .
find the first and third quartile in the set of numbers
3,4,8,8,8,11,13,14,15,18 is it q1=3 and q3 = 18
• math115 -
No.
• math115 -
There are different formulas for finding the quartiles.
Can you dig out the one used by your teacher and post it here so we can help you check if you did it correctly?
• math115 -
Rewrite the given data set in ascending order.
Step 2 Find the median.
Step 3 Make a list of only those numbers (in order) that are to the left of the
median.
Step 4 Find the median of the list created in Step 3; this is the first quartile Q1.
Step 5 Repeat steps 3 and 4 with those numbers that are to the right of the
median; this gives the third quartile Q3.
• math115 -
find the first and third quartile in the set of numbers
3,4,8,8,8,11,13,14,15,18 is it q1=3 and q3 = 18
math115 - Ms. Sue, Friday, October 16, 2009 at 12:05pm
No.
math115 - MathMate, Friday, October 16, 2009 at 12:07pm
There are different formulas for finding the quartiles.
Can you dig out the one used by your teacher and post it here so we can help you check if you did it correctly?
math115 - p, Friday, October 16, 2009 at 3:51pm
Rewrite the given data set in ascending order.
Step 2 Find the median.
Step 3 Make a list of only those numbers (in order) that are to the left of the
median.
Step 4 Find the median of the list created in Step 3; this is the first quartile Q1.
Step 5 Repeat steps 3 and 4 with those numbers that are to the right of the
median; this gives the third quartile Q3.
• math115 -
Have you done step 2? What is the median of the 10 numbers?
Now can you find out how to find the median, according to your teacher, so that we can follow that step by step?
• math115 -
11 is the median
• math115 -
So what is the rule from your teacher to find the median?
• math115 -
The median is the number for which there are as many instances that are above
that number as there are instances below it. To find the median, follow these
steps:
Step 1 Rewrite the numbers in order from smallest to largest.
Step 2 Count from both ends to find the number in the middle.
Step 3 If there are two numbers in the middle, add them together and find
their mean.
• math115 -
Can you review the steps to see if you followed step 3?
Remember there are ten numbers in all.
• math115 -
8, 11 add together then divide by 2=9.2
• math115 -
Good, but I suppose you meant 9.5 which is (8+11)/2.
Perfect!
Now that you have found the median correctly according to your teacher's instructions, we can continue to find the quartiles. The number 9.5 can be thought of as between the fifth and the sixth number, or the 5.5th number. Can you now do step 3 and 4 for the quartiles. Tell me what you found.
"Step 3 Make a list of only those numbers (in order) that are to the left of the median.
Step 4 Find the median of the list created in Step 3; this is the first quartile Q1. "
• math115 -
q1=8 q3=14
• math115 -
Perfect!!! 100%
Congratulations, you got it!
• math115 -
yeah thank you you are totally awesome!!!!! my problem is I have more to be checked not quartiles but others boo me
• math115 -
You're welcome!
You work well and did well following your teacher's instructions.<|endoftext|>
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# Dimensions of a rectangleWhich are the dimensions of the original rectangle if it's length is 2 times the width and if the length is decreased by 5 units and the width is increased by 5 units, the...
Dimensions of a rectangle
Which are the dimensions of the original rectangle if it's length is 2 times the width and if the length is decreased by 5 units and the width is increased by 5 units, the area is increased by 75 square units?
giorgiana1976 | Student
Let's establish the dimensions of the original rectangle:
x=width of the rectangle, in umits;
2x=length of the rectangle, in units.
We'll put into equation the enunciation: The area of the first rectangle + 75 is equal to the area of the second rectangle.
2x*x + 75 = (2x-5)(x+5)
2x^2+75=2x^2+10x-5x-25
5x-25-75=0
5x=100
x=20
2x=40
So the dimensions of the original rectangle are: the width x=20 units and the length is 2x=40units.
neela | Student
Let the breadth and length of the rectangle be x and 2x.
The its area = 2x^2.
After increasing the breadth by 5 and decreasing the length by 5 unita the are = (x+5)(2x-5).
Given that the area (x+5)(2x-5) = 2x^2+75. Or
2x^2+5x-25 = 2x^2+75. Or
5x = 75+25 = 100. Or
x = 100/5 =20 is the width and the length = 2x =2*20 = 45.<|endoftext|>
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# College Algebra Tutorial 3
College Algebra
Tutorial 3: Scientific Notation
Step 1: Move the decimal point so that you have a number that is between 1 and 10.
*Decimal is at the end of the number
*Move decimal to create a number between 1 and 10
Step 2: Count the number of decimal places moved in Step 1 .
How many decimal places did we end up moving?
We started at the end of the number 123400 and moved it between the 1 and 2. That looks like a move of 5 places.
What direction did it move?
Looks like we moved it to the left.
So, our count is +5.
Step 3: Write as a product of the number (found in Step 1) and 10 raised to the power of the count (found in Step 2).
Note how the number we started with is a bigger number than the one we are multiplying by in the scientific notation. When that is the case, we will end up with a positive exponent.
*Move the decimal 6 to the left
*When mult. like bases you add your exponents: 7 + (-2) = 5
*Move the decimal 5 to the right
*When div. like bases you subtract your exponents: -4 - 9 = -13
*Move the decimal 13 to the left<|endoftext|>
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A geological study has revealed that the massive ice sheet has fixed the landscape in place, rather than destroying it.
Ice is usually really good at scouring stuff away – it doesn’t take a scientist to tell you that. But according to this new study, some of the sub-glacial landscape may have remained unchanged for almost 3 million years, ever since the island became completely ice-covered, according to researchers funded by the National Science Foundation (NSF).
They analyzed samples from a 3,000 meters long core and concluded that “pre-glacial landscapes can remain preserved for long periods under continental ice sheets.” Seriously, that’s just how awesome geology is – we dig holes in the ground, and we find out stuff.
“The soil has been preserved and only slowly eroded, implying that an ancient landscape underlies 3,000 meters of ice at Summit, Greenland,” they conclude. They add that “these new data are most consistent with [the concept of] a continuous cover of Summit… by ice … with at most brief exposure and minimal surface erosion during the warmest or longest interglacial [periods].”
By now, you’re probably wondering how on Earth were they able to find that out by drilling. Well, just to clarify things up, an ice core is cylinder of ice in which individual layers of ice, compacted from snowfall, going back over millennia can be observed and sampled. As the ice forms from the incremental build up of annual layers of snow, lower layers are older than upper, and an ice core contains ice formed over a range of years. The properties of this ice and recrystallized inclusions can be studied to reconstruct a climatic record normally through isotopic analysis.
The scientists looked at the proportions of the elements carbon, nitrogen and Beryllium-10, the source of which is cosmic rays, in sediments taken from the bottom 13 meters (42 feet) of the GISP2 ice core. They also compared it to similar samples from Alaska, to test how old it is.
Their discovery is also supported by the fossils they found in northern Greenland. Contrary to what you might expect, they found fossils of forest dwelling creatures, suggesting that Greenland was, in fact, green – at least a couple of million years ago. Even after ice sheet started to form, the center of Greenland remained stable, allowing the landscape to be locked away, unmodified, capped out under ice through cycles of warming and cooling.
“Rather than scraping and sculpting the landscape, the ice sheet has been frozen to the ground, like a giant freezer that’s preserved an antique landscape”, said Paul R. Bierman, of the Department of Geology and Rubenstein School of the Environment and Natural Resources at the University of Vermont and lead author of the paper.
But understanding how Greenland evolved is not just ancient history – it might help us understand how it might behave in the future, especially considering the global warming we are currently causing. As global temperatures continue to rise, scientists are worried about how the ice sheets will react, and figuring out how they behaved through (natural) cycles of warming and cooling might prove to be very valuable.<|endoftext|>
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Historical Overview Section
The Patrician Roman departure from Britain followed a decline across the empire, as attacks from barbarian tribes ate into the manpower and resources of the Roman state, forcing a recall of troops to protect the center. The withdrawal was gradual, but many date it formally to 410. The archaeological records of the final decades of Patrician Roman rule show undeniable signs of decay across Britain as pottery shards are not present in levels dating past 400, and coins minted past 402 are rare. The garrison in Britain came under increasing pressure from barbarian attack on all sides towards the end of the 4th century, and "Roman" troops were too few to mount an effective defence. The army in Britain became a focus for rebellion, after elevating two disappointing leaders at the end of the 4th Century, finally chose a soldier, Constantine III, to become their emperor in 407. When he took his army across into Gaul he was defeated, and it is unclear how many troops remained or ever returned, or whether a commander-in-chief in Britain was ever reappointed by Rome.
A Saxon incursion in 408 was apparently repelled by the Britons, and in 409 Zosimus records that the natives expelled the Roman civilian administration. With the higher levels of the military and civil government gone, administration and justice fell to municipal authorities, and small warlords gradually emerged all over a Britain which was still aspiring to Roman ideals and conventions. The change was not sudden however, and more recent scholars uphold a view - endorsed by this list - that there was considerable continuity from the British tribes in the pre-Roman and Roman periods to the kingdoms that formed in the post-Roman period.
The first wave of Middle Anglo Saxons were reputedly invited by Vortigern to assist in fighting the Later Pictish and Later Scots-Irish, however Germanic migration into Roman Britannia may well have begun much earlier even than that. There is recorded evidence, for example, of Germanic Roman auxiliaries being brought to Britain in the first and second centuries to support the legions, and when these new arrivals rebelled they plunged the country into a series of wars that eventually led to the Middle Anglo Saxon occupation of Lowland Britain by 600. Around this time many Britons fled to Brittany (hence its name). A significant date in sub-Roman Britain is the famous Groans of the Britons, an unanswered appeal to Aëtius, leading general of the western Empire, for assistance against the Middle Anglo Saxon invasion in 446; another is the Battle of Dyrham in 577, after which the significant cities of Bath, Cirencester and Gloucester fell and the Saxons reached the western sea. Non-Anglo-Saxon kingdoms soon began appearing in western Britain, largely based around Roman administrative structures, indicating further continuity, but also drawing on outside later Scots-Irish and Early Welsh influences as well. In the north kingdoms of the Hen Ogledd, the "Old North", arose, and fifth and sixth century repairs along Hadrian's Wall have been uncovered to further suggest a continuing Romanized influence.
King Arthur is a legendary British leader who, according to medieval histories and romances, led the defence of Britain against the Saxon invaders in the early 6th century. The details of Arthur's story are mainly composed of folklore and literary invention, and his historical existence is debated and disputed by modern historians.
Using the army in ADLG
- +3 command with some LH is just about getting into territory where you could plan to play a terrain game, which would allow you to go mob-handed with the MD Spearmen, but in reality its more likely that this is a fairly solid HF Spear army
- 4 Irish Swordsmen plus a couple of MF Spears makes a decent rough terrain force, to counteract the relative lack of mounted punch in the Non-Arthurian version of the army.
- Those same Knights of the Round Table are pretty good in theme, especially when supported by the 4-8 normal cavalry. Unlike most Impact Cv in this theme they benefit from being non-Impetuous
- The 2 Saxon mercenaries may be tempting, but they do then mess with your command and control as that sort of necessitates having at least a Brilliant commander with their command, and prevents the "3 Competent" option. It'd debatable whether their additional cutting edge is sharp enough to justify the cost - but they do make an otherwise-stodgy army a bit more intriguing.
User-contributed links about this army:
- Roman Britain.org history site
- Postroman.org Web project focused on Britain, Ireland and the offshore islands between 350AD and 850AD.
- Sub-Roman British DBA Figure Gallery for this army - from Fanaticus
- Romano-Brits DBA Figure Gallery for this army - from Fanaticus
Sample army lists for this army
101 Romano-British Michigan Brothers
4 Arthurian Cavalry Heavy cavalry impact Elite
2 Briton Cavalry Heavy cavalry
1 Scouts Light cavalry javelin Mediocre
6 Spearmen Heavy spearmen
2 Spearmen Medium spearmen
1 Light Foot Light infantry javelin
1 Martyrs Levy expendable Mediocre
3 Spearmen Medium spearmen
2 Bowmen Bowmen
1 Skirmishers Light cavalry javelin Mediocre
2 Britons Cavalry Heavy cavalry<|endoftext|>
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Tangent Circles
(Assignment 7)
by
J. Matt Tumlin, Cara Haskins, Robin Kirkham, and Audrey Simmons
Let us consider the problem of having two circles whereas one is inscribed within the other, and a third circle is constructed such that it is tangent to both the original circles.
We are given :
Two circles
A point on one of the circles
Let us start with two circles below shown in green.
We add the red circle inside the large green circle keeping it outside of the smaller green circle. The red circle has been constructed using the point on the outer circle.
When adding the purple circle tangent to the larger green circle we observe that there is no tangency to the smaller circle. The purple circle is constructed to circumscribe the smaller green circle.
To use the tangent circle script click HERE!
## Loci of the centers of the tangent circles
If we look at the loci of the centers of the red tangent circle, what shape would be graphed? To see click HERE!
If we look at the loci of the centers of the purple tangent circle, what shape would be graphed? To see click HERE!
Each set of loci makes a separate ellipse.
Loci of the base of the isosceles triangle
Review the construction of the red tangent circle, the one that was externally tangent to the small green circle (above).
Segment AB is the base of an isosceles triangle. It is apparent that all points on the dotted red line are equidistant from both point A and B since the line is the perpendicular bisector of AB.
Can you guess what shape would be made with the trace of the loci of the midpoint of the base of the isosceles triangle? Instead of making an ellipse, the loci of the midpoints is a ________?
(Try this and see)
To look at the locus of the midpoint of the segment, click HERE!
Notice that the locus of the midpoints of the purple circle is also the same.
The consistency of the arrangement of the points is quite interesting.
Conclusion: In summary, the loci of the centers of the tangent circles form ellipses. The loci of the midpoints of the base of the isosceles triangle (while under construction) form circles.<|endoftext|>
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# How Will Gerard Butler Fare Today? (12/23/2019)
How will Gerard Butler do on 12/23/2019 and the days ahead? Let’s use astrology to complete a simple analysis. Note this is not at all guaranteed – take it with a grain of salt. I will first work out the destiny number for Gerard Butler, and then something similar to the life path number, which we will calculate for today (12/23/2019). By comparing the difference of these two numbers, we may have an indication of how well their day will go, at least according to some astrology people.
PATH NUMBER FOR 12/23/2019: We will take the month (12), the day (23) and the year (2019), turn each of these 3 numbers into 1 number, and add them together. This is how it’s calculated. First, for the month, we take the current month of 12 and add the digits together: 1 + 2 = 3 (super simple). Then do the day: from 23 we do 2 + 3 = 5. Now finally, the year of 2019: 2 + 0 + 1 + 9 = 12. Now we have our three numbers, which we can add together: 3 + 5 + 12 = 20. This still isn’t a single-digit number, so we will add its digits together again: 2 + 0 = 2. Now we have a single-digit number: 2 is the path number for 12/23/2019.
DESTINY NUMBER FOR Gerard Butler: The destiny number will take the sum of all the letters in a name. Each letter is assigned a number per the below chart:
So for Gerard Butler we have the letters G (7), e (5), r (9), a (1), r (9), d (4), B (2), u (3), t (2), l (3), e (5) and r (9). Adding all of that up (yes, this can get tiring) gives 59. This still isn’t a single-digit number, so we will add its digits together again: 5 + 9 = 14. This still isn’t a single-digit number, so we will add its digits together again: 1 + 4 = 5. Now we have a single-digit number: 5 is the destiny number for Gerard Butler.
CONCLUSION: The difference between the path number for today (2) and destiny number for Gerard Butler (5) is 3. That is smaller than the average difference between path numbers and destiny numbers (2.667), indicating that THIS IS A GOOD RESULT. But this is just a shallow analysis! As mentioned earlier, this is not scientifically verified. If you want a reading that people really swear by, check out your cosmic energy profile here. Go see what it says for you now – you may be absolutely amazed. It only takes 1 minute.
### Abigale Lormen
Abigale is a Masters in Business Administration by education. After completing her post-graduation, Abigale jumped the journalism bandwagon as a freelance journalist. Soon after that she landed a job of reporter and has been climbing the news industry ladder ever since to reach the post of editor at Tallahasseescene.
#### Latest posts by Abigale Lormen (see all)
Abigale Lormen
Abigale is a Masters in Business Administration by education. After completing her post-graduation, Abigale jumped the journalism bandwagon as a freelance journalist. Soon after that she landed a job of reporter and has been climbing the news industry ladder ever since to reach the post of editor at Tallahasseescene.<|endoftext|>
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Semantic memory is the individual's concept of meanings. It includes factual knowledge and general knowledge, right or wrong. This is the raw material of the vocabulary and expression, the basis of formative usage. The 'memory' analogy is an apt description of the quality of this information, which may be blurred confused or skewed by perception of information or translation from communication. (Malapropisms are a good example of where the semantic memory goes off the rails.) It's the quality of memory which defines the semantics.
Examples of Semantic Memory:
Basic: A dog looks, acts and barks like a dog.|
Confused: A cat makes violins by disemboweling itself.
Malapropism: Getting better is a natural regurgative process
Addtree reconstruction of semantic memory organization for the category 'animals' using verbal fluency data from children age 7-8 based on crowe and prescott (2003).<|endoftext|>
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# How to Calculate the Diameter of a Rectangle
••• kkant1937/iStock/GettyImages
Print
A rectangle is any flat shape with four straight sides and four 90 degree angles, or right angles. Each side of a rectangle joins with two right angles. The diameter of a rectangle is the length of a diagonal, or either of two long lines that join opposite corners. A diagonal divides a rectangle into two identical right angle triangles. In mathematics, the diagonal of a right angle triangle is called the hypotenuse. Use the Pythagorean theorem:
H^2 = A^2 + B^2
to determine the length of the diagonal and thus calculate the diameter of a rectangle.
Examine the T-square and make sure the two pieces meet at a 90 degree angle.
Draw any rectangle that fills about half a sheet of paper. Use the T-square as a guide to make all four angles right angles. Ensure that the opposite sides of your rectangle are parallel and of equal length.
Draw a diagonal between two opposite corners using the T-square.
Measure the length of each side to highest precision using the T-square, and write the values near the respective sides. Label the sides: mark any side "A," label the adjacent side (opposite the hypotenuse) "B," and make the hypotenuse "H."
Calculate the triangle's hypotenuse (diagonal) length using the equation:
H = \sqrt{A^2 + B^2}
derived from the Pythagorean theorem, to calculate the hypotenuse of the triangle. Square the values of A and B, then add the squares together. Calculate the value of H by using a calculator to find the square root of the resulting sum. The value of H, the length of the diagonal, is also the diameter of the rectangle formed by the two triangles.
Measure the length of the hypotenuse with the T-square and compare the measurement with the calculated value.
## Example Calculation
Example calculation: if A = 5.5 inches and B = 7.7 inches, then:
\begin{aligned} H^2 &= 5.5^2+ 7.7^2 \\ &= 30.25 + 59.29 \\ &= 89.54 \end{aligned}
Therefore:
\begin{aligned} H &= \sqrt{89.54} \\ &= 9.46 \text{ inches} \end{aligned}
Any difference between the lengths you obtain by measuring and those you calculate will reflect the precision of your drawing and measuring.
#### Things You'll Need
• Pencil
• Paper
• T-square or try square
• Calculator<|endoftext|>
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<meta http-equiv="refresh" content="1; url=/nojavascript/"> Exponents and Exponential Functions | CK-12 Foundation
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# Chapter 8: Exponents and Exponential Functions
Created by: CK-12
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## Introduction
Exponential functions occur in daily situations; money in a bank account, population growth, the decay of carbon-14 in living organisms, and even a bouncing ball. Exponential equations involve exponents, or the concept of repeated multiplication. This chapter focuses on combining expressions using the properties of exponents. The latter part of this chapter focuses on creating exponential equations and using the models to predict.
## Summary
This chapter begins by talking about exponential properties involving products and quotients. It also discusses zero and negative exponents, as well as fractional exponents. Next, scientific notation is covered, and evaluating scientific notation with a calculator is outlined in detail. Instruction is then given on exponential growth and decay, and geometric sequences and exponential functions are highlighted. Finally, the chapter concludes by giving examples of how to solve real-world problems.
### Exponents and Exponential Functions Review
Define the following words.
1. Exponent
2. Geometric Sequence
Use the product properties to simplify the following expressions.
1. $5 \cdot 5 \cdot 5 \cdot 5$
2. $(3x^2y^3) \cdot (4xy^2)$
3. $a^3 \cdot a^5 \cdot a^6$
4. $(\gamma^3)^5$
5. $(x \cdot x^3 \cdot x^5)^{10}$
6. $(2a^3 b^3)^2$
Use the quotient properties to simplify the following expressions.
1. $\frac{c^5}{c^3}$
2. $\frac{a^6}{a}$
3. $\frac{a^5b^4}{a^3b^2}$
4. $\frac{x^4 y^5 z^2}{x^3 y^2 z}$
Simplify the following expressions.
1. $\frac{6^5}{6^5}$
2. $\frac{\gamma^2}{\gamma^5}$
3. $\frac{7^3}{7^6}$
4. $\frac{2}{\chi^3}$
5. $\sqrt[4]{\alpha^3}$
6. $\left(a^{\frac{1}{3}}\right)^2$
7. $\left(\frac{x^2}{y^3}\right)^{\frac{1}{3}}$
Write the following in scientific notation.
1. 557,000
2. 600,000
3. 20
4. 0.04
5. 0.0417
6. 0.0000301
7. The distance from the Earth to the moon: 384,403 km
8. The distance from Earth to Jupiter: 483,780,000 miles
9. According to the CDC, the appropriate level of lead in drinking water should not exceed 15 parts per billion (EPA’s Lead & Copper Rule).
Write the following in standard notation.
1. $3.53 \times 10^3$
2. $89 \times 10^5$
3. $2.12 \times 10^6$
4. $5.4 \times 10^1$
5. $7.9 \times 10^{-3}$
6. $4.69 \times 10^{-2}$
7. $1.8 \times 10^{-5}$
8. $8.41 \times 10^{-3}$
Make a graph of the following exponential growth/decay functions.
1. $\gamma=3 \cdot (6)^x$
2. $\gamma=2 \cdot \left(\frac{1}{3}\right)^x$
3. Marissa was given 120 pieces of candy for Christmas. She ate one-fourth of them each day. Make a graph to find out in how many days Marissa will run out of candy.
4. Jacoby is given \$1500 for his graduation. He wants to invest it. The bank gives a 12% investment rate each year. Make a graph to find out how much money Jacoby will have in the bank after six years.
Determine what the common ratio is for the following geometric sequences and finish each sequence.
1. 1, 3, __, __, 81
2. __, 5, __, 125, 625
3. 7, __, 343, 2401, __
4. 5, 1.5, __, 0.135, __
5. The population of ants in Ben’s room increases three times daily. He starts with only two ants. Make a geometric graph to determine how many ants will be in Ben’s room at the end of a 30-day month if he does not take care of the problem.
### Exponents and Exponential Functions Test
Simplify the following expressions.
1. $x^3 \cdot x^4 \cdot x^5$
2. $(a^3)^7$
3. $(y^2 z^4)^7$
4. $\frac{a^3}{a^5}$
5. $\frac{x^3 y^2}{x^6 y^4}$
6. $\left(\frac{3x^8 y^2}{9x^6 y^5}\right)^3$
7. $\frac{3^4}{3^4}$
8. $\frac{2}{x^3}$
9. $\sqrt[3]{5^6}$
Complete the following story problems.
1. The intensity of a guitar amp is 0.00002. Write this in scientific notation.
2. Cole loves turkey hunting. He already has two after the first day of hunting season. If this number doubles each day, how many turkeys will Cole have after 11 days? Make a table for the geometric sequence.
3. The population of a town increases by 20% each year. It first started with 89 people. What will the population of the town be after 15 years?
4. A radioactive substance decays 2.5% every hour. What percent of the substance will be left after nine hours?
5. After an exterminator comes to a house to exterminate cockroaches, the bugs leave the house at a rate of 16% an hour. How long will it take 55 cockroaches to leave the house after the exterminator comes?
#### Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9618.
Basic
8 , 9
Feb 24, 2012<|endoftext|>
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# Alg2 lesson 6-5
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### Alg2 lesson 6-5
1. 1. Quadratic Formula:<br />If ax2 + bx + c = 0, then<br />
2. 2. page 313<br />
3. 3. Solve by using the Quadratic Formula.<br />or<br />Two real solutions (both rational)<br />Example 5-1a<br />
4. 4. The x-intercept(s) of the graph of<br />f(x) = ax2 + bx + c <br />are also the solution(s) of the equation<br />ax2 + bx + c = 0 <br />
5. 5. Solve by using the Quadratic Formula.<br />or<br />Two real solutions (both rational)<br />Example 5-1a<br />
6. 6. Solve by using the Quadratic Formula.<br />or<br />Two real solutions (both rational)<br />Example 5-1a<br />
7. 7. Solve byusingtheQuadraticFormula.<br />=<br />One real solution<br />Example 5-2a<br />
8. 8. Solve by using the Quadratic Formula.<br />The two solutions are the complex<br />numbers and<br />Example 5-4a<br />
9. 9. In the quadratic formula, b2 – 4ac is called the discriminant<br />
10. 10. Find the value of the discriminant for . Then describe the number and type of roots for the equation.<br />Answer: The discriminant is 0, so there is one rational root.<br />Example 5-5a<br />
11. 11. Find the value of the discriminant for . Then describe the number and type of roots for the equation.<br />Answer: The discriminant is negative, so there are two complex roots.<br />Example 5-5a<br />
12. 12. Find the value of the discriminant for . Then describe the number and type of roots for the equation.<br />Answer: The discriminant is 80, which is not a perfect square. Therefore, there aretwo irrational roots.<br />Example 5-5a<br />
13. 13. Find the value of the discriminant for . Then describe the number and type of roots for the equation.<br />Answer: The discriminant is 81, which is a perfect square. Therefore, there are tworational roots.<br />Example 5-5a<br /><|endoftext|>
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Voltage and Current Phase Relationships in an Inductive Circuit
Figure 2a: V-I Relationship of the Circuit R. Capacitors and Inductors are called Reactive Components, in which the Voltage and Current are "Out-of-Phase" with . The mnemonic "ELI the ICE man" can be helpful in keeping track of the phase between the voltage and current in an AC circuit. In a circuit with only an inductor . Expressed mathematically, the relationship between the voltage dropped across the inductor and rate of current change through the inductor is as such.
This opposition to current change is called reactance, rather than resistance. Expressed mathematically, the relationship between the voltage dropped across the inductor and rate of current change through the inductor is as such: The inductance L is in Henrys, and the instantaneous voltage eof course, is in volts. Figure below Pure inductive circuit: Inductor current lags inductor voltage by 90o.
- Series Resistor-Inductor Circuits
- Phase Shift
If we were to plot the current and voltage for this very simple circuit, it would look something like this: Figure below Pure inductive circuit, waveforms. Remember, the voltage dropped across an inductor is a reaction against the change in current through it. Therefore, the instantaneous voltage is zero whenever the instantaneous current is at a peak zero change, or level slope, on the current sine waveand the instantaneous voltage is at a peak wherever the instantaneous current is at maximum change the points of steepest slope on the current wave, where it crosses the zero line.
This results in a voltage wave that is 90o out of phase with the current wave. If you are using a calculator that has the ability to perform complex arithmetic without the need for conversion between rectangular and polar forms, then this extra documentation is completely unnecessary.Phase difference in E and I in A C
Since this is a series circuit, we know that opposition to electron flow resistance or impedance adds to form the total opposition: Just as with DC, the total current in a series AC circuit is shared equally by all components.
This is still true because in a series circuit there is only a single path for electrons to flow, therefore the rate of their flow must uniform throughout. Consequently, we can transfer the figures for current into the columns for the resistor and inductor alike: And with that, our table is complete.
The exact same rules we applied in the analysis of DC circuits apply to AC circuits as well, with the caveat that all quantities must be represented and calculated in complex rather than scalar form. So long as phase shift is properly represented in our calculations, there is no fundamental difference in how we approach basic AC circuit analysis versus DC.
Now is a good time to review the relationship between these calculated figures and readings given by actual instrument measurements of voltage and current. The figures here that directly relate to real-life measurements are those in polar notation, not rectangular!
In other words, if you were to connect a voltmeter across the resistor in this circuit, it would indicate 7. To describe this in graphical terms, measurement instruments simply tell you how long the vector is for that particular quantity voltage or current.
Rectangular notation, while convenient for arithmetical addition and subtraction, is a more abstract form of notation than polar in relation to real-world measurements. As I stated before, I will indicate both polar and rectangular forms of each quantity in my AC circuit tables simply for convenience of mathematical calculation.
This effective resistance is known as the inductive reactance. This is given by: The unit of inductance is the henry. As with capacitive reactance, the voltage across the inductor is given by: Where does the energy go?
AC Inductor Circuits
One of the main differences between resistors, capacitors, and inductors in AC circuits is in what happens with the electrical energy. With resistors, power is simply dissipated as heat. In a capacitor, no energy is lost because the capacitor alternately stores charge and then gives it back again. In this case, energy is stored in the electric field between the capacitor plates. The amount of energy stored in a capacitor is given by: In other words, there is energy associated with an electric field.
In general, the energy density energy per unit volume in an electric field with no dielectric is: With a dielectric, the energy density is multiplied by the dielectric constant. There is also no energy lost in an inductor, because energy is alternately stored in the magnetic field and then given back to the circuit.
The energy stored in an inductor is: Again, there is energy associated with the magnetic field. The energy density in a magnetic field is: RLC Circuits Consider what happens when resistors, capacitors, and inductors are combined in one circuit. The overall resistance to the flow of current in an RLC circuit is known as the impedance, symbolized by Z.
The impedance is found by combining the resistance, the capacitive reactance, and the inductive reactance. Unlike a simple series circuit with resistors, however, where the resistances are directly added, in an RLC circuit the resistance and reactances are added as vectors.
This is because of the phase relationships.
Series Resistor-Inductor Circuits | Reactance and Impedance -- Inductive | Electronics Textbook
In a circuit with just a resistor, voltage and current are in phase. When all three components are combined into one circuit, there has to be some compromise.
To figure out the overall effective resistance, as well as to determine the phase between the voltage and current, the impedance is calculated like this. The impedance, Z, is the sum of these vectors, and is given by: The phase relationship between the current and voltage can be found from the vector diagram:<|endoftext|>
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Have the laws of physics stayed the same throughout the history of the cosmos? It’s an interesting question because even minute changes to physical constants could imply the existence of extra dimensions, of the sort posited by string theorists. But that’s a big ‘could’, because despite earlier controversial findings, at least one cornerstone constant — the ratio of a proton’s mass to that of an electron — looks to be exactly the same in a galaxy some 6 billion light years away as it is when we measure it on Earth. A study led by Michael Murphy (Swinburne University) presents the result in a recent issue of Science.
The constant, known as mu, determines the value of the strong nuclear force, so it has everything to do with how atomic nuclei hold themselves together. No one can say why the mass of a proton should be 1836 times that of an electron. All we know is that it is. To be more precise, the value is 1836.15. The recently published research studied light from the quasar B0218+367, examining how it was partially absorbed by ammonia gas in an intervening galaxy on its way to Earth-based astronomers. It’s a useful measurement because the wavelengths at which ammonia absorbs energy from the quasar turn out to be quite sensitive to mu.
Take a look at the image below to get an idea of how key gravitational lensing has proven to be in this work. The quasar B0218+367 is about 7.5 billion years away. Two things are happening to its light as it moves toward us. First, its wavelength is being stretched, making it redder the farther it travels. And usefully for us, the light is being gravitationally lensed by the intervening galaxy six billion light years away. The result: Two quasar images and one extremely helpful set of data.
Image: Radio contour map of the quasar B0218+367 at about 7.5 billion light years distance. The galaxy containing absorbing ammonia molecules lies about 6 billion light years away and, though it is not seen in this radio map, gravitationally lenses the background quasar light to produce two bright quasar images on the sky (big red circles). The physical size of the image (at the distance of the absorbing galaxy) is about 19,000 light years across. Credit: Andi Biggs (MERLIN Image).
Christian Henkel (Max Planck Institute for Radio Astronomy) sees a clear result: “By comparing the ammonia absorption with that of other molecules, we were able to determine the value of the proton-electron mass ratio in this galaxy, and confirm that it is the same as it is on Earth.”
While we tend to assume that the laws of physics are the same everywhere, it’s an assumption that has to be verified by observations of different times and places in the cosmos. For that matter, the four fundamental forces of nature — gravity, electromagnetism, and the strong and weak nuclear forces — can’t be predicted from our theories, but can only be measured by experiment. That points to a major hole in our understanding of how physical constants govern the universe.
Finding out whether these constants remain the same is a prerequisite for deepening our understanding of how they emerge. And despite the earlier claim (in a Dutch study of 2006) that small differences in mu were observable, this new work is the first to use ammonia molecules, which turn out to be ten times more sensitive than any previous method (the Dutch team used molecular hydrogen). What’s next is to move the investigation beyond the confines of a single galaxy to weigh the value of mu in other eras, but as the details of the lensing involved in this experiment make clear, finding the right targets is a significant challenge.
The paper is Murphy et al., “Strong Limit on a Variable Proton-to-Electron Mass Ratio from Molecules in the Distant Universe,” Science Vol. 320. No. 5883 (20 June 2008), pp. 1611-1613 (abstract).<|endoftext|>
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## What is a quadratic function in factored form?
Quadratic functions can be written in three forms. Factored form, the product of a constant and two linear terms: or. The parameters and are the roots of the function (the x-intercepts of the graph ). Converting a quadratic function to factored form is called factoring.
## What is a factored form?
Factored form refers to the form of a number or algebraic expression when it has been broken down into a product of its factors.
## What is factored form example?
A factored form is negative unate in x, if x’ appears in F, but x does not. F is unate in x if it is either positive or negative unate in x, otherwise F is binate in x. Example: (a+b’)c+a’ is positive unate in c, negative unate in b, and binate in a. 12.
## What is quadratic standard form?
The standard form and the general form are equivalent methods of describing the same function. The general form of a quadratic function is f(x)=ax2+bx+c where a, b, and c are real numbers and a≠0. The standard form of a quadratic function is f(x)=a(x−h)2+k. The vertex (h,k) is located at h=–b2a,k=f(h)=f(−b2a).
## What does R and S mean in factored form?
The vertex of the parabola is the maximum if the parabola opens downward and the minimum if it opens upward. • In factored form, the zeros are r and s. The vertex of the parabola is halfway between the two zeros. To find the x coordinate of the vertex, add the two zeros together and divide by two.
## How do you simplify quadratic equations?
Completing the squarePut the equation into the form ax 2 + bx = – c.Make sure that a = 1 (if a ≠ 1, multiply through the equation by. before proceeding).Using the value of b from this new equation, add. Find the square root of both sides of the equation.Solve the resulting equation.
You might be interested: Atc equation
In other words, the quadratic formula is simply just ax^2+bx+c = 0 in terms of x. So the roots of ax^2+bx+c = 0 would just be the quadratic equation, which is: (-b+-√b^2-4ac) / 2a. Hope this helped!
## How do you write an equation in standard form?
The standard form for linear equations in two variables is Ax+By=C. For example, 2x+3y=5 is a linear equation in standard form. When an equation is given in this form, it’s pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations.
### Releated
#### Convert to an exponential equation
How do you convert a logarithmic equation to exponential form? How To: Given an equation in logarithmic form logb(x)=y l o g b ( x ) = y , convert it to exponential form. Examine the equation y=logbx y = l o g b x and identify b, y, and x. Rewrite logbx=y l o […]
#### H2o2 decomposition equation
What does h2o2 decompose into? Hydrogen peroxide can easily break down, or decompose, into water and oxygen by breaking up into two very reactive parts – either 2OHs or an H and HO2: If there are no other molecules to react with, the parts will form water and oxygen gas as these are more stable […]<|endoftext|>
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"Travelers' diseases" is a broad term for infections that may be acquired when traveling away from home, especially from a developed or industrialized area to a less developed area. Every travel destination and each geographical location has its health risks. If you are planning a trip abroad or to another part of the country, you should educate yourself about your destinations and discuss with your healthcare practitioner:
- Any diseases known to be prevalent in the areas you will be visiting
- How long you plan to stay in any particular location
- What activities you plan to do during your visit
With the proper care, many travelers’ diseases can be prevented by:
- Avoiding or taking special precautions in environments where disease-carrying insects or animals are present
- Avoiding risky behaviors that could result in the spread of disease
- Taking care with food and water
- Getting recommended vaccines and/or completing a course of preventive medications
Some diseases are found throughout the world and, unless prevented through vaccination, frequently cause childhood illnesses. In some cases, these illnesses can lead to lifelong complications. Many nations have vaccination programs to decrease the number of people who contract conditions such as measles, rubella (German measles), mumps, and polio. In areas that are unable to uniformly vaccinate their populations, these conditions can be endemic and/or there may be epidemics of the disease. Travelers who are not protected through previous vaccinations, young children who have not been fully immunized, and people with weakened immune systems may be at an increased risk of contracting one of these infections.
What are some common causes of travelers' diseases and how are they acquired?
Travelers' diseases caused by microbes such as bacteria, viruses, and parasites can be acquired in a variety of ways, such as through contaminated food or water, from animal droppings or animal bites, and from soil. Physical contact with infected animals or animal hides can also put a person at risk. Some diseases are carried by insects, such as mosquitoes, flies, fleas, and ticks. Some can be acquired while swimming in freshwater or walking with bare feet. Others are passed from person to person through contact with blood or other body fluids.
For a more complete list of diseases related to travel, including information on the way they are transmitted, visit the CDC's web page on Travelers' Health: Diseases.<|endoftext|>
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# Given the first term and the common dimerence er an arithmetic sequence how do you find the 52nd term and the explicit formula: a_1=12, d=-20?
Dec 7, 2017
${a}_{52} = - 1008$
#### Explanation:
We can write the nth term of the sequence:
${a}_{n} = {a}_{1} + d \cdot \left(n - 1\right)$
So, we have ${a}_{n} = 12 - 20 \cdot \left(n - 1\right)$.
To find the 52nd term we now substitute 52 for n:
${a}_{52} = 12 - 20 \cdot \left(52 - 1\right) =$
$= 12 - 20 \left(51\right) = 12 - 1020 = - 1008$<|endoftext|>
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## Real MATHEMATICS – Strange Worlds #15
Human Knot Game
Inside a classroom divide students into groups such that each group has at least 5 students. Groups should stand up and form a circle before following the upcoming instructions:
1. If there is an even number of student in a group:
• Have each student extend his/her right hand and take the right hand of another student in the group who is not adjacent to him/her.
• Repeat the same thing for the left hand.
2. If there is an odd number of student in a group:
• Have each student (except one of them) extend his/her right hand and take the right hand of another student in the group who is not adjacent to him/her.
• Then take the extra student and have him/her extend his/her right hand so that the extra student can hold the left hand of another student who is not adjacent to him/her.
• Finally, repeat the process for the students whose left hands are free.
In the end, students will be knotted.
Now, each group should find a way to untangle themselves without letting their hands go. To do that, one can use Reidemeister moves.
Reidemeister Moves
Back in 1926 Kurt Reidemeister discovered something very useful in knot theory. According to him, in the knot theory one can use three moves which we call after his name. Using these three moves we can show if two (or more) knots are the same or not.
For example, using Reidemeister moves, we can see if a knot is an unknot (in other words, if it can be untangled or not).
What are these moves?
Twist
First Reidemeister move is twist. We can twist (or untwist) a part of a knot within the knot theory rules.
Poke
Second one is poke. We can poke a part of a knot as long as we don’t break (or cut) the knot.
Slide
Final one is slide. We can slide a part of a knot according to Reidemeister.
One wonders…
After you participating in a human knot game, ask yourselves which Reidemeister moves did you use during the game?
M. Serkan Kalaycıoğlu
## Real MATHEMATICS – Strange Worlds #14
In my youth, I would never step outside without my cassette player. Though, I had two knotting problems with my cassette player: the First one was about the cassette itself. Rewinding cassettes was a big issue as sometimes the tape knotted itself. Whenever I was lucky, sticking a pencil would solve the whole problem.
The second knotting problem was about the headphone. Its cable would get knotted so bad, it would take me 10 minutes to untangle it. Most of the time I would bump into a friend of mine and the whole plan about listening to music would go down the drain.
The funny thing is I would get frustrated and chuck the headphone into my bag which would guarantee another frustration for the following day.
A similar tangling thing happens in our body, inside our cells, almost all the time.
### DNA: A self-replicating material that is present in nearly all living organisms as the main constituent of chromosomes. It is the carrier of genetic information.
DNA has a spiral curve shape called “helix”. Inside the cell, the DNA spiral sits at almost 2-meters. Let me give another sight so that you can easily picture this length in your mind: If a cell’s nucleus was at the size of a basketball, the length of the DNA spiral would be up to 200 km!
You know what happens when you chuck one-meter long headphones into your bags. Trying to squeeze a 200 km long spiral into a basketball?! Lord; knots everywhere!
Knot Theory
This chaos itself was the reason why mathematicians got involved with knots. Although, knot-mathematics relationships existed way before DNA researches started. In the 19th century, a Scottish scientist named William Thomson (a.k.a. Lord Kelvin) suggested that all atoms are shaped like knots. Though soon enough Lord Kelvin’s idea was faulted and knot theory was put aside for nearly 100 years. (At the beginning of the 20th century, Kurt Reidemeister’s work was important in knot theory. I will get back to Reidemeister in the next post.)
Is there a difference between knots and mathematical knots?
For instance the knot we do with our shoelaces is not mathematical. Because both ends of the shoelace are open. Nevertheless a knot is mathematical if only ends are connected.
###### The left one is a knot, but not mathematically. The one on the right is mathematical though.
Unknot and Trefoil
In the knot theory we call different knots different names with using the number of crosses on the knot. A knot that has zero crossing is called “unknot”, and it looks like a circle:
Check out the knots below:
They look different from each other, don’t they? The one on the left has 1 crossing as the one on the right has 2:
But, we can use manipulations on the knot without cutting (with turning aside and such) and turn one of the knots look exactly like the other one!
This means that these two knots are equivalent to each other. If you take a good look you will see that they are both unknots. For example, if you push the left side of the left knot, you will get an unknot:
Is there a knot that has 1 crossing but can’t be turned into an unknot?
The answer is: No! In fact, there are no such knots with 2 crossiongs either.
We call knots that have 3 crossings and that can’t be turned into unknots a “trefoil”.
Even though in the first glance you would think that a trefoil could be turned into an unknot, it is in fact impossible to do so. Trefoil is a special knot because (if you don’t count unknot) it has the least number of crossings (3). This is why trefoil is the basis knot for the knot theory.
One of the most important things about trefoils is that their mirror images are different knots. In the picture, knot a and knot b are not equivalent to each other: In other words, they can’t be turned into one another.
Möbius Strip and Trefoil
I talked about Möbius strips and its properties in an old post. Just to summarize what it is; take a strip of paper and tape their ends together. You will get a circle. But if you do it with twisting one of the end 180 degrees, you will get a Möbius band.
Let’s twist one end 3 times:
Then cut from the middle of the strip parallel to its length:
We will get a shape like the following:
After fixing the strip, you can see that it is a trefoil knot:
To be continued…
One wonders…
1. In order to make a trefoil knot out of a paper strip, we twist one end 3 times. Is there a difference between twisting inwards and outwards? Try and observe what you end up with.
2. If you twist the end 5 times instead of 3, what would you get? (Answer is in the next post.)
M. Serkan Kalaycıoğlu<|endoftext|>
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Ants and Ecosystem3
The abundance of ants on earth is legendary. They live almost everywhere except very cold places such as Antarctica and Greenland. A worker is less than one-millionth the size of a human being, yet ants taken collectively rival people as dominant organisms on the land. Lean against a tree almost anywhere and the first creature that crawls on you will probably be an ant. Stroll down a suburban sidewalk with your eyes fixed on the ground, counting the different kinds of animals you see. The ants will win hands down. The British entomologist4 C. B. Williams once calculated that the number of insects alive on earth at a given moment is one million trillion, 1018. If, to take a conservative figure, one percent of this host is ants, their total population is ten thousand trillion. Individual workers weigh on average between one to five milligrams, according to the species. When combined, all ants in the world taken together weigh about as much as all human beings. But being so finely divided into tiny individuals, this biomass5 saturates the terrestrial environment.6
Ants absolutely dominate in rainforests, which are the most biologically diverse ecosystems on earth. Rainforests are so diverse that in a single leguminous tree (a relative to beans and peas) in Peru, 43 species of ants belonging to 26 genera 7 were found, about equal to the ant fauna8 of the British Isles. In a single square mile of tropical forest in Peru or Brazil, there may be 1,500 or more species of butterflies-twice the total number found in the United States and Canada combined.9 In Amazon rainforests ants and termites together compose nearly a third of the animal biomass. In other words, when all kinds of animals, large and small, from jaguars to monkeys down to roundworms and mites, are weighed, nearly a third of the weight consists of the flesh of ants and termites.
All of the ants, composing in formal taxonomic classification the family Formicidae of the order Hymenoptera, contain about 9,500 species known to science and at least twice that number of species remaining to be discovered, most of which are confined to the tropics. The total number of species of social insects is about 13,500 out of a grand total of 750,000 insect species that have been recognized to date by biologists. These numbers show that social insects seem to constitute 2 percent of all insects yet, in terms of biomass, social insects are half or more of all insects. Why are ants and other social insects so successful in the terrestrial environment? Their strength comes from their social organization. 10 In addition to the question of why ants and other highly social insect species have been so successful, it is also important to understand how such a large collection of individuals maintains order and collectively accomplishes tasks without producing chaos. With potentially thousands of individual ants to coordinate, how do they make decisions regarding who does what and when, especially critical decisions regarding reproduction?11 These questions become even more intriguing when you realize that ants have quite limited sensory devices to experience the world. They also have relatively simple nervous systems that process only a limited number of stimuli and are aware of only a few minutes to a few hours into the past.12
What is Collective Intelligence? 13
Intelligence can be defined simply as the ability to solve problems. One system is more intelligent than another system if in a given time interval it can solve more problems, or find better solutions to the same problems. A group can then be said to exhibit collective intelligence if it can find more or better solutions than the whole of all solutions that would be found by its members working individually.
All organizations, whether they are firms, institutions or sporting teams, are created on the assumption that their members can do more together than they could do alone. Yet, most organizations have a hierarchical structure, with one individual at the top directing the activities of the other individuals at the levels below. Although no president, chief executive or general can oversee or control all the tasks performed by different individuals in a complex organization, one might still suspect that the intelligence of the organization is somehow merely a reflection or extension of the intelligence of its hierarchical head. This is no longer the case in small, closely interacting groups such as soccer or football teams, where the “captain” rarely gives orders to the other team members. The movements and tactics that emerge during a soccer match are not controlled by a single individual, but result from complex sequences of interactions. Still, they are simple enough for an individual to comprehend, and since soccer players are intrinsically intelligent individuals, it may appear that the team is not really more intelligent than its members.
With the growing interest in complex adaptive systems, artificial life, swarms, and simulated societies, the concept of “collective intelligence” is coming more and more to the fore. The basic idea is that a group of individuals (e.g. people, insects, robots etc.) can be smart in a way that none of its members is. Complex, apparently intelligent behavior may emerge from the synergy created by simple interactions between individuals that follow simple rules.
How do ants succeed?
Now we have lots of questions to ask about the success of ants as a group. How do they govern? Who is the ruler? How do they foresee the future? How do they elaborate plans and preserve equilibrium? These, indeed, are puzzling questions. Every single ant in a colony seems to have its own agenda, and yet an insect colony looks so organized. The seamless integration of all individual activities does not seem to require a supervisor. For example, leafcutter ants cut leaves from plants and trees to grow fungi. Workers forage for leaves hundreds of meters away from the nest, literally organizing highways to and from their foraging sites. Weaver ant workers form chains of their own bodies, allowing them to cross wide gaps and pull stiff leaf edges together to form a nest. Several chains can join to form a bigger one over which workers run back and forth. In their moving phase, army ants organize impressive hunting raids, involving up to 200,000 workers, during which they collect thousands of prey.14
A harvester ant colony performs many tasks: It must collect and distribute food, build a nest, and care for the eggs, larvae, and pupae. It lives in a changing world to which it must respond. When there is a windfall of food, more foragers are needed. When the nest is damaged, extra effort is required for quick repairs. Task allocation is the process that results in certain workers engaged in specific tasks, in numbers appropriate to the current situation. Task allocation is a solution to a dynamic problem and thus it is a process of continual adjustment. It operates without any central or hierarchical control to direct individual ants into particular tasks. Although “queen” is a term that reminds us of human political systems, the queen is not an authority figure. She lays eggs and is fed and cared for by the workers. She does not decide which worker does what. In a harvester ant colony, many feet of intricate tunnels and chambers and thousands of ants separate the queen, surrounded by interior workers, from the ants working outside the nest and using only the chambers near the surface. It would be physically impossible for the queen to direct every worker’s decision about which task to perform and when. Consider the commercially available ant farms being sold. Since it’s forbidden to transfer ant queens, in the US ant farms are sold with only worker ants. Still they work in harmony. They build their nest, they build bridges, they collect food and they defend their colony. They do all these things without a queen. The absence of central control may seem counterintuitive, because we are accustomed to hierarchically organized social groups in many aspects of human societies, including universities, businesses, governments, orchestras and armies. This mystery underlies the ancient and pervading fascination of social insect colonies.
No ant is able to assess the global needs of the colony, or to count how many workers are engaged in each task and decide how many should be allocated differently. The capacity of an individual is limited. It cannot make complicated assessments. It probably cannot remember anything for very long. Its behavior is based on what it perceives in its immediate environment. Each worker needs to make only fairly simple decisions. There is abundant evidence, throughout physics, the social sciences and biology that such simple behavior by individuals can lead to predictable patterns in the behavior of the group. It should be possible to explain task allocation in a similar way, as the consequence of simple decisions by individuals.
Though ant colonies must respond to changing conditions, the response does not have to be perfect. It is not like clockwork, or an army, each unit snapping into place so the whole system ticks on without a hitch. There must be enough ants to collect food, often enough for the colony to survive and grow. The appropriate range of numbers should be allocated over a set of similar occasions. If the colony did not get enough food today, perhaps it will tomorrow. The process results in more or less the right number of ants engaged in the appropriate task, often enough for the colony to carry on.
Maximizing the number of ants that perform each task may not always be best for the colony. A task allocation problem for a human city is how to get the right number of firefighters to the scene of a fire. It may be a waste to have too many firefighters on the city payroll. Too many ants allocated to each task may be expensive for a colony if the excess ants could be doing something more useful than waiting around when they are not needed.
The most difficult thing to grasp about task allocation is that it is not a deterministic process even at the individual level. An ant does not respond the same way every time to the same stimulus; nor do colonies. Some events influence the probabilities that certain ants will perform certain tasks, and this regularity leads to predictable tendencies rather than perfectly deterministic outcomes. The ant is jostled in a stream of events that send it sometimes into one task, sometimes another. Task allocation is not a system in which each ant awaits the crucial event that defines its status forever. Like a twig in a turbulent river, an ant may tend to go in one direction, but there are many places it could get washed ashore, to be picked up and then swept in another direction altogether.
Stories about totalitarian societies, inexorable armies, and voracious monsters are often told as stories about ants. But ants have no dictators, no generals and no evil masterminds. In fact, there are no leaders at all.
In short, the basic mystery about ant colonies is that there is no management. A functioning organization with no one in charge is so unlike the way humans operate as to be virtually inconceivable. There is no central control. No insect issues commands to another or instructs it to do things in a certain way. No individual is aware of what must be done to complete any colony task. Each ant scratches and prods its way through the tiny world of its immediate surroundings. Ants meet each other, separate, go about their business. Somehow these small events create a pattern that drives the coordinated behavior of colonies.15
Elements of Collective Intelligence16
More is different. This old slogan of complexity theory actually has two meanings that are relevant to our ant colonies. First, the statistical nature of ant interaction demands that there is a critical mass of ants for the colony to make intelligent assessments of its global state. Ten ants roaming across the desert floor will not be able to accurately judge the overall need for foragers or nest-builders, but two thousand will do the job admirably. Individual ants do not know that they are prioritizing pathways between different food sources when they lay down a pheromone 17 gradient near a pile of nutritious seeds. In fact, if we only studied individual ants in isolation, we’d have no way of knowing that those chemical secretions were part of an overall effort to create a mass distribution line, carrying comparatively huge quantities of food back to the nest. It is only by observing the entire system at work that the global behavior becomes apparent.
Ignorance is usually useful for ants. The simplicity of the ant language-and the relative stupidity of the individual ants-is, as the computer programmers say, a feature but not a bug. Emergent systems can grow unwieldy when their component parts become excessively complicated. Better to build a densely interconnected system with simple elements, and let the more sophisticated behavior trickle up. That is why an ant does not respond to all stimuli around her, namely she ignores until she decides that the stimulus is strong enough to be responded to.
Encourage random encounters. Decentralized systems such as ant colonies rely heavily on the random interactions of ants exploring a given space without any predefined orders. Their encounters with other ants are individually arbitrary, but because there are so many individuals in the system, those encounters eventually allow individuals to gauge and alter the state of the colony itself. Without those haphazard encounters, the colony would not be capable of stumbling across new food sources or of adapting to new environmental conditions.
Look for patterns in the signs. While the ants do not need an extensive vocabulary and are capable of syntactical formulations, they do rely heavily on patterns in the semiochemicals they detect. A gradient in a pheromone trail leads them toward a food source, while encountering a high ratio of nest-builders to foragers encourages them to switch tasks. This knack for pattern detection allows meta-information to circulate through the colony mind: signs about signs. Smelling the pheromones of a single forager ant means little, but smelling the pheromones of fifty foragers imparts information about the global state of the colony.
Pay attention to your neighbors. This may well be the most important lesson that the ants have to give us, and the one with the most far-reaching consequences. You can restate it as “Local information can lead to global wisdom.” The primary mechanism of swarm logic is the interaction between neighboring ants in the field: ants stumbling across each other, or each other’s pheromone trails, while patrolling the area around the nest. Adding ants to the overall system will generate more interactions between neighbors and will consequently enable the colony to solve problems and regulate itself more effectively. Without neighboring ants stumbling across one another, colonies would be just a senseless assemblage of individual organisms-a swarm without logic.
Ants, first of all, have something to teach us about how nature works. Any system whose behavior arises from the interactions of its components has something in common with ant colonies. Using ants and other social insects as models, computer scientists have developed software agents that cooperate to solve complex problems, such as the rerouting of traffic in a busy telecom network or internet. Another example, the famous traveling salesman problem, in which a salesman tries to find the shortest and fastest route between many cities, is almost impossible to solve definitively. But with the methods inspired by ants the problem can be solved at least approximately, because ants are very good at finding the shortest path between the food and the nest collectively. Collective robotics borrowed from collective intelligence in ant colonies is being used to manage systems composed of lots of robots in synchronization.
Nature is a book to be read by the people who approach it to live in harmony, not to dominate. We are not the owners of the beautiful things around us, but observers searching for signs which reveal the wisdom behind them.
1 Haskins C.P., Of Ants and Men, Prentice-Hall Inc., 1939.
2 Huxley J., Ants, AMS Press, 1969.
3 An ecosystem is a grouping of plants, animals, and other organisms interacting with each other and with the environment in such a way as to perpetuate the grouping more or less indefinitely.
4 The scientific discipline in which ants are studied is called myrmecology and it is one of the branches of the study of insects, entomology.
5 Biomass is the total weight of all living organisms in a biological environment.
6 Holldobler B. and Wilson E.O., Journey to the Ants, Harvard University Press, 1994.
7 The word Genera is the plural of genus. Genus is a taxonomic category ranking below a family and above a species and generally consisting of a group of species exhibiting similar characteristics.
8 Fauna (Flora) is the animals (plants) of a particular region or period, considered as a group.
10 Holldobler and Wilson, ibid.
11 Bonabeau E., Dorigo M., and Theraulaz G., Swarm Intelligence: From Natural to Artificial System, Santa Fe Institute Studies in the Sciences of Complexity, Oxford University Press, NY:1999.
12 Holldobler and Wilson, ibid.
13 Heylighen, F. “Collective Intelligence and its Implementation on the Web: Algorithms to Develop a Collective Mental Map,” Computational & Mathematical Organization Theory. 1999, Vol. 5, no. 3, pp. 253-280.
14 Bonabeau et al, ibid.
15 Gordon D., Ants at Work, W. W. Norton. 1999.
16 Johnson S., Emergence Simon & Schuster. 2001.
17 The pheromone is the semiotic chemical ants use to communicate with each other and with other colonies. Every colony has its own odor. That is why ants can recognize their sisters from the same colony easily.<|endoftext|>
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How do you find the midpoint of ( 3, 8 ) and ( 5, 4 )?
May 24, 2018
$\left(4 , 6\right)$
Explanation:
$\text{given coordinates of endpoints say}$
x_1,y_1)" and "(x_2,y_2)" then"
$\text{midpoint } = \left[\frac{1}{2} \left({x}_{1} + {x}_{2}\right) , \frac{1}{2} \left({y}_{1} + {y}_{2}\right)\right]$
$\text{let "(x_1,y_1)=(3,8)" and } \left({x}_{2} , {y}_{2}\right) = \left(5 , 4\right)$
$\text{midpoint } = \left[\frac{1}{2} \left(3 + 5\right) , \frac{1}{2} \left(8 + 4\right)\right]$
$\textcolor{w h i t e}{\text{midpoint }} = \left(4 , 6\right)$
May 24, 2018
The midpoint equal:
$M = \left({x}_{\text{midpoint"),y_("midpoint}}\right) = \left(4 , 6\right)$
Explanation:
Let ;
$A = \left({x}_{1} , {y}_{1}\right) = \left(3 , 8\right)$
$B = \left({x}_{2} , {y}_{2}\right) = \left(5 , 4\right)$
To find the ${x}_{\text{midpoint}}$
${x}_{\text{midpoint}} = \frac{{x}_{1} + {x}_{2}}{2} = \frac{3 + 5}{2} = \frac{8}{2} = 4$
To find the ${y}_{\text{midpoint}}$
${y}_{\text{midpoint}} = \frac{{y}_{1} + {y}_{2}}{2} = \frac{8 + 4}{2} = \frac{12}{2} = 6$
The midpoint equal:
$M = \left({x}_{\text{midpoint"),y_("midpoint}}\right) = \left(4 , 6\right)$
Show the sketch below:<|endoftext|>
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## University Calculus: Early Transcendentals (3rd Edition)
(a) $$f^{-1}(x)=-x+1$$ The graphs are shown in the image below. The line $y=-x+1$ and the line $y=x$ intersect at $90^\circ$. (b) $$f^{-1}(x)=-x+b$$ The line $y=-x+b$ makes with the line $y=x$ an angle of $90^\circ$. (c) The inverses of those functions, whose graphs are perpendicular to the line $y=x$, are themselves.
(a) $$y=f(x)=-x+1\hspace{1cm}$$ - To find its inverse: 1) Solve for $x$ in terms of $y$: $$y=-x+1$$ $$x=-y+1$$ 2) Interchange $x$ and $y$: $$y=-x+1$$ Therefore, $$f^{-1}(x)=-x+1$$ The graphs are shown in the image below. From the graphs, we can see that the line $y=-x+1$ and the line $y=x$ intersect at $90^\circ$. (b) $$y=f(x)=-x+b\hspace{1cm}$$ - To find its inverse: 1) Solve for $x$ in terms of $y$: $$y=-x+b$$ $$x=-y+b$$ 2) Interchange $x$ and $y$: $$y=-x+b$$ Therefore, $$f^{-1}(x)=-x+b$$ We can deduce from (a), or notice that the slope of $y=-x+b$ is $-1$ and the slope of $y=x$ is $1$ and since $-1\times1=-1$, these two lines would make an angle of $90^\circ$ with each other. (c) From (a) and (b), we notice that the formula of the inverse are exactly the same as the formula of the original functions. That means, their graphs must overlap each other, or in other words, the inverse of those functions, whose graphs are lines perpendicular to the line $y=x$, are themselves.<|endoftext|>
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# Equation
A mathematical statement with an ‘equal to’ symbol between two expressions with equal values is called an ‘Equation’— for example, 5x+5=15. Linear, quadratic, cubic, etc are various kinds of equations.
## What is an Equation?
Two algebraic expressions on both sides of an ‘equal to (=)’ sign are called an Equation. It shows the relationship of equality between the expression composed on the left side with the expression composed on the right side. In each equation in maths, we have, L.H.S = R.H.S (left hand side = right hand side). Equations can be addressed to find the worth of an obscure variable addressing an obscure amount. Assuming that there is no ‘equal to’, it implies it’s anything but an equation. It will be considered as an expression.
## Different Parts Of an Equation
Various pieces of an equation incorporate coefficients, variables, operators, constants, terms, expressions, and an equal sign. At the point when we compose an equation, it is obligatory to have an “=” sign, and terms on the two sides. The two sides ought to be equal. An equation doesn’t have to have numerous terms on both sides, having variables, and operators. An equation can be shaped without these too, for instance, 5=5=10. This is a maths equation without any variables. Rather than this, an equation with variables is a mathematical equation.
## Steps to solve an Equation
An equation resembles a weighing offset with equal loads on the two sides. Assuming we add or deduct a similar number from the two sides of an equation, it holds. Additionally, if we duplicate or separate similar numbers into the two sides of an equation, it fits.
The steps to solve the basic equation with one variable are given below:
• Stage 1: Bring every one of the terms with variables on one side and every one of the constants on the opposite side of the equation by applying math procedure on the two sides.
• Stage 2: Combine every single like term (terms containing a similar variable with a similar type) by adding/deducting them.
• Stage 3: Simplify it and find the solution.
## Types Of Equations
Equations are basically classified into three sections:
1. Linear Equations: Equations with 1 as the degree are known as straight equations in maths. In such equations, 1 is the most noteworthy type of term. These can be additionally grouped into straight equations in a single variable, two-variable direct equations, with three variables, and so on. The standard type of a straight equation with variables X and Y is aX + bY – c = 0, where an and b are the coefficients of X and Y separately and c is the constant.
2. Quadratic Equations: Equations with degree 2 are known as quadratic equations. The standard type of a quadratic equation with variable x is ax2 + bx + c = 0, where a ≠ 0. These equations can be tackled by parting the centre term, finishing the square, or by the discriminant strategy.
3. Cubic Equations: Equations with degree 3 are known as cubic equations. Here, 3 is the most elevated exponent of no less than one of the terms. The standard type of a cubic equation with variable x is ax3 + bx2 + cx + d = 0, where a ≠ 0.
## Difference Between an Equation and an Expression
#### Make Math Stronger with Logimath - Personlized Live Math Classes
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Math Olympiad is a competitive exam that assesses students for...<|endoftext|>
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a series of radioactive metallic elements in Group 3 of the periodic table. Members of the series are often called actinides, although actinium (at. no. 89) is not always considered a member of the series. The series always includes the 14 elements with atomic numbers 90 through 103. The other members are (in order of increasing atomic number) thorium, protactinium, uranium, neptunium, plutonium, americium, curium, berkelium, californium, einsteinium, fermium, mendelevium, nobelium, and lawrencium. Thorium and uranium are the only actinides found in the earth's crust in appreciable quantities, although small amounts of neptunium and plutonium have been found in uranium ores. Actinium and protactinium are found in nature as decay products of some thorium and uranium isotopes. All the others have only been synthesized in small quantities (see synthetic elements).
Study of the properties of the actinides is hampered by their radioactive instability. It is known, however, that all members of the series resemble actinium and each other in their chemical properties and that they have a strong chemical resemblance to their homologs in the lanthanide series. The actinides are reactive and assume a number of different valences in their compounds. As the atomic number increases in this series, added electrons enter the 5f electron orbital. Elements in this series with atomic numbers greater than that of uranium (92) are called transuranium elements. Elements with atomic numbers greater than 103 are not members of the actinide series; element 104 (rutherfordium) is the first of the transactinide elements.
Group of radioactive elements with similar chemical properties. Their atomic numbers range from 89 to 103. Each element is analogous to the...
(actinoid series). The group of radioactive elements starting with actinium and ending with element 105. All are classed as metals. Those with atomi
Any of the chemical elements after uranium in the periodic table (with atomic numbers greater than 92). All are radioactive (see radioactivity), wi<|endoftext|>
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We are going to demonstrate the configuration of the RIP protocol, which belongs to the distance vector category. So let us analyze it according to our three criteria.
Distance Vector Routing Protocol
The first criteria is how routers discover each other and the answer to that lies in the fact that distance vector protocols will rely on neighbors to tell them the direction or vector toward destinations and how far away those destinations are, and so a routing protocol like RIP will advertise to its neighbors and obtain the information directly from its neighbors, those routers running the same protocol. In that sense, they rely on sort of a chain reaction.
In example, router would need to wait his neighbour routers to advertise to each other in chain. This makes them slower in understanding network changes and it also makes them rely on secondhand information, which also makes them inaccurate. They become even slower, when you see how they exchange information with their neighbors. They will periodically pass copies of the entire routing table. Some of them, like RIP version 1, will actually broadcast this information which is also a very inefficient method.
Source of Information and Discovering Routes
In terms of how distance vector protocols learn the information then, we were talking about periodic advertisements of the entire routing table coming from directly connected neighbors. Some of the information though is remote and so that is why they rely on secondhand information. The figure illustrates the chain reaction that would have to happen for A to learn about networks known to C. The exchange is periodic and it includes the complete routing table, so even if the topology does not change, routers keep telling each other about the same thing.
Other features of RIP result from our third criteria, how the protocols adjust to changes in the topology. The answer for a distance vector protocol like RIP is very slowly convergence timed or the amount of time it takes for RIP to know about a topology change and select a different path is a matter of minutes sometimes. Also, in order to be more accurate, it tends to be a very conservative protocol and so in the presence of a redundant topology like this one with loops in order to avoid packets from traveling around the network. Following the loop it will set a maximum cost of 15 for any path in the network; 16 is considered unreachable.
Those costs are measured in hop counts or the number of routers to go through along a certain path. This makes it less effective in selecting the best path in the presence of this redundant topology. The one hop path across that slower link will be selected, whereas the three-hop path would be faster but would not be selected. Even though a maximum of 16 equal cost paths can be selected, which is good in terms of load balancing, this routing protocol suffers from some inherent features that make it a less efficient, for example, periodically advertising every 30 seconds.
RIPv1 and RIPv2 Comparison
In an effort to improve RIP, designers came up with a second version of the protocol, RIP version 2. Several improvements were made to make this a classless routing protocol, which means it supports variable length subnet masks because the masks are advertised along with the routing update. This also makes RIP version 2 a protocol that can summarize routes. Other efforts were made to make it more efficient and so the advertisements are made on a multicast address and not as a broadcast.
With more security in mind, the protocol also supports router-to-router authentication, which means routers will not exchange routing information unless they share a common secret. Still, RIP version 2 is a distance vector protocol and it suffers from similar convergence issues. It still frequently advertises every 30 seconds and it still suffer from the maximum hop count of 16 for any network.
IP Routing Configuration Tasks
One of the good news about routing protocols is that their configuration is fairly consistent across the board in Cisco IOS routers. They follow similar steps. You first need to select the routing protocol and enable it in global configuration mode and then define the networks that you want this routing protocol to advertise on and receive advertisements through.
In the end, this is going to become advertisements coming in and out of interfaces, but the configuration is based on the network numbers configured on those interfaces. Remember, some routing protocols are classful and so they will not understand subnets, and this means that by enabling the routing protocol in a certain major network, you may be enabling the routing protocol on multiple interfaces that are simply subnets of the same major network.
So as we see here the basic commands to enable something like RIP in certain networks; similar steps will be necessary for other routing protocols. The command router enables the routing protocol process and at this time we are using RIP, but you could be using OSPF or EIGRP with the same command. With that, you enter routing protocol configuration mode.
Enter configuration commands, one per line. End with CNTL/Z.
For RIP, we are enabling RIP version 2 just to make sure that we are dealing with a classless routing protocol and to obtain the advantages of the more efficient RIP version 2.
The third command defines the networks directly connected to this router that will be participating in the RIP process. Even though RIP version 2 is a classless protocol, its configuration follows a classful criteria and so the network you specify there is a major classful network number.
RIP Configuration Example
So it becomes your job to look at the router’s interfaces, understand the network IDs attached or assigned to those interfaces, and then enable the protocol on those networks. In this example, for router A, this device is attached to networks 10 and 172.16 and so we enable the protocol on those two classful networks. Notice that B is connected only to networks that belong to the major classful network 10 and so we need only one command there, network 10.0.0.0, and that is going to enable the protocol on all interfaces that belong to that network 10, in this case, both serial interfaces.
There is a similar configuration for router C. Now in that router C, we had another link on network 10 and we do not want RIP running there; we would be running it with that network statement. The meaning of the network statement then is to enable the protocol on any and all interfaces that match the major network specified in the statement.
Verifying the RIP Configuration
Several commands are available to verify the configuration. Other than show running, you can see more live definitions of how RIP is configured and working by using the show IP protocols command. The output shows general information on the current timers being used by RIP and any routing filter that you may have applied. In terms of the network statement, the routing for networks section will display the exact networks that you configured the routing protocol for. So this means that we have two network statements, one for network 10 and one for network 172.16. The impact of that in the case of router A is that those two interfaces are now advertising and receiving advertisement for RIP. Fa0/0 belongs to network 172.16 and S0/0 belongs to network 10; both are sending and receiving RIP version 2 advertisements.
RouterA#show ip protocols
Routing Protocol is "rip"
Outgoing update filter list for all interfaces is not set
Incoming update filter list for all interfaces is not set
Sending updates every 30 seconds, next due in 15 seconds
Invalid after 180 seconds, hold down 180, flushed after 240
Default version control: send version 2, receive version 2
Interface Send Recv Triggered RIP Key-chain
FastEthernet0/0 2 2
Serial0/0 2 2
Automatic network summarization is in effect
Maximum path: 4
Routing for Networks:
Routing Information Sources:
Gateway Distance Last Update
10.1.1.2 120 00:00:24
Distance: (default is 120)
Displaying the IP Routing Table and Troubleshooting
The main responsibility of any routing protocol is to populate the routing table, and so displaying the routing table is a good troubleshooting and verification approach. The output here belongs to router A again and we can see all the entries or routes being learned via RIP by looking at the first column. This piece will have information on the protocol that advertised the route.
RouterA#show ip route
Codes: C - connected, S - static, R - RIP, M - mobile, B - BGP
D - EIGRP, EX - EIGRP external, O - OSPF, IA - OSPF inter area
N1 - OSPF NSSA external type 1, N2 - OSPF NSSA external type 2
E1 - OSPF external type 1, E2 - OSPF external type 2
i - IS-IS, su - IS-IS summary, L1 - IS-IS level-1, L2 - IS-IS level-2
ia - IS-IS inter area, * - candidate default, U - per-user static route
o - ODR, P - periodic downloaded static route
Gateway of last resort is not set
172.16.0.0/24 is subnetted, 1 subnets
C 172.16.1.0 is directly connected, FastEthernet0/0
10.0.0.0/24 is subnetted, 2 subnets
R 10.2.2.0 [120/1] via 10.1.1.2, 00:00:16, Serial0/0
C 10.1.1.0 is directly connected, Serial0/0
R 192.168.1.0/24 [120/2] via 10.1.1.2, 00:00:16, Serial0/0
If you look at the legend up there, then you will see the different letters associated to each protocol. Through this output, we also know that the administrative distance of RIP is 120 and that the cost to get to that particular destination is the second number within brackets. 10.2.2.0 is one hop away, while 192.168.1.0 is two hops away. The timer there indicates the amount of time since the route was updated. Remember, RIP will advertise every 30 seconds and if it misses one route in one of the advertisements, it will flag it as possibly down.
If you want to see live RIP advertisements, you can use the debug IP RIP command. You can see how in this example for router A the router is sending advertisements on both interfaces to a broadcast destination. It is also receiving updates coming from B on 10.1.1.2 through the serial interface. In both cases, the routes being learned and advertised are part of the output of the command as well as the cost to reach each destination. This is a very powerful tool to verify whether your neighbors are running the protocol or whether they have filters that may be blocking certain networks or whether you made mistakes in enabling the protocols on certain interfaces.
RouterA#deb ip rip
RIP protocol debugging is on
00:16:59.871: RIP: received v2 update from 10.1.1.2 on Serial0/0
00:16:59.875: 10.2.2.0/24 via 0.0.0.0 in 1 hops
00:16:59.875: 192.168.1.0/24 via 0.0.0.0 in 2 hops
00:17:00.747: RIP: sending v2 update to 220.127.116.11 via Serial0/0 (10.1.1.1)
00:17:00.747: RIP: build update entries
00:17:00.747: 172.16.0.0/16 via 0.0.0.0, metric 1, tag 0
00:17:22.779: RIP: sending v2 update to 18.104.22.168 via FastEthernet0/0 (172.16.1.1)
00:17:22.779: RIP: build update entries
00:17:22.779: 10.0.0.0/8 via 0.0.0.0, metric 1, tag 0
00:17:22.783: 192.168.1.0/24 via 0.0.0.0, metric 3, tag 0
00:17:28.907: RIP: received v2 update from 10.1.1.2 on Serial0/0
00:17:28.911: 10.2.2.0/24 via 0.0.0.0 in 1 hops
00:17:28.911: 192.168.1.0/24 via 0.0.0.0 in 2 hops
All possible debugging has been turned off<|endoftext|>
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How do you solve u/5+6=2?
$u = - 20$
Explanation:
To solve this, let's first get the term with $u$ on one side and the terms without $u$ on the other by subtracting 6 from both sides:
$\frac{u}{5} + 6 = 2$
$\frac{u}{5} + 6 - 6 = 2 - 6$
$\frac{u}{5} = - 4$
Now we can multiply both sides by 5:
$\frac{u}{\cancel{5}} \cancel{5} = - 4 \left(5\right)$
$u = - 20$<|endoftext|>
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Amino acids are small molecules of the form
where is a side chain called the -group. There are 20 different amino acids found in proteins, each characterized by its -group.
Peptides and proteins are chains of amino acids. Proteins are long such chains, whereas peptides and polypeptides are shorter ones. The amino acids are linked together by peptide bonds:
In 3d, the chains each lie in their respective plane:
Put together, they form the protein backbone.
The 3 dimensional conformation of the protein backbone is determined by the angles in the bonds around the -groups . These angles are denoted by and .
Proteins tend to stay in relatively stable configurations. Thus every protein has an associated 3 dimensional structure which is the spatial position of each atom in the protein. This structure, including the amino acids specific -groups, determines the protein’s function.
The structure prediction problem is to predict the 3 dimensional structure of a protein, given the sequence of amino acids that composes it. Of help is the Protein Data Bank (PDB), an online repository containing the experimentally determined 3d structure of thousands of proteins.
Decomposition of the problem
The structure of a protein is decomposed into 4 levels.
- Primary structure. The (ordered) sequence of amino acids that composes the protein.
- Secondary structure. The local structure of the of the protein. Many local patterns are classified. These include helices and sheets.
- Tertiary structure. The overall structure of the protein. When the protein is made of many protein subunits (which are polypeptides), the tertiary structure describes their individual conformation.
- Quaternary structure. The global structure of the protein (including how subunits are arranged together).
Figure 1. The structure of a protein, the polymerase basic protein 2, from the PDB. On the left is its ball-and-stick representation; on the right is highlighted its secondary structure in a cartoony style: notice the helices (pink) and sheets (yellow) joined by irregular segments (white).
Figure 2. The protein’s backbone.
Characterizing the phi-psi angles distributions
In a 1963 short article entitled Stereochemistry of Polypeptide Chain Configurations and in related works, Ramanchandran, Ramakrishnan and Sasisekharan conventionalized the use of the and angles to describe the conformation of a polypeptide backbone. They also predicted ranges of « allowed » and « disallowed » regions for the angles, representing physical constraints, and associated with these regions notable patterns such as the -helices and -sheets.
Fourty years later, Hovmoller et al compared their prediction with empirical data from the Protein Data Bank. For each of the 20 amino acids, they plotted the distribution of their angles as they appeared in 1042 protein subunits. These plots are called Ramachandran plots and the empirical results were surprisingly close to Ramanchandran’s predictions.
Figure 3. The alanyne (c) and glycine (d) Ramachandran plots reproduced from .
The alanyne plot is very similar to most of the other amino acids’ Ramachandran plots. The angles in the upper left cluster of the alanyne plot are in the -sheet region. Angles in the lower cluster tend to be highly concentrated and are in the -helix region. This association of angular regions with secondary structure pattern is not exact, but was found to correspond up to 95-99% to the DSSP secondary structure classification. (The DSSP is a standard tool for secondary structure assignment from protein 3d structure.)
Predicting phi-psi angles
The angles around a given amino acid residue in a protein can be predicted using emprical data from similar protein structure in the PDB. The predictions can then be suggested to more sophisticated structure prediction search algorithms.
In , it was suggested to predict the “half-angles” instead of the “whole-angles” , as these were associated to two amino acid residues and therefore were more specific. However, the datasets of angles are correspondingly much smaller. They therefore used a Dirichlet process mixture (with bivariate von Mises distributions) to model the angles distribution and predicted angular values from the posterior predictive distribution.
Their statistical model is clear and well suited to the task. Most of the parameters are easily interpretable and allow for the necessary prior information incorporation (some parameters are more difficult to interpret and are there to give flexibility to the model). They clearly show that the use of “half-angles” yields better prediction than the use of “whole-angles”
Watson, Baker, Bell, Gann, Levine and Losick, Molecular Biology of the Gene, sixth edition.
Richardson, J.S., The Anatomy and Taxonomy of Protein Structure, http://www.sciencedirect.com/science/article/pii/S0065323308605203
Ramanchandran et al., Stereochemistry of Polypeptide Chain Configurations, J. Mol. Bio. 1963
Hovmoller, S., Tuping, Z. and Thomas, O. Conformations of amino acids in proteins, Biological Crystallography, 2002.
Lennox, K., et al., Density Estimation for Protein Conformation Angles Using a Bivariate von Mises Distribution and Bayesian Nonparametrics, Journal of the American Statistical Association, 2009.<|endoftext|>
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Child Psychology in Early Childhood and Primary Settings, with Dr Melanie Woodfield
This course will provide a unique and interesting perspective on how core psychological ideas relate to the classroom environment and more importantly, how this knowledge can be utilised to support children to learn and connect to the best of their ability.
Specifically, this course will provide:
- An introduction to typical child development and how to identify development that’s different, including a spotlight on autism spectrum disorder
- An overview of the concept of intelligence and how it is assessed by psychologists, including what is usually meant by terms such as giftedness, intellectual disability, and specific learning difficulties
- An overview of common mental health issues in children and adolescents, how they impact on learning and relating to others, and how to support students with these needs.
Particular attention will be paid to ADHD.
- Tips and techniques to enhance communication with students, drawn from strategies therapists use to engage young people.
This lively day is filled with practical strategies and real world examples and will be of interest to teachers and professionals working with young children.<|endoftext|>
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### P(-1.96 z < 1.96) = .95
```P(-1.96 < z < 1.96) = .95
Shape of t Distributions for
n = 3 and n = 12
Degrees of Freedom (dof)
Example:
If 10 students take a quiz with a mean of 80, we can freely assign values to
the first 9 students, but the 10th student will have a determined score.
We know the sum of all 10 has to be 800; so when we add up the first 9
that sum will establish what 10th score has to be.
Because the first 9 could be freely assigned we say that there are 9
degrees of freedom.
For applications in this section the Degrees of Freedom is simply
the sample size minus one : dof = n-1
Properties of the Chi-Square Distribution
The chi-square distribution is not symmetric, unlike the normal
and t distributions.
As the number of degrees of freedom increases, the
distribution becomes more symmetric.
Chi-Square Distribution
Chi-Square Distribution for df = 10 and df = 20
A symmetric distribution is convenient, we only have to find value for a z-score (or tscore) because we know the other side will be the simply the positive or negative.
Confidence Interval for σ
(n 1)s 2
,
2
/2
(n 1)s 2
2
1 /2
(n 1)s 2
,
2
R
(n 1)s 2
2
L
• Since the distribution is not symmetric we have to look up
2 separate scores for a confidence interval.
• The area we find will be to the right of the score.
2
•
2 / 2 R2
•
12 / 2 L2
will be the larger value.
will be the smaller value.
•
X
z
Confidence Interval for μ with known σ:
/2
n
• Find sample size for mean: n z /2
m
2
s
• Confidence Interval for μ with unknown σ: X t /2
n
•
Confidence Interval for p: pˆ Z / 2
• Find sample size for proportion: n z
ˆˆ
pq
n
2
/2
ˆˆ
pq
E2
OR
z 2 / 2
n
4E 2
• Confidence Interval for σ :
( n 1) s 2
,
2
/2
( n 1) s 2
12 / 2
( n 1) s 2
,
2
R
( n 1) s 2
L2
```<|endoftext|>
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Q:
# How do I read Roman numerals?
A:
In order to read Roman numerals, make sure to know what each symbol means, when numbers add or subtract, and how to handle somewhat-larger numbers.
Know More
## Keep Learning
Credit: Joelle Icard Photodisc Getty Images
1. Know each symbol
In Roman numerals, M equals 1,000, D equals 500, C equals 100, L equals 50, X equals 10, and I equals one.
2. Add numbers when bigger is followed by smaller
When a larger number is followed by a smaller number, such as XI, the numbers are added. Thus, XI = 10 + 1 = 11.
3. Subtract numbers when smaller is followed by bigger
When it is the opposite, such as IX, the smaller number is subtracted from the bigger number. So, IX = 10 - 1 = 9.
4. Use only three of the same symbol together
When using multiple numbers of the same value, no more than three of the same symbol should appear together. If four are needed, it should be that value subtracted from the next-largest value. Examples include XL = 40, IV = 4 and CD = 400.
5. Add a bar on top if multiplying by 1,000
For each value above one, placing a bar or vinculum above the number multiplies it by 1,000. For instance, an X with a vinculum is 10,000.
Sources:
## Related Questions
• A:
The Roman numerals for one to five are I, II, III, IV and V. The Roman numeral system assigned number values to certain letters in the Roman alphabet. By combining these letters according to their own placement rules and applying simple arithmetic, the ancient Romans were able to represent a large range of numbers.
Filed Under:
• A:
The equivalent of the number 93 in Roman numerals is XCIII. The Roman numeral system uses letters instead of numbers. The basic Roman numerals up through 100 are as follows: I equals one, V equals five, X equals 10, L equals 50 and C equals 100.
Filed Under:
• A:
The Arabic number "17" is written as "XVII" in Roman numerals. The "X" represents a quantity of ten, the "V" represents a quantity of five, and "I" represents a quantity of one. The sum of the Roman numerals identifies its numeric value.
Filed Under:
• A:
The number 666 is written as DCLXVI in Roman numerals. The letters stand for 500, 100, 50, five and one respectively. There are only seven symbols in Roman numerals.<|endoftext|>
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# 11.5 Substructure of the nucleus (Page 3/16)
Page 3 / 16
$r={r}_{0}{A}^{1/3},$
where ${r}_{0}=\text{1.2 fm}$ and $A$ is the mass number of the nucleus. Note that ${r}^{3}\propto A$ . Since many nuclei are spherical, and the volume of a sphere is $V=\left(4/3\right){\mathrm{\pi r}}^{3}$ , we see that $V\propto A$ —that is, the volume of a nucleus is proportional to the number of nucleons in it. This is what would happen if you pack nucleons so closely that there is no empty space between them.
Nucleons are held together by nuclear forces and resist both being pulled apart and pushed inside one another. The volume of the nucleus is the sum of the volumes of the nucleons in it, here shown in different colors to represent protons and neutrons.
## How small and dense is a nucleus?
(a) Find the radius of an iron-56 nucleus. (b) Find its approximate density in $kg/{m}^{3}$ , approximating the mass of ${}^{56}\text{Fe}$ to be 56 u.
Strategy and Concept
(a) Finding the radius of ${}^{56}\text{Fe}$ is a straightforward application of $r={r}_{0}{A}^{1/3},$ given $A=56$ . (b) To find the approximate density, we assume the nucleus is spherical (this one actually is), calculate its volume using the radius found in part (a), and then find its density from $\rho =\mathrm{m/V}$ . Finally, we will need to convert density from units of $u/{fm}^{3}$ to $kg/{m}^{3}$ .
Solution
(a) The radius of a nucleus is given by
$r={r}_{0}{A}^{1/3}.$
Substituting the values for ${r}_{0}$ and $A$ yields
$\begin{array}{lll}r& =& {\left(1.2 fm\right)\left(56\right)}^{1/3}=\left(1.2 fm\right)\left(3.83\right)\\ & =& 4.6 fm.\end{array}$
(b) Density is defined to be $\rho =\mathrm{m/V}$ , which for a sphere of radius $r$ is
$\rho =\frac{m}{V}=\frac{m}{\left(4/3\right){\mathrm{\pi r}}^{3}}.$
Substituting known values gives
$\begin{array}{lll}\rho & =& \frac{56 u}{\left(1.33\right)\left(3.14\right){\left(4.6 fm\right)}^{3}}\\ & =& 0.138 u/{fm}^{3}.\end{array}$
Converting to units of $kg/{m}^{3}$ , we find
$\begin{array}{lll}\rho & =& \left(0.138 u/{fm}^{3}\right)\left(1.66×{10}^{–27}\phantom{\rule{0.25em}{0ex}}\text{kg/u}\right)\left(\frac{1 fm}{{10}^{–15}\phantom{\rule{0.25em}{0ex}}\text{m}}\right)\\ & =& 2.3×{10}^{17}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}.\end{array}$
Discussion
(a) The radius of this medium-sized nucleus is found to be approximately 4.6 fm, and so its diameter is about 10 fm, or ${10}^{–14}\phantom{\rule{0.25em}{0ex}}\text{m}$ . In our discussion of Rutherford’s discovery of the nucleus, we noticed that it is about ${10}^{–15}\phantom{\rule{0.25em}{0ex}}\text{m}$ in diameter (which is for lighter nuclei), consistent with this result to an order of magnitude. The nucleus is much smaller in diameter than the typical atom, which has a diameter of the order of ${10}^{–10}\phantom{\rule{0.25em}{0ex}}\text{m}$ .
(b) The density found here is so large as to cause disbelief. It is consistent with earlier discussions we have had about the nucleus being very small and containing nearly all of the mass of the atom. Nuclear densities, such as found here, are about $2×{10}^{14}$ times greater than that of water, which has a density of “only” ${10}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ . One cubic meter of nuclear matter, such as found in a neutron star, has the same mass as a cube of water 61 km on a side.
## Nuclear forces and stability
What forces hold a nucleus together? The nucleus is very small and its protons, being positive, exert tremendous repulsive forces on one another. (The Coulomb force increases as charges get closer, since it is proportional to $1/{r}^{2}$ , even at the tiny distances found in nuclei.) The answer is that two previously unknown forces hold the nucleus together and make it into a tightly packed ball of nucleons. These forces are called the weak and strong nuclear forces . Nuclear forces are so short ranged that they fall to zero strength when nucleons are separated by only a few fm. However, like glue, they are strongly attracted when the nucleons get close to one another. The strong nuclear force is about 100 times more attractive than the repulsive EM force, easily holding the nucleons together. Nuclear forces become extremely repulsive if the nucleons get too close, making nucleons strongly resist being pushed inside one another, something like ball bearings.
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!<|endoftext|>
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BOOK ONLINE OR CALL (647) 479-8808 FOR AN APPOINTMENT
What is Measles?
Measles is caused by a virus. The virus can live in your nose, mouth, eyes and on your skin. It is highly contagious, which means it spreads very easily. Measles is a highly contagious disease and is a leading cause of vaccine-preventable deaths in children worldwide.
What is my risk?
Your risk depends on several factors: destination, length of trip, and immunization history. Speak with one of our Travel Health Specialists to understand the risk of measles for your trip.
If you have not been vaccinated and have never had measles, you are at risk of infection. Measles is very contagious and easy to catch when you have contact with someone who is infected with the virus.
Travellers who are not vaccinated may bring measles into Canada. As a result, outbreaks may occur, especially in communities where people do not vaccinate their children.
How is it transmitted?
The measles virus spreads through:
- direct contact
- through the air, such as when an infected person coughs or sneezes
- on objects that were recently exposed to infected mucous or saliva (e.g. shared utensils, cups, tissues)
What are the symptoms?
Symptoms begin 7 to 18 days after exposure. You can spread the virus to others from 4 days before the rash starts until 4 days after the rash appears. The virus is most often spread when people first get sick or before they know they have measles.
Initial symptoms include:
- runny nose
- red eyes
- irritability (feeling cranky or in a bad mood)
Small, white spots may also show up inside the mouth and throat.
After 3 to 7 days, a red blotchy rash develops on the face and spreads down the body.
Can measles be treated?
There is no specific treatment for measles since it is caused by a virus. Most people fully recover within 2 or 3 weeks.
Your health care provider will likely:
- give you medication (like pain relievers) to reduce your fever
- tell you to drink plenty of fluids, eat healthy foods and get lots of rest
If you have measles, you should stay at home until 4 days after the rash appeared. This will help to limit the spread of the virus.
Speak with one of our Travel Health Specialists at least 6 weeks before your trip.
Get vaccinated if you are not yet immunized. As measles occurs in many parts of the world, travellers may be at increased risk.
The measles vaccine is part of the measles-mumps-rubella (MMR) or measles-mumps-rubella-varicella (MMRV) immunization.<|endoftext|>
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A Mile of Pi
If your class did not get a chance to see this video, please take a few minutes to watch it. Remember that irrational numbers (like Pi) really do go on forever. They don’t end after a million digits or even a billion digits.
Number Theory and Sets
I wanted to include the diagram of number sets from today’s class on the website along with clarification of what numbers are included in various sets. Don’t forget to ask if you need clarification. Questions during class usually help everyone learn!
To review some older number theory, remember that you first learned your counting numbers:
$\left\{ 1,2,3,4,5... \right\}$
Then, you added zero, and got whole numbers:
$\left\{ 0,1,2,3,4,5... \right\}$
Eventually, you added the complexity of negative numbers and got integers:
$\left\{ ...,-3,-2,-1,0,1,2,3,... \right\}$
Rational numbers include any number that can be written as the ratio of two integers such as:
$\left\{ 0,-1,37,\frac{14}{3},\frac{3}{2} \right\}$
Rational numbers when written in decimal form will always have decimals that either terminate or repeat.
$\begin{array}{l}\frac{1}{4}=0.25\\\frac{1}{6}=0.1\bar{6}\\\frac{1}{9}=0.\bar{1}\end{array}$
Irrational numbers cannot be written as a ratio of two integers. A rational number can’t be irrational, and an irrational can’t be rational. Two famous irrational numbers are $\pi$ and $e$. Irrationals are also found when taking the square root of numbers that are not perfect squares like:
$\left\{ \sqrt{27},-\sqrt{31} \right\}$
Note that it’s perfectly ok to take the negative of the square root of 37, but if you try to find the square root of negative 37, you encounter a pegasus unicorn, or an Imaginary number. You’ll learn more about those in future coursework.
Real numbers (often shown using the symbol: $\mathbb{R}$) include all of the irrational, rational, integers, whole, and counting numbers.<|endoftext|>
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kuvitia9f
2021-12-31
Evaluate the integrals.
$\int {x}^{3}{e}^{{x}^{4}}dx$
ambarakaq8
Step 1
Solution -
Given integral -
$y=\int {x}^{3}{e}^{{x}^{4}}dx$
Let,
$t={x}^{4}$
differentiating on both sides w.r.t x,
$\frac{dt}{dx}=4{x}^{3}$
$dt=4{x}^{3}dx$
$\frac{dt}{4}={x}^{3}dx$
Step 2
Now substituting these values in the given integral,
$y=\frac{1}{4}\int {e}^{t}dt$
$y=\frac{1}{4}\left[{e}^{t}\right]+C$
where C is the constant.
Philip Williams
Step 1
This problem can be solved using a u-substitution. Let $u={x}^{4}$. Then $du=4{x}^{3}dx$.
$\int {x}^{3}{e}^{{x}^{4}}dx=\int \frac{{e}^{u}}{4}du=\frac{{e}^{u}}{4}+c=\frac{{e}^{{x}^{4}}}{4}+c$
Result:
$\frac{{e}^{{x}^{4}}}{4}+c$
Vasquez
Step 1
Substitute P
$\int {x}^{3}{e}^{{x}^{4}}dx=\frac{1}{4}\int {e}^{u}du=\frac{1}{4}{e}^{u}+C$
Substitute back $u={x}^{4}$
$=\frac{1}{4}{e}^{{x}^{4}}+C$
Result:
$\frac{1}{4}{e}^{{x}^{4}}+C$
Do you have a similar question?<|endoftext|>
| 4.5 |
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Montana: Northern Cheyenne Reservation
About the Northern Cheyenne: The Northern Cheyenne originally lived in the Great Lakes Region. During the 15th century, they migrated westward, shifting their subsistence living from fishing to agriculture. The original tribe split into two bands, and the Northern Cheyenne band moved to the Plains.
History of the Reservation: The government attempted to force the Northern Cheyenne to merge with their traditional enemies the Crow, but eventually the U.S. gave the Northern Cheyenne their own reservation. An Executive Order in November of 1884 identified a tract of land west of the Tongue River in Southeastern Montana as the Northern Cheyenne Reservation. The reservation is bounded on the east by the Crow Reservation.
Life on the Reservation: The Northern Cheyenne economy is supported by farming, ranching and small businesses. The largest employers on the reservation are the federal government, tribal government, power companies and construction companies. The reservation hosts several small textile factories.
Northern Cheyenne on the map: Southeastern Montana.<|endoftext|>
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# 113 grams equals how many cups? Conversion table from 113 grams to cups for popular ingredients in baking
Converting between cups and grams is a common challenge for bakers. Baking is an art form that takes precision and practice to perfect. One small mistake can ruin an entire dish.
In this blog, Jan Cranitch will show you know to the conversion table from 113 grams to cups for popular ingredients in baking. With this knowledge, you don’t take the guesswork out of 113 grams equals how many cups. Now, let’s get started!
## 113 grams equals how many cups?
Perhaps, this is a common question that many people ask. So, have you ever wondered about this question? The answer is 0.48 cups ( the result is rounded). To put it in another way, 1 cup equals approximately 226 grams. Therefore, to convert 113 grams into cups, divide the number of grams by 226 and the result should be 0.48 cups or just under half a cup.
Remember that the exact conversion depends on the density of your ingredient. If as different ingredients have different densities.
For example, fors flour, 1 cup will weigh approximately 120 grams; however, if the ingredient is denser such as sugar or rice, then 1 cup may be around 200g. Therefore, it is important to pay attention to the weight listed in your recipe.
By understanding the conversion, you can make sure that your recipe turns out as expected. Now that you have this information, you should be able to measure ingredients with ease!
## How many practical methods are there for converting 113 grams to cups?
The answer is simple that are 2 methods including a math formula and a conversion calculator.
### Method 1: Use a math formula to convert 113 g to cups
To convert 113 grams to cups, divide the weight by 236.588 and times the density of the ingredient. Therefore, we have a specific formula as follows:
Cups = Grams / 236.588236 × ingredient density
This will give you the number of cups in 113 grams. For water, we found the ingredient density is 1 g/cm3. Therefore, when converting 113 g to cups of water, the result will be equal to 0.48 cups or you can say that the result is approximately 1/2 cup of water.
### Method 2: Convert 113 g to cups with a conversion calculator
Another practical method for converting 113 grams to cups would be to use a measurement conversion calculator. Such calculators can easily and quickly convert 113 grams to cups in a matter of seconds. With either method, the answer is 0.48164 cups. Therefore, when it comes to converting 113 grams to cups, it is simple and easy to do so either by performing the math or using a measurement calculator.
The end result is that 113 grams equal 0.48164 cups. This can be used when baking, cooking, or performing any other activity where measurements of weight and volume are necessary. This same method can be applied to convert any other weight into cups as well. Knowing how to convert grams to cups is a useful skill to have!
No matter the application, it is important to be able to accurately measure weight and volume for recipes and other tasks. Being able to convert grams to cups makes this easy.
Volume is the measure of the 3-dimensional space occupied by matter, or enclosed by a surface, measured in cubic units. The SI unit of volume is the cubic meter (m3), which is a derived unit.
Source: https://www.nist.gov/
## How many cups is 113 g of flour?
To answer this question, we first need to know how much 113 g of flour weighs. One cup of all-purpose flour (APF) weighs approximately 120g.
Therefore, it is safe to assume that 113 g of APF would make slightly less than one cup. To be more precise, it would take 3/4 cups or 11/12 cups of APF to make 113 g. It is important to note that different ingredients can have different weights, so the answer may vary depending on what you are measuring.
To be sure, it is best to consult a weight-to-volume conversion chart specific to that ingredient. Here is a conversion chart to help you easily convert 113 g of flour to cups:
## Conversion table from 113 grams to cups for popular ingredients in baking
The conversion from grams to cups varies depending on the ingredient. Therefore, when converting 113 g to cups, you need to know the specific ingredients. Understanding this pain for many people, we will point out the conversion table 113 grams to cups for common ingredients in baking below:
### 113 g to cups for a butter conversion table
Grams of Butter Cups 57 grams 1/4 cup 76 grams 1/3 cup 113 grams 1/2 cup 227 grams 1 cup
### 113 g to cups for a dry goods conversion table
Grams of dry goods Cups 16 g 1/8 c (2 tbsp) 32 g 1/4 cup 43 g 1/3 cup 64 g 1/2 cup 85 g 2/3 cup 96 g 3/4 cup 113 g 1 cup
### 113 g to cups for a rolled oats conversion table
Grams of rolled oats Cups 21 g 1/4 cup 28 g 1/3 cup 43 g 1/2 cup 85 g 1 cup 113 g 1.5 cup
### 113 g to cups for a white sugar conversion table
Grams of white sugar Cups 25 g 2 tbsp 50 g 1/4 cup 67 g 1/3 cup 100 g 1/2 cup 113 g 2/3 cup 150 g 3/4 cup 201 g 1 cup
### 113 g to cups for a packed brown sugar conversion table
Grams of packed brown sugar Cups 55 g 1/4 cup 73 g 1/3 cup 110 g 1/2 cup 113 g 2/3 cup 220 g 1 cup
### 113 g to cups for a honey, molasses & syrupconversion table
Grams of honey, molasses & syrup Cups 43 g 2 tbsp 85 g 1/4 cup 113 g 1/3 cup 170 g 1/2 cup 227 g 2/3 cup 255 g 3/4 cup 340 g 1 cup
### 113 g to cups for a milkconversion table
Grams of milk Cups 113 g 1/2 cup 143 g 3/4 cup 284 g 1 cup
### 113 g to cups for a vegetable oilconversion table
Grams of vegetable oil Cups 113 g 1/2 cup 227 g 1 cup 454 g 2 cup
### 113 g to cups for a cocoa powderconversion table
Grams of cocoa powder Cups 18 g 1/4 cup 27 g 1/3 cup 45 g 1/2 cup 113 g 1 1/8 cup
## Crucial note should remember to convert 113 g to cups
When it comes to baking, weighing out dry ingredients (like flour, butter, cocoa powder, etc) by gram yields more precise results than traditional measurements like cups. 113 grams is a good starting point for most recipes; however, bear in mind that due to factors such as room temperature and ingredient quality, this could vary slightly. Since the weight measurements are precise, you can rest easy knowing that you’ll always get tasty dishes.
In addition, it’s important to keep in mind that 113 grams are not equivalent to one cup. To convert grams (g) into cups, divide the number of grams by 236.588 Therefore, 113 g would be equal to 0.48 cups. This is an easy way to ensure your recipes are always accurate and delicious!
## FAQs: 113 grams equals how many cups?
### How to convert 113 grams to cups of chocolate?
To convert 113 grams of chocolate into cups, you first need to know the density of the particular type of chocolate. Depending on the type of chocolate – dark, milk, or white – its density will vary. Generally speaking, a cup of chocolate chips weighs around 150-180 grams.
Taking this into consideration and for the purpose of this tutorial, we can assume a cup of chocolate chips weigh 160 grams. Therefore, 113 grams of chocolate is equal to 0.446 cups of chocolate chips.
### What are 113 grams in cups of cottage cheese?
There are 1/2 cups of cottage cheese in 113 grams.
### How do you convert 113 grams to ml?
To convert 113 grams to milliliters, you will need to know the density of the ingredient in question. Take the density of the ingredient (in g/ml) and multiply it by 113. The result is the equivalent amount in ml. For water, 113 grams equals precisely 113 ml.
### How many oz are in 113 grams?
113 grams equal 3.9859577003 ounces.
## Conclusion
Ukpfna.com hopes you found this conversion table from grams to cups for popular ingredients in baking helpful your answer of 113 grams equals how many cups. While there are many factors that can affect the outcome of your bake, using accurate measurements is a great place to start.
So, if you need to convert 113 g to cups, don’t miss our article. Baking is an exact science so be sure to measure carefully and enjoy the process!<|endoftext|>
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# Lesson 9-1: Area of 2-D Shapes 1 Lesson 9-1 Area of 2-D Shapes.
## Presentation on theme: "Lesson 9-1: Area of 2-D Shapes 1 Lesson 9-1 Area of 2-D Shapes."— Presentation transcript:
Lesson 9-1: Area of 2-D Shapes 1 Lesson 9-1 Area of 2-D Shapes
Lesson 9-1: Area of 2-D Shapes 2 Squares and Rectangles s s A = s² 6 6 A = 6² = 36 sq. units L W A = LW 12 5 A = 12 x 5 = 60 sq. units Example: Area of Rectangle: A = LW Area of Square: A = s²
Lesson 9-1: Area of 2-D Shapes 3 Circles and Sectors r 9 cm A = (9)² = 81 sq. cm Area of Circle: A = r² arc r B C A 120° Example: 9 cm
Lesson 9-1: Area of 2-D Shapes 4 Triangles and Trapezoids h h h bb b1b1 b2b2 h is the distance from a vertex of the triangle perpendicular to the opposite side. h is the distance from b1 to b2, perpendicular to each base
Lesson 9-1: Area of 2-D Shapes 5 Example: Triangles and Trapezoids 7 6 8 12 6
Lesson 9-1: Area of 2-D Shapes 6 Parallelograms & Rhombi Area of Parallelogram: A = b h 6 9 A = 9 x 6 = 54 sq. units 8 10 A = ½ (8)(10) = 40 sq units h b Example:
Lesson 9-1: Area of 2-D Shapes 7 Area of Regions 8 10 12 414 8 The area of a region is the sum of all of its non-overlapping parts. A = ½(8)(10) A= 40 A = (12)(10) A= 120 A = (4)(8) A=32 A = (14)(8) A=112 Area = 40 + 120 + 32 + 112 = 304 sq. units
Lesson 9-1: Area of 2-D Shapes 8 Areas of Regular Polygons Perimeter = (6)(8) = 48 apothem = Area = ½ (48)( ) = sq. units 8 If a regular polygon has an area of A square units, a perimeter of P units, and an apothem of a units, then A = ½ (a)(p).
Download ppt "Lesson 9-1: Area of 2-D Shapes 1 Lesson 9-1 Area of 2-D Shapes."
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The far side of the moon: What is it, why we might grow potatoes there
Multiple spacecrafts have touched down on the near side of the moon, which faces Earth, but this is the first-ever landing on the dark side. USA TODAY
It's sometimes referred to as the "dark side of the moon." Sorry, Pink Floyd fans, but there is plenty of light on what is typically known as the moon's far side.
On Thursday, China's growing space program said it landed a spacecraft on the far side of the moon for the first time ever.
"This is a first for humanity and an impressive accomplishment," wrote NASA administrator Jim Bridenstine on Twitter.
A photo taken by the spacecraft shows a small crater and barren surface, and there's a lot more researchers hope to learn about this elusive spot on the moon.
Here's what we know about the moon's so-called dark side:
Why do we have a far side of the moon?
According to astronomers at Cornell University, the moon is tidally locked to earth, which means it exerts tidal forces strong enough to cause the moon to only show one side as it orbits. Also, there is no such thing as a dark side of the moon, says NASA. Both sides are alternately lit as it rotates.
How much do we know about the far side?
Not a lot, which is why astronomers are so excited about China landing a spacecraft in the area. In 1959, the Soviet Luna 3 spacecraft shared the first images of the moon's far side, said NASA. Since then, several NASA missions have captured additional images with more detail. In 2015, a camera aboard the Deep Space Climate Observatory (DSCOVR) satellite captured a view of the fully-illuminated far side.
What makes it different?
The far side features far more craters, some of which are the size of small countries, reports Space.com. The side we see shows more basaltic plains called "mare," said the report, which were created by volcanic activity billions of years ago.
Wait, we might grow potatoes on the moon?!?!
The BBC reports the Chinese lander is carrying six live species from Earth: cotton, rapeseed, fruit fly, yeast, a flowering plant named arabidopsis and potatoes. The goal is to try to form a mini-biosphere.
Follow Brett Molina on Twitter: @brettmolina23<|endoftext|>
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The Fifteenth Amendment to the United States Constitution was ratified in 1870, just a few years after the end of the Civil War. This Amendment prohibits both federal and state governments from infringing on a citizen’s right to vote “on account of race, color, or previous condition of servitude.” The Fifteenth Amendment is the third of three “Reconstruction Amendments” ratified in the aftermath of the Civil War. The other two are the Thirteenth Amendment that abolished slavery, and the 14th Amendment granted citizenship to all persons, “born or naturalized in the United States.”
Prior to the Fifteenth Amendment, the states were empowered to set the qualifications for the right to vote. The Fifteenth Amendment essentially transferred this power to the federal government. Its ratification, however, had little effect for nearly a century. It had practically no effect in southern states, which devised numerous ways such as poll taxes and grandfather clauses to keep blacks from voting. Over time, federal laws and Supreme Court judicial opinions eventually struck down voting restrictions for blacks. Eventually, Congress passed the Civil Rights Act of 1957 which established a commission to investigate voting discrimination. And in 1965 the Voting Rights Act was passed to increase black voter registration by empowering the Justice Department to closely monitor voting qualifications.<|endoftext|>
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Keratoconus – Cross Linking
What is Keratoconus?
Keratoconus is the leading cause of blindness especially among the youth in East Africa.
A normal cornea (see definition of cornea below) is shaped like a watch glass, but sometimes it starts thinning and bulging; this is known as Keratoconus (Kerato meaning Cornea and Conus like a cone).
In such cases, the cornea cannot perform its main function of focusing light rays entering the eye on to the retina due to its distorted shape.
Eye with Keratoconus
Here the middle part of the cornea becomes weak and thin. As a result, it cannot withstand the pressure of the eye causing the cornea to bulge.
The exact cause of Keratoconus is not known, but it is believed that it could be due to a malfunction in the enzymes, which make the stroma (please see cross-section of the cornea) weak. It could also be associated with allergy and there is some evidence that it may be hereditary.
Keratoconus is also linked with over exposure to sunlight, improper contact lens fitting and constant eye rubbing. If there is progressive stretching in the Descemet’s membrane, it leads to rupturing of the membrane and can cause hydrops, where the cornea becomes completely cloudy as shown in the picture below.
Cross-section of the cornea
Hydrops – Here the cornea has become competently cloudy
- Diagnosis of Keratoconus is done through Topography and Tomography, which is a gold standard for Keratoconus but your optician, will suspect Keratoconus if there are following changes:-
- Frequently change in the astigmatism of the eye and despite of giving the correct glasses the vision of the quality is not very good and there is difference in glasses prescription between your two eyes.
Topography showing Normal cornea
Topography showing Keratoconus
Tomography with the cutting edge Precisio machine
The Topography Machine
Precisio Showing Keratoconus
Diagnosis of Keratoconus by Epithelial Mapping
Optical Coherence Tomography test: This test detects early signs of Keratoconus
Normal Tomography test
Tomography showing early signs of Keratoconus
At Laser Eye Centre we have the following protocol for treating Keratoconus
- Early to moderate stages we do central corneal regularization followed by Cross-Linking
- Fairly advanced stage only Cross-Linking
- Very advanced stage Corneal Transplant
CENTRAL CORNEAL REGULARIZATION FOLLOWED BY CROSS-LINKING
- FIRST STAGE CENTRAL CORNEAL REGULARIZATION
This is done by reshaping the cornea; we reduce the size of the cone by very specialized laser which takes only 34 seconds.
- SECOND STAGE COLLAGEN CROSSLINKING:
This is the latest treatment for Keratoconus and was invented by Prof Seiler (Switzerland), and Prof Spoerl (Germany). Cross-Linking has become the standard treatment to treat Keratoconus
- FIRST STAGE CENTRAL CORNEAL REGULARIZATION
CENTRAL CORNEAL REGULARIZATION
- Keratoconus gives rise to irregular irregular Astigmatism and patient’s vision is not improving with glasses and because of atopic conjunctiva contact lenses is a challenge.
- The principle of Corneal Regularization is to reshape the cornea and make the Keratoconus cone as flat as possible and convert the irregular irregular Astigmatism into regular Astigmatism.
- In CCR, the surgeon treats the Keratoconus cone by shifting the cone in the Centre.
- With 1000 Hz frequency, IVIS laser will reshape the top of the cone and the surrounding area.
- This is achieved by performing laser on the side of the cornea which is thicker. Therefore, not damaging the cornea that is already thin.
CENTRAL CORNEAL REGULARIZATION IS DONE IN 34 SECONDS.
CENTRAL CORNEAL REGULARIZATION IS ONLY AVAILABLE AT LASER EYE CENTRE
HOW IS IT DONE?
Checking the mapping of the cornea and diagnosing Keratoconus
Exporting data on to the laser machine
Step 3 :
Central Corneal Regularization in progress
Clinical Results Before & After Central Corneal Regularization
What is Cross linking?
- This is a mini-invasive treatment, which has been proven to strengthen the weak corneal tissues.
- Riboflavin eye drops (vitamin B12) are applied to the patient’s eye for half an hour.
- The Riboflavin is then activated by Ultra Violet light. An interaction between the ultra violet light and the Riboflavin soaked cornea leads to the strengthening and stiffening of the collagen fibers in the cornea.
- This does not allow Keratoconus to progress further. In our clinical practice, cross linking is helpful for 90% of Keratoconus patients but it does not work if the Keratoconus is very advanced.
- The standard length of treatment is 10 minutes.
IF KERATOCONUS IS DIAGNOSED AT AN EARLY STAGE, CROSS-LINKING IS THE TREATMENT OF CHOICE AS IT IS NON–INVASIVE AND RELATIVELY SIMPLE.
New breakthrough Cross linking treatment:
- At Laser Eye Center, it is now possible to reduce the cross-linking procedure time to 5 minutes.
- We can do this by incorporating our C-Ten (1000Htz) laser and the CCL-365 platform.
- The overall result is better vision than cross-linking alone in a fraction of the time using cutting edge and safe technology.
Corneal Strip Before and After Cross linking
After cross linking the cornea has become stiff
CROSS-LINKING IN PROGRESS
Examining Patient before starting Cross-Linking
Instilling Riboflavin eye drops to the patient’s eyes
Cross-Linking Treatment in Progress performed by Dr. Mukesh Joshi
CCL 365 10 minute cross linking with 18mW/cm3
UV radiation Light
WHO CAN DO CROSS-LINKING?Any ophthalmologist can do Cross-Linking as long as they have been certified to perform Cross-Linking
Side Effects of Cross-linking:-
- If Crosslinking has not been done by an experienced doctor, chances are the crosslinking may not be accurate.
- 90% of crosslinking’s have been successful; however the 10% may require repeating the procedure or may lead to corneal opacity crosslinking.
Spectacles and Contact Lenses:
- With spectacles, vision cannot be corrected due to the irregular shape of the cornea.
- A hard contact lens (rigid and gas permeable) is sometimes a good solution. Here as the lens is made of a hard material it helps to correct the irregular shape of Keratoconus.
- Fitting of these specialised lenses is delicate and time consuming.
- Over time, it can give rise to progressive thinning of the cornea and patients with allergies may have difficulty in contact lens fitting.
If the fitting of contact lens is guided with the help of Topography, the results are excellent.
CCL-365 10 Minute Cross-Linking
Laser Eye Centre is the first centre in Asia and Africa to offer cross linking treatment in any form and is also the first and only centre to offer the 10 minute Cross-Linking / CCL-365 platform.
- Dr. Joshi was appointed as a global expert on Keratoconus during the 5th International meeting European Society of Cataract and Refractive Surgeons (Germany 2010).
- Dr.Joshi has also been invited by various international societies to share his experience on crosslinking; we at Laser Eye centre were the first to start crosslinking in Africa and Asia.
- An ophthalmologist must undergo training for crosslinking certification so that he/she can do this treatment.
In December 2014, Dr. Joshi was invited as a guest speaker alongside with Prof. Theo Seiler to chair a session at the 10th International Cross-Linking Society meeting.
Corneal Transplant / Keratoplasty
- Corneal Transplant (also known as Keratoplasty) is usually carried out if the patient’s Keratoconus is at a very advanced stage where contact lenses or cross-linking are not possible.
- The best solution to such cases would be corneal transplant or Keratoplasty (please see the Keratoplasty section of the website).<|endoftext|>
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# 12 /02/2014 Properties of Special Parallelograms - PowerPoint PPT Presentation
12 /02/2014 Properties of Special Parallelograms. Pg 420. Warm Up Solve for x . 1. 16 x – 3 = 12 x + 13 2. 2 x – 4 = 90 ABCD is a parallelogram. Find each measure. 3. CD 4. m C. 4. 47. 104°. 14. Objectives. Prove and apply properties of rectangles, rhombuses, and squares.
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### 12/02/2014 Properties of Special Parallelograms
Pg 420
Solve for x.
1.16x – 3 = 12x + 13
2. 2x – 4 = 90
ABCD is a parallelogram. Find each measure.
3.CD4. mC
4
47
104°
14
Prove and apply properties of rectangles, rhombuses, and squares.
Use properties of rectangles, rhombuses, and squares to solve problems.
rectangle
rhombus
square
A second type of special quadrilateral is a rectangle. A rectangleis a quadrilateral with four right angles.
Since a rectangle is a parallelogram by Theorem 6-4-1, a rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2.
rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2.diags. bisect each other
Example 1: Craft Application
A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM.
Rect. diags.
KM = JL = 86
Def. of segs.
Substitute and simplify.
Check It Out! rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2. Example 1a
Carpentry The rectangular gate has diagonal braces.
Find HJ.
Rect. diags.
HJ = GK = 48
Def. of segs.
Check It Out! rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2. Example 1b
Carpentry The rectangular gate has diagonal braces.
Find HK.
Rect. diags.
Rect. diagonals bisect each other
JL = LG
Def. of segs.
JG = 2JL = 2(30.8) = 61.6
Substitute and simplify.
A rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2.rhombus is another special quadrilateral. A rhombusis a quadrilateral with four congruent sides.
Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses.
Example 2A: Using Properties of Rhombuses to Find Measures apply the properties of parallelograms to rhombuses.
TVWX is a rhombus. Find TV.
WV = XT
Def. of rhombus
13b – 9=3b + 4
Substitute given values.
10b =13
Subtract 3b from both sides and add 9 to both sides.
b =1.3
Divide both sides by 10.
Example 2A Continued apply the properties of parallelograms to rhombuses.
TV = XT
Def. of rhombus
Substitute 3b + 4 for XT.
TV =3b + 4
TV =3(1.3)+ 4 = 7.9
Substitute 1.3 for b and simplify.
Example 2B: Using Properties of Rhombuses to Find Measures apply the properties of parallelograms to rhombuses.
TVWX is a rhombus. Find mVTZ.
mVZT =90°
Rhombus diag.
Substitute 14a + 20 for mVTZ.
14a + 20=90°
Subtract 20 from both sides and divide both sides by 14.
a=5
Example 2B Continued apply the properties of parallelograms to rhombuses.
Rhombus each diag. bisects opp. s
mVTZ =mZTX
mVTZ =(5a – 5)°
Substitute 5a – 5 for mVTZ.
mVTZ =[5(5) – 5)]°
= 20°
Substitute 5 for a and simplify.
Check It Out! apply the properties of parallelograms to rhombuses. Example 2a
CDFG is a rhombus. Find CD.
CG = GF
Def. of rhombus
5a =3a + 17
Substitute
a =8.5
Simplify
GF = 3a + 17=42.5
Substitute
CD = GF
Def. of rhombus
CD = 42.5
Substitute
Check It Out! apply the properties of parallelograms to rhombuses. Example 2b
CDFG is a rhombus.
Find the measure.
mGCH if mGCD = (b + 3)°
and mCDF = (6b – 40)°
Def. of rhombus
mGCD + mCDF = 180°
b + 3 + 6b –40 = 180°
Substitute.
7b = 217°
Simplify.
b = 31°
Divide both sides by 7.
Check It Out! apply the properties of parallelograms to rhombuses. Example 2b Continued
mGCH + mHCD = mGCD
Rhombus each diag. bisects opp. s
2mGCH = mGCD
Substitute.
2mGCH = (b + 3)
Substitute.
2mGCH = (31 + 3)
Simplify and divide both sides by 2.
mGCH = 17°
A apply the properties of parallelograms to rhombuses.square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three.
Helpful Hint apply the properties of parallelograms to rhombuses.
Rectangles, rhombuses, and squares are sometimes referred to as special parallelograms.
Example 3: Verifying Properties of Squares apply the properties of parallelograms to rhombuses.
Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other.
Step 1 apply the properties of parallelograms to rhombuses. Show that EG and FH are congruent.
Since EG = FH,
Example 3 Continued
Step 2 apply the properties of parallelograms to rhombuses. Show that EG and FH are perpendicular.
Since ,
Example 3 Continued
Step 3 apply the properties of parallelograms to rhombuses. Show that EG and FH are bisect each other.
Since EG and FH have the same midpoint, they bisect each other.
Example 3 Continued
The diagonals are congruent perpendicular bisectors of each other.
SV = TW = 122 so, SV apply the properties of parallelograms to rhombuses.@ TW .
1
slope of SV =
11
slope of TW = –11
SV ^ TW
Check It Out! Example 3
The vertices of square STVW are S(–5, –4), T(0, 2), V(6, –3) , and W(1, –9) . Show that the diagonals of square STVW are congruent perpendicular bisectors of each other.
Step 1 apply the properties of parallelograms to rhombuses. Show that SV and TW are congruent.
Since SV = TW,
Check It Out! Example 3 Continued
Step 2 apply the properties of parallelograms to rhombuses. Show that SV and TW are perpendicular.
Since
Check It Out! Example 3 Continued
Step 3 apply the properties of parallelograms to rhombuses. Show that SV and TW bisect each other.
Since SV and TW have the same midpoint, they bisect each other.
Check It Out! Example 3 Continued
The diagonals are congruent perpendicular bisectors of each other.
Given: ABCD is a rhombus. E is the midpoint of , and F is the midpoint of .
Prove: AEFD is a parallelogram.
|| Proofs
Example 4 Continued
Check It Out! Proofs Example 4
Given: PQTS is a rhombus with diagonal
Prove:
2. Proofs
4.
5.
6.
7.
Check It Out! Example 4 Continued
1. PQTSis a rhombus.
1. Given.
2. Rhombus → each
diag. bisects opp. s
3. QPR SPR
3. Def. of bisector.
4. Def. of rhombus.
5. Reflex. Prop. of
6. SAS
7. CPCTC
Lesson Quiz: Part I Proofs
A slab of concrete is poured with diagonal spacers. In rectangle CNRT, CN = 35 ft, and NT = 58 ft. Find each length.
1.TR2.CE
35 ft
29 ft
Lesson Quiz: Part II Proofs
PQRS is a rhombus. Find each measure.
3.QP4. mQRP
42
51°
Lesson Quiz: Part III Proofs
5. The vertices of square ABCD are A(1, 3), B(3, 2), C(4, 4), and D(2, 5). Show that its diagonals are congruent perpendicular bisectors of each other.
6. ProofsGiven:ABCD is a rhombus.
Prove:
Lesson Quiz: Part IV<|endoftext|>
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… Found in the parched Sonoran desert of southern Arizona and northern Mexico, the senita moth depends on a single plant species — the senita cactus — both for its food and for a place to lay eggs. The senita cactus is equally dependent upon the moth, the only species that pollinates its flowers. Senita cacti and senita moths have a rare, mutually dependent relationship, one of only three known dependencies in which an insect actively pollinates flowers for the purpose of assuring a food resource for its offspring.. . .
The problem is that the moths lay their eggs inside the cacti’s flowers immediately after pollination, and when the eggs hatch the moth larvae eat the fruit, destroying the flowers’ chances to produce seeds. Historic theory predicts extreme ecological instability for this relationship; as moth populations increase, more flowers are destroyed, fewer new cacti appear, and the spiral continues until both species disappear.
Yet that hasn’t happened, and Holland, assistant professor of ecology and evolutionary biology, spends several months each year observing moths and cacti in the Mexican desert to document why.
Many orchids have also developed relationships like this. Most orchid flowers have evolved to attract a very specific insect. I hadn’t realized some cactus did this as well.
Which came first, the moth or the cactus?<|endoftext|>
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# Waves/Waves in One Dimension
Waves : 1 Dimensional Waves
1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13
Examples - Problems - Solutions - Terminology
## The Mathematics Of Waves
We start our discussion of waves by taking the equation for a very simple wave and describing its characteristics. The basic equation for such a wave is
${\displaystyle y=a\ \sin \left({\frac {2\pi x}{\lambda }}-2\pi ft+\alpha \right)}$
where ${\displaystyle y}$ is the height of the wave at position ${\displaystyle x}$ and time ${\displaystyle t}$. This equation describes a fairly simple wave, but most complex waves are just sums of simpler ones. If we freeze this equation in time at ${\displaystyle t=0}$, we get
${\displaystyle y=a\ \sin \left({\frac {2\pi x}{\lambda }}+\alpha \right)}$
which looks like this: [TODO - Add a Graph]
From the graph we can see that each of the three parameters has a meaning. ${\displaystyle a}$ is the amplitude of the wave, how high it is. ${\displaystyle \lambda }$ is the wavelength, the distance from a part of the wave in one cycle to the same part of the wave in the next cycle. ${\displaystyle \alpha }$ is the phase of the wave, which shifts the wave to the left or right. The wavelength is a distance, and is usually measured in meters, millimeters or even nanometers depending on the wave. Phase is an angle, measured in radians.
Now that we have mapped out the wave in space, let's instead set ${\displaystyle x=0}$ and see how the wave changes over time
${\displaystyle y=a\ \sin(-2\pi ft+\alpha )}$
Amplitude ${\displaystyle a}$ and phase ${\displaystyle \alpha }$ remain, but the wavelength is gone and a new quantity has appeared: ${\displaystyle f}$, which is the frequency, or how rapidly the wave moves up and down. Frequency is measured in units of inverse time: in a fixed period of time, how many times does the wave move up and down? The unit usually used for this is the hertz, or inverse second.
Now let's combine these two pictures and see how the wave moves. Figure 3 is a diagram of how the wave looks when you plot it in both space and time. The straight lines are the places where the simple wave reaches a maximum, minimum, or zero (where it crosses the x axis).
We can look at the zeros to determine the phase velocity of the wave. The phase velocity is how fast a part of the wave moves. We can think of it as the speed of the wave, but for more complicated waves it is only one type of speed - more on that in later sections.
We can get an equation for the zeros by setting our equation to zero.
${\displaystyle 0=a\ \sin \left({\frac {2\pi x}{\lambda }}-2\pi ft+\alpha \right)}$
${\displaystyle 0={\frac {2\pi x}{\lambda }}-2\pi ft+\alpha }$
${\displaystyle x=f\lambda t-{\frac {\alpha \lambda }{2\pi }}}$
You see here that we have the equation for a straight line, describing a point that is moving at velocity ${\displaystyle f\lambda }$. This gives us the equation for the phase velocity of the wave, which is
${\displaystyle {\mbox{velocity}}={\mbox{frequency}}\times {\mbox{wavelength}}\quad v=f\lambda }$
Waves : 1 Dimensional Waves
1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13
Examples - Problems - Solutions - Terminology<|endoftext|>
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# How do you integrate int e^(-x)ln 3x dx using integration by parts?
Feb 5, 2018
color(blue)( int " " e^(-x)" "ln (3x)" " dx = -e^(-x) ln(3x) - E_1(x)+C
#### Explanation:
Given:
color(green)( int " " e^(-x)" "ln (3x)" " dx
Integration by Parts Method must be used to solve the problem.
The formula is
color(brown)(int f*g' = f*g-int f'*g
For our problem: $\int \text{ " e^(-x)" "ln (3x)" } \mathrm{dx} ,$
color(red)(f = ln(3x) and g'=(e^-x)
color(green)(Step.1
We will differentiate:
color(red)(d/(dx)[ ln(3x)]
$\Rightarrow \left[\frac{1}{3 x}\right] \cdot \frac{d}{\mathrm{dx}} \left[3 x\right]$
$\Rightarrow \frac{3 \cdot \frac{d}{\mathrm{dx}} \left[x\right]}{3 x}$
$\Rightarrow \frac{1}{x}$
color(brown)[:.d/(dx)[ ln(3x)] = 1/x
color(green)(Step.2
We will next integrate color(red)(e^(-x)*dx
color(red)(int " "e^(-x)*dx
Substitute color(green)(u = -x
$\Rightarrow \mathrm{dx} = - \mathrm{du}$
$\Rightarrow - \int \text{ } {e}^{u} \cdot \mathrm{du}$
$\Rightarrow - {e}^{u}$
$\Rightarrow - {e}^{- x}$ Substitute back $u = - x$
$\therefore \int \text{ } {e}^{- x} \cdot \mathrm{dx} = - {e}^{- x} + C$
color(green)(Step.3
We are Given:
color(green)( int " " e^(-x)" "ln (3x)" " dx
Refer to the formula
color(brown)(int f*g' = f*g-int f'*g
Now we can write our final solution as:
$\Rightarrow - \int - \frac{{e}^{- x}}{x} \mathrm{dx} \cdot - {e}^{- x} \cdot \ln \left(3 x\right)$
Note that,
$\int \text{ "-(e^(-x))/x " } \mathrm{dx}$
is a special integral and is also an exponential integral
This can be written as color(red)(E_1 (x)
Hence, our final solution can be re-written as
color(blue)( int " " e^(-x)" "ln (3x)" " dx = -e^(-x) ln(3x) - E_1(x)+C<|endoftext|>
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# Links from the Interview to Mathematics Developmental Continuum - Working Mathematically
From Term 1 2017, Victorian government and Catholic schools will use the new Victorian Curriculum F-10. Curriculum related information is currently being reviewed and may be subject to change.
The Victorian Curriculum F–10 - VCAA
When a student is successful on these questions, learning can be continued using suggested activities such as these:
## Working mathematically
When student is successful on:
• First year of schooling detour' Q II (b), (c), (d)
Identify, copy, continue a pattern of coloured teddies
Simple Patterns (0.5):
• Activity 1: Sorting
• Activity 2: Forming and extending patterns
When student is successful on:
• Question 1
Estimate and count teddies scooped into a cup (at least 20)
Making better estimates (1.0):
• Activity 1: How many marbles?
• Activity 3: Estimating calculations
When student is successful on:
• Question 9 (a) and (b)
Use a calculator to record and say numbers
Using a calculator (1.5):
• Activity 2: Supporting learning of place value
When student is successful on:
• Question 2 (a) (b)
Count by 1s
• Question 8 (a)
• Question 9 (a) (b)
Use calculator to record and say numbers to 2 digits
• Question 10 (a) (b)
Order 1 and 2 digit sets of number cards
• Question 11
Bundling task - Make 36 using bundles of 10 sticks and single sticks
• Question 12
Identify missing number in 100 chart
• Question 18
I have 9 teddies and you have 4, how many teddies do we have altogether? (using counting all strategy)
• Question 21
Basic strategies (using counting on strategy)
Recognise and use pattern (1.5):
• Activity 1: Using the repeat function on the calculator
• Activity 3: Patterns in a modified hundreds chart
• Activity 4: What is my counting pattern?
When student is successful on:
• Question 18
I have 9 teddies and you have 4, how many teddies do we have altogether? (using counting on strategy)
• Question 21
Basic strategies (using counting on strategy)
• Question 28
Sharing teddies on a mat (by sharing using skip counting or known fact)
• Question 29
Tennis balls (by known fact or skip counting)
• Question 30
Dots arrays partly hidden (by known fact or skip counting)
Using diagrams and models (2.75):
• Activity 2: Empty number line: explaining with a model
• Activity 4: Multiplication of tens and ones: a mental image for an algorithm<|endoftext|>
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## Sieve of Eratosthenes
This is a simple and ancient algorithm for finding all prime numbers up to a given number, say n. Although it is a very, very old algorithm, it works fairly efficiently for numbers under 10 million. As the name suggests, this algorithm was created by a Greek mathematician by the name of Eratosthenes…
The following is the algorithm he suggested:
Simply make a list of consecutive numbers from 2 up to the number n. Now, take each number from the front of the list (first 2, then 3 and so on) and remove the multiples of each of these numbers from the rest of the list starting from the square of the number. At the end of this procedure, all remaining numbers are prime…
Therefore, in essence, we keep removing elements over and over using one number at a time to choose the numbers that must be removed.
Here’s a working version of the algorithm. Say we want all prime numbers from 1 to 20.
Step 1: Make a list of all numbers from 2 to 20
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Step 2: Take the first number i.e. 2 and remove all its multiples from the list starting from 4.
2 3 5 7 9 11 13 15 17 19
Step 2: Take the next number (2nd number) i.e. 3 and remove all its multiples from the list starting from 9
2 3 5 7 11 13 17 19
Step 3: Take the next number (3rd number) i.e. 5 and remove all its multiples from the list starting from 25. But, since 25 is greater than our n = 20, the process stops at this step.
Therefore, our final list of prime numbers is:
2 3 5 7 11 13 17 19
And yay! that’s the right answer… and computationally easy and efficient!
## Squaring Numbers Close to 100
So, here’s another math trick and this one may be a little less intuitive to follow. It’s about squaring numbers close to 100 without that piece of machine called the calculator.
Example: Squaring 108 i.e. Calculating (108) * (108)
Now, we know that 108 differs from 100 by a +8. We make use of this and add this 8 to the number we have.
108+8 = 116
Now, we leave 116 as the left most digits of the answer and take the square of just 8. We get 8*8 = 64.
Therefore, our answer is : 11664
Damn, I love math tricks.
## Lavenshtein Distance – Distance Between Words
I recently had a conversation with a friend about the distance between words. Yes, it sounds absurd but it is a rather interesting concept and ended up being my “CS fact of the day”
Lavenshtein Distance is the most commonly used technique to determine the distance between two words and it basically measures the minimum number of changes required to transform one word into the other.It can be used to check for possible spelling errors etc. by minimizing the distance.
Let’s say word A is ‘meteor’ and word B is ‘enter’. There isn’t any spelling mistake here per se, but we can still measure the distance.
Transforming A into B involves identifying the minimum number of changes. In this case, it is LD(A,B) = 3.
This is because we can make the following changes:
meteor
eteor (delete m)
enteor (insert n)
enter (delete o)
There are obviously infinite number of sequences that will go from A to B, but the objective here is to calculate the minimum one (sort of like the linear distance concept because linear distance is the shortest)
Interestingly enough, this LD (or Lavenshtein Distance) happens to form a metric space. That is, if LD(A, B) = x and LD(B, C) = y, then LD(A, C) = z and (x+y) >= z. This is basically the triangle inequality. So, it enables you to build spacial partitioning trees.
Pretty neat, huh?
So, there’s more than a difference between words… there is a distance… 😀
## Squaring of Numbers Close to 100
So, here’s just another math trick:
Suppose, we have a number like 106 and we want to square it. So,we’re taking 100 as the number we compare to and 106 = 100 + (6)
Now, we add 6 to 106. This gets us 106+6 = 112
Now, we keep 112 as the leftmost three digits of the answer.
We square 6 and get 6*6 = 36.
36 will be the right most 2 digits of the answer.
Now, we’ll try the same thing with a number less than 100. Let’s take 92 for an example.
100 + (-8) = 92 So, the difference from 100 is that of (-8). So this is the number we will play with.
We add -8 to 92 : 92 + (-8) = 84. Now, we take these as the left most digits and then add the square of (-8) as the right more TWO digits.
This is applicable for numbers much greater than 110 as well. But, with a difference.
Suppose we take 111 and try to square it. We will need to add 11 to it first and we get 122.
For the next step when we square 11, we get 121. But, we can only take upto 2 right-most digits. So:
122 and 121 combined together when we can only take 2 digits from 121 gives us:
122 + 1 (from the 121 carrying over) and add the 21 to the right. = 12321
## Squaring Numbers Ending With 5
Here’s a really easy way to calculate the squares of numbers ending with a 5… suppose we are taking the square of 85:
85 * 85 = 7225
Now, the easiest way of doing this is by taking the part of the number without the last “5” in it.. which in this example would be simply 8. Now, multiply 8 by the next number, which is 9. We get a 72… just add a 25 at the end of it and you get the answer: 7225
Therefore, as a general equation we have:
105 * 105 = (10 * 11) with a 25 at the end = 11025<|endoftext|>
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There are a variety of eating disorders, including anorexia nervosa, bulimia nervosa, binge eating disorder, pica, and avoidant/restrictive food intake disorder. They range in exact behaviors, but all are behavior disorders also linked to mental health issues like depression, anxiety, and obsessive-compulsive disorder.
Definition & Facts
Eating disorders are mental disorders that have negative effects on a person's mental, emotional, and physical health. Bulimia is a condition where a person eats a large amount of food (called binging) and then takes drastic action to avoid weight gain caused by these binges. Common strategies for bulimics include vomiting, taking laxatives, excessive exercise and restricting food (fasting).
Anorexia nervosa is a condition where a person eats very little and maintains an unnaturally low body weight. Binge eating disorder is when a person eats a large amount of food in a short time, but does not purge like bulimics.
Symptoms & Complaints
Patients with anorexia nervosa suffer a variety of symptoms both while sick and after recovery. Low bone density can lead to broken bones or osteoporosis later in life. Anorexics complain of feeling cold, even in warm settings. Low body weight can lead to lanugo which is the growth of fine hairs all over the body in the body's attempt to stay warm.
People with bulimia can face a variety of gastrointestinal problems from forced vomiting. This can range from damage to teeth from stomach acid, to fatal esophagus tears, and stomach ruptures while binging. Repeated binging and purging can cause an imbalance of electrolytes in the body, causing fatal heart attacks.
Those with anorexia will be underweight for their height and age, while those with bulimia can be underweight, normal weight, or overweight. Obesity characterizes about half of those with binge eating disorder.
There is no single known cause of eating disorders in patients. These conditions are co-morbid with conditions like depression and anxiety, which also do not have definitive causes. Cultural factors are often cited, such as the cultural emphasis on thinness and small waistlines.
While women are often depicted as the population that deals with eating disorders, men also experience eating disorders. They are especially common among college athletes, or those participating in sports with a large emphasis on speed or light body weight.
Additionally, poor self-esteem and body image issues are a large contributor. These can stem from peer pressure, internal emotional issues, or poor parenting. Social pressure isn't the only cause. There is evidence that a propensity towards eating disorders may be genetic.
Diagnosis & Tests
There are a variety of tests used in the Diagnostic and Statistical Manual of Mental Disorders (DSM) to diagnose eating disorders. Due to the psychological nature of eating disorders, these diagnoses are determined primarily through the behaviors and thoughts expressed by the patients. Other mental health issues such as depression, anxiety, or substance abuse disorder are commonly diagnosed among the affected patients.
The DSM defines bulimia as a condition where the patient consumes a large amount of food in a discrete period of time, often with a sense of being "out of control" during these episodes. This is then followed by an inappropriate compensatory behavior to avoid weight gain (purging, compulsive exercise).
The DSM criteria for anorexia nervosa includes a patient's refusal or inability to maintain an adequate body weight in the context of age and sex. Additionally, the patient will have an intense fear of gaining weight or increased body fat. Previous iterations of the DSM included a loss of menstruation for women as a diagnostic criterion; however, the most recent edition of the DSM has dropped this qualification.
Treatment & Therapy
There are a variety of treatments and therapy approaches for addressing eating disorders. Psychiatric medication is often prescribed to treat depression and anxiety. Common medications prescribed include SSRI's and MAOI anti-depressants, and a variety of anti-anxiety medications.
However, medication alone is insufficient to address the deep-seated roots of eating disorders in patients. Patients will also benefit from psychotherapy performed by a licensed therapist in a clinical setting. Treatment for intense eating disorders can be a residential treatment, where patients live at the facility for the duration of their treatment. Other options include intensive outpatient programs, known as IOP.
One form of psychotherapy used to treat eating disorders is cognitive behavioral therapy (CBT), which has a strong track record for treating conditions like depression, anxiety, and obsessive-compulsive disorder. Cognitive behavioral therapy is an approach in which therapists and patients address the interplay between thoughts, behaviors, and emotions in order to help patients develop more appropriate coping mechanisms for negative emotions.
Patients with eating disorders are not comfortable with their emotions and use food behaviors in order to avoid strong feelings. Through CBT, patients are taught that their constant negative thoughts are a symptom of their disorder, not an accurate reflection of reality. They are given methods to fight these thoughts with the goal of reducing symptoms and improving overall emotional health.
Another therapy for treating eating disorders is mindfulness-based cognitive therapy, which incorporates aspects of CBT and mindfulness meditation. Through this approach, patients are taught to analyze their thoughts and emotions as well as to perform meditation techniques to clear their minds and remain psychologically centered.
Prevention & Prophylaxis
Preventive programs may be targeted towards either a universal audience with an emphasis on societal changes and public policy goals, a specific demographic vulnerable to developing eating disorders (young girls), or specific individuals who have already exhibited behaviors that may be indicative of eating disorders. Research suggests a strong link between early intervention and positive treatment outcomes, which indicates the importance of prevention.<|endoftext|>
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# Ex.3.1 Q2 Pair of Linear Equations in Two Variables Solution - NCERT Maths Class 10
Go back to 'Ex.3.1'
## Question
The coach of a cricket team buys $$3$$ bats and 6 balls for $$₹\,3900.$$ Later, she buys another bat and $$3$$ more balls of the same kind for ₹ $$1300.$$ Represent this situation algebraically and geometrically.
Video Solution
Pair Of Linear Equations In Two Variables
Ex 3.1 | Question 2
## Text Solution
What is Known?
(i) Three bats and six balls for ₹ $$3900$$
(ii) One bat and three balls for ₹ $$1300$$
What is Unknown?
Represent the situation geometrically and algebraically
Reasoning:
Assuming the cost of one bat as ₹ $$x$$ and the cost of one ball as ₹ $$y,$$ two linear equations can be formed for the above situation.
Steps:
The cost of $$3$$ bats and 6 balls is ₹ $$3900.$$
Mathematically:
\begin{align}3x + 6y&= 3900\\3(x + 2y)& = 3900\\x + 2y& = 1300\end{align}
Also, the cost of $$1$$ bat and $$3$$ balls is ₹ $$1300.$$
Mathematically:
$x + 3y = 1300$
Algebraic representation where $$x$$ and $$y$$ are cost of bat and ball respectively.
\begin{align}x + 2y &= 1300 \qquad(1)\\x + 3y &= 1300\qquad(2)\end{align}
Therefore, the algebraic representation for equation $$1$$ is:
\begin{align}x + 2y &= 1300\\2y &= 1300-x\\y &= \frac{{1300 - x}}{2}\end{align}
And, the algebraic representation for equation $$2$$ is:
\begin{align}x + 3y &= 1300\\3y &= 1300-x\\y &= \frac{{1300 - x}}{3}\end{align}
Let us represent these equations graphically. For this, we need at least two solutions for each equation. We give these solutions in table shown below.
$$x$$ $$700$$ $$500$$ $$y = \frac{{1300 - x}}{2}$$ $$300$$ $$400$$
$$x$$ $$400$$ $$700$$ $$y = \frac{{1300 - x}}{3}$$ $$300$$ $$200$$
The graphical representation is as follows.
Unit: $$1\,\rm{cm} =$$$$100.$$
The answer is $$(1300, \,0)$$
Learn math from the experts and clarify doubts instantly
• Instant doubt clearing (live one on one)
• Learn from India’s best math teachers
• Completely personalized curriculum<|endoftext|>
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# 2.5: Finding the Equation of Perpendicular Lines
Difficulty Level: At Grade Created by: CK-12
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In most cities, it is very common for the streets to be laid out in a grid format. Below is an example. (Assume the lower left corner is the origin.) Are any of the streets perpendicular? What are the slopes of each street?
### Guidance
When two lines are perpendicular, they intersect at a \begin{align*}90^\circ,\end{align*} or right, angle. The slopes of two perpendicular lines, are therefore, not the same. Let’s investigate the relationship of perpendicular lines.
#### Investigation: Slopes of Perpendicular Lines
Tools Needed: Pencil, ruler, protractor, and graph paper
1. Draw an \begin{align*}x-y\end{align*} plane that goes from -5 to 5 in both the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} directions.
2. Plot (0, 0) and (1, 3). Connect these to form a line.
3. Plot (0, 0) and (-3, 1). Connect these to form a second line.
4. Using a protractor, measure the angle formed by the two lines. What is it?
5. Use slope triangles to find the slope of both lines. What are they?
6. Multiply the slope of the first line times the slope of the second line. What do you get?
From this investigation, the lines from #2 and #3 are perpendicular because they form a \begin{align*}90^\circ\end{align*} angle. The slopes are 3 and \begin{align*}- \frac{1}{3},\end{align*} respectively. When multiplied together, the product is -1. This is true of all perpendicular lines.
The product of the slopes of two perpendicular lines is -1.
If a line has a slope of \begin{align*}m,\end{align*} then the perpendicular slope is \begin{align*}- \frac{1}{m}\end{align*}.
#### Example A
Find the equation of the line that is perpendicular to \begin{align*}2x -3y = 15\end{align*} and passes through (6, 5).
Solution: First, we need to change the line from standard to slope-intercept form.
\begin{align*}2x - 3y &= 15\\ -3y &=-2x + 15\\ y &= \frac{2}{3}x - 5\end{align*}
Now, let’s find the perpendicular slope. From the investigation above, we know that the slopes must multiply together to equal -1.
\begin{align*}\frac{2}{3} \cdot m &= -1\\ \xcancel{\frac{3}{2} \cdot \frac{2}{3}} \cdot m &= -1 \cdot \frac{3}{2}\\ m &= - \frac{3}{2}\end{align*}
Notice that the perpendicular slope is the opposite sign and reciprocals with the original slope. Now, we need to use the given point to find the \begin{align*}y-\end{align*}intercept.
\begin{align*}5 &= - \frac{3}{2}(6) + b\\ 5 &= -9 + b\\ 14 &= b\end{align*}
The equation of the line that is perpendicular to \begin{align*}y = \frac{2}{3}x - 5 \end{align*} is \begin{align*}y = - \frac{3}{2}x + 14\end{align*}.
If we write these lines in standard form, the equations would be \begin{align*}2x - 3y = 15\end{align*} and \begin{align*}3x + 2y = 28,\end{align*} respectively.
#### Example B
Write the equation of the line that passes through (4, -7) and is perpendicular to \begin{align*}y = 2\end{align*}.
Solution: The line \begin{align*}y = 2\end{align*} does not have an \begin{align*}x-\end{align*}term, meaning it has no slope and a horizontal line. Therefore, to find the perpendicular line that passes through (4, -7), it would have to be a vertical line. Only need the \begin{align*}x-\end{align*}coordinate. The perpendicular line would be \begin{align*}x = 4\end{align*}.
#### Example C
Write the equation of the line that passes through (6, -10) and is perpendicular to the line that passes through (4, -6) and (3, -4).
Solution: First, we need to find the slope of the line that our line will be perpendicular to. Use the points (4, -6) and (3, -4) to find the slope.
\begin{align*}m = \frac{-4-(-6)}{3-4} = \frac{2}{-1} = -2\end{align*}
Therefore, the perpendicular slope is the opposite sign and the reciprocal of -2. That makes the slope \begin{align*}\frac{1}{2}\end{align*}. Use the point (6, -10) to find the \begin{align*}y-\end{align*}intercept.
\begin{align*}-10 &= \frac{1}{2}(6) + b\\ -10 &= 3 + b \\ -7 &= b\end{align*}
The equation of the perpendicular line is \begin{align*}y = \frac{1}{2}x - 7\end{align*}.
Intro Problem Revisit Using the graphed endpoints of each street, we find the slope of A St. is \begin{align*}- \frac{1}{2}\end{align*}, 7th Ave. is \begin{align*}2\end{align*}, and the slope is Lincoln Blvd. is \begin{align*}\frac{1}{2}\end{align*}. Therefore, because they are negative reciprocals of each other, A St. and 7th Ave. are perpendicular. Lincoln Blvd. is neither perpendicular nor parallel to 7th Ave. or A Street.
### Guided Practice
1. Find the equation of the line that is perpendicular to \begin{align*}x - 2y = 8\end{align*} and passes through (4, -3).
2. Find the equation of the line that passes through (-8, 7) and is perpendicular to the line that passes through (6, -1) and (1, 3).
3. Are \begin{align*}x - 4y = 8\end{align*} and \begin{align*}2x + 8y = -32\end{align*} parallel, perpendicular or neither?
1. First, we need to change this line from standard form to slope-intercept form.
\begin{align*}x -2y &= 8\\ -2y &= -x + 8\\ y &=\frac{1}{2}x - 4\end{align*}
The perpendicular slope will be -2. Let's find the new \begin{align*}y-\end{align*}intercept.
\begin{align*}-3 &= -2(4) + b\\ -3 &= -8 + b \\ 5 &= b\end{align*}
The equation of the perpendicular line is \begin{align*}y = -2x + 5\end{align*} or \begin{align*}2x + y = 5\end{align*}.
2. First, find the slope between (6, -1) and (1, 3).
\begin{align*}m = \frac{-1-3}{6-1} = \frac{-4}{5} = -\frac{4}{5}\end{align*}
From this, the slope of the perpendicular line will be \begin{align*}\frac{5}{4}\end{align*}. Now, use (-8, 7) to find the \begin{align*}y-\end{align*}intercept.
\begin{align*}7 &= \frac{5}{4}(-8) + b\\ 7 &= -10 + b \\ 17 &= b\end{align*}
The equation of the perpendicular line is \begin{align*}y = \frac{5}{4}x + 17\end{align*}.
3. To determine if the two lines are parallel or perpendicular, we need to change them both into slope-intercept form.
\begin{align*}x - 4y &= 8 \qquad \qquad \qquad 2x + 8y = -32\\ -4y &= -x + 8 \quad and \qquad \quad \ 8y = -2x -32\\ y &=\frac{1}{4}x - 2 \qquad \qquad \qquad \ y = - \frac{1}{4}x - 4\end{align*}
Now, just look at the slopes. One is \begin{align*}\frac{1}{4}\end{align*} and the other is \begin{align*}- \frac{1}{4}\end{align*}. They are not the same, so they are not parallel. To be perpendicular, the slopes need to be reciprocals, which they are not. Therefore, these two lines are not parallel or perpendicular.
### Vocabulary
Perpendicular
When two lines intersect to form a right, or \begin{align*}90^\circ\end{align*}, angle. The product of the slopes of two perpendicular lines is -1.
### Practice
Find the equation of the line, given the following information. You may leave your answer in slope-intercept form.
1. Passes through (4, 7) and is perpendicular to \begin{align*}x - y = -5\end{align*}.
2. Passes through (-6, -2) and is perpendicular to \begin{align*}y = 4\end{align*}.
3. Passes through (4, 5) and is perpendicular to \begin{align*}y = - \frac{1}{3}x - 1\end{align*}.
4. Passes through (1, -9) and is perpendicular to \begin{align*}x = 8\end{align*}.
5. Passes through (0, 6) and perpendicular to \begin{align*}x - 4y = 10\end{align*}.
6. Passes through (-12, 4) and is perpendicular to \begin{align*}y = -3x + 5\end{align*}.
7. Passes through the \begin{align*}x-\end{align*}intercept of \begin{align*}2x - 3y = 6\end{align*} and perpendicular to \begin{align*}x + 6y = -3\end{align*}.
8. Passes through (7, -8) and is perpendicular to \begin{align*}2x + 5y = 14\end{align*}.
9. Passes through (1, 3) and is perpendicular to the line that passes through (-6, 2) and (-4, 6).
10. Passes through (3, -10) and is perpendicular to the line that passes through (-2, 2) and (-8, 1).
11. Passes through (-4, -1) and is perpendicular to the line that passes through (-15, 7) and (-3, 3).
Are the pairs of lines parallel, perpendicular or neither?
1. \begin{align*}4x + 2y = 5\end{align*} and \begin{align*}5x - 10y = -20\end{align*}
2. \begin{align*}9x + 12y = 8\end{align*} and \begin{align*}6x + 8y = -1\end{align*}
3. \begin{align*}5x - 5y = 20\end{align*} and \begin{align*}x + y = 7\end{align*}
4. \begin{align*}8x -4y = 12\end{align*} and \begin{align*}4x - y = -15\end{align*}
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TermDefinition
Perpendicular Perpendicular lines are lines that intersect at a $90^{\circ}$ angle. The product of the slopes of two perpendicular lines is -1.
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For decades, the United States and the Soviet Union were locked in a tense race against time where only one question mattered: Who would be the first to dominate space exploration? This competition would become known as the “Space Race
On October 3, 1942, Nazi Germany launched the V-2 rocket, developed by German engineer Wernher von Braun. Flying faster than 3,500 miles per hour, the V-2 shot upward for 60 miles, escaped the Earth’s atmosphere and became the first man-made object to successfully reach the edge of space
The Space Race roared to life on October 4, 1957, when the Soviets mounted a satellite onto a rocket and launched it into orbit. It was named Sputnik, meaning “fellow traveler of Earth,” and it circled the globe every 92 minutes at a speed of 18,000 miles per hour
Soviet Union finished first at almost everything in the Space Race! They sent the first animal (a dog called Laika) and the first human (Yuri Gagarin) into orbit. They launched the first multi-person crew. They made the first space walk. They were the first to achieve unmanned orbit of the moon. They were even the first to land an unmanned capsule on the moon!
But in 1968, the Americans staged a spectacular surprise victory. The astronauts of Apollo 8 became the first humans to orbit the moon. And, in 1969, Americans Neil Armstrong and Buzz Aldrin of the Apollo 11 mission became the first humans to pilot a craft to the moon, land, and step onto its surface.
On July 17, 1975, the American Apollo and the Soviet Soyuz spacecrafts met high above the Earth and docked. Floating in space, an American astronaut and a Soviet cosmonaut reached through the open hatches of their joined ships and shook hands<|endoftext|>
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Genetic Drift: the random change in the frequency of an allele (gene) within a population (Masel, 2011).
Genetic Drift is one of the four basic mechanisms of evolution. Since genetic drift is entirely based on chance it does not lead to adaptations (University of California Museum of Paleontology, n.d.). Unlike natural selection that is differential reproductive success that happens for a reason; genetic drift is differential reproductive success that just happens (Freeman and Herron, 2007)
The high incidence of Ellis-Van Creveld Syndrome in the Old Order Amish in Pennsylvania is an example of genetic drift within a human population. The practice of marrying and reproducing within their small community, (which remained isolated from the general population) caused an increased frequency of this normally rare genetic disease to occur. This is a special type of population bottleneck called a "Founder Effect" because the inflated number of cases of EVC Syndrome is not representative of a normal population where gene flow occurs (pbs.org).
Freeman, S. & Herron, J. (2007). Evolutionary Analysis 4th Ed. Form and Function, pp 363-396.
Masel, J. (2011). Genetic Drift. Current Biology 21 (20).
University of California Museum of Paleontology. (n.d.) Genetic Drift. retrieved from: http://evolution.berkeley.edu/evosite/evo101/IIIDGeneticdrift.shtml<|endoftext|>
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BETA
Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
be references to the narrative flow of the book that I have yet to remove.
This work is under development and has not yet been professionally edited.
If you catch a typo or error, or just have a suggestion, please submit a note here. Thanks!
# Derivatives and Partial Derivatives
A critical feature of any function is how the output changes with the changes to its inputs.
## Derivatives
For a function of one variable $f(x)$, the derivative at some value $x$ may be written as $df/dx$: $${df \over dx} = \lim_{\Delta x \rightarrow 0} {f(x + \Delta x) - f(x) \over \Delta x}$$ The reason we use this notation is that the $df$ refers to the vertical distance measured in the numerator, and $dx$ represents the horizontal distance in the denominator.
Visually, this means a line connecting $(x, f(x))$ and $(x + \Delta x, f(x + \Delta x))$ converges to a line tangent to the function at $(x, f(x))$ as $\Delta x \rightarrow 0$. The following diagram illustrates this for a few functions. Use the slider to bring $\Delta x$ to zero; you can also change the value of $x$ to see how the derivative changes (or doesn’t) as $x$ changes.
## Partial derivatives
A partial derivative of a multivariable function is defined in much the same way. For a function of two variables (say, $x$ and $y$) we can proceed in the same way as above, comparing the value of the function at $f(x,y)$ as we change the values of $x$ and $y$ by small amounts. The building blocks of our analysis are the partial derivatives of the function, which measure how the output of the function changes when one variable is increased while the other(s) are held constant.
In the case of a function of two variables, $x$ and $y$, we can define the partial derivative “with respect to $x$” as $\partial f/\partial x$, where $${\partial f \over \partial x} = \lim_{\Delta x \rightarrow 0} {f(x + \Delta x, y) - f(x, y) \over \Delta x}$$ Visually, this has the same interpretation as above, except now the two points are points along the surface plot of the function $f(x,y)$, as as $\Delta x \rightarrow 0$ the line is tangent to the surface, not just a curve:
The partial derivative with respect to $y$ is defined similarly: holding $x$ constant, it measures the rate at which $f(x,y)$ changes when $y$ increases by a $\Delta y$. In the limit as $\Delta y \rightarrow 0$, it may be represented as a line tangent to the surface plot of the function, pointing in the $y$ direction.
You may have noticed that the graphs of the various $f(x)$ functions in the first example were related to the later multivariable examples. That’s because they were: in fact, the three functions shown were the same as the three functions illustrating $f(x,y)$ with $y$ fixed at 3. Indeed, one helpful way of thinking about partial derivatives is as the derivative of the function implied by holding the other variables of a multivariable function constant. (In economics, this often means imposing a “ceteris paribus” assumption that all other variables are held constant.)
To see what this means visually in this case, we can plot the (two-dimensional) function $f(x | y = \overline y)$. For illustrative purposes, we can see side-by-side what this looks like in three dimensions and two dimensions:
Indeed, if you look carefully, you can see that the two-dimensional graph is exactly the same as the graph of the intersection of the surface with the plane at $y = \overline y$.
## Calculating derivatives and partial derivatives
You should review the various rules for calculating derivatives (a nice summary is here). For taking the partial derivative, the key thing to remember is that when you take the derivative with respect to a single variable, you’re holding all other variables constant; so all variables except the variable of interest are treated as constants for the purposes of differentiation.
For example, with the univariate function $f(x) = 12x^{1 \over 2}$, by the exponent rule we have $${df \over dx} = \tfrac{1}{2}\times 12x^{ {1 \over 2} - 1} = 6x^{-{1 \over 2} }$$ For the multivariable function $f(x,y) = 4x^{1 \over 2}y$, the $y$ is treated as a constant when taking the partial derivative: $${\partial f \over \partial x} = \tfrac{1}{2}\times 4x^{ {1 \over 2} - 1}y = 2x^{-{1 \over 2} }y$$ It’s easy to see that when $y = 3$, these two expressions are identical.
Next: The Chain Rule
Copyright (c) Christopher Makler / econgraphs.org<|endoftext|>
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Spider Mites in the Landscape and Nursery
There are many types of mites, but spider mites, eriophyid mites (many being gall-inducing mites) and predatory mites are the most important groups when plant care is involved. In this article, spider mites and their natural predators will mostly be addressed.
Much of what we know about mites has been discovered just within the past 60 years for two reasons: mites are so small that they go unnoticed and they were rarely a problem for plants until the advent of chemical pesticides, which disrupted the natural check and balance system between spider mites and their predators, thus allowing for outbreaks. Mites are not insects, of course, and are more closely related to spiders and ticks. Adult mites have four pairs of legs and their mouth is comprised of a pair of fangs (chelicerae) that puncture the tissue of the host plant, creating a tiny wound that exudes sap. The mite then drinks the sap. Each individual puncture results in a tiny yellow spot. Repeated puncturing by a multitude of spider mites can quickly deplete foliage of their chlorophyll, giving the leaf a bleached appearance. However, when examined closely, thousands of individual 'piercing-sucking' wounds can be identified.
Mites, in general, are usually unseen and forgotten until the plant damage that some of them are responsible for creating suddenly becomes all too obvious. Oftentimes, this damage is the direct result of human mismanagement of other arthropods; this phenomenon is known as a secondary pest outbreak and is caused by the use of chemical pesticides. In fact, spider mites were rarely a problem in agriculture and horticulture until the invention and widespread use of chemical insecticides after World War II. In most cases, spider mites were kept at low numbers on plants and in large-scale plantings due to natural controls, specifically predatory mites.
Within the animal world, there are two basic strategies for survival of offspring: K-strategy and r-strategy. K-strategy animals produce very few offspring per individual and invest large amounts of care into those offspring. Humans are a great example of K-strategy. Those animals, such as insects and spider mites, that use r-strategy produce vast numbers of offspring and invest almost nothing in their survival other than depositing the eggs on or near a suitable plant host. Also, such species usually go through their generational time quite quickly and often have multiple generations per year.
Such r-strategy species are generally those that we need to pay extra attention to in agriculture and horticulture, given their ability to reproduce quickly, pass along genes for pesticide resistance quickly, and become major pests that are difficult to control. Many of the chemical insecticides of the past (and some in the present) have selected for spider mite resistance, as well as acted to annihilate the predatory mites, which are not well designed to develop resistance to pesticides. Given that some spider mite species can go from the egg stage to reproductive adult within 5 days under optimal weather conditions, the prevalence of resistance to pesticides and the sheer numbers of plant-damaging mites can increase almost overnight. Therefore, many spider mite species are the quintessential r-strategy plant pests in the landscape and nursery.
In addition to spider mites being well-adapted and quick to develop pesticide resistance, often within a growing season, humans also worsen the problem with the very same pesticides by destroying the predatory mite populations, thus eliminating the natural controls on the spider mite species and allowing them even more opportunity to explode in population numbers.
In the world of spider mites (family: Tetranychidae), there are two basic types that relate to their activity times: cool-season mites and warm-season mites. An example of a cool-season species is the spruce spider mite here in New England. This mite becomes active in the spring from over-wintering eggs that were deposited on the host plant the previous fall. These mites feed, develop and reproduce into June but then go dormant as the annual hot and dry conditions of summer begin. Then, as temperatures cool in late summer/early autumn, they re-appear and begin feeding and reproducing again. If the fall remains atypically warm, these mites can become problematic in September-October, a time when few are thinking about insect and mite problems, so such problems are often not noticed until the damage has reached unacceptable visible levels.
Warm-season mites, such as the two-spotted spider mite, also become active in the spring but continue to develop and reproduce at increasing rates as daily temperatures become warmer, reaching peak reproductive and developmental capabilities during the hottest time of the summer. When temperatures are in the 90º–100º F range, such mite species can go from egg to reproductive adult within 5 days, with the potential for each female to produce more than 100 eggs.
Many of the extremely important predatory mite species responsible for keeping spider mite numbers below damaging levels do not function well in such high temperature environments and often leave the plant for leaf litter or other cooler and more protected areas and go dormant. Some may stay but their foraging and feeding activity is greatly reduced, thus providing another opportunity for warm-season spider mite species to increase in numbers virtually unimpeded. Lastly, host plants for spider mites often do not perform well during the hot and dry periods that favor spider mites. Drought especially puts pressure on plants to fend off insects and mites. Water is involved with the natural production of defensive compounds that plants make, often in the roots, to defend themselves from invertebrate herbivores such as spider mites. At the onset of drought, plants usually have a sudden increase in the production and translocation of these defensive compounds, but as drought continues, this production and movement drops off considerably, thus leav-ing the plant more susceptible to damage from mites and insects. Drought stressed plants generally experience greatly reduced growth, thus prohibiting them from producing new foliage to 'outgrow' damage created by spider mites.
Management of Spider Mites
Plant-care professionals need to be aware of those spider mite species that are inherent to specific plants, as well as recognizing the potential presence of mite population build-up: fine yellow stippling, the presence of fine silk, and the mites themselves, usually on the foliage undersides. One can monitor for spider mites by gently shaking host plant foliage over a piece of paper and examining closely (with a hand lens) what falls onto the paper. In general, spider mites are oval and predatory mite species have more of a teardrop shape, but not always.
A general rule of thumb for a naturally managed spider mite population is a ratio of 1 predatory mite to every 10 spider mites. However, this varies given the time of year (warm-season mites are very prolifi c during hot and dry times and predators are not actively feeding). Also, if large numbers of spider mites are present, despite the presence of predators, it may indicate a situation where the natural controls are now incapable of reducing the pest species below damaging levels.
Insecticides and miticides are readily available but the risk of pesticide resistance remains strong with such compounds. Some of the newer miticides are growth regulators and disrupt some normal biological process of the mites, such as reducing the ability to molt properly, sterilizing female mites, killing the embryo within the egg, or forcing the molting process when the mite is not yet ready; all methods being lethal to the mite. Some growth regulators are so specific that they kill spider mites but not predatory mites. Some are only available as restricted-use compounds. The properties of each of these mite growth regulators varies widely and labels should be thoroughly read, understood and used properly.
Horticultural oil sprays are very effective for many mite species but need to be targeted to where the mites are feeding in order to be effective, usually the foliage undersides. Repeated applications may be necessary for warm-season mites. Overall, the secret to sound management is to avoid mite build-up in the first place and the development of pesticide resistance. Understanding how mites become a problem and avoiding the human factors that contribute so strongly to problems with spider mites goes a long way in the management of these tiny yet potentially serious pests.
Written by: Robert Childs<|endoftext|>
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Atomic and Molecular Structure
C.1. The periodic table displays the elements in increasing atomic number and shows how periodicity of the physical and chemical properties of the elements relates to atomic structure. As a basis for understanding this concept:
C.1.e. Students know the nucleus of the atom is much smaller than the atom yet contains most of its mass.
C.11. Nuclear processes are those in which an atomic nucleus changes, including radioactive decay of naturally occurring and human-made isotopes, nuclear fission, and nuclear fusion. As a basis for understanding this concept:
C.11.c. Students know some naturally occurring isotopes of elements are radioactive, as are isotopes formed in nuclear reactions.<|endoftext|>
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Online resources can increase the range of materials teachers have access to for planning, and the ‘just in time’ (Wilson & Stacey, 2003, p. 546) sourcing of online resources can provide support for teachers’ activities including their planning. The Science Learning Hub (SLH) provides teachers with access to a range of quality-assured resources, but their impact in the classroom relies on a teacher’s expertise.
Shulman’s (1987) notion of pedagogical content knowledge comes into play here. Teachers require sound pedagogical content knowledge to know how to plan and teach specific disciplinary ideas and skills to particular students. Student learning is also affected by how teaching gives continuity and coherency of meaning (Mercer, 2008). Science topics are often made up of a series of activities, and so planning with attention to pedagogical link-making (Scott, Mortimer & Ametller, 2011) is essential if students are to experience their learning as connected and cumulative.
This project was undertaken over one term to investigate how teachers at different levels of schooling used and adapted SLH resources for their science teaching. The project involved six teachers from four schools – one year 3–4 teacher, one year 4, one year 5–6, two year 7–8 and one year 9–10. Two teachers were specialist science teachers.
The project was initiated with a 2-day teacher workshop when the teachers planned a science topic using SLH resources. Data was collected from audiotapes, observations, field notes, interviews and teaching materials.
The focus of this research brief is on the processes teachers used to incorporate the SLH resources into their science planning.
When planning lessons, the project teachers worked iteratively between different facets of planning to develop and maintain coherency. First, they came to the project workshop with a topic in mind. This served as their science focus. Then they sourced the big idea that fitted their topic using Harlen (2010). Choosing an overarching big idea helped confirm the science focus and provided an overview for their planning and the unit activities. Next, the teachers consulted Science in the New Zealand Curriculum (Ministry of Education, 2007) to identify and develop the science ideas/learning intentions that nested within their big idea. They also checked that their planning was at the relevant level(s).
At the same time as crafting learning goals, the teachers planned the unit science activities. They designed these as a connected sequence intended to scaffold their students towards their learning intentions and the unit big idea. They considered how they would link activities in a cohesive manner, mindful that they would be guiding student learning. Finally, the teachers moved to the SLH and searched for supporting resources and detailed New Zealand-based information. They slotted the SLH materials into the activity sequence and adapted the materials to suit their students. Throughout this process, teachers kept in mind all the facets of the planning process. They juggled each aspect to develop a coherent plan driven by the big idea and the related learning intentions. They were able to do this because they had formed an overview of their unit idea and sequence and they knew their students.
It was excellent having that SLH base and all the background information to build my programme around.
Using the SLH
The SLH played a support role in planning. For example, teachers sourced learning intentions, associated activities and materials that would fit their topic including videos, animations and images. Teachers used the SLH throughout the 2-day workshop to research and learn more about the science they would be teaching, to check their science understandings and to refine their plans. As one teacher commented, “It was excellent having that SLH base and all the background information to build my programme around.” Discussions about SLH materials and activities that had already worked for other teachers were beneficial.
Having a topic in mind helps to focus teacher planning as does iteratively moving between the big idea, the intended learning outcomes and associated activities. A coherent plan is formed when teachers nestle and weave all aspects together. The SLH can support teacher planning as it is a source of science ideas for teaching and it provides examples that have worked for other teachers. Pedagogical content knowledge comes into play as teachers use the SLH to make decisions about how to craft a unit tailored to specific subject ideas and to their students. Pedagogical linkmaking skills are also required, as the different facets of the SLH are not necessarily linked. An extended time for planning, along with a focus on science, contributes to teachers creating effective plans.
Harlen, W. (2010). Principles and big ideas of science education. Hatfield, Herts: Association for Science Education.
Mercer, N. (1995). Guided construction of knowledge: Talk amongst teachers and students. United Kingdom: Multilingual Matters.
Ministry of Education. (2007). The New Zealand Curriculum. Wellington, NZ: Learning Media.
Scott, P., Mortimer, E. & Ametller, J. (2011). Pedagogical link-making: A fundamental aspect of teaching and learning scientific conceptual knowledge. Studies in Science Education, 47(1), 3–36.
Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1–22.
Wilson, G. & Stacey, E. (2003). Online interaction impacts on learning: Teaching the teachers to teach online. In G. Crisp, D. Thiele, I. Scholten, S. Baker & J. Baron (Eds.), Interact, integrate, impact: Proceedings of the 20th Annual Conference of the Australasian Society for Computers in Learning in Tertiary Education (ASCILITE), Adelaide, SA, 7–10 December, 541–551.<|endoftext|>
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1.18 Why secularism? (Part 2) - Viewpoints
This resource covers a wide range of political and philosophical figures and their views on secularism. Through this resource students will gain an understanding of the diversity of secularist thought and stimulate the formation of their own viewpoints.
Politics | Religion & Belief Education | Citizenship | SMSC | Philosophy
- Who is a secularist?
- Why do people support or oppose secularism?
- Where does secularism come from?
Students should demonstrate they can:
- Understand, compare and contrast a range of viewpoints on secularism.
In addition to the basic learning outcomes, students should demonstrate they can:
- Critically reflect on a range of viewpoints on secularism.
- Extrapolate from the source viewpoints why the range of authors might support or oppose forms of secularism.<|endoftext|>
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# Modular Arithmatic - Solving congruences
I'm sure this is pretty basic but I'm struggling to understand how to go about solving this problem for my homework. The question states "Solve the following congruences for x". The first problem is $2x+1\equiv 4\pmod 5$.
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Have you studied modular inverses, and how to compute them? – Bill Dubuque Jul 5 '14 at 16:47
Probably about 100 years ago... :) Are modular inverses key to solving these types of problems? If so, I can start studying them! – aneorddot Jul 5 '14 at 17:01
Yes, if you wish to learn how to solve linear modular equations (congruences) then you should learn about modular inverses. See any textbook on elementary number theory. – Bill Dubuque Jul 5 '14 at 17:11
@Zlatan Please be more careful with your edits (which changed the modulus) – Bill Dubuque Jul 5 '14 at 17:11
@BillDubuque thank you, I'll look it up now as I see JMac31 referenced inverses below. – aneorddot Jul 5 '14 at 17:16
There are many ways to solve the problem. The conceptually simplest, but most tedious, is to test one by one the possibilities $x\equiv 0\pmod{5}$, $x\equiv 1\pmod{5}$, and so on up to $x\equiv 4\pmod{5}$. Quickly we find that $x\equiv 4\pmod{5}$. (This approach would become quite unpleasant if $5$ were replaced by $97$.)
It is simpler to use some algebra. So rewrite as $2x\equiv 3\pmod{5}$. Since $3\equiv 8\pmod{5}$, it is convenient to rewrite the congruence as $2x\equiv 8\pmod{5}$. Then since $2$ and $5$ are relatively prime, we can divide by $2$, getting $x\equiv 4\pmod{5}$.
A fancier version is to start from $2x\equiv 3\pmod{5}$. Now multiply both sides by $3$ (the modular inverse of $2$). We get $6x\equiv 9\pmod{5}$. But $6\equiv 1\pmod{5}$ and $9\equiv 4\pmod{5}$, so we conclude that $x\equiv 4\pmod{5}$.
Remark: We used congruence notation throughout, since it is very important to get accustomed to it. But $2x\equiv 3\pmod{5}$ means that $5$ divides $2x-3$. So we want to solve $2x-3=5k$, that is, $2x=3+5k$. So we want to find a $k$ such that $3+5k$ is divisible by $2$. It is clear that $k=1$ works, giving $2x=8$ so $x=4$. Any number congruent to $4$ modulo $5$ will also work, giving answer $x\equiv 4\pmod{5}$.
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Okay, your remark clarification helped clear up where I was missing the point. It seems simple right now because like you said it's mod 5. If the mod 5 were replaced with a larger number, the same concept works, correct? Would there ever be a case where there's more than one value of x? – aneorddot Jul 5 '14 at 17:07
Yes, it can happen that there are no solutions, or several solutions. The issue arises when we are looking at $ax\equiv b\pmod{m}$, where $a$ and $m$ are not relatively prime. For example, look at the congruence $4x\equiv 0\pmod{8}$. This has $4$ solutions modulo $8$, namely $x\equiv 0\pmod{8}$, $x\equiv 2$, $x\equiv 4$, and $x\equiv 6$. However, when $a$ and $m$ are relatively prime, there is always a unique (modulo $m$) solution of the linear congruence $ax\equiv b\pmod{m}$. – André Nicolas Jul 5 '14 at 17:11
Okay, so first step would be to determine whether a and m are relatively prime. If they are, then I know I'm only looking for one solution. If they are not, then I know there are either no solutions or multiple solutions. Thank you for your help! – aneorddot Jul 5 '14 at 17:14
You are welcome. For large $m$, and $a$ relatively prime to $m$, the modular inverse approach is probably best, with the modular inverse of $a$ computed using the Extended Euclidean Algorithm. But at this early stage, the exercises are for getting familiarization with the congruence notation. – André Nicolas Jul 5 '14 at 17:18
Hint $\ {\rm mod}\,\ 2k\!-\!1\!:\,\ 2k\!-\!1\equiv 0\,\Rightarrow\, \color{#c00}{2k\equiv 1},\$ so $\, k\equiv 2^{-1}.\,$ Therefore, as usual, we can solve
the linear equation $\ 2x\equiv b\$ by scaling it by $\, 2^{-1}\equiv k\,$ to get $\, x\equiv (\color{#c00}{2k})x \equiv kb.$
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Solve it like you would any linear equation expect you figure out what is $2^{-1}$ the multiplicative inverse of $2 \pmod 5$. That is find $a$ such that $2a \equiv 1 \pmod 5$.
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By multiplaying with 3, we get: $$6x+3=12\mod 5,$$ from were is: $$x=9 \mod 5=4\mod 5$$.
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The idea for the current radiation warning symbol (the Trefoil) was developed in 1946 by a small team of scientists at the University of California Radiation Laboratory. Led by Nelson Garden, the head of the Health Chemistry Group at the Radiation Laboratory, the group designed a small solid circle with what looks like three propeller blades equally spaced around the circle.
The Japanese Battle Flag as Inspiration
he reason for their choice of symbol has remained a mystery over time. In a letter written in 1995 by Paul Frame, a doctor at Oak Ridge Institute for Science and Education, he describes his belief that the symbol takes after the Japanese battle flag, which has a circle representing the sun in the middle with rays radiating out from it, because, "after all, [the radiation warning symbol] came along within a couple of years of WWII". Others believe that it is just a variation of the many symbols used to depict radiation before that time. Some of those early symbols were very similar to this three bladed design. One had a small circle in the center and then four lightning bolts surrounding it pointing out from the center much like the propellers. Others were just small lightning bolts in a circular formation. In a letter written in 1952, Nelson Garden did describe his first thought when his group produced the idea, "a design which was supposed to represent activity radiating from an atom," though it is still unknown if this is the true reason for its development.
Not only did the group of scientists have to think of an idea for this symbol that was not supposed to be similar to anything else, but they also had to come up with a color scheme, one that would stand out to everyone.
The Color Correction
Their original idea was a magenta symbol placed on a blue background. This received criticism because many believed that the blue would not be visible enough. However, Garden said in a letter he wrote in 1948, "The use of a blue background was selected because there is very little blue color used in most of the areas where radioactive work would be carried out. "He then explained his reasoning for not using yellow, classic for a warning sign because of its easy visibility, "the very fact that . . . the high visibility yellow stands out most prominently has led to extensive use of this color and it is very common." Garden continued to press for the use of the blue background, but in 1948, majority of workers denied it, complaining that the blue faded out very quickly during the daytime and was not visible during the night.
The design was then taken to Oak Ridge National Laboratory where various color schemes were produced and tested in order to see which scheme would be the most visible at a distance of twenty feet. The color plan they finally decided on was keeping the circle and propellers magenta-with the option of making them black instead (standard for a warning sign) and changing the blue background to the classic yellow. In their tests the yellow background was clearly the most visible, ending the search for a radiation warning symbol. In the end, the workers who needed the warning symbol felt that it mattered much more to them to have their symbol be as visible as possible than to have it be an original color plan. Thus, the modern symbol emerged.
Supplementing Safety, Reducing Risk with New Symbol
The trefoil symbol did not become iconic in the public mind, even as it became ubiquitous in fields where radiation exposure is a hazard.
This led to calls for reform. In December 2000, at an international conference of national experts on radiation safety and security, participants noted a worldwide increase in number of deaths and serious injuries caused by exposure to International Atomic Energy Agency (IAEA) category 1, 2 (and later 3) sources of radiation. They concluded that the existing trefoil symbol was insufficient to warn individuals, such as children, who were untrained in radiation safety.
As such they recommended a new, universal warning for identifying large sources of ionizing radiation. They proposed that this warning should clearly indicate to anyone, anywhere in the world, that they must leave the object alone. The graphics should convey that one needs to "run away."
In 2001, the International Organization for Standardization (ISO) started working to develop a standard for this.
In 2005, the IAEA designed various symbols with different shapes and colors and tested its effectiveness with over 1600 people from different cultures in 11 different countries. These people had little or no technical education of radiation safety.
In 2007, the IAEA and the ISO launched the new symbol depicting radiation waves, a skull and crossbones, and a running person.
The new warning symbol for Ionizing radiation is a "supplement" and not a replacement to the Trefoil symbol.
According to a press release by the IAEA,
- The symbol should be placed on the device encasing the source of radiation to prevent a person from disassembling the device or going closer to it. The symbol will be visible only when someone dismantles it.
- It should not be placed on building access doors, shipping packages or containers.<|endoftext|>
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Marsupial Rats and Mice
Marsupial rats and mice are a diverse group of about 40 species of small, native carnivores of Australia, Tasmania, and New Guinea, in the family Dasyuridae. The young of marsupial rats and mice, as with those of all marsupials, are born while still in a tiny, embryonic stage of development. The almost helpless babies migrate to the belly of their mother, where they fix on a nipple and suckle until they are ready to lead an independent life. Unusual among the marsupials, the females of some species of marsupial rats and mice do not have a belly pouch (or marsupium) that encloses their nipples and protects their young. Other species do have a permanent pouch, or they have one that develops only during the breeding season.
Marsupial rats and mice are small mammals with a uniformly dark, brownish coat, often with a whitish belly, and they have a superficial resemblance to placental rats and mice. Most species are nocturnal predators that feed on insects and other small prey. The larger species, such as marsupial rats, feed on smaller marsupials, birds, and reptiles, and introduced rodents. In a sense, marsupial mice fill the ecological roles played by the smallest placental predators on other continents, for example, shrews, while the marsupial rats are ecologically similar to larger small predators, such as weasels.
The brush-tailed marsupial mice or brush-tailed tuans (Phascogale spp.) are two species that occur in extreme southern Australia. The broad-footed marsupial mice (Antechinus spp.) are 10 species that occur in Australia, Tasmania, and New Guinea. The fat-skulled marsupial mice (Planigale spp.) are three species that live in Australia and New Guinea. The crested-tailed marsupial mouse (Dasycercus cristicauda) occurs in dry habitats of central Australia. The narrow-footed marsupial mice or pouched mice ( Sminthopsis spp.) are about ten species that occur in various types of habitats in Australia, Tasmania, and New Guinea. The long-legged jumping marsupials or jerboa marsupial mice (Antechinomys spp.) are two rare species of sandy deserts and savannas of Australia.
Many species of marsupial rats and mice have declined greatly in abundance due to habitat loss and the deadly effects of introduced placental mammalian predators, such as cats and foxes. Numerous species are now endangered, and special conservation measures must be taken if they are to survive.<|endoftext|>
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Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
# Find the Value of X for Which the Points (X, −1), (2, 1) and (4, 5) Are Collinear. - Mathematics
Find the value of x for which the points (x, −1), (2, 1) and (4, 5) are collinear.
#### Solution
Let the given points be A (x, −1), B (2, 1) and C (4, 5).
Slope of AB = $\frac{1 + 1}{2 - x} = \frac{2}{2 - x}$
Slope of BC = $\frac{5 - 1}{4 - 2} = \frac{4}{2} = 2$
It is given that the points (x, −1), (2, 1) and (4, 5) are collinear.
$\therefore$ Slope of AB = Slope of BC
$\Rightarrow \frac{2}{2 - x} = 2$
$\Rightarrow 1 = 2 - x$
$\Rightarrow x = 1$
Hence, the value of x is 1.
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Exercise 23.1 | Q 18 | Page 14<|endoftext|>
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The think, pair, share strategy is a cooperative learning technique that encourages individual participation and is applicable across all grade levels and class sizes. Students think through questions using three distinct steps:
Students think independently about the question that has been posed, forming ideas of their own.
Students are grouped in pairs to discuss their thoughts. This step allows students to articulate their ideas and to consider those of others.
Student pairs share their ideas with a larger group, such as the whole class. Often, students are more comfortable presenting ideas to a group with the support of a partner. In addition, students’ ideas have become more refined through this three-step process.<|endoftext|>
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# Difference between revisions of "2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4"
## Problem
Find the smallest and largest possible distances between the centers of two circles of radius $1$ such that there is an equilateral triangle of side length $1$ with two vertices on one of the circles and the third vertex on the second circle.
## Solution
The smallest distance would be found if the two circles were externally tangent, so testing that and messing around with it yields: $[asy] draw(circle((0,0),1)); draw(circle((2,0),1)); draw((1,0)--(0.5,0.86602),linewidth(.5)); draw((0,0)--(0.5,0.86602),linewidth(.5)); draw((1,0)--(1.5,0.86602),linewidth(.5)); draw((2,0)--(1.5,0.86602),linewidth(.5)); draw((0.5,0.86602)--(1.5,0.86602),linewidth(.5)); dot((0,0)); dot((2,0)); dot((1,0)); dot((0.5,0.86602)); dot((1.5,0.86602)); label("P",(1,0),SW); label("1",(0.25,0.43301),NW); label("1",(1.75,0.43301),NE); label("1",(0.75,0.43301),SW); label("1",(1.25,0.43301),SE); label("1",(1,0.86602),N); [/asy]$ Where $P$ is the point of tangency. This clearly works, so the smallest distance would be $2*1=\boxed{2}$
The largest distance would be found by first finding the closest place to the edge of a circle to place a line segment with side length $1$ (a side of the triangle), then adding the other two sides outwards like shown:
```draw(circle((0,0),1));
draw(circle((2.73205,0),1));
draw((0.86602,0.5)--(0.86602,-0.5),linewidth(.5));
draw((0.86602,0.5)--(1.73205,0),linewidth(.5))
draw((1.73205,0)--(0.86602,-0.5),linewidth(.5));
draw((0,0)--(0.86602,-0.5),linewidth(.5));
draw((0,0)--(0.86602,0.5),linewidth(.5));
dot((0.86602,0.5));
dot((0.86602,-0.5));
dot((1.73205,0));
(Error compiling LaTeX. d856f942c24e46cd45e1bf8b0904d19a39cae561.asy: 9.1: syntax error
error: could not load module 'd856f942c24e46cd45e1bf8b0904d19a39cae561.asy')```<|endoftext|>
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Quick Math Homework Help
Master the 7 pillars of school success that I have learned from 25 years of teaching.
## Finding the Volume of a Cube
cube is a three dimensional shape that has six equal sides. The volume is found by using the following formula
The Base Area of a cube = Length * Width
Example problem from video
Find the volume of cube 1 that has
sides of 5 units.
Step 1 Find the base area
Length * Width
5 * 5 = 25
Step 2 Multiple base area * height
25 * 5 = 125 units^3
# How to find the volume of a cube
Hi welcome to MooMooMath. Today we are going to look at the volume of a cube. Over here we have a cube and a cube is like the Rubik’s cube, it has 6 side, 6 faces, and 4 sides on the base and they are all equal. This cube has sides of 5. This is a prism so we will look at the prism formula and simplify it down to make it specific to the cube. The base area of a prism times the height gives us the volume, so we need to find the area of the base. The base is just a square of 5 so 5 times 5 equals a base area of 25. So I will plug a twenty five in four base area. I will multiple it by the height of the cube, which is 5, so 5 (the height) times 25(the area of a base) is 125 and since this is volume it is units cubed because it is three dimensional. Now let’s simplify this down because we do have a special case. We have a cube that all sides are the same length. Since we have to find the base area and the height and they are all the same we can simplify that to S times S for the base area times S for the height or S cubed. So that is how you find the volume of a cube. So let’s review the rules for the volume of a cube it will be S times S times S or S cubed the units will also be cubed so let’s add that .It is derived from the prism formula which is base area time’s height to get the volume. The base area of a square is S times S times the height which is also S gives us the volume of a cube. Hope this video was helpful in finding the volume of a cube.
5
5
5
Because all the sides of a cube are equal you can also find the volume of a cube by cubing one side.
Formula for volume of a cube = Side^3
5^3 = 125 units ^3
Cube 1
Problem 2. This problem is slightly more challenging.
Find the volume of a cube with a diagonal length of 6 units.
Step 3. Because you have the hypotenuse length you can find the side length by dividing hypotenuse/ √2 = 6/√2
Step 4. Rationalize. =
Step 5a. Plug the side length into the volume formula "side^3" which becomes (3√2 )^3 = 3*3*3 *√2*√2*√2
45
45
6
Step 5b. 3*3*3 *√2*√2*√2 = 27*2√2
Step 6. Volume of Cube 27*2√2 = 54√(2 ) units^3
Step 2. Use your 45-45-90 triangle rules to find the length of one side.
Step 1. The diagonal cuts the cube into two 45 degree angles.
The length of the hypotenuse equals leg length√2
The length of one leg of a 45-45-90 triangle equals hypotenuse/√2
### Volume Formula for a cube equals: Base Area * Height | Ba * h
Related Sites..
Cubes/Illuminations Use this interactive cube creator to help understand the volume of a cube, You can also calculate the volume of different cubes, and it will check your work.
Calculating the volume of a Cube/WikiHow Step by step directions for calculating the volume of a cube with pictures.<|endoftext|>
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Edit Article
# How to Add Square Roots
You can perform all the usual mathematical operations on square roots, including addition, subtraction, division and multiplication. But because the radical sign over the square root represents a mathematical operation already in place, the rules for adding square roots are a little different than the rules you may be used to with integers. To add square roots, you must first understand how to simplify them.
### Part 1 Simplifying Square Roots
1. 1
Factor each radicand into prime numbers.[1] An easy way to factor a number is by creating a factor tree diagram. Read Do a Factor Tree for complete instructions.
• A radicand is the number under the radical sign.
• A prime number is a number that can only be divided evenly by 1 and itself,[2] for example, 2, 3, 5, 7, 11, etc.
• You do NOT need to factor any coefficients. A coefficient is a number in front of the radical sign.
• Let’s say, for example, that you want to add ${\displaystyle {\sqrt {20}}+4{\sqrt {45}}+{\sqrt {5}}+{\sqrt {7}}.}$
To do this, you need to factor ${\displaystyle 20}$ as ${\displaystyle 2\times 2\times 5}$. You also need to factor ${\displaystyle 45}$ as ${\displaystyle 3\times 3\times 5}$.
• If a radicand is already a prime number, it does not need to be factored. For example, since ${\displaystyle 5}$ and ${\displaystyle 7}$ are already prime numbers, ${\displaystyle {\sqrt {5}}}$ and ${\displaystyle {\sqrt {7}}}$ do not need to be factored.
2. 2
Rewrite the expression. Keep all the factors under the radical sign.
• For example, after factoring the radicands, the example expression would be${\displaystyle {\sqrt {2\times 2\times 5}}+4{\sqrt {3\times 3\times 5}}+{\sqrt {5}}+{\sqrt {7}}.}$
3. 3
Circle pairs of like factors under each radical. Since you are finding a square root, by pairing up like factors, you can easily simplify the expression.
• For example, ${\displaystyle {\sqrt {2\times 2\times 5}}}$ has a pair of 2s, so draw a circle around them. ${\displaystyle 4{\sqrt {3\times 3\times 5}}}$ has a pair of 3s, so draw a circle around them.
4. 4
Factor out coefficients by identifying paired factors under each radical. The square root of any pair of factors will equal the factor, because ${\displaystyle x\times x=x^{2}}$ and ${\displaystyle {\sqrt {x^{2}}}=x}$. Place this number in front of the radical sign. If the expression already has a coefficient, multiply the two numbers.[3]
• For example:
${\displaystyle {\sqrt {2\times 2\times 5}}}$
${\displaystyle ={\sqrt {4}}{\sqrt {5}}}$
${\displaystyle =2{\sqrt {5}}}$
So, ${\displaystyle {\sqrt {20}}}$ simplifies to ${\displaystyle 2{\sqrt {5}}}$.
• ${\displaystyle 4{\sqrt {3\times 3\times 5}}}$
${\displaystyle =4\times {\sqrt {9}}{\sqrt {5}}}$
${\displaystyle =(4\times 3){\sqrt {5}}}$
${\displaystyle =12{\sqrt {5}}}$
So, ${\displaystyle 4{\sqrt {45}}}$simplifies to ${\displaystyle 12{\sqrt {5}}}$.
5. 5
Rewrite your problem, using the simplified terms. This will make the adding process much easier.
• For example:
${\displaystyle {\sqrt {20}}+4{\sqrt {45}}+{\sqrt {5}}+{\sqrt {7}}}$ simplifies to
${\displaystyle 2{\sqrt {5}}+12{\sqrt {5}}+{\sqrt {5}}+{\sqrt {7}}}$
### Part 2 Adding Square Roots
1. 1
Place a 1 in front of any square root that doesn’t already have a coefficient. The 1 is always understood, and so is rarely written. However, when adding, writing the 1 can help you keep track of coefficients.
• A coefficient is the number in front of the radical sign.
• For example, write ${\displaystyle {\sqrt {5}}}$ as ${\displaystyle 1{\sqrt {5}}}$.
2. 2
Check for square roots with the same radicand. You can only add square roots that have the same radicand.
• The radicand is the number underneath the radical sign.
• For example, you can add the first three terms in the expression
${\displaystyle 2{\sqrt {5}}+12{\sqrt {5}}+{\sqrt {5}}+{\sqrt {7}}}$, because they all have the same radicand (5).
3. 3
Add the coefficients. Only add the coefficients for terms that have the same radicand. Do NOT add the radicands.
• For example, ${\displaystyle 2{\sqrt {5}}+12{\sqrt {5}}+1{\sqrt {5}}=15{\sqrt {5}}}$.
4. 4
Add any unlike radicands to the expression. These cannot be simplified any further, and cannot be added to any other terms. The result will be your final, simplified answer.
• For example, ${\displaystyle 15{\sqrt {5}}+{\sqrt {7}}}$.
## Community Q&A
Ask a Question
If this question (or a similar one) is answered twice in this section, please click here to let us know.
## Article Info
Categories: Addition and Subtraction
Thanks to all authors for creating a page that has been read 11,301 times.
Did this article help you?<|endoftext|>
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# 45000 in Words
45000 in words is Forty-five Thousand. The number name of any number can be written using the ones, tens, hundreds, and thousands place of a number. Thus, the place value chart helps to write the number 45000 in words.
45000 in Words: Forty-five Thousand
In this article, let us learn how to write the number 45000 in words, and look at the solved examples in detail.
## How to Write 45000 in Words?
The number 45000 in words can be written using the place value of a number. To write the number name of 45000, first, determine the place value of each digit of the given number. In this case, the number name is 45000. Thus, for the number 45000,
Digit in 1’s position = 0
Digit in 10’s position = 0
Digit in 100’s position = 0
Digit in 1000’s position = 45
Thus, the place value chart for the number 45000 in words is:
1000’s Place 100’s Place 10’s Place 1’s Place 45 0 0 0
Therefore, the number 45000 in words is Forty-five Thousand.
### Examples
Example 1:
Find the value of 50000 – 5000. Describe the value in words.
Solution:
Given expression: 50000 – 5000
⇒ 50000 – 5000 = 45000
So, the value of 50000 – 5000 is 45000.
Hence, 45000 in words is forty-five thousand.
Example 2:
Determine the value of forty thousand plus five thousand in words.
Solution:
Forty thousand = 40000
Five thousand = 5000
Forty thousand plus five thousand = 40000 + 5000 = 45000
Hence, the value of forty thousand plus five thousand is forty-five thousand in words.
## Frequently Asked Questions on 45000 in Words
Q1
### Write 45000 in words.
45000 in words is forty-five thousand.
Q2
### Express the value of 47000 – 2000 in words.
Simplifying 47000 – 2000, we get 45000. Hence, 45000 in words is forty-five thousand.
Q3
### How to write forty-five thousand in numbers?
Forty-five thousand in numbers is 45000.<|endoftext|>
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DIY robots help marine biologists discover new deep-sea dwellers
While the cold and airless deep sea is inhabitable for humans, it is filled with delicate organisms that thrive in its harsh environment. Studying those organisms requires specialized equipment mounted on a remotely operated vehicle (ROV); any other type of equipment could literally crack under pressure. A multidisciplinary group of engineers, marine biologists, and roboticists have developed a sampling device that is soft, flexible, and customizable, which allows scientists to gently collect different types of organisms from the sea without harming them. It also allows 3D-print modifications to the device overnight without the need to return to a land-based laboratory. An upcoming paper in PLOS One takes a deeper dive into this research.
The "soft gripper" devices that the team designed have two to five "fingers" made of polyurethane and other squishy materials that open and close via a low-pressure hydraulic pump system that uses seawater to drive their movement. "Many of the animals we encounter in the deep-sea are new species and these soft robots allow us to delicately interact and study a more diverse suite of fauna," said co-author David Gruber, Presidential Professor of Biology and Environmental Science at the City University of New York's Baruch College. The grippers are attached to a wooden ball that is held and manipulated using an ROV's, hard claw-like tools, controlled by a human operator on the ship to which the ROV is tethered.
The team put the latest version of their soft grippers to the test on a voyage aboard R/V Falkor in the remote Phoenix Islands Protected Area in the South Pacific. Being in such an isolated environment meant that obtaining new parts for the grippers would be nearly impossible, so they brought two 3D printers for creating new components on-the-fly. "By 3D printing at sea, we can innovate, on-the-fly, and come up with soft robotics to interact with soft and delicate animals that were previously unexamined – as they were too fragile," said Gruber, who is also a 2017-2018 Radcliffe Fellow and National Geographic Explorer.
"Being on a ship for a month meant that we had to be able to make anything we needed, and it turns out that the 3D printers worked really well for doing that on the boat. We had them running almost 24/7, and we were able to take feedback from the ROV operators about their experience using the soft grippers and make new versions overnight to address any problems," said Daniel Vogt, a Research Engineer at the Wyss Institute.
The soft grippers were able to collect sea slugs, corals, sponges, and other marine life much more effectively and with less damage than traditional underwater sampling tools. Based on input from the ROV operators, the team 3D-printed "fingernail" extensions that could be added to the gripper's fingers to help them get underneath samples that were sitting on hard surfaces. A flexible mesh was also added to each finger to help keep samples contained within the fingers' grip. Another, two-fingered version of the grippers was also created based on ROV pilots' familiarity with controlling existing two-fingered graspers, and their request that the two fingers be able to hold samples with both a "pinch" grasp (for small objects) and a "power" grasp (for large objects).
The team is in the process of further developing the grippers, hoping to add sensors that can indicate to the ROV operator when the grippers come into contact with an organism, "feel" how hard or soft it is, and take other measurements. Ultimately, their goal is to be able to capture sea creatures in the deep ocean and obtain full physical and genetic data without taking them out of their native habitats.
This research was supported by the National Oceanographic and Atmospheric Association, the Schmidt Ocean Institute, the National Science Foundation, the National Academy of Sciences, the PIPA Conservation Trust, the PIPA Scientific Committee, and the Wyss Institute at Harvard University.
The City University of New York is the nation's leading urban public university. Founded in New York City in 1847, the University comprises 24 institutions: 11 senior colleges, seven community colleges, and additional professional schools. The University serves nearly 275,000 degree-credit students and 218,083 adults, continuing and professional education students.
For more information, please contact: Shante Booker or visit http://www.cuny.edu/research<|endoftext|>
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Firewood: Quality Fuel Is Crucial To Successful Wood-Burning Appliance Operation & An Enjoyable Fire
You may not think that the type of firewood you choose to burn or the way you stack or store it much matters, but it does! In order to understand why, it’s important that you have a basic understanding of the combustion process.
What’s Needed For Combustion To Occur?
While the process of combustion is complex and technical, I can break it down simply like this: Fuel, air, and heat must be present for combustion to occur, regardless of what fuel is used or what appliance is used.
In a solid fuel-burning heater (like a wood stove), heat breaks up wood molecules, causing carbon and hydrogen to mix with oxygen from the incoming air, turning it into more heat, chemicals, and gases.
If any of the three required elements (fuel, air, heat) is removed from the equation, combustion stops. If all three occur in the proper ratio, combustion is self-sustaining.
- If complete combustion occurs, only water vapor (H2O), carbon dioxide, heat, and ash are produced.
- If incomplete combustion occurs, carbon monoxide, volatile hydrocarbons (creosote), as well as H2O and ash are produced.
The Stages Of Combustion
For the purpose of burning wood, there are four main stages of combustion:
- Stage I: Drying — As the fuel is heated, any moisture it contains is evaporated. Evaporation absorbs heat. Green wood has a higher water content, which means more heat is needed for evaporation to occur. This is heat that is lost.
- Stage II: Pyrolysis — When the wood is dry enough, the temperature will rise to the point where molecules making up the chemical structure of the wood will break down and vaporize, turning to hydrocarbon gases. This mixture of tar fog and combustible gases is not yet hot enough to burn and heat is still being absorbed at this point.
- Stage III: Gas Burning — As the temperature increases, the tar fog and hydrocarbon gases reach ignition temperature in the presence of oxygen, oxidizing and burning.
- Stage IV: Charcoaling — At the point during combustion when everything except carbon has burned off, charcoal remains. This carbon must come into direct contact with oxygen in order to burn — burning with little or no flame. Water vapor continues to be produced because of the mixture of hydrogen and oxygen from the combustion air.
Start With Good Wood
Since the first two stages of combustion actually absorb heat, I can see how crucial good, dry wood (wood with a moisture content of no more than 20%) is to heat production. But good, dry wood will also reduce creosote buildup and chance of chimney fire. Water vapor condensing in the flue and unburned volatile hydrocarbons sticking to that water vapor is what creates glazed creosote, the main cause of all chimney fires. So start with good wood and enjoy more heat, a better fire, and a cleaner, safer chimney system!
This barkless log looks dry
The end of the log appears dry
The moisture meter with the probes only halfway inserted reveal over 20% moisture
When I probe the middle of the log I get 30% moisture.
- Good logs, not rotting and punky, should be cut, split, stacked, and allowed to season (dry) for at least a year.
- Wood should be stacked where it will get long days of sun and good airflow going through the stack.
- Wood stacks should be covered on top or kept out of the weather.
Keep in mind that stacks should never be wrapped with tarps or plastic, as this will not allow moisture to escape, and will shield the stack from good airflow.
Have more questions about what type of firewood to buy or how to properly season and store wood? Feel free to give me a call at 208-550-8474. I am here to be a helpful local resource!<|endoftext|>
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Math Pacing - PowerPoint PPT Presentation
Math Pacing
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Presentation Transcript
1. Solving Open Sentences Involving Absolute Value | | | | | | | | | | | | | | | | | | | | – 3 – 2 – 1 0 1 2 3 4 5 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 Math Pacing
2. Solving Open Sentences Involving Absolute Value There are three types of open sentences that can involve absolute value. Solving Open Sentences Involving Absolute Value Consider the case | x | = n. | x | = 5 means the distance between 0 and x is 5 units If | x | = 5, then x = – 5 or x = 5. The solution set is {– 5, 5}.
3. Solving Open Sentences Involving Absolute Value When solving equations that involve absolute value, there are two cases to consider: Solving Open Sentences Involving Absolute Value Case 1 The value inside the absolute value symbols is positive. Case 2 The value inside the absolute value symbols is negative. Equations involving absolute value can be solved by graphing them on a number line or by writing them as a compound sentence and solving it.
4. means that the distance between b and –6 is 5 units. To find b on the number line, start at –6 and move 5 units in either direction. Answer: The solution set is Solve an Absolute Value Equation Example 5-1a Method 1 Graphing The distance from –6 to –11 is 5 units. The distance from –6 to –1 is 5 units.
5. Write as or Original inequality Case 1 Case 2 Subtract 6 from eachside. Simplify. Answer: The solution set is Solve an Absolute Value Equation Example 5-1a Method 2 Compound Sentence
6. Solve an Absolute Value Equation Example 5-1b Answer: {12, –2}
7. So, an equation is . Write an Absolute Value Equation Example 5-2a Write an equation involving the absolute value for the graph. Find the point that is the same distance from –4 as the distance from 6. The midpoint between –4 and 6 is 1. The distance from 1 to –4 is 5 units. The distance from 1 to 6 is 5 units.
8. Check Substitute –4 and 6 into Answer: Write an Absolute Value Equation Example 5-2a
9. Answer: Write an Absolute Value Equation Example 5-2b Write an equation involving the absolute value for the graph.
10. Solving Open Sentences Involving Absolute Value Consider the case | x | < n. | x | < 5 means the distance between 0 and x is LESS than 5 units Solving Open Sentences Involving Absolute Value If | x | < 5, then x > – 5 andx < 5. The solution set is {x| – 5 < x < 5}.
11. Solving Open Sentences Involving Absolute Value When solving equations of the form | x | < n, find the intersection of these two cases. Solving Open Sentences Involving Absolute Value Case 1 The value inside the absolute value symbols is less than the positive value of n. Case 2 The value inside the absolute value symbols is greater than negative value of n.
12. Then graph the solution set. Write as and Case 2 Case 1 Original inequality Add 3 to each side. Simplify. Answer: The solution set is Solve an Absolute Value Inequality (<) Example 5-3a
13. Then graph the solution set. Answer: Solve an Absolute Value Inequality (<) Example 5-3b
14. Solving Open Sentences Involving Absolute Value Consider the case | x | > n. | x | > 5 means the distance between 0 and x is GREATER than 5 units Solving Open Sentences Involving Absolute Value If | x | > 5, then x < – 5 orx > 5. The solution set is {x| x < – 5 or x > 5}.
15. Solving Open Sentences Involving Absolute Value When solving equations of the form | x | > n, find the union of these two cases. Solving Open Sentences Involving Absolute Value Case 1 The value inside the absolute value symbols is greater than the positive value of n. Case 2 The value inside the absolute value symbols is less than negative value of n.
16. Then graph the solution set. Write as or Case 2 Case 1 Original inequality Add 3 to each side. Simplify. Divide each side by 3. Simplify. Solve an Absolute Value Inequality (>) Example 5-4a
17. Answer: The solution set is Solve an Absolute Value Inequality (>) Example 5-4a
18. Then graph the solution set. Answer: Solve an Absolute Value Inequality (>) Example 5-4b
19. Solving Open Sentences Involving Absolute Value In general, there are three rules to remember when solving equations and inequalities involving absolute value: Solving Open Sentences Involving Absolute Value • If then or(solution set of two numbers) • If then and(intersection of inequalities) • If then or(union of inequalities)<|endoftext|>
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Revolution is a word usually reserved for significant political or social changes. In science, there have been several revolutions of ideas (paradigm shifts) that have forced scientists to re-examine their entire field. Darwin’s On the Origin of Species in 1859, Mendel’s discovery of genetics in 1866, and the discovery of DNA by James Watson, Francis Crick, and Rosalind Franklin in the 1950s did that for biology. Albert Einstein’s relativity and quantum mechanics concepts in the early twentieth century did the same for Newtonian physics. Plate tectonics was just as revolutionary for geology. Plate tectonics, the idea that the outer part of the Earth moves and causes earthquakes, mountains, and volcanoes, is the lens through which geologic study must be viewed because all earth processes make more sense in this context. Its importance in understanding how the world works is why it is the first topic of discussion in this text.
4.1 Continental Drift Hypothesis
Alfred Wegener (1880-1930) was a German scientist who specialized in meteorology and climatology. He had a knack for questioning accepted ideas, and this started in 1910 when he disagreed with isostasy (vertical land movement due to the weight being removed or added) as the explanation for the Bering Land Bridge. After literary reviews, he published a hypothesis stating the continents had moved in the past. While he did not have the precise mechanism worked out, he had a long list of evidence that backed up his hypothesis of continental drift.
Early Evidence for Continental Drift
The first piece of evidence is that the shape of the coastlines of some continents fit together like pieces of a jigsaw puzzle. Since the first world map, people have noticed the similarities in the coastlines of South America and Africa, and the continents being ripped apart had even been mentioned as an explanation. Antonio Snider-Pellegrini even did preliminary work on continental separation and matching fossils in 1858.
What Wegener did differently than others was synthesized a significant amount of data in one place, as well as use the shape of the continental shelf, the actual edge of the continent, instead of the current coastline, which fit even better than previous efforts. Wegener also compiled and added to evidence of similar rocks, fossils, and glacial formations across the oceans.
For example, the primitive aquatic reptile Mesosaurus was found on the separate coastlines of the continents of Africa and South America, and the reptile Lystrosaurus was found on Africa, India, and Antarctica. These were land-dwelling creatures that could not have swam across an entire ocean; thus this was explained away by opponents of continental drift by land bridges. The land bridges, which, in the hypothesis of proponents, had eroded away, allowed animals and plants to move between the continents. However, some of the presumed land bridges would have had to have stretched across broad, deep oceans.
Mountain ranges with the same rock types, structures, and ages are now on opposite sides of the Atlantic Ocean. The Appalachians of the eastern United States and Canada, for example, are just like mountain ranges in eastern Greenland, Ireland, Great Britain, and Norway. Wegener concluded that they formed a single mountain range that was separated as the continents drifted.
Another significant piece of evidence was climate anomalies. Late Paleozoic glacial evidence was found in widespread, warm areas like southern Africa, India, Australia, and the Arabian subcontinent. Wegener himself had found evidence of tropical plant fossils in areas north of the Arctic Circle. According to Wegener, the simpler explanation that fit all the climate, rock, and fossil observations, mainly as more data were collected, involved moving continents.
Grooves and rock deposits left by ancient glaciers are found today on different continents very close to the equator. This would indicate that the glaciers either formed in the middle of the ocean and/or covered most of the Earth. Today glaciers only form on land and nearer the poles. Wegener thought that the glaciers were centered over the southern land mass close to the South Pole and the continents moved to their present positions later on.
Proposed Mechanism for Continental Drift
Wegener’s work was considered a fringe theory for his entire life. One of the most significant apparent flaws and easiest dismissals of Wegener’s hypothesis was a mechanism for movement of the continents. The continents did not appear to move, and extraordinary evidence would need to be provided to change the minds of the establishment, including a mechanism for movement. Other pro-continental drift followers had used expansion, contraction, or even the origin of the Moon as ideas to how the continents moved. Wegener used centrifugal forces and precession to explain the movement, but that was proven wrong. He had some speculation about seafloor spreading, with hints of convection within the earth, but these were unsubstantiated. As it turns out, convection within the mantle has been revealed as a significant force in driving plate movements, according to current knowledge.
4.2 Development of the Theory of Plate Tectonics
Wegener died in 1930 on an expedition in Greenland. In his lifetime, he was poorly respected, and his ideas of moving continents seemed destined to be lost to history as a fringe idea. However, starting in the 1950s, evidence started to trickle in that made continental drift more viable. By the 1960’s, there was enough evidence supporting Wegener’s missing mechanism, seafloor spreading, allowing the hypothesis of continental drift to develop into the Theory of Plate Tectonics. Widespread acceptance among scientists has transformed Wegener’s hypothesis to a Theory. Today, GPS and earthquake data continue to back up the theory. Below are the pieces of evidence that allowed the transformation.
Mapping the Ocean Floors
Starting in 1947 and using an adaptation of SONAR, researchers began to map a poorly-understood topographic, and thermal high in the mid-Atlantic . Bruce Heezen and Marie Tharp were the first to make a detailed map of the ocean floor, and this map revealed the mid-Atlantic Ridge, a basaltic feature, unlike the continents. Initially, this was thought to be part of an expanding Earth or a mechanism for the growth of the ocean. Transform faults were also added to explain movements more completely. When it was later realized that earthquake epicenters were also located within this feature, the idea that this was part of continental movement took hold.
Another way the seafloor was mapped was magnetically. Scientists had long known of strange magnetic anomalies (magnetic values that differ from expected values) associated with the ocean floor. This tool was adapted by geologists later for further study of the ocean depths, including strange alternating symmetrical stripes on both sides of a feature (which would be discovered later as the mid-ocean ridge) showing reversing magnetic pole directions. By 1963, these magnetic stripes would be explained in concordance with the spreading model of Hess and others.
Seafloor sediment was also an important feature that was measured in the oceans, both with dredging and with drilling. Sediment was believed to have been piling up on ocean floors for a very long time in a static model of accumulation. Initial studies showed less sediment than expected, and initial results were even used to argue against continental movement. With more time, researchers discovered thinner sediment close to ridges, indicating a younger age.
As the video below explains, today scientists are also able to use satellite imagery to map the ocean floor.
Around the same time that mid-ocean ridges were being investigated, ocean trenches and island arcs were also being linked to seismic action, thus explaining the opposite sides of the movement of plates. A zone of deep earthquakes that lay along a plane trending from the surface near the trenches to inside the Earth beneath the continents and island arcs were recognized independently by several scientists. Today called the Wadati-Benioff zone; it was an essential piece of the puzzle.
Magnetic field mapping, as mentioned above, was not the only way magnetism was used in the development of plate tectonics. In fact, the first new hard evidence that supported plate motion came from paleomagnetism. Paleomagnetism is the study of magnetic fields frozen within rocks, basically a fossil compass. This is typically most useful with igneous rocks where magnetic minerals like magnetite crystallizing in the magma align with the Earth’s magnetic field and in the solid rock point to the paleo-magnetic north. The earth’s magnetic field creates flux lines surrounding the magnetic north and south poles (like a bar magnet) which are both close to the Earth’s rotational north and south poles. In igneous rocks, magnetic minerals align parallel with these flux lines as shown in the figure. Thus both magnetic inclination, related to latitude, and declination related to magnetic north are preserved in the rocks.
Scientists had noticed for some time that magnetic north, to which many rocks pointed, was nowhere close to current magnetic north. This was explained by implying the magnetic north pole moved over time. Eventually, scientists started to realize that moving continents explained the data even better than moving the pole around alone.
Seafloor Spreading Hypothesis
World War II gave scientists the tools to find the mechanism for continental drift that had eluded Wegener. Maps and other data gathered during the war allowed scientists to develop the seafloor spreading hypothesis. This hypothesis traces oceanic crust from its origin at a mid-ocean ridge to its destruction at a deep sea trench and is the mechanism for continental drift.
During World War II, battleships and submarines carried echo sounders to locate enemy submarines. Echo sounders produce sound waves that travel outward in all directions, bounce off the nearest object, and then return to the ship. By knowing the speed of sound in seawater, scientists calculate the distance to the object based on the time it takes for the wave to make a round-trip. During the war, most of the sound waves ricocheted off the ocean bottom. This animation shows how sound waves are used to create pictures of the seafloor and ocean crust.
After the war, scientists pieced together the ocean depths to produce bathymetric maps, which reveal the features of the ocean floor as if the water were taken away. Even scientists were amazed that the seafloor was not completely flat. What was discovered was a large chain of mountains along the deep seafloor, called mid-ocean ridges. Scientists also discovered deep-sea trenches along the edges of continents or in the sea near chains of active volcanoes. Finally, large, flat areas called abyssal plains we found. When they first observed these bathymetric maps, scientists wondered what had formed these features.
Scientists brought these observations together in the early 1960s to create the seafloor spreading hypothesis. In this hypothesis, hot buoyant mantle rises up a mid-ocean ridge, causing the ridge to rise upward. The hot magma at the ridge erupts as lava that forms new seafloor. When the lava cools, the magnetite crystals take on the current magnetic polarity and as more lava erupts, it pushes the seafloor horizontally away from ridge axis.
The magnetic stripes continue across the seafloor. As oceanic crust forms and spreads, moving away from the ridge crest, it pushes the continent away from the ridge axis. If the oceanic crust reaches a deep sea trench, it sinks into the trench and is lost into the mantle. Scientists now know that the oldest crust is coldest and lies deepest in the ocean because it is less buoyant than the hot new crust.
The Unifying Theory of Plate Tectonics
Using all of the evidence mentioned, the theory of plate tectonics took shape. In 1966, J. Tuzo Wilson was the first scientist to put the entire picture together of an opening and closing ocean. Before long, models were proposed showing the plates moving concerning each other with clear boundaries between them, and scientists had also started to piece together complicated tectonic histories. The plate tectonic revolution had taken hold.
Seafloor and continents move around on Earth’s surface, but what is actually moving? What portion of the Earth makes up the “plates” in plate tectonics? This question was also answered because of technology developed during the Cold War. The tectonic plates are made up of the lithosphere. During the 1950s and early 1960s, scientists set up seismograph networks to see if enemy nations were testing atomic bombs. These seismographs also recorded all of the earthquakes around the planet. The seismic records could be used to locate an earthquake’s epicenter, the point on Earth’s surface directly above the place where the earthquake occurs. Earthquake epicenters outline these tectonic plates. Mid-ocean ridges, trenches, and large faults mark the edges of these plates along with where earthquakes occur.
The lithosphere is divided into a dozen major and several minor tectonic plates. The plates’ edges can be drawn by connecting the dots that mark earthquakes’ epicenters. A single plate can be made of all oceanic lithosphere or all continental lithosphere, but nearly all plates are made of a combination of both. Movement of the plates over Earth’s surface is termed plate tectonics. Plates move at a rate of a few centimeters a year, about the same rate fingernails grow.
4.3 Layers of the Earth
To understand the details of plate tectonics, one must first understand the layers of the Earth. Humankind has insufficient first-hand information regarding what is below; most of what we know is pieced together from models, seismic waves, and assumptions based on meteorite material. In general, the Earth can be divided into layers based on chemical composition and physical characteristics.
The Earth has three main divisions based on their chemical composition, which means chemical makeup. Indeed, there are countless variations in composition throughout the Earth, but it appears that only two significant changes take place, leading to three distinct chemical layers.
The outermost chemical layer and the layer humans currently reside on is known as the crust. The crust has two types: continental crust, which is relatively low density and has a composition similar to granite, and oceanic crust, which is relatively high density (especially when it is cold and old) and has a composition similar to basalt. In the lower part of the crust, rocks start to be more ductile and less brittle, because of added heat. Earthquakes, therefore, generally occur in the upper crust.
At the base of the crust is a substantial change in seismic velocity called the Mohorovičić Discontinuity, or Moho for short, discovered by Andrija Mohorovičić (pronounced mo-ho-ro-vee-cheech) in 1909 by studying earthquake wave paths in his native Croatia. It is caused by the dramatic change in composition that occurs between the mantle and the crust. Underneath the oceans, the Moho is about 5 km down. Under continents, the average is about 30-40 km, except near a sizeable mountain-building event, known as an orogeny, where that thickness is about doubled.
The mantle is the layer below the crust and above the core, and is the most substantial layer by volume, extending from the base of the crust to a depth of about 2900 km. Most of what we know about the mantle comes from seismic waves, though some direct information can be gathered from parts of the ocean floor that are brought to the surface, known as ophiolites. Also, carried within magma are xenoliths, which are small chunks of lower rock carried to the surface by eruptions. These xenoliths are mainly made of the rock peridotite, which on the scale of igneous rocks is ultramafic. We assume the majority of the mantle is made of peridotite.
The core of the Earth, which has both liquid and solid components, is made mostly of iron and nickel and possibly minor oxygen. First discovered in 1906 by looking into seismic data, it took the union of modeling, astronomical insight, and seismic data to arrive at the idea that the core is mostly metallic iron. Meteorites contain much more iron than typical surface rocks, and if meteoric material is what made the Earth, the core would have formed as dense material (including iron and nickel) sank to the center of the Earth via its weight as the planet formed, heating the Earth intensely.
The Earth can also be broken down into five distinct physical layers based on how each layer responds to stress. While there is some overlap in the chemical and physical designations of layers, specifically the core-mantle boundary, there are significant differences between the two systems.
The lithosphere, with ‘litho’ meaning rock, is the outermost physical layer of the Earth. Including the crust, it has both an oceanic component and a continental component. Oceanic lithosphere, ranging from a thickness of zero (at the forming of new plates on the mid-ocean ridge) to 140 km, is thin and relatively rigid. Continental lithosphere is considerably more plastic in nature (especially with depth) and is overall thicker, from 40 to 280 km thick. Most importantly, the lithosphere is not continuous. It is broken into several segments that geologists call plates. A plate boundary is where two plates meet and move relative to each other. It is at and near plate boundaries where the real action of plate tectonics is seen, including mountain building, earthquakes, and volcanism.
The asthenosphere, with ‘astheno’ meaning weak, is the layer below the lithosphere. The most distinctive property of the asthenosphere is movement. While still solid, over geologic time scales it will flow and move because it is mechanically weak. It is in this layer that movement, partly driven by convection of intense interior heat, allows the lithospheric plates to move. Since certain types of seismic waves pass through the asthenosphere, we know that it is solid, at least at the very short time scales of the passage of seismic waves. The depth and occurrence of the asthenosphere are dependent on heat and can be very shallow at mid-ocean ridges and very deep in plate interiors and beneath mountains.
The mesosphere, or lower mantle as it is sometimes called, is more rigid and immobile than the asthenosphere, though still hot. This can be attributed to increased pressure with depth. Between approximately 410 and 660 km depth, the mantle is in a state of transition as minerals with the same composition are changed to various forms, dictated by the conditions of increasing pressure. Changes in seismic velocity show this, and this zone also can be a physical barrier to movement. Below this zone, the mantle is relatively uniform and homogeneous, as no major changes occur until the core is reached.
The outer core is the only liquid layer found within Earth. It starts at 2,890 km (1,795 mi) depth and extends to 5,150 km (3,200 mi). Inge Lehmann, a Danish geophysicist, in 1936, was the first to prove that there was an inner core that was solid within the liquid outer core based on analyzing seismic data. The solid inner core is about 1,220 km (758 mi) thick, and the outer core is about 2,300 km (1,429 mi) thick.
It seems like a contradiction that the hottest part of the Earth is solid, as high temperatures usually lead to melting or boiling. The solid inner core can be explained by understanding that the immense pressure inhibits melting, though as the Earth cools by heat flowing outward, the inner core grows slightly larger over time. As the liquid iron and nickel in the outer core moves and convects, it becomes the most likely source for Earth’s magnetic field. This is critically important to maintaining the atmosphere and conditions on Earth that make it favorable to life. Loss of outer core convection and the Earth’s magnetic field could strip the atmosphere of most of the gases essential to life and dry out the planet; much like what has happened to Mars.
4.4 Plate Tectonic Boundaries
Places, where oceanic and continental lithospheric tectonic plates meet and move relative to each other, are called active margins (e.g., the western coasts of North and South America). A location where continental lithosphere transitions into oceanic lithosphere without movement is known as a passive margin (e.g., the eastern coasts of North and South America). This is why tectonic plates may be made of both oceanic and continental lithosphere. In the process of plate tectonics, the lithospheric plates movement is the primary force that causes the majority of features and activity on the Earth’s surface that can be attributed to plate tectonics. This movement occurs (at least partially) via the drag of motion within the asthenosphere and because of density.
As they move, the tectonic plates interact with each other at the boundaries between the tectonic plates. These interactions are the primary drivers of mountain building, earthquakes, and volcanism on the planet. In a simplified plate tectonic model, plate interaction can be placed in one of three categories. In places where plates move toward each other, the boundary is known as convergent. In places where plates move apart, the boundary is known as divergent. In places where the plates slide past each other, the boundary is known as a transform boundary. The next three subchapters will explain the details of the movement at each type of boundary.
Convergent boundaries, sometimes called destructive boundaries, are places where two or more tectonic plates have a net movement toward each other. Convergent boundaries, more than any other, are known for orogenesis, the process of building mountains and mountain chains. The key to convergent boundaries is understanding the density of each plate involved in the movement. Continental lithosphere is always lower in density and is buoyant when compared to the asthenosphere. Oceanic lithosphere, on the other hand, is denser than continental lithosphere and, when old and cold, may even be denser than the asthenosphere. When plates of different density converge, the more dense plate sinks beneath, the less dense plate, a process called subduction.
Subduction is when oceanic lithosphere descends into the mantle due to its density. The average rate of subduction of oceanic crust worldwide is 25 miles per million years, about a half inch per year. Continental lithosphere can partially subduct if attached to sinking oceanic lithosphere, but its buoyancy does not allow it to subduct fully. As the tectonic plate descends, it also pulls the ocean floor down in a feature known as a trench. On average, the ocean floor is around 3-4 km deep. In trenches, the ocean can be more than twice as deep, with the Mariana Trench approaching a staggering 11 km.
Within the trench is a feature called the accretionary wedge, sometimes known as melange or accretionary prism, which is a mix of ocean floor sediments that are scraped and compressed at the boundary between the subducting plate and the overriding plate. Sometimes pieces of continental material, like microcontinents, riding with the subducting plate will become sutured to the accretionary wedge, forming a terrane. In fact, large portions of California are comprised of accreted terranes.
When the subducting plate, known as a slab, submerges into the depths of the mantle, the heat and pressure are so immense that lighter materials, known as volatiles, like water and carbon dioxide are pushed out of the subducting plate into an area called the mantle wedge above. The volatiles are released mostly via hydrated minerals that revert to non-hydrated forms in these conditions. These volatiles, when mixed with asthenospheric material above the tectonic plate, lower the melting point of the material. At the temperature of that depth, the material melts to form magma. This process of magma generation is called flux melting. Magma, because of its lower density, migrates toward the surface, creating volcanism. This forms a curved chain of volcanoes, due to many boundaries being curved on a spherical Earth, a feature called an arc. The overriding plate which contains the arc can be either oceanic or continental, where some features are different, but the general architecture remains the same.
How subduction initiates is still a matter of some debate. Presumably, this would start at passive margins where oceanic and continental crust meet. At the current time, there is oceanic lithosphere that is denser than the underlying asthenosphere on either side of the Atlantic Ocean that is not currently subducting. Why has it not turned into an active margin? Firstly, there is strength in the connection between the dense oceanic lithosphere and the less dense continental lithosphere it is connected to, which needs to be overcome. Gravity could cause the denser oceanic plate to force itself down, or the plate can start to flow ductility at a low angle. There is evidence that new subduction is starting off the coast of Portugal. Large earthquakes, like the 1755 Lisbon Earthquake, may even have something to do with this process of creating a subduction zone, though it is not definitive. Transform boundaries that have brought areas of different densities together are also thought to start subduction possibly.
Besides volcanism, subduction zones are also known for the largest earthquakes in the world. In places, the entire subducting slab can become stuck, and when the energy has built up too high, the entire subduction zone can slide at once along a zone extending for hundreds of kilometers along the trench, creating enormous earthquakes and tsunamis. The earthquakes can not only be large, but they can be deep, outlining the subducting slab as it descends. Subduction zones are the only places on Earth with fault surfaces large enough to create magnitude nine earthquakes. Also, because the faulting occurs beneath seawater, subduction also can create giant tsunamis, such as the 2004 Indian Ocean Earthquake and the 2011 Tōhoku Earthquake in Japan.
Subduction, which is a convergent motion, can have varying degrees of convergence. In places that have a high rate of convergence, mostly due to young, buoyant oceanic crust subducting, the subduction zone can create faulting behind the arc area itself, known as back-arc faulting. This faulting can be tensional, or this area is subject to compressional forces. A modern example of this occurs in the two ‘spines’ of the Andes Mountains. In the west, the mountains are formed from the volcanic arc itself; in the east, thrust faults have pushed up another, non-volcanic mountain range still part of the Andes. This type of thrusting can typically occur in two styles: thin-skinned, which only faults surficial rocks, and thick-skinned, which thrusts deeper crustal rocks. Thin-skinned deformation notably occurred in the western U.S. during the Cretaceous Sevier Orogeny. Near the end of the Sevier Orogeny, thick-skinned deformation also occurred in the Laramide orogeny.
The Laramide Orogeny is also known for another subduction feature: flat slab subduction. When the slab subducts at such a low angle, there is an interaction between the slab and the overlying continental plate. Magmatic activity can give rise to mineral deposits, and deformation can occur well into the interior of the overriding plate. All subduction zones have a forearc basin, which is an area between the arc and the trench. This is an area of a high degree of thrust faulting and deformation, seen mostly within the accretionary wedge. There are also places where the convergence shows the results of tensional forces. A variety of causes have been proposed for this, including slab roll-back due to density or ridge migration. This causes extension behind the volcanic or island arc, known as a back-arc basin. These can have so much extension that rifting and divergence can develop, though they can be more asymmetric than their mid-ocean ridge counterparts.
Oceanic-continental subduction occurs when an oceanic plate dives below continental plates. This boundary has a trench and mantle wedge, but the volcanoes are expressed in a feature known as a volcanic arc. A volcanic arc is a chain of mountain volcanoes, with famous examples including the Cascades of the Pacific Northwest (map) and the Andes of South America (map).
Oceanic-oceanic subduction zones have two significant differences from boundaries that have continental lithosphere. Firstly, each plate in an ocean-ocean plate boundary is capable of subduction. Therefore, it is typical that the denser, older, and colder of the two plates is the one that subducts. Secondly, since both plates are oceanic, volcanism creates volcanic islands instead of continental volcanic mountain ranges. This chain of active volcanoes is known as an island arc. There are many examples of this on Earth, including the Aleutian Islands off of Alaska (map), the Lesser Antilles in the Caribbean (map), and several island arcs in the western Pacific.
In places where two continental plates converge toward each other, subduction is not possible. This occurs where an ocean basin closes, and a passive margin is attempted to be driven down with the subducting slab. Instead of subducting beneath the continent, the two masses of continental lithosphere slam into each other in a process known as a collision. Collision zones are known for tall mountains and frequent, massive earthquakes, with little to no volcanism. With subduction ceasing with the collision, there is not a process to create the magma for volcanism.
Continental plates are too low density to subduct, which is why the process of collision occurs instead of subduction. Unlike the dense subducting slabs that form from oceanic plates, any attempt to subduct continental plates is short lived. An occasional exception to this is obduction, in which a part of a continental plate is caught beneath an oceanic plate, formed in collision zones or with small plates caught in subduction zones. This imbalance in density is solved by the continental material buoying upward, bringing oceanic floor and mantle material to the surface, and is the primary source of ophiolites. An ophiolite consists of rocks of the ocean floor that are moved onto the continent, which can also expose parts of the mantle on the surface.
Foreland basins can also develop near the mountain belt, as the lithosphere is depressed due to the mass of the mountains themselves. While subduction mountain ranges can cause this, collisions have many examples, with possibly the best modern example being the Persian Gulf, a feature only there due to the weight of the nearby Zagros Mountains. Collisions are powered by the subducting oceanic lithosphere, and eventually stop as the continental plates combine into a larger mass. In truth, a small portion of the continental crust can be driven down into the subduction zone, though due to its buoyancy, it returns to the surface over time. Because of the relative plastic nature of continental lithosphere, the zone of deformation is much broader. Instead of earthquakes located along a narrow boundary, collision earthquakes can be found hundreds of miles from the suture between the land masses.
The best modern example of this process occurs concurrently in many locations across the Eurasian continent and includes mountain building in the Pyrenees (the Iberian Peninsula converging with France, map), Alps (Italy converging into central Europe, map), Zagros (Arabia converging into Iran, map), and Himalayan (India converging into Asia, map) ranges. Eventually, as ocean basins close, continents join together to form a massive accumulation of continents called a supercontinent, a process that has taken place in hundreds of million-year cycles over earth’s history.
Divergent boundaries, sometimes called constructive boundaries, are places where two or more plates have a net movement away from each other. They can occur within a continental plate or an oceanic plate, though the typical pattern is for divergence to begin within continental lithosphere in a process known as “rift to drift,” described below.
Because of the thickness of continental plates, heat flow from the interior is suppressed. The shielding that supercontinents provide is even stronger, eventually causing upwelling of hot mantle material. This material uplift weakens overlying continental crust, and as convection beneath naturally starts pulling the material away from the area, the area starts to be deformed by tensional stress forming a valley feature known as a rift valley. These features are bounded by normal faults and include tall shoulders called horsts, and deep basins called grabens. When rifts form, they can eventually cause linear lakes, linear seas, and even oceans to form as divergent forces continue.
This breakup via rifting, while initially seeming random, actually has two influences that dictate the shape and location of rifting. First of all, the stable interiors of some continents, called a craton, are seemingly too strong to be broken apart by rifting. Where cratons are not a factor, rifting typically occurs along the patterns of a truncated icosahedron, or “soccer ball” pattern. This is the geometric pattern of fractures that requires the least amount of energy when expanding a sphere equally in all directions. Taking into account the radius of the Earth, this includes ~110 km segments of deformation and volcanism which have 120 degree turns, forming something known as failed rift arms. Even if the motion stops, a minor basin can develop in this weak spot called an aulacogen, which can form long-lived basins well after tectonic processes stop. These are places where extension started but did not continue. One famous example is the Mississippi Valley Embayment, which forms a depression through which the upper end of the Mississippi River flows. In places where the rift arms do not fail, for example, the Afar Triangle, three divergent boundaries can develop near each other forming a triple junction.
Rifts come in two types: narrow and broad. Narrow rifts contain concentrated stress or divergent action. The best active example is the East African Rift Zone, where the horn of Africa near Somalia is breaking away from mainland Africa (map). Lake Baikal in Russia is also an active rift (map). Broad rifts distribute the deformation over a wide area of many fault-bounded locations, like in the western United States in a region known as the Basin and Range (map). The Wasatch Fault, which created the Wasatch Range in Utah, marks the eastern edge of the Basin and Range (map).
Earthquakes, of course, do occur at rifts, though not at the severity and frequency of some other boundaries. Volcanism is also frequent in the extended, faulted, and thin lithosphere found at rift zones due to decompressional melting and faults acting as conduits for the lava reaching the surface. Many relatively young volcanoes dot the Basin and Range, and very strange volcanoes occur in East Africa like Ol Doinyo Lengai in Tanzania, which erupts carbonatite lavas, relatively cold liquid carbonate.
As rifting and volcanic activity progress, the continental lithosphere becomes more mafic and thinner, with the eventual result transforming the plate under the rifting area into the oceanic lithosphere. This is the process that gives birth to a new ocean, much like the narrow Red Sea (map) emerged with the movement of Arabia away from Africa. As the oceanic lithosphere continues to diverge, a mid-ocean ridge is formed.
A mid-ocean ridge, also known as a spreading center, has many distinctive features (map). They are the only places on Earth where the new oceanic lithosphere is being created, via slow oozing volcanism. As the oceanic lithosphere spreads apart, rising asthenosphere melts due to decreasing pressure and fills in the void, making the new lithosphere and crust. These volcanoes produce more lava than all the other volcanoes on Earth combined, and yet are not usually listed on maps of volcanoes due to the vast majority of mid-ocean ridges being underwater. Only rare locations, such as Iceland, are the volcanism and divergent characteristics seen on land. Technically, these places are not mid-ocean ridges, because they are above the surface of the seafloor. The video below is drone imagery of the Icelandic Lava River.
Alfred Wegener even hypothesized this concept of mid-ocean ridges. Because the lithosphere is very hot at the ridge, it has a lower density. This lower density allows it to isostatically ‘float’ higher on the asthenosphere. As the lithosphere moves away from the ridge by continued spreading, the plate cools and starts to sink isostatically lower, creating the surrounding abyssal plains with lower topography. Age patterns also match this idea, with younger rocks near the ridge and older rocks away from the ridge. Sediment patterns also thin toward the ridge, since the steady accumulation of dust and biologic material takes time to accumulate.
Another distinctive feature around mid-ocean ridges is magnetic striping. Called the Vine-Matthews-Morley Hypothesis, it states that as the material moves away from the ridge, it cools below the Curie Point, which is the temperature at which the magnetic field is imprinted on the rock as the rock freezes. Over time, the Earth’s magnetic field has flipped back and forth, and it is this change in the field that causes the stripes. This pattern is an excellent record of past ocean-floor movements and can be used to reconstruct past tectonics and determine rates of spreading at the ridges.
Mid-ocean ridges also are home to some of the unique ecosystems ever discovered, found around hydrothermal vents that circulate ocean water through the shallow oceanic crust and send it back out rich with chemical compounds and heat. While it was known for some time that hot fluids could be found on the ocean floor, it was only in 1977 when a team of scientists using the Diving Support Vehicle Alvin discovered a thriving community of organisms, including tube worms bigger than people. This group of organisms is not at all dependent on the sun and photosynthesis but instead relies on chemical reactions with sulfur compounds and heat from within the Earth, a process known as chemosynthesis. Before this discovery, the thought in biology was that the sun was the ultimate source of energy in ecosystems; now we know this to be false. Not only that, some have suggested it is from this that life could have started on Earth, and it now has become a target for extraterrestrial life (e.g., Jupiter’s moon Europa).
A transform boundary, sometimes called a strike-slip or conservative boundary, is a place where the motion is of the plates sliding past each other. They can move in either dextral fashion with the side opposite moving toward the right or a sinistral fashion with the side opposite moving toward the left. Most transform boundaries can be viewed as a single fault or as a series of faults. As stress builds on adjacent plates attempting to slide them past each other, eventually a fault occurs and releases stress with an earthquake. Transform faults have a shearing motion and are common in places where tectonic stresses are transferred. In general, transform boundaries are known for only earthquakes, with little to no mountain building and volcanism.
The majority of transform boundaries are associated with mid-ocean ridges. As spreading centers progress, these aseismic fracture zone transform faults accommodate different amounts of spreading due to Eulerian geometry that a sphere rotates faster in the middle (Equator) than at the top (Poles) than along the ridge. However, the more significant transform faults, in the eyes of humanity, are the places where the motion occurs within continental plates with a shearing motion. These transform faults produce frequent moderate to large earthquakes. Famous examples include California’s San Andreas Fault (map), both the Northern and Eastern Anatolian Faults in Turkey (map), the Altyn Tagh Fault in central Asia (map), and the Alpine Fault in New Zealand (map).
Transpression and Transtension
In places where transform faults are not straight, they can create secondary faulting. Transpression is defined as places where there is an extra component of compression with shearing. In these restraining bends, mountains can be built up along the fault. The southern part of the San Andreas Fault has a large area of transpression known as the “big bend” and has built, moved, and even rotated many mountain ranges in southern California.
Transtension is defined as places where there is an extra component of extension with shearing. In these releasing bends, depressions and sometimes volcanism are formed along the fault. The Dead Sea and California’s the Salton Sea are examples of basins formed by transtensional forces.
A piercing point is a feature that is cut by a fault, and thus can be used to recreate past movements along the fault. While this can be used on all faults, transform faults are most adapted for this technique. Normal and reverse faulting and divergent and convergent boundaries tend to obscure, bury, or destroy these features; transform faults generally do not. Piercing points usually consist of unique lithologic, structural, or geographic patterns that can be matched by removing the movement along the fault. Detailed studies of piercing points along the San Andreas Fault has shown over 225 km of movement in the last 20 million years along three different active traces of the fault.
4.5 Wilson Cycle
The Wilson Cycle, named for J. Tuzo Wilson who first described it in 1966, outlines the origin and subsequent breakup of supercontinents. This cycle has been operating for the last billion years with supercontinents Pangaea and Rodinia, and possibly billions of years before that. The driving force of this is two-fold. The more straightforward mechanism arises from the fact that continents hold the Earth’s internal heat much better than the ocean basins. When continents congregate together, they hold more heat in which more vigorous convection can occur, which can start the rifting process. Mantle plumes are inferred to be the legacy of this increased heat and may record the history of the start of rifting. The second mechanism for the Wilson Cycle involves the destruction of plates. While rifting eventually leads to drifting continents, a few unanswered questions emerge:
- Does their continued movement result from a continuation of the ridge spreading and underlying convection, known as ridge push?
- Do the tectonic plates move because of the weight of the subducting slab sinking via its density, known as slab pull?
- Alternatively, does the height of the ridge pushing down, known as gravitational sliding?
To be sure, these are all factors in plate movement and the Wilson Cycle. It does appear, in the current best hypothesis, that there is a more significant component of slab pull than ridge push. Plate tectonic models are beginning to detail the next supercontinent, called Pangea Proxima, that will form 250 million years.
While the Wilson Cycle can give a general overview of plate motions in the past, another process can give more precise, but mainly recent, plate movement. A hot spot (map) is an area of rising magma, causing a series of volcanic centers which form volcanic islands in the ocean or craters/mountains on land. There is not a plate tectonic process, like subduction or rifting, that causes this volcanic activity; it seems as if disconnected to plate tectonics processes. Also first postulated by J. Tuzo Wilson, in 1963, hot spots are places that have a continual source of magma with no earthquakes, besides those associated with volcanism. The classic idea is that hot spots do not move, though some evidence has been suggested that the hot spots do move as well. Even though hotspots and plate tectonics seem independent, there are some relationships between them, and they have two components: Firstly, there are several hot spots currently and several others in the past that are believed to have begun at the time of rifting. Secondly, as plate tectonics moves the plates around, the assumed stationary nature of hot spots creates a track of volcanism that can measure past plate movement. By using the age of the eruptions from hot spots and the direction of the chain of events, one can identify a specific rate and direction of movement of a plate over the time the hot spot was active.
Hot spots are still very mysterious in their exact mechanism of magma generation. The main camps on hotspot mechanics are opposed. Some claim deep sources of heat, from as deep as the core, bring heat up to the surface in a structure called a mantle plume. Some have argued that not all hot spots are sourced from deep within the planet, and are sourced from shallower parts of the mantle. Others have mentioned how difficult it has been to image these deep features. The idea of how hot spots start is also controversial. Usually, divergent boundaries are tabbed as the start, especially during supercontinent break up, though some question whether extensional or tectonic forces alone can explain the volcanism. Subducting slabs have also been named as a cause for hotspot volcanism. Even impacts of objects from space have been used to explain plumes. However they are formed, there are dozens found throughout the Earth. Famous examples include the Tahiti, Afar Triangle, Easter Island, Iceland, the Galapagos Islands, and Samoa. The United States has two of the largest and best-studied examples: Hawai’i and Yellowstone.
Hawaiian Hot Spot
The big island of Hawai’i (map) is the active end of the Hawaiian-Emperor seamount chain, which stretches across the Pacific for almost 6000 km. The evidence for this hot spot goes back at least 80 million years, and presumably, the hot spot was around before then, but rocks older than that in the Pacific Plate had already subducted. The most striking feature of the chain is a significant bend that occurs about halfway through the chain that occurred about 50 million years ago. The change in direction has been more often linked to a plate reconfiguration, but also to other things like plume migration. While it is often assumed that mantle plumes do not move, much like the plumes themselves, this idea is under dispute by some scientists.
3D seismic imaging, called tomography, has mapped the Hawaiian mantle plume at depths including the lower mantle. Within the Hawaiian Islands, there is clear evidence of the age of volcanism decreasing, including island size, rock age, and even vegetation. Hawai’i is one of the most active hotspots on Earth. Kilauea, the main active vent of the hot spot eruption, has continually erupted since 1983.
The Yellowstone Hot Spot (map) is formed from rising magma, much like Hawai’i. The big difference is Hawai’i sits on a thin oceanic plate, which makes the magma easily come to the surface. Yellowstone, however, is on a continental plate. The thickness of the plate causes the generally much more violent and less frequent eruptions that have carved a curved path in the western United States for over 15 million years (see figure). Some have speculated an even earlier start to the hotspot, tying it to the Columbia River flood basalts and even 70 million-year-old volcanism in Canada’s Yukon.
The most recent significant eruption formed the current caldera and the Lava Creek Tuff. This eruption threw into the atmosphere about 1000 cubic kilometers of magma erupted 631,000 years ago. Ash from the eruption has been found as far away as Mississippi. The next eruption, when it occurs, should be of similar size, causing a massive calamity to not only the western United States, but also the world. These so-called “supervolcanic” eruptions have the potential for volcanic winters lasting years. With so much gas and ash filling the atmosphere, sunlight is blocked and unable to reach Earth’s surface as well as usual, which could drastically alter global environments and send worldwide food production into a tailspin.<|endoftext|>
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## Friday, January 16, 2009
### Sudoku techniques
Sudoku grid consists of 81 squares divided into nine columns marked a through i, and nine rows marked 1 through 9. The grid is also divided into nine 3x3 sub-grids named boxes which are marked box 1 through box 9.
## Scanning techniques
The easiest way starting a Sudoku puzzle is to scan rows and columns within each triple-box area, eliminating numbers or squares and finding situations where only a single number can fit into a single square. The scanning technique is fast and usually sufficient to solve easy puzzles all the way to the end. The scanning technique is also very useful for hard puzzles up to the point where no further progress can be made and more advanced solving techniques are required. Here are some ways of using scanning techniques:
#### 1. Scanning in one direction:
In our first example we will focus on box 2, which like any other box in Sudoku must contain 9. Looking at box 1 and box 3 we can see there are already 9s in row 2 and in row 3, which excludes the two bottom rows of box 2 from having 9. This leaves square e1 as the only possible place into which 9 can fit in.
#### 2. Scanning in two directions:
The same technique can be expanded by using information from perpendicular rows and columns. Let’s see where we can place 1 in box 3. In this example, row 1 and row 2 contain 1s, which leaves two empty squares in the bottom of box 3. However, square g4 also contains 1, so no additional 1 is allowed in column g. This means that square i3 is the only place left for 1.
#### 3. Searching for Single Candidates:
Often only one number can be in a square because the remaining eight are already used in the relevant row, column and box. Taking a careful look at square b4 we can see that 3, 4, 7 and 8 are already used in the same box, 1 and 6 are used in the same row, and 5 and 9 are used in the same column. Eliminating all the above numbers leaves 2 as the single candidate for square b4.
#### 4. Eliminating numbers from rows, columns and boxes:
There are more complex ways to find numbers by using the process of elimination. In this example the 1 in square c8 implies that either square e7 or square e9 must contain 1. Whichever the case may be, the 1 of column e is in box 8 and it is therefore not possible to have 1 in the centre column of box 2. So the only square left for 1 in box 2 is square d2.
#### 5. Searching for missing numbers in rows and columns:
This method can be particularly useful when rows (and columns) are close to completion. Let’s take a look at row 6. Seven of the nine squares contain the numbers 1, 2, 3, 4, 5, 8 and 9, which means that 6 and 7 are missing. However, 6 cannot be in square h6 because there is already 6 in that column. Therefore the 6 must be in square b6.
## Analyzing techniques
As Sudoku puzzle levels get harder you will find the simple scanning methods described above are not enough and more sophisticated solving techniques must be used. Hard puzzles require deeper logic analysis which is done with the aid of pencilmarks. Sudoku pencilmarking is a systematic process writing small numbers inside the squares to denote which ones may fit in. After pencilmarking the puzzle, the solver must analyze the results, identify special number combinations and deduce which numbers should be placed where. Here are some ways of using analyzing techniques:
#### 1. Eliminating squares using Naked Pairs in a box:
In this example, squares c7 and c8 in box 7 can only contain 4 and 9 as shown with the red pencilmarks below. We don’t know which is which, but we do know that both squares are occupied. In addition, square a6 excludes 6 from being in the left column of box 7. As a result the 6 can only be in square b9. Such cases where the same pair can only be placed in two boxes is called Disjoint Subsets, and if the Disjoint Subsets are easy to see then they are called Naked Pairs.
#### 2. Eliminating squares using Naked Pairs in rows and columns:
The previous solving technique is useful for deducing a number within a row or column instead of a box. In this example we see that squares d9 and f9 in box 8 can only contain 2 and 7. Again we don’t know which is which, but we do know that both squares are occupied. The numbers which remain to be placed in row 9 are 1, 6 and 8. However, 6 can’t be placed in square a9 or in square i9, so the only possible place is square c9.
#### 3. Eliminating squares using Hidden Pairs in rows and columns:
Disjoint Subsets are not always obvious to see at first sight, in which case they are called Hidden Pairs. If we take a very close look at the pencilmarks in row 7 we can see that both 1 and 4 can only be in square f7 and square g7. This means that 1 and 4 are a Hidden Pair, and that square f7 and square g7 cannot contain any other number. Using the scanning technique we see that 7 can only be in square d7.
#### 4. Eliminating squares using X-Wing:
The X-Wing technique is used in rare situations which occur in some extremely difficult puzzles. Scanning column a we see that 4 can only be in square a2 or square a9. Similarly, 4 can only be in square i2 or square i9. Because of the X-Wing pattern where boxes are in the same row (or column), a new logic constraint occurs: it is obvious that in row 2 the 4 can only be either in square a2 or in square i2, and it cannot be in any other square. Therefore 4 is excluded from square c2, and square c2 must be 2.
http://www.conceptispuzzles.com
# History and curiosities
– This is the budget price alternative Arlanda has needed for quite a long time and also a new landmark at Arlanda offering a unique experience for the guests, says hostel owner and the man behind Jumbo Hostel, Oscar Diös.
Oscar Diös previously owns and operates the hostel Uppsala Vandrarhem och Hotell.
– I was getting ready to expand my hostel business in 2006 when I heard about an old wreck of an aircraft for sale at Arlanda. Since I had for a long time wanted to establish my business at Arlanda I didn’t hesitate for a second when this opportunity struck, Oscar Diös explains.
The airplane, a decommissioned model 747-200 jumbo jet built in 1976, was last operated by Transjet, a Swedish airline that went bankrupt in 2002. It was originally built for Singapore Airlines and later served with legendary Pan Am.
In December 2007, Sigtuna authorities granted a building permit for establishing Jumbo Hostel at the entrance to Arlanda airport. In January 2008, the aircraft was moved to a construction site parking where the first phase of the conversion has begun with the dismantling of the old interior, new paint and new decorations for the rooms. 450 seats are taken out and the plane is sanitized in its entirety. The hostel is built like any house, subjected to the same demands on climate control and isolation. It adheres to all common energy standards. Heating is achieved with an air-air inverter.
Summer 2008 the plane was towed to its final destination at the entrance to Arlanda where it was placed on a concrete foundation with the landing gear secured in two steel cradles. Here, Jumbo Hostel are a spectacular landmark as a portal to Arlanda offering a view of the landing strip. No visitor to Arlanda will miss the new hostel!
http://www.jumbohostel.com
## Elevate the experience with the Washlet® S400. It’s the first Washlet seat that combines the convenience of a hands-free flush with an automatic open & close lid.
The Washlet S400, designed to work exclusively with select TOTO toilets, is our first intuitive Washlet. Experience the ultimate in clean comfort with an automatic, hands-free flushing system and a sensor-activated lid that automatically lifts as you approach the toilet and lowers as you walk away. A convenient, easy-to-use remote control affords you effortless operation of our most luxurious Washlet model to date. The S400 offers the following features:
• Auto Flush Activated by Sensors or the Simple Touch of a Button
• Automatic Open / Close Lid Activated by Sensors, or the Simple Touch of a Button on the Remote Control
• Gentle Aerated Warm Water
• Front and Rear Washing
• Massage Feature
• Warm Air Drying with Variable Three-Temperature Setting
• Automatic Air Purifier
• Heated Seat with Temperature Control
• Convenient Wireless Remote Control with Large LCD Panel
• Docking Station for Easy Cleaning & Installation
• Cleaner, Sleeker Look
• Reinforced Base Plate for Enhanced Durability
## How does the Washlet work?
The Washlet is designed to introduce you to a level unprecedented comfort, while delivering on the promise of maximum cleanliness. At your command, an integrated, self-cleaning nozzle extends to release a warm, soothing stream of aerated water to provide the ultimate in personal cleansing.
http://www.totousa.com/Default.aspx?tabid=209
Automatic Open / Close Lid Front and Rear Washing Warm Air Dryer Heated Seat Self-Cleaning Wand<|endoftext|>
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There are two different uses of the word ‘see’. You can see an object, a picture or a drawing for instance, where you see what is in front of you, as a blueprint. But you can also see a likeness in someone’s face. There’s a categorial difference between the two objects of sight.
Other examples of this seeing-as are seeing a geometrical drawing as a glass cube, seeing a triangle as a mountain, wedge or arrow, or the well known rabbit-duck drawing, in which you can recognize a rabbit as well as a duck.
The Austrian-British philosopher Ludwig Wittgenstein calls this experience ‘noticing an aspect’, or aspect perception (Aspektenwechsel): you see that the object you are looking at has not changed, and yet you see it differently.
Aspect perception is an unstable state. Only in the change of aspect does one become conscious of the aspect. In general, it involves noticing a similarity. You can say that you experience a comparison, an object is seen as a variation, or derivation, or copy, of another one.
When you see a tree, for example, and realize that it is a tree, you see its resemblance with other trees. Seeing is comparing, seeing is interpreting. Everything we perceive, we perceive in its relevant aspects: in a picture we immediately see what it represents and respond to it accordingly.
When a child plays with a box and cries out that it is a house, the box immediately becomes a house. To see this aspect means to see something that cannot be shown.
Seeing an aspect means recognizing an aspect that you hadn’t seen like that before. The picture doesn’t change, and our seeing doesn’t change either. It’s our interpretation that changes. This raises the question whether a picture is seen or thought of.<|endoftext|>
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Home > Statistical Methods calculators
AtoZmath.com - Homework help (with all solution steps) Educational Level Secondary school, High school and College Program Purpose Provide step by step solutions of your problems using online calculators (online solvers) Problem Source Your textbook, etc
1.1 Ungrouped data 85,96,76,108,85,80,100,85,70,95
1. Mean, Median and Mode for ungrouped data
2. Quartiles, Deciles and Percentiles for ungrouped data
3. Population Variance, Standard deviation and coefficient of variation for ungrouped data
4. Sample Variance, Standard deviation and coefficient of variation for ungrouped data
5. Population Skewness, Kurtosis for ungrouped data
6. Sample Skewness, Kurtosis for ungrouped data
1.2 Grouped data
Class 50-55 45-50 40-45 35-40 30-35 35-30 20-25 f 25 30 40 45 80 110 170
1. Mean, Median and Mode for grouped data
2. Quartiles, Deciles and Percentiles for grouped data
3. Population Variance, Standard deviation and coefficient of variation for grouped data
4. Sample Variance, Standard deviation and coefficient of variation for grouped data
5. Population Skewness, Kurtosis for grouped data
6. Sample Skewness, Kurtosis for grouped data
1.3 Mixed data
Class 1 2 5 6-10 10-20 20-30 30-50 50-70 70-100 f 3 4 10 23 20 20 15 3 2
1. Mean, Median and Mode for mixed data
2. Population Variance, Standard deviation and coefficient of variation for mixed data
3. Sample Variance, Standard deviation and coefficient of variation for mixed data
2. Missing frequency for
1. Ungrouped data The Mean of the observations 18,14,15,19,15,a,12,15,16 is 16. Find missing frequency a
2. Grouped data The mean of a frequency distribution of 40 persons is 16.5. Find the missing frequencies.
Class 0 - 5 5 - 10 10 - 15 15 - 20 20 - 25 25 - 30 30 - 35 f 1 7 11 ? ? 4 2
3. Statistics Word Problem The mean of 10 observations is 35 . While calculating the mean two observations were by mistake taken as 35 instead of 25 and 30 instead of 45 . Find the correct mean.
4. Statistics Graph
1. Histogram
2. Frequency Polygon
3. Frequency Curve
4. Less than type cumulative frequency curve
5. More than type cumulative frequency curve
The frequency distribution of the marks obtained by 100 students in a test of Mathematics carrying 50 marks is given below. Draw Histogram, Frequency Polygon, Frequency Curve, Less than type cumulative frequency curve and More than type cumulative frequency curve of the data.
Marks obtained 0 - 9 10 - 19 20 - 29 30 - 39 40 - 49 number of students 8 15 20 45 12
5. Arithmetic Mean, Geometric Mean, Harmonic Mean 1. Find Arithmetic mean, Geometric mean, Harmonic mean Find Arithmetic mean, Geometric mean, Harmonic mean for ungrouped data like 2,3,4,5,6 2. Find X from Arithmetic mean, Geometric mean, Harmonic mean Find X where Arithmetic mean=3.5 for ungrouped data 2,3,X,5 3. Find Mean or Median or Mode from other two's Find Mode when Mean=3 and Median=4
6. Combined mean and
combined standard deviation
Find the combined standard deviation from the following data.
A B number of Observations 40 60 Average 10 15 S.D. 1 2
7. Mean deviation about mean for
1. Ungrouped data 85,96,76,108,85,80,100,85,70,95
2. Grouped data Calculate mean deviation about mean for the following distribution.
Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 No. of student 6 5 8 15 7 6 3
8. Pearson's Correlation Coefficient
1. for X & Y or Class-X & Y Calculate correlation coefficient for following weights fathers X and their sons Y
X 65 66 67 67 68 69 70 72 Y 67 68 65 68 72 72 69 71
2. for bivariate grouped data Calculate the correlation coefficient
Class-YClass-X 90-100 100-110 110-120 120-130 50-55 4 7 5 2 55-60 6 10 7 4 60-65 6 12 10 7 65-70 3 8 6 3
9. Spearman's Rank Correlation Coefficient (RHO) 1. Ten participants in a contest are ranked by two judges as follows
X 1 6 5 10 3 2 4 9 7 8 Y 6 4 9 8 1 2 3 10 5 7
2. Obtain the rank correlation coefficient for the following data.
X 68 64 75 50 64 80 75 40 55 64 Y 62 58 68 45 81 60 68 48 50 70
10. Regression line equations
1. Find the equation of two regression lines, also estimate Find the equation of regression line equations and correlation coefficient from the following data.
X 28 41 40 38 35 33 46 32 36 33 Y 30 34 31 34 30 26 28 31 26 31
2. Find Correlation Coefficient from two Regression line equations The regression equation of two variables are 5y = 9x - 22 and 20x = 9y + 350. Find the means of x and y. Also find the value of r.
3. Find Regression line equations using mean, standard deviation and correlation correlation (r) The following information is obtained form the results of examination
Marks in Stats Marks in Maths Average 39.5 47.5 S.D. 10.8 16.8
The correlation coefficient between x and y is 0.42. Obtain two regression line equations and estimate y for x = 50 and x for y = 30.
4. Find Regression line equations from sum x, sum y, sum x^2, sum y^2, sum xy, n The following information is obtained for two variables x and y. Find the regression equations of y on x.
sum xy = 3467 sum x = 130 sum x^2 = 2288 n = 10 sum y = 220 sum y^2 = 8822
11. Regression line equations for bivariate grouped data Find Regression Lines
Class-YClass-X 90-100 100-110 110-120 120-130 50-55 4 7 5 2 55-60 6 10 7 4 60-65 6 12 10 7 65-70 3 8 6 3
12. Curve Fitting - Method of Least Squares
1. Fitting a straight line (y=a+bx)
2. Fitting a second degree parabola (y=a+bx+cx^2)
3. Fitting a cubic equation (y=a+bx+cx^2+dx^3)
Fit a straight line, second degree parabola, cubic equation for the following data on production.
Year 1996 1997 1998 1999 2000 Production 40 50 62 58 60
13. Permutation & combination 1. Find n! 2. Find (n!)/(m!) 3. Find {::}^nP_r 4. Find {::}^nC_r 5. How many words can be formed from the letters of the word daughter? 6. How many ways a committee of players can be formed ? 7. Permutation, Combination List
14. Probability 1. Coin 2. Dice 3. Cards 4. Balls<|endoftext|>
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PROSEA, Introduction to Vegetables
- 1 Definition and species diversity
- 2 Importance of vegetables and vegetable growing
- 3 Botany
- 4 Ecology
- 5 Agronomy
- 5.1 Production systems
- 5.2 Planting materials
- 5.3 Husbandry
- 5.4 Crop protection
- 6 Harvesting and post harvest handling
- 7 Utilization and processing
- 8 Genetic resources and breeding
- 9 Prospects
- 10 Authors
Definition and species diversity
Choice of species
Vegetables form a large and diverse commodity group. They are considered a distinct group, not because they have botanical features in common, but largely because of the way in which they are grown and their produce is used. Vegetables are usually cultivated intensively in "gardens" and consequently are part of horticulture. They are usually consumed in combination with starchy staple foods, sometimes used in small quantities to contrast with other foods in taste or to add flavour to a meal.
A vegetable as a product or commodity may be defined as a usually succulent plant or portion of a plant which is consumed as a side dish with the starchy staple. All vegetable crops share certain common characteristics but very few completely fit any definition.
In this volume approximately 100 important vegetable species (sometimes subdivided into cultivar groups as in Allium cepa L., Brassica oleracea L. and Brassica rapa L.) are described in 86 papers in Chapter 2, and summary data are given on 125 minor vegetables in Chapter 3.
The selection of species is somewhat arbitrary because it is impossible to define vegetables in a way that clearly sets the boundaries with other commodity groups. Extensionists, farmers and agronomists generally group all crops in the category "vegetables" which show similarities in cultivation methods with the familiar vegetables. For example, melon and watermelon are commonly classified as vegetables because of the resemblance to cucumber; traders and consumers, however, generally classify them as fruits. In other cases such as garlic, chives or capsicum pepper, the borderline with "spices" is vague, and inclusion in the vegetables is a matter of convention or convenience.
The choice of species for this volume, however, is above all a function of the commodity grouping adopted for the Prosea handbook (Jansen et al., 1991). Important leguminous vegetables (French bean, kidney bean, lablab, mungbean, pea, soyabean) are not described in this volume; because the mature, dry seeds of these species are also used, they have been assigned to Prosea volume 1: "Pulses" (1989). Immature fruits of jackfruit and papaya, and the leaves of Gnetum gnemon L. are important vegetables, but these species are described in Prosea volume 2: "Edible fruits and nuts" (1991). Sesbania grandiflora (L.) Poiret can be found in Prosea volume 4: "Forages" (1992). Rattans and bamboos are dealt with in volumes 6 and 7. For many root and tuber crops with vegetable uses (cassava, potato, sweet potato, taro, yam bean) reference is made to Prosea volume 9: "Plants mainly producing carbohydrates". Zea mays L., and thus vegetable maize (including baby corn, sweet corn and young corn cobs) is described in Prosea volume 10: "Cereals", and Centella asiatica (L.) Urb. in Prosea volume 12: "Medicinal plants". Coriander and mint are included in Prosea volume 13: "Spices". Prosea volume 15: "Lower plants" deals with edible fungi, ferns and other lower organisms. Many plant species other than those mentioned above yield vegetables as a by product, though they themselves used primarily for other purposes. Approximately 800 such species are listed in Chapter 4 of this volume, with a tentative reference to the volume where more detailed information, including the vegetable aspects, can be found.
Domestication and introduction
It is estimated that in the course of time and on a worldwide scale, 1500-2000 plant species have been used as supplementary food. For South-East Asia, the number is close to 1000 species (Terra, 1966; Grubben, 1977; Siemonsma & Aarts van den Bergh, 1989; Jansen et al., 1991).
Originally they were gathered from the indigenous vegetation, but soon some form of "in situ" protection led to primitive cultivars in at least 500 species. The most suitable ones (about 200 species) were moved closer to dwellings and cultivated in home gardens or mixed with field crops to obtain a more reliable supply for home consumption. About 80 of these 200 cultivated species proved to be sufficiently profitable for labour intensive market garden production. Only 20 species proved suitable for the highly intensified, protected cultivation systems as practised in western countries.
Among the 225 "primary use" vegetables described in this volume, the species cultivated for the market or for home consumption (about 120) figure prominently. Nevertheless, more than 100 wild species are described as well, including a large number of weedy companions of the field crops.
The above description of the domestication process suggests a harmonious situation with a well balanced, well adapted assortment of vegetables consisting of the best elements of the indigenous vegetation, as a result of a long process of selection and elimination. It may, therefore, come as a surprise that more than 80 of the 130 cultivated species have been introduced - deliberately or accidentally - into the South East Asian region. Most of them only occur in cultivation, although some introductions from South and Central America have proven to be well adapted and have become naturalized (e.g. Cosmos caudatus Kunth, Limnocharis flava (L.) Buchenau). About 50 species from the Asian mainland and other tropical areas have crept into the region, brought along by missionaries, merchants and settlers. These vegetables have been assimilated in a gradual process during many centuries.
The colonial past led to the introduction of about 30 species of temperate origin, more or less suitable for cultivation in tropical highland areas (e.g. white cabbage). Some have succeeded in obtaining a strong position in the commercial sector in South-East Asian countries. New introductions still emerge from time to time. Popular Japanese vegetables such as "gobo" (Arctium lappa L.), "mitsuba" (Cryptotaenia canadensis (L.) DC.) and "fuki" (Petasites japonicus (Sieb. & Zucc.) Maxim.) now occupy small niches in South East Asian highland areas, satisfying the demand of a foreign clientele but with the potential to be assimilated by the well to do part of the local population as exotic vegetables with a high social value. The question arises whether one is witnessing here a harmonious addition to the assortment of vegetables in South-East Asia or the symptoms of an unequal battle between the highlands and the lowlands, between temperate and tropical vegetables.
From ancient times vegetables have been produced in the vicinity of human dwellings because in contrast to cereals and pulses it is difficult to transport and store these bulky and perishable products. In modern times, as a result of improved roads to consumer markets in urban centres, vegetable production areas have developed where land facilities and climatic conditions are good.
The cultivation of vegetables from temperate areas, often called "exotic", "European" or "highland" vegetables, has become very widespread in the recent past. For instance, about half of the registered vegetable production in Java (Indonesia) consists of highland vegetables. These highland vegetables have partly replaced the traditional "tropical" vegetables in local diets, especially in urban areas. Important production areas for highland vegetables can be found in Indonesia at Puncak, Sukabumi and Lembang for the urban centres of Jakarta and Bandung (West Java); at Dieng for Yogyakarta and Semarang (Central Java); at Tretes and Batu for the markets of Surabaya and Malang (East Java); further at Brastagi for the market of Medan (North Sumatra) and for export to Singapore. Similar highland vegetable production can be found in Malaysia in the Cameron Highlands, in Thailand at Chiang Mai, in the Philippines at Baguio, and in Vietnam at Dalat.
In the lowlands, large concentrated areas of vegetable production have also developed. Often these production areas are situated close to big cities, e.g. Jakarta, Surabaya, Bangkok, HoChiMinh City, and include the more traditional "tropical" vegetable types, especially the easily perishable leafy vegetables. In some cases they are situated hundreds of kilometres from urban markets. The shallots and capsicum peppers grown in Tegal/Brebes (Central Java) are traded all over Java. However, in general the production of vegetables in the lowlands is more thinly spread over larger areas, and this impedes extension to and organization of farmers, and forms a handicap for traders.
The high productivity of temperate vegetables such as white cabbage, carrot and tomato, led to the prejudice that they are superior to indigenous species, whereas this is simply the result of prolonged and expert selection by the western based horticultural breeding industry during a period that the research and development efforts in South East Asian countries were still allocated to other priorities, mainly to self sufficiency in staple foods. With the explosive development of large urban centres, these productive temperate vegetables are a convenient answer to the increased dependence on market products. However, there are disadvantages as well: they are often cultivated in environmentally sensitive areas (slopes, watersheds); their cultivation depends more heavily on fertilizers, pesticides and a seed supply from abroad; the highland vegetables are on average more expensive, and the most popular highland vegetable, i.e. white cabbage, lacks the vitamin A which makes traditional green leafy vegetables so nutritious. It is generally acknowledged that of all crop plants in the tropics, indigenous vegetables have been long neglected in agricultural research, and much remains to be done to collect and study their diversity and to improve yield and quality. This is especially true for the lowland humid tropics. Even now there is more "adaptation" breeding work going on to try to invade the lowland areas with heat tolerant "highland" vegetables including Irish potato than to develop the lowland vegetables in their natural surroundings.
Advantage should be taken of the potential to grow temperate vegetables (enrichment of assortment, off season export), but substitution and the disappearance of traditional vegetables will be regretted in the long run. The South East Asian countries are still at the lower end of the recommended vegetable consumption. Demand for vegetables shows high income elasticity, and with increased economic performance, demand, not least for more diversity, is expected to grow sharply. Most South East Asians live in lowland areas, most indigenous vegetables are lowland species.
Importance of vegetables and vegetable growing
As the product of an intensive form of agriculture, vegetables are becoming increasingly important as cash crops for urban and export markets, with a great potential to improve nutrition and health of the rural and urban poor, as well as to increase their incomes and provide better employment opportunities.
Statistics on vegetable production are scarce and not very reliable. Most countries only record the acreage and production of the major commercial vegetables and ignore the many minor commercial crops and the very important part produced for home consumption. FAO statistics which are based upon country statistics are misleading in this regard. Food consumption surveys offer the clearest picture of the types and quantities consumed. A comparison between countries is also misleading because the crops or food commodities considered as "vegetables" differ from country to country. Statistics are often based on monthly observations on planted and/or harvested area, and consequently the acreage of vegetables with repeated harvests over several months is much overestimated and the yields are underestimated when these monthly records are computed into an annual list. To give a striking example: 20 ha of kangkong which is ratooned once per month during 8 months, yielding 120 t per ratoon, may appear in the statistics as: kangkong 160 ha; 960 t (6 t/ha), which should be corrected to: kangkong 20 ha; 960 t (48 t/ha). Another problem in collecting statistics is the estimate of areas per crop in the case of mixed cropping.
Based on recent inventories for this book of a large number of urban markets in South East Asian countries, an attempt has been to indicate the occurrence and relative importance of all major "primary use" vegetables described in this volume (Table 1). The most important "secondary use" vegetables (out of the 800 listed in Chapter 4) are added at the end of the list.
In 1989, the world production of vegetables reported by FAO (1990) was 433 940 000 t, the South East Asian production (Indonesia, Malaysia, Papua New Guinea, the Philippines, Thailand, Vietnam) was 10 674 000 t. On a global scale, production per capita per year rose during the 1980s from 78.6 to 83.4 kg, in South East Asia it fell from 32.1 to 28.1 kg. It is a matter of considerable concern that in many tropical countries, South East Asian countries included, vegetable production seems to be lagging behind the rate of population growth. For some countries such as the Philippines and Thailand, figures even indicate a decrease in absolute terms over the last decade. Urbanization and changes in landuse at the periphery of the big cities, and a backward market system might be the main causes of this phenomenon. In food deficit areas in Vietnam, the difficulty of supplying minimum energy requirements through supplementary imports from food surplus areas has contributed to a strong emphasis on rice self sufficiency. This has been accompanied by a decline in the production of other subsidiary food crops such as vegetables, which are particularly important in improving the dietary balance and variety (FAO, 1988). The FAO production estimate (kg/capita per year) for Indonesia is rather low (18.8 kg in 1989) compared with other countries in the region. This suggests the existence of an important informal production circuit (small family gardens) not covered by the FAO data. Household expenditure surveys (BPS, 1983) suggested that in 1980 the total production (and consumption) of vegetables in Indonesia was at least twice the FAO statistics.
Vegetables as a group account for 5-6% of the total value of agricultural production in South East Asia, exceeding the value of major secondary food crops such as maize, cassava, groundnut, soyabean, sweet potato or mungbean.
Although the productivity of most vegetable crops is still rather low, incomes per unit area are usually relatively high. Net revenues per hectare from shallot, pepper and tomato in Indonesia have been reported to be 3-5 times higher than those from rice. The import and export of fresh vegetables in South East Asia is still very limited as a percentage of total production. Trade of fresh produce between countries within the region is mainly linked to the supply of large urban centres like Singapore (e.g. cabbage from Brastagi, North Sumatra, Indonesia, and from Cameron Highlands, Malaysia), whereas there is some off season production of temperate vegetables for the Japanese market (e.g. Arctium lappa L. and bulb onions from northern Thailand).
Vegetables are consumed because they are tasty and healthy. They add variety and flavour to the diet. There is little chance of malnutrition occurring in families consuming enough vegetables. Vegetable consumption may, therefore, be considered as an important economic factor in a society because it improves health and working capacity. Vegetables are generally low in energy and dry matter content, but most important as sources of protective nutrients, especially vitamins and minerals. Vegetables (together with fruits) are the most important source of vitamin A, which is deficient throughout South East Asia where rice based diets predominate, blinding thousands of children annually (Oomen & Grubben, 1978). Although the protein content of vegetables is considered unimportant in developed countries, it appears to be highly significant in countries with an overall deficiency in proteins. Vegetables also provide fibre in the form of cellulose which aids the digestion of other foods and stimulates and cleans the intestinal canal.
The nutritive value of vegetables varies but is usually greatest in those eaten raw. In Indonesia and Malaysia, but also in other parts of the region, it is customary to eat a large variety of vegetables raw. This so called "lalab" or "ulam" can also be eaten after being blanched (immersed in boiling water for several minutes), making the vegetables softer but without losing their consistency. They are often consumed in combination with a groundnut sauce or with coconut milk. Also in Chinese cuisine, vegetables are not boiled thoroughly, but usually stewed or fried in oil (strongly enhancing the availability of carotene). In New Guinea, vegetables are often wrapped in banana leaves and baked in hot ashes or between hot stones. The disadvantages of certain preparation techniques leading to overcooking can easily be overcome by slightly increased consumption, and therefore emphasis should be on averting underconsumption instead of on preparation methods.
Consumption in South East Asia can be roughly put at 80% of the production in kg/capita per year, and (after correcting the production data for Indonesia) averages about 90 g/capita per day or 33 kg/capita per year. This compares favourably with consumption levels in Africa and South America, but is only 45% of the consumption in the developed countries. It is also much lower than the daily intake of 150 g/capita recommended by nutritionists as a target, provided that one third of this amount derives from leafy vegetables.
A FAO/World Bank study of the agricultural and food situation in Vietnam (FAO, 1988) revealed that the nutritional status of the Vietnamese people is extremely low. Vietnam ranks among the most deprived countries in Asia. Overall there is a high prevalence of protein energy malnutrition (PEM), endemic goitre, iron deficiency anaemia, vitamin A deficiency and other micronutrient deficiencies. The nutritional status is most severe in the northern and central coastal regions and in hilly and mountainous areas, not necessarily in areas of the highest population density.
Nutritional needs are subject to regional differences. According to FAO/WHO standards for East Asia (FAO, 1972), the average daily requirements of an adult man (55 kg) are: energy 10 600 kJ, protein 46 g, carotene (pro vitamin A) 1.5 mg, thiamine (vitamin B1) 1.0 mg, riboflavine (vitamin B2) 1.5 mg, niacin 17 mg, vitamin C 30 mg, Ca 500 mg, Fe 9 mg. This information can be related to the data on the nutritional composition of individual vegetables as given in the entries in this book, but it must be stressed that the composition of foodstuffs varies widely as a result of environmental factors, varietal differences, cultural practices, harvesting stage of the plant, methods of storage, processing and preparation. Data are expressed on a fresh weight basis, and are therefore strongly influenced by the dry matter content.
Characteristics of the vegetable sector per country
The situation of vegetable production and consumption in six South East Asian countries is outlined briefly below.
The Lembang Horticultural Research Institute (LEHRI) prepared a profile of the vegetable production and consumption situation of Indonesia, based upon the available statistics (van Lieshout, 1990). Production statistics are defined by the commercial production of 18 major vegetable types. The commercial production of 1988 is presented in Table 2. Subsequently it was calculated that the total commercial area needed for the year 2000 would be 1 148 000 ha, based on the following assumptions:
- 7% annual growth of commercial demand as a consequence of population growth (2%), increase of incomes (3.5%) and urbanization (1.5%);
- 7% annual growth of commercial supply as a consequence of better technology (2%) and area enlargement (5%).
The seven most important vegetables in terms of total production in 1988 (each more than 200 000 t) were respectively cabbage, hot capsicum pepper, potato, shallot, cucumber, yard long bean, and caisin/Chinese cabbage. The total farm gate production value was about 1250 billion Rupiah or US$ 735 million. The commercial production of 4 170 400 t mentioned in Table 2 is only 47% of the consumed quantity, which leads to the conclusion that the many non registered commercial vegetables, each of them of minor importance, together with the numerous species for home consumption, account for 53% of total consumption.
The annual export (1986: 22 900 t, mainly potato and cabbage, valued at US$ 10.4 million) and import (1986: 3400 t, mainly garlic, shallot planting material and dried hot pepper, valued at US$ 1.5 million) amount to less than 1% of the total production. The gross intake per capita per year is 46.7 kg, which means 37.4 kg net if 20% is deducted for waste between farm gate and consumption. A daily intake of 102 g per head is reasonably high if compared with other tropical countries. But the consumption is not evenly spread. High income classes consume more vegetables than low income classes, and consumption in West Indonesia, especially West Java, is much higher than in the eastern part.
At present the production for home consumption is still very important, but it is expected that in the coming decades it will partly be replaced by commercial production. The gross urban consumption in 1990 (43.7 kg/capita) was lower than recorded in 1987 (47.7 kg/capita) whereas the gross rural consumption increased slightly (from 47.4 to 48.0 kg/capita). The most likely explanation is reduced availability in urban areas, possibly caused by an obsolete marketing system.
The data of 1990 reveal that boiled leafy vegetables (kangkong, amaranth, caisin, cabbage, cassava leaves, etc.) constitute 40% of the total quantity consumed; boiled non leafy vegetables (yard long bean, eggplant, chayote and other cucurbits, carrot, young kidney beans, potato, bean sprouts, young jackfruit) constitute 41%; spice vegetables (shallot, garlic, hot capsicum pepper) constitute 13% and raw (salad or "lalab") vegetables such as cucumber and many leafy vegetables constitute 6%.
From the scarce statistics available it may be deduced that the range of vegetable crops lies between Thailand and Indonesia. Highland vegetables are produced in the Cameron Highlands, and lowland vegetable production is scattered everywhere in the coastal plains. The seven most important crops in acreage in Peninsular Malaysia registered in 1988 are watermelon 2400 ha, hot capsicum pepper 1400 ha, cucumber 1300 ha, cabbage 800 ha, leaf mustard 800 ha, kangkong 700 ha and amaranth 500 ha. The annual export of vegetables (mostly highland) to Singapore is considerable: about 120 000 t valued at 61 million Malaysian Ringgit or about US$ 24 million (1989). Onions, shallots and garlic are imported. No statistics are available on vegetable production for home consumption.
Papua New Guinea
The vegetable types grown in Papua New Guinea are essentially the same as those in Irian Jaya and other provinces of East Indonesia. Popular highland vegetables are garlic, potato, tomato, cabbage, carrot and welsh onion. Cucumber is very popular and is grown both in the highlands and in the lowlands. Winged bean is important for its tubers, seeds and young pods. Popular leafy vegetables are aibika, chayote tops, sweet potato tops, amaranth and rungia. There is no significant import or export of vegetables. Numerous indigenous species are collected or grown in home gardens but no statistics are available.
Inventories of vegetable markets in Luzon, the Philippines, give the impression that fruit vegetables (cucurbit and solanaceous fruits) are in ample supply, but that leafy vegetables play a smaller role in the diet than in other South East Asian countries. The registered area of vegetables in 1987 was 142 348 ha (including potato and ginger, excluding dry beans and peas). The recorded production was 1 008 960 t, the average yield 7.1 t/ha. The most important vegetables in cultivated area are tomato (18 000 ha), eggplant (16 000 ha), onion and shallot (7000 ha). The most important vegetables in tonnage of yield are watermelon, tomato, eggplant, cabbage, and pumpkin. The most important fresh vegetable for export is onion (7400 t with a value of US$ 2.6 million) but many other fresh and preserved vegetables are exported.
Two main vegetable production areas are distinguished. In the hot humid lowlands, hot season vegetables are produced the year around. During the rainy season the types most vulnerable to heavy rains like shallot, tomato and Chinese cabbage are grown to a lesser extent. In the northern part, the cool dry winter season is suitable for the more temperate types such as cabbage, tomato, and garlic. Area, yield and production of 22 major commercial vegetable species are presented in Table 3.
The import of vegetables is insignificant, the export is becoming increasingly important. In 1989 the export of garden crop products amounted to 233 225 t with a value of 3.6 billion Bahts or US$ 144 million. The most important of the many vegetable commodities exported were baby corn and bamboo shoots. No statistics are available on vegetable production for home consumption or on consumption per commodity.
The Ministry of Agriculture recorded an area of 242 800 ha for vegetables in 1988 (private 208 900 ha; cooperative 33 300 ha; state 600 ha) with an average yield of 12.0 t/ha. The total production was 2 909 000 t. The projected annual growth rate for the period 1986-1990 was 6% for acreage, 2.7% for yield and 8.9% for production. However, production statistics are unreliable. Perhaps half of the vegetables for home consumption and sale in local markets are grown on private plots, not monitored by the state (FAO, 1988). There are two distinct production seasons, the "winter" season (October - February) and the "summer" season (April - August). Yields of vegetables reach 20 t/ha in Lam Dong and some Mekong Delta provinces, 18 t/ha in HoChiMinh City (15 000 ha) and 13-14 t/ha in Hanoi (13 000 ha). In the past, the government has taken little interest in small scale private sector production, which receives almost no attention from research, extension or other officially sponsored support services. Nevertheless, horticulture accounts for a large part of the value of agricultural production, consumption and cash income for farm families. Fresh or processed vegetables are a potentially important export crop. The export of fresh vegetables (cabbage, carrot, kohlrabi, cucumber, onion, cauliflower) and of preserved products (pickled cucumber) was about 12 000 t in 1988 (with a value of US$ 1.9 million). The export of watermelon (1988: 11 300 ton with a value of US$ 1.7 million) is as important as all other vegetables together.
Priority is given to expansion of production during the winter/spring season, especially in the Red River Delta. Input requirements and transport and marketing difficulties are the major constraints. Vegetables reportedly absorb about one half of pesticide imports. Insecticide resistance has become a problem in the production of leafy vegetables. The major issues for vegetable production of the big cities Hanoi and HoChiMinh City are local self sufficiency and reducing the seasonality of production. Before 1975 HoChiMinh City had only 10% of its present vegetable area and depended on shipments mainly from Dalat, 300 km away in the highlands of Lam Dong Province. This trade declined as transport deteriorated and private marketing activities were suppressed, and HoChiMinh City was forced to strive for local self sufficiency in lowland vegetables. The cool climate of Dalat is suited for temperate type vegetables (cabbage, kohlrabi, carrot, potato, tomato, onion, garlic, etc.) especially during the off season (April, May, September and October). Similar opportunities exist for cool season vegetables during the summer period in the mountainous areas north west of Hanoi.
The heterogeneity of the commodity group vegetables is well illustrated by the fact that the 225 species described belong to approximately 60 different plant families: vegetables occur throughout the plant kingdom. Nevertheless, some families figure prominently with respect to number of vegetable species as well as economic importance: Compositae and Cruciferae, mainly as leafy vegetables, Cucurbitaceae and Solanaceae, predominantly as fruit vegetables.
The proper naming of plants is extremely important, because it enables repeatability and use of scientific methods. Taxonomy provides such a naming service and is therefore an important biological science. Many taxonomical problems are still unsolved. Few major genera of economically important vegetable families have been revised in their entity within the last 50 years. But in genera of minor importance such as Spilanthes Jacquin or Sonchus L., the lack of linkage of data to well defined taxa also makes much of the information useless.
The vegetables comprise some genera, Brassica L. in particular, showing the most bewildering array imaginable of types created by man and nature over centuries during the selection of cultivated plants. Brassica vegetables span a range of morphotypes comprising succulent modifications of leaves, stems, roots, buds and floral parts. Taxonomically they have been variously classified, leading to much confusion because the classic taxonomy is primarily intended for wild taxa. Closely related types were often classified as subspecies or varieties (formal classification under the International Code of Botanical Nomenclature), but the most recent approach is to distinguish cultivars and to group these in cultivar groups (the informal "open" classification guided by the International Code of Cultivated Plants). Where workable cultivar group classifications have been developed (Allium cepa L., Brassica oleracea L., B. rapa L.), these are being followed or even promoted in this volume. Because of its illustrative value, the nine contributions on particular Brassica crops are preceded by a genus article (Brassica L.) outlining the taxonomic and cytogenetic basis of the present day classification.
The taxonomy of cultivated plants is developing: as yet there is no worldwide accepted system for naming and classification, but proposals are being discussed.
Vegetables are often classified according to morphological criteria such as plant parts used or growth habit. Plant parts used as vegetables comprise anything from whole leaves (kangkong, amaranth, welsh onion) to petioles (rhubarb), fruits (cucurbits, solanaceous fruits, leguminous pods), flowers (cauliflower, broccoli), stems (asparagus, bamboo shoots) and stem tubers (above ground such as kohlrabi, or underground as in yam bean, potato), seeds (cucurbits, young leguminous seeds), storage roots (carrot, edible burdock) or bulbs, which are swollen leaf sheaths (onion, garlic, shallot). In one case, the stem tuber is the result of a symbiosis of plant host (Manchurian wild rice) and a fungal parasite. If the 225 or so vegetable species described in this volume are classified according to plant parts used, leafy vegetables are by far the most frequent (about 60%), whereas fruit, flower, stem, root and seed vegetables account for about 15% of the species. The remaining 25% consist of multipurpose vegetables, characterized by more than one edible part. The record is probably held by the winged bean (Psophocarpus tetragonolobus (L.) DC.) with edible young pods, young seeds, flowers, leaves and tubers. Multipurpose vegetables may be useful for home gardens, but it should be kept in mind that the use of one part often precludes, or at least negatively affects the yield of the other; multiple use of the same plant is therefore not very common in commercial production.
In areas with cold winters, annual crops dominate the agricultural scene, but as one moves from higher latitudes towards the equator, woody perennials become more important as sources of food, and this also applies to vegetables (Cannell, 1989). About 25% of the species treated in this volume have a woody growth habit, shrubs being more frequent than trees. They play a relatively important role in home garden production as they are a more permanent and flexible source of supplementary food, and can serve other purposes simultaneously.
Growth and development
Knowledge of crop growth and development and insight in the eco physiology should help the grower to manipulate the crop and its environment so as to achieve the optimum yield of the desired plant part. Most vegetable crops do not pass through their complete life cycle under field conditions because they are grown for their vegetative parts, or young, immature generative parts. For leaf vegetables, flowering and fruiting are to be avoided or delayed. For bulb and tuber vegetables, the formation of the storage organs is a delicate phase in the succession of growth and development processes. When flowers or fruits are the useful product, the objective is to channel as much energy to these generative plant parts. Although flowering and fruiting are often undesirable in vegetable production, the potential to complete the life cycle is important for seed production.
Information on the growth and development of vegetables is very limited except for a number of species of temperate origin. The sequence of germination, vegetative growth, development of storage organs or generative development (flower initiation, flowering, pollination, fruiting) is highlighted in the species treatments.
The area covered by Prosea lies between 20°N and 10°S. It consists mainly of tropical lowlands, but also has large areas at medium to high altitudes.
In agriculture, the choice of crops and cropping systems is mainly determined by interactions of ecological factors (climate, soil) and management variables. Horticulture is an intensive form of agriculture, usually on small acreages, in which restrictions imposed by adverse climatic factors and poor soils can often be overcome by intensive management practices. Commercial vegetables are not always grown on the site and at the time which are ecologically the most appropriate, because the ultimate motive for the farmer is profit, not yield. A short distance to the market or a high price during the off season often compensate for a lower yield or higher production costs. In Indonesia the altitude of the production area and the distance to the city markets (transport facilities) appear to be by far the most important factors determining which species farmers choose to grow.
The climates of South East Asia are of the monsoon type. Monsoons are seasonal winds blowing moist air from the sea to the heated land mass bringing heavy rains during the hot season, and blowing air from the land to the sea during the cold season. In Indonesia and Malaysia, situated close to the equator, the dry south east trade wind from Australia causes a dry spell from May to October. This wind turns to the north above the equatorial zone and takes up much moisture above the ocean. It is known as the south west monsoon in Thailand and neighbouring countries and causes the rainy season from May to October ("summer"). The inverse happens from December to February when the north east monsoon causes the dry season ("winter") in Thailand but brings rain as the west monsoon in Indonesia.
Temperature is the most important climatic factor for vegetable production. In the lowlands near the equator the average daily temperatures are generally about 27°C the year around, the differences between a hot and a cold season becoming more pronounced northwards. In northern Vietnam the average temperature from November to April is only 16°C. In these areas the summers, from May to October, are very hot and subject to typhoons.
In mountainous areas the temperature drops by about 1°C per 160 metre increase of elevation and the difference between day and night temperature broadens. The occurrence of micro climates is quite common. Large variation in rainfall, temperature, radiation and wind may be observed between areas at relatively short distances from each other. A large part of the vegetable production is in the highlands. Since temperature is the most important factor determining the choice of vegetables that can be grown at a certain altitude and since the transition between highlands and lowlands is gradual, there is a need for a practical classification into ecological zones. An example is the empirical classification made by the Lembang Horticultural Research Institute in Java, Indonesia (Buurma & Basuki, 1989). This classification, derived from a statistical databank of 18 commercial vegetable crops, is fashioned in such a way that over 70% of the area of the typical lowland vegetables (cucumber, kangkong, yard long bean) comes into the lowland zone and over 70% of the typical highland vegetables (cabbage, carrot, potato) in the highland zone, with only a minimum of overlap in the medium elevation zone (Fig. 1). Depending on the precision desired, the classification may comprise three zones (lowland < 200 m; medium land 200-700 m; highland >700 m), or an even finer classification into four zones by subdivision of the medium land zone into a low medium (200-450 m) and a medium high zone (450-700 m). This method of defining the ecological zones is useful for the interpretation of statistical data and the results of multilocational cultivar trials.
The lowland and low medium zones of Java (below 450 m) are characterized by high maximum day temperatures (30-27°C) and night temperatures (25-22°C) and high light intensity. The main soil types are alluvial clay in the coastal plains and latosol at higher altitudes. The medium high and highland zones (above 450 m) are characterized by maximum temperatures below 27°C, a larger variation between day and night temperature, a lower light intensity because of cloudy weather and a high air humidity. The main soil types are andosols and grumosols.
Most vegetables are dominantly present in one of the ecological zones, but they overlap. Welsh onion, red kidney beans or tomato apparently have a large optimal temperature range, since they are grown from sea level up to high altitudes. Hot capsicum pepper is a typical lowland vegetable (79% < 450 m) but yet it occurs also in the highlands, even up to 1800 m. Typical highland cultivars of capsicum pepper do not perform well in the lowlands and vice versa. The distinction between highland and lowland cultivars, or, in the higher latitudes of northern Thailand, Philippines and Vietnam, between "summer" (hot season) and "winter" (cool season) cultivars is well known for some important commercial vegetables such as cauliflower, white cabbage, Chinese cabbage, tomato, and capsicum pepper.
The variation in daylength is a less important climatic factor in the area close to the equator, but it is of increasing importance further north. At 10°N (southern part of the Philippines, Thailand and Vietnam) the daylength varies from about 11.30 h to 12.40 h and at 20°N (northern part of the Philippines, Thailand and Vietnam) from about 10.50 h to 13.20 h. The distinction between a "summer" and a "winter" becomes tangible above 10°N, by variations in the photoperiod and in the total daily radiation. Some crops are very sensitive to daylength variations, a nice example being okra (Siemonsma, 1982). These daylength effects will be dealt with in the species treatments.
In South East Asia vegetables can be grown year round, provided that enough water is available. As a rule of thumb, actively growing leafy vegetables need 6 mm (6 liter per m2) daily and other vegetables 4 mm. In the absence of rain or irrigation, the moment growth will be retarded and drought damage will occur depends on the type of crop, the soil properties and the cultural practices. In general, the yields obtained from commercial vegetable production are higher in the dry season with irrigation than in the rainy season without irrigation. The reasons for this yield depression during the rainy season are the deficient radiation by cloudy weather and the damage resulting from diseases.
Farmers are bound to a certain land area and normally have little opportunity to choose a soil type suitable for a certain crop. In general, a good soil for vegetable crops should have the following properties:
- Good structure: this means that the soil must be durably friable and stable, providing adequate water retention and aeration. A good soil is a proper medium for high microbiological activity and for undisturbed root development to a depth of at least 60 cm.
- High chemical fertility: the soil should contain reserves of essential nutrients, a sufficient amount of which must be readily available in the soil moisture. The rate of growth and production depend on the element in shortest supply ("critical element").
Ideal soil properties are identical for almost any vegetable crop. However, vegetable species and even cultivars of the same species differ in their yielding ability under adverse conditions, e.g. shallow, wet or dry, acid or saline soils. A high soil salinity (electrical conductivity > 3.0 mmho/cm) is a serious restriction for satisfactory yields of most vegetables; however, tomato, broccoli, cucumber and pumpkin are reasonably salt tolerant. Vegetables with a shallow root system like onion, cabbage and kangkong are much more susceptible to drought than deep rooting species like tomato, watermelon and asparagus.
The predominant soil types in South East Asia are andosol and latosol (both of the sandy loam type) and alluvial clays. Light soils have the advantage of easy tillage, adequate drainage and aeration, provided that the organic matter content is sufficiently high. Clay soils have the advantage of a better water holding capacity and higher natural fertility. With good cultural practices, most vegetables will give satisfactory results on a wide range of soil types. Yet some crops such as cabbage and garlic prefer a heavy soil, whereas others like asparagus, carrot and radish prefer a light soil. Table 4 gives characteristics of two typical soil types used for vegetable production in the lowlands of Indonesia (Titulaer, 1991).
In intensive vegetable cultivation, the lack of chemical fertility is not perceived as the most serious limiting factor because amendments with manure and/or inorganic fertilizer are relatively easy. Lacking adequate recommendations, most farmers in South East Asia rely on their own experience in the application of manure and mineral fertilizer. Unfortunately, in many cases their practice is injudicious and unbalanced, supplying too much of one element and not enough of another. Many farmers use heavy N dressings in the form of cheap urea, which causes fast vegetative growth, making the plants susceptible to diseases and damage. Too much fertilizer means high costs of inputs and pollutes the environment. Sustainable soil conditions should be aimed at by establishing a sufficiently high level of basic fertility, and giving an appropriate fertilizer gift to compensate for the expected uptake per crop.
The pH of the soil influences the availability of nutrients and also the soil structure. If the soil is very acid (pH water < 5.5), the choice of the crop will be limited to only a very few species such as shallot or watermelon, and will certainly not be suitable for cabbage. Crops on acid soils often suffer from Mg, Ca or P deficiencies, or from Mn and Al toxicity. Liming with slakes (or preferably with dolomite) is useful at a rate of about 2 t/ha per crop until a level of pH 6-6.5 has been reached. On acid soils, it is recommended not to use too much of acidifying fertilizers such as ammonium sulphate or urea.
Physical soil properties and organic manure
The volume of an ideal soil profile consists for one third of each of the three elements: solid mass, moisture and air. This constitution will guarantee adequate aeration, water holding capacity, drainage and biological activity. Organic matter has the characteristic that it reduces the compactness of heavy soils and increases the water holding capacity of light soils. Light sandy soils should contain at least 4% organic matter, which corresponds to 2% C. For heavy clay, about 2% organic matter content (1% C) is needed. At these levels the yearly losses of organic matter are approximately 5 t/ha. This loss can be compensated by an application of about 10 t/ha of manure, but higher doses (up to 80 t/ha) are often practised for intensive vegetable production.
The low productivity of many vegetable crops in the tropics is largely due to the lack of research attention, aggravated by agro economic constraints such as insufficient farm capital, inadequate transportation, and dramatic price fluctuations. Increased agronomic research attention for vegetable production translates itself into:
- knowledge of appropriate production systems;
- improved cultivars and the availability of high quality seeds;
- appropriate cultural and management techniques;
- adequate control of diseases and pests.
The production systems for vegetables in South East Asia fall into four main groups, based on land use and the level of inputs. In this section the relative importance of each of these for home consumption and for marketing is estimated and expressed as a percentage. However, in each country and region the situation will be different.
Collection of weeds and wild plants
The picking and gathering of vegetables from the wild vegetation (mostly leafy vegetables but also berries, roots, etc.) is still important in rural areas. While weeding or gathering firewood, the women often pick these edible plants (pot herbs) for the preparation of their meals at home. In the fields they also practise selective weeding, the useful weed plants being spared.
About 100 species of the 225 primary use vegetables described in this volume are weeds or wild plants. Possibly they account for about 15% of home consumed and 5% of the marketed volume. With the increase in the population, the urbanization and the specialization in professional activities, this type of food collection will decrease further.
A considerable part of the vegetable production, estimated at about 30% of the home consumed and 10% of the market vegetables, is derived from the compounds close to the houses. Annual vegetable types often encountered in the compounds are amaranth, caisin, kangkong, yard long bean, lablab, winged bean, cucumber, bottle gourd, pumpkin, bitter gourd, chayote, capsicum pepper, eggplant, cassava (leaves). Climbing types of leguminous species or cucurbits are important because of their ability to occupy open places. However, trees and shrubs dominate the home gardens almost everywhere because they are a more permanent and more flexible source of supplementary food than annuals, they have been less successful in migrating to more commercial market gardens than herbaceous crops, and they serve many other purposes better than annual herbs such as providing shade in the compound (Parkia speciosa Hassk., Archidendron jiringa (Jack) Nielsen, Ficus spp.), serving as a hedge (Sauropus androgynus (L.) Merrill, Polyscias spp.), providing living support for other plants (Moringa oleifera Lamk), as ornamentals (Polyscias spp.), medicinal plants (Gynura procumbens (Loureiro) Merrill), or as a source of forage (Sesbania grandiflora (L.) Poiret) or fuelwood (Moringa oleifera Lamk).
Home gardens are characterized by a great diversity of useful plant species growing in a herbaceous layer near the ground, and various layers of canopies of shrubs and trees. Traditionally, home gardens in rural areas are very rich in fruits, spices and medicinal plants. Vegetables constitute a relatively modest part (in East Java about 17% of the cash value) of the home garden products, and most of these are the perennial types (Laumans et al., 1985). Also very important are the starchy tuber crops as a buffer for periods of scarcity of the main staples (mostly rice or maize). The leaves of cassava and sweet potato are important vegetables as well.
Another characteristic of the home garden is the very low input of capital (no agro chemicals, no special tools needed, planting material at hand) and the use of cheap family labour mostly in spare time. The soil is kept fertile by the household debris and manure. In this natural biological ecosystem with its great diversity of plants, the incidence of diseases and pests is generally very low (Soemarwoto, 1985; Landauer & Brazil, 1990).
Amidst all the praise for the home garden as a cropping system whose strength lies in stability rather than peak performance, it has become clear that home gardens are well suited to feed the family, but that commercial market gardens have to cater to the millions.
Extensive field production
The field production of vegetables for home consumption needs only a little space. It is common practice to use open places in the food crops and on field borders instead of planting entire plots with vegetables. Very common vegetables on dikes along rice fields are yard long bean, winged bean, lablab, kangkong, amaranth, caisin, eggplant, and pumpkin.
Numerous vegetables, each usually represented by a few plants only, can be found in the fields of food crops. Hence it is a form of mixed intercropping and constitutes the vegetable part of the subsistence farm system. Not much care is given to these vegetables. They do not get separate chemical spraying or fertilizers, but may profit from the treatment given to the main crop. The great diversity of species is the best guarantee for continuous production. About 50% of the vegetables for home consumption may come from this type of vegetable production.
A considerable part of the cash crop vegetables, possibly 20%, is also produced in a very extensive way, characterized by rain dependent production with low inputs of pesticides, fertilizers and labour and by low yield levels and poor quality with low prices. This explains the very low average yield level of many crops in national statistics, e.g. 3.2 t/ha for capsicum pepper in Indonesia. Upland fields used for the production of cash crop vegetables during the rainy season may also be planted with vegetables during the dry season, provided that sufficient irrigation water is available. Vegetables are also planted as dry season crops after rice, often on residual moisture. There is no abrupt border with the intensive production system described below; there is a gradation of intermediate cultural practices, from very extensive to very intensive.
Intensive market gardening
Intensive market gardening accounts for at least 65% of all vegetables marketed in South East Asia. Only a very small part (5%) of the harvested products is used for own consumption. The main features of this category are the high costs for labour and inputs (seed, fertilizers, pesticides), the professional application of cultural practices and the tendency to offer improved quality products for the organized marketing sector. Within this category of intensive gardening, a distinction can be made between:
- upland vegetable production. Rainfed or irrigated vegetable crops in permanent production or in rotation with other upland food crops, e.g. maize, soyabean, groundnut, or with sugar cane.
- wet field vegetable production. Vegetable production during the dry season, after the wet season paddy, is commonest, but permanent cultivation of vegetables also occurs. The advantage of growing vegetables after rice is that soilborne diseases are eliminated by the flooding of the fields; the disadvantage is the amount of labour needed for soil tillage.
Because many vegetables are short duration crops, they can often profitably be fitted into cropping systems based on food or industrial crops, in order to improve the cropping intensity of agricultural land. Therefore, a large proportion of the vegetables are not produced in sole cropping but in mixed intercropping. Which of these systems is chosen depends on many factors, e.g. the tradition of the farmer, the type of crops, the cost of labour, and potential for mechanization. In many cases, mixed intercropping takes the form of relay cropping, in which the growing period of a younger and an older crop, or of a long duration and a short life crop overlap. Farmers use mixed intercropping instead of sole cropping for economic reasons in order to:
- reduce the risk of losses. If one crop fails, the second or third crop growing at the same time in the same field may give a profit;
- make better use of the land. Young plants do not cover the land area completely and the sunlight is underutilized. Intercropping with other plants traps the available light as efficiently as possible. Relay cropping shortens the time in between harvests. A good example is capsicum pepper in Indonesia, planted on thousands of hectares in between shallots one month before the shallots are harvested.
- economize on production inputs (fertilizers, pesticides). Cabbage and capsicum pepper are planted between tomato and they profit from the pesticides and fertilizers applied to tomato.
Several other advantages from the agronomical and environmental point of view can be mentioned:
- pathogens, pest populations (thrips, aphids, mites etc.) and virus infections may be kept at a lower level, perhaps below the damage threshold. Tomato plants repel diamond back moth in cabbage. Maize plants give protection to the predators (natural enemies) of pests of capsicum pepper.
- the dense vegetation in mixed intercropping reduces soil erosion by heavy rainfall.
- minerals are better used and leaching is reduced.
- weeds are suppressed.
- bamboo or wooden poles for tomato or cucumber are used again by other climbing vegetables (loofah, bitter gourd, beans).
- the shade of the earlier crop (maize) is profitable for younger crops such as capsicum pepper.
But mixed intercropping certainly also has disadvantages:
- spraying with pesticides is no longer selective. Farmers use wide spectrum pesticides, they spray routine wise and do not respect the safety period needed before harvesting the earliest species in the mixture.
- crop rotation for reduction of soilborne diseases and pests is difficult. In Java a common relay cropping system lasting one year is tomato/cabbage (white or Chinese); harvested cabbage is replaced by capsicum pepper, French bean is planted against tomato sticks. In this system, the soilborne diseases club root, bacterial wilt and root knot nematodes will be maintained.
- manual and mechanized weed control are difficult.
Farmers know by experience which crops combine well. They combine plants with a certain tolerance of shade (capsicum pepper, welsh onion, Chinese cabbage) with tall crops (maize) or climbing species (leguminous vegetables, cucurbits). However, farmers have less knowledge of the crop rotation required to avoid soilborne diseases.
Some of the woody species in the vegetable assortment may have a role to play in agroforestry systems. This is a relevant possibility since an important issue for the future is how to combine agriculture and forestry in order to achieve sustained production of food, fuel and timber.
Although the use of improved commercial seed is rapidly increasing in South East Asia, a large part of the marketed vegetables is harvested from crops of local cultivars, landraces or farmers' selections. Self pollinated crops (e.g. tomato) breed true to type and the grower can easily obtain next season's sowing seed from a limited number of healthy plants. Maintaining the identity of cross pollinated crops (hot pepper, cucurbits) is more complicated. The grower has to remove plants that are off types in an early stage and he then has to take the seed from the best plants in the middle of his field. If homogeneity is really important for the grower then he could better leave the seed production to the professional seed growers.
The lack of modern selected cultivars and of an efficient seed supply system is a serious hindrance to the improvement of commercial vegetable cultivation. In most South East Asian countries a local seed industry is gradually emerging, and a start has been made to establish independent official control of the genetic integrity and physical quality of vegetable seeds. Imported seeds of European type vegetables are not usually adapted to tropical conditions and are often inadequately protected from the effects of high temperatures and humidity.
If the private sector is to invest in the development of improved cultivars, the crops concerned must cover sufficient area and it must be possible to cover the costs of development; seed firms therefore try to develop and market F1 hybrid cultivars, whose seed has to be renewed each season unless the lower yields through inbreeding depression in successive generations are accepted.
A great advantage of hybrid cultivars is that the period required to combine useful characters, e.g. for resistance to disease, is much shorter than in conventional cultivars. The creation of hybrid cultivars is relatively easy, especially in the case of solanaceous vegetables and cucurbits. Because the hybrid seed is so expensive, farmers often harvest the seed from their F1 hybrid crop for the next planting, but their experience with the segregating F2 hybrid material is in most cases very frustrating. The inbreeding depression is reflected in a lower yield and loss of uniformity and quality. In some cases this depression is relatively slight, e.g. for certain hybrids of onion, tomato, capsicum pepper. In watermelon the depression is so large that the product of the F2 is no longer marketable.
Modern seed companies strive to bring a complete assortment of vegetable seeds of the region into the seed merchants. This includes the many OPs, i.e. the open pollinated cultivars which might easily be reproduced by the farmers themselves. In practice, however, farmers gradually realize that it pays to buy all seed from a dealer, provided that the price is reasonable (Groot et al., 1988). The production of healthy and viable seed which is true-to-type is a skilled business.
The vegetatively propagated crops (garlic, shallot) are less attractive for seed companies because it is easy for the farmer himself to renew the expensive planting material. Yet for these crops too, professional seed producers will gradually take over from the farmers because they will supply superior virus free planting material. For vegetatively propagated vegetables like aibika, the kangkong type reproduced by cuttings and star gooseberry, farmers must rely on their own planting material.
Table 5 gives some data on the seed of commercial vegetables.
In the future crop husbandry will have to concentrate on the efficient use of resources and approaches to recycling. Better agronomic practices can reduce soil erosion and lower chemical inputs. Because of the high value of vegetable crops and their adaptability to different cropping systems, manuring and recycling of plant nutrients can be promoted.
Fertilizer recommendations can be arrived at in two ways:
- Soil analyses combined with field trials result in response curves for macro nutrients. If the farmer has soil samples analysed before planting, these curves can be used to translate soil analysis data into precise recommendations for his own field, but this seldom occurs. Normally, these data are used by agronomists and extensionists to prepare general average recommendations for the farmers, taking the soil and crop type into consideration but not the condition of the specific field. Table 6 gives a valuation of soil analysis data.
- Plant analyses and yield data give information about the amounts of nutrients taken up by the crop. The best indication gives the analysis of the total biomass of the harvested plant parts (removal of nutrients) and the plant parts remaining in the field (temporarily immobilized nutrients), the total being the nutrient uptake of the crop. Some examples of the uptake of macro nutrients are presented in Table 7.
Assuming that the non harvested plant parts will remain in the field, it still must be realized that the uptake or absorption, and thus immobilization, of minerals is much greater than the actual removal from the field in the harvested part. The uptake of minerals seems to vary greatly between species, but there is much conformity for crops within the same species. Variations are caused by growing conditions, varietal differences and soil properties. A higher than normal uptake (luxury consumption) is caused by too high a supply, e.g. of N or K.
In practice, the yield expected from a good crop under local conditions is often taken as criterion for a calculation of the fertilizer recommendation. When estimating the adequate fertilizer gift based upon the amounts taken up, losses by leaching or immobilization have to be compensated for. For example, it has been found that the uptake of a shallot crop producing 15 t/ha of bulbs in Indonesia was 80 kg N, 15 kg P (35 kg P2O5) and 90 kg K (105 kg K2O). The recovery (utilization rate, efficiency factor) was estimated at respectively 60%, 40% and 70% of the nitrogen, phosphate and potassium fertilizer, leading to a recommendation of 80/0.6 = 130 kg N, 35/0.4 = 90 kg P2O5 and 105/0.7 = 150 kg K2O.
Usually, skilled vegetable farmers apply manure, compost or other organic fertilizer whenever it is available, in quantities from 10-30 t/ha or more. Apart from improving the physical properties, this manure will amend the soil with considerable amounts of nutrients. For instance, 10 t of cow dung (LEHRI, West Java) contains 260 kg N, 45 kg P (corresponding to 100 kg P2O5) and 130 kg K (corresponding to 160 kg K2O). These minerals are partly fixed in the organic material and are therefore released gradually. Thus, in the shallot example mentioned above, 10 t of cow dung might cover the entire uptake of these macro nutrients. Unfortunately, in shallot production areas (Brebes Tegal), insufficient farm manure is available and all nutrients are applied as mineral fertilizer.
The plant needs different quantities of nutrients during its lifetime. In order to reduce losses by leaching, especially in the rainy season, it is good practice to supply the N and K fertilizer, or at least the nitrogen, in split applications. The phosphate should preferably be given during ploughing or tillage, together with the organic manure, because it is not leached.
Many vegetable growers use foliar sprays of mineral fertilizers, often mixed with pesticides in the same sprayer. The quantity of macro nutrients which can be applied with this method is very limited. The most profitable is the application of urea: with a 5% solution and 500 l/ha, the rate is 25 kg/ha of urea or only 12 kg N. Many types of foliar sprays with N, P, K, Mg and micronutrients are on sale. Apart from being expensive, they are superfluous in normal growing conditions. The drawbacks of foliar application are the risk of scorching the plant, the possible interference with the action of pesticides, and the corrosion of the sprayer apparatus. But foliar application may be justified to cure an apparent deficiency of a micro nutrient, e.g. borium or iron deficiency on alkaline soils.
In practice, many types of organic waste material and all types of manure including nightsoil are used for vegetable production. If the C/N ratio of the material is above 15, as in the case of rice straw and bran, an addition of N fertilizer (7 kg per t straw) is recommended to avoid N deficiency. The high soil temperature in the lowlands ensures organic material decays fast. Crops such as amaranth can be grown successfully on fresh or only partly decomposed town waste, although there is a risk that this waste will pollute the soil with plastics and heavy metals.
Another way of increasing the organic matter content in soil is to grow a cover crop which is ploughed in before planting the main vegetable crop. A leguminous plant (e.g. Crotalaria) which fixes nitrogen through Rhizobium bacteria is normally used for this green manure. Although highly recommended by researchers, green manure is infrequently used by vegetable growers, for economic reasons. Mulching of vegetables with straw, usually rice straw, is a very common practice among vegetable growers in South East Asia. Apart from reducing the growth of weeds, it limits sun burning of the organic material, impedes soil erosion, and keeps the soil cool and moist. The straw mulch gradually decays and becomes available to the soil as organic manure.
Vegetables in general are succulent crops and attractive to pests and disease organisms. In the international terminology, the word "pest" is interpreted in two senses. In the broad sense it means any organism that hampers the crop: weeds, insects, mites, snails and slugs, rodents, birds, nematodes, fungi, bacteria, viruses. The word "pest" in Integrated Pest Management (IPM) fits in this concept. In the more usual terminology all animal causes of plant damage except nematodes are called "pests", whereas the microorganisms including nematodes are grouped as "diseases" and the noxious plants competing with the crop are referred to as "weeds".
Exact information on the economic level of crop losses is limited. It is difficult to assess the losses caused by a single pest or disease. Crop health is the complicated outcome of the attack by several organisms trying to proliferate on plants with a genetically determined constitution, which is strongly influenced by the ecology. Overall yield losses in the vegetable sector may amount to 25%, which is higher than for all other categories of crops. For the farmer, the costs incurred for the chemical control of pests and diseases are very high, often between 100 and 400 US$ per ha or 10-40% of the variable costs (material inputs and labour). Diseases and pests cause a downgrading of the market quality and consequently of the farm gate prices, and reduce the export chances.
Crop protection has evolved along with the crops and the cropping systems. In home gardens, fences were constructed to protect the vegetables from larger animals. Other simple control measures were the manual removal of caterpillars or repelling the insects with wood ash (a kind of chemical control). With the production of vegetables for the market, pest control measures became more urgent, in order to achieve the highest possible yield of products undamaged by pests or diseases. A number of non indigenous pests such as the diamond back moth (Plutella xylostella) and club root (Plasmodiophora) on cruciferous vegetables have been introduced into South East Asia with planting material or otherwise and have become extremely troublesome (Eveleens & Vermeulen, 1976). Compared with temperate countries, little is known of diseases and pests of vegetables in the tropics. The diseases and pests of individual vegetable crops are mentioned in the species descriptions. This section is restricted to some general observations about their control.
Although very costly, the application of pesticides has become the most common and easiest way of pest control in vegetables. Although there is a growing awareness of the dangers of toxic substances, the use of these biocides is still increasing and has reached levels at which there are health risks to growers and consumers, and considerable damage to the environment. The preventive spraying of pesticides has become routine, especially on the commercial highland vegetables of foreign origin, which lack the internal defence mechanisms of indigenous vegetables.
Encouraged by a sometimes rather aggressive sales promotion by chemical companies and by ineffective governmental control on toxicity or residual effects and by the lack of know how among farmers and extensionists, the use of pesticides often leads to the intoxication of the people handling the pesticides and to noxious effects for the health of the consumers. In many places in South East Asia the environment, the land and the water for drinking or fishing, has become polluted.
Chemical control has a serious negative side effect; it destroys predators, parasites or natural enemies of the pest. This disturbance of the natural balance leads to a further intensification of the chemical treatments. Many pests have developed resistance to pesticides, forcing the farmers to spray more frequently and with stronger concentrations. The control of the diamond back moth on cabbage is a notorious example. Yet another negative effect which is rarely recognized by the farmer is that many pesticides are phytotoxic. Crop damage often occurs when pesticides are sprayed in higher concentrations than prescribed, especially during the dry season. The concentration of fungicides on the leaves of e.g. tomato and capsicum pepper is often so high that the stomata are blocked and photosynthesis is hampered.
Thus, chemical treatments should be applied only when the economic threshold for damage is surpassed, when no other control measures are effective and when precautions are taken for safe use.
The use of natural enemies to control a pest, i.e. predators, parasites or diseases, is called biological control. Many predators of insect pests on vegetables have already been found in South East Asia. The rearing and release of egg parasitoids of the genus Trichogramma and larval parasitoids of the genus Diadegma for the control of diamond back moth (Plutella xylostella) on cabbage has already been tried out in the highlands of the Philippines and Malaysia, apparently with some success (Talekar, 1992). The spraying of a bio insecticide produced by the bacteria Bacillus thuringiensis, often called BT, is a promising microbial control method. Several strains of BT are known. This microbial insecticide is sometimes used against caterpillars in vegetable crops. The disadvantages of BT are that it is rather costly and that the insect population gradually develops resistance.
Control with cultural practices
Depending on the type of pest, the crop and the environment, the damage caused by pests can be kept at a low level with the right cultural practices. Whether these cultural practices are economic depends strongly on the local conditions and the skill of the farmer. Several cultural practices are known to reduce pest incidence and damage. "Crop rotation" is effective in the control of soilborne diseases and sometimes also against insect pests. With the right "time of planting", also called "timing or planning of crop production", certain important pests may be avoided; insects are often more abundant in the dry season, whereas fungal diseases are worse during the rainy season. "Mixed intercropping" sometimes reduces the pest incidence, e.g. tomato plants and garlic are known to repel insect pests of cabbage or carrot. "Disinfection by heating" of nursery soil against pathogens like Pythium (damping off) is sometimes practised, e.g. for raising tomato or capsicum pepper transplants. A "balanced fertilization" with a reduction of the often too high dose of nitrogen makes the crop stronger and less attractive to pathogens. The "soil structure and pH" are very important for crop health. Organic manure improves the soil structure and reduces bacterial wilt and nematodes. "Liming" of acid soils reduces club root disease of crucifers. Good "drainage" reduces bacterial wilt and fungus diseases of many crops. "Mulching" with rice straw or plastic is a method of weed control and reduces soil erosion and bacterial wilt. Plastic mulch is reported to reduce thrips and aphid populations. In some areas, farmers raise nursery plants (cabbage, capsicum pepper, tomato) and even whole crops (cabbage) under fine mesh insect proof "nylon netting". Netting whole fields of Chinese kale (kailan) is used against diamond back moth in Thailand. Good "sanitation" is another helpful practice. This means the removal of crop residues and of infected plants or planting material (roguing). In shallot growing in Indonesia it is common practice to pick off all Spodoptera caterpillars and egg clusters by hand and to destroy them.
Control with resistant cultivars
The cheapest and most practical control method is to use resistant cultivars. Landraces generally possess high "horizontal resistance", a genetically determined level of tolerance, which means the plants are attacked but do not suffer very much. This is in contrast to many resistances in modern cultivars, which are narrowly based on one or a few genes. These resistances are often broken in a short time, by the pathogen evolving and forming new strains or races. Plant breeders have developed hundreds of cultivars of the more important commercial vegetables with resistant genes against fungal or bacterial diseases, nematodes and viruses, but resistance to insects or mites is very rare. The existence of resistant cultivars is mentioned in the species treatments.
Integrated Pest Management (IPM) is a worldwide accepted control method for diseases and pests. It is a combination of non chemical control measures (resistant cultivars, cultural practices, biological control) with a minimum use of indispensable pesticides based upon threshhold observations. It is mostly practised for insect pests and often concentrates on a single major pest such as diamond back moth of cabbage.
If the overall health condition of the crop is taken as the major issue, a more holistic approach to integrated control is the Integrated Crop Management (ICM), which takes the coherence and relationship between human and environmental factors into consideration. ICM is defined as "a system whereby all interacting crop production and pest control tactics aimed at maintaining and protecting plant health are harmonized in the appropriate sequence to achieve optimum crop yield and quality and maximum net profit, in addition to stability in the agro ecosystem, benefiting society and mankind" (El Zik & Frisbie, 1985).
Harvesting and post harvest handling
First of all it is important to realize that pre harvest choices such as cultivar and cultural practices, strongly influence the quality obtained at harvesting. Size, form, colour, firmness, taste and other internal and external product qualities are genetically determined. During cultivation, all measures which assure good health of the crop also have an impact on the post harvest quality. The plant density influences the product size and form. A common mistake is to apply too much nitrogen fertilizer, which makes the harvested product more watery, weaker and more susceptible to damage and rotting.
Most vegetables are very perishable products. The losses of product and deterioration of quality caused by inappropriate harvesting and post harvest handling are considerable. Losses of one third of the harvested product are not exceptional. Harvested vegetables are still living parts of plants, which remain very susceptible to damage until ultimate consumption. Vegetables have a high water content (70-95%) and the leafy types in particular will wilt easily because of continuing respiration after the harvest. Some recommendations regarding the correct handling of harvested vegetables to minimize losses are given below.
Two types of harvesting methods may be distinguished. Once over harvesting is the harvest of all the useful parts or of all plants at once. This is practised e.g. on carrot, radish, cabbage, onion, garlic. More common is the repeated harvesting of the plant parts desired in several picking rounds, e.g. for capsicum pepper, cucumber, tomato, yard long bean, asparagus. In many cases the grower himself can choose which type of harvest will be applied. Many leafy vegetables (amaranth, kangkong) can be once over harvested by uprooting or cutting the whole plants, or they can be harvested repeatedly by successive cuts. In the latter case, the cultural practices will be different; in the example of amaranth, the amount of nitrogen fertilizer, the height and frequency of cutting, the plant spacing and their interactions will strongly influence the ultimate yield and quality, in particular through their effect on flowering (Grubben, 1976).
The maturity stage of the product wanted is greatly influenced by the time and frequency of harvesting. For example in tomato, the farmer has to consider the maturity stage requested by the dealer. If harvested immature green, the tomatoes will not taste good. Mature fruits have the best taste but will not tolerate several days of transport and storage. The farmer will try to compromise by harvesting at the mature green stage to let the fruits ripen in transit or storage before marketing. To deliver high quality products, the farmer must have a good knowledge of harvest indices: size, colour, firmness.
Post harvest handling has the objective of bringing the harvested product to the consumer with a minimum of quality deterioration. A first step is a sorting into various quality classes or gradings. The principle is a two way sorting, i.e. by appearance and size. An example is presented in Table 8, giving the prescriptions for the eight quality classes used in the trade of shallot in Indonesia (Schoneveld, 1992).
Packing and transport
Proper packaging is aimed at avoiding mechanical damage by pressure and at avoiding warming up from respiration by inadequate ventilation. Excessive ventilation is also undesirable, as it results in wilting and weight loss. The packing materials used for vegetables are very diverse: net bags, bamboo baskets, wooden or plastic crates, cardboard boxes and plastic bags, and also loose on the truck. The choice is purely economic: cheap packing materials generally lead to more deterioration of quality and a lower price. A suitable packing unit is 20 kg. Bags with dried products such as garlic and shallot may contain 40 kg. Packing units are often made too large and too heavy (sometimes 100 kg baskets or bags) for easy handling. They lack sufficient ventilation, and heating of the product can easily cause serious rotting. The main cause of post harvest losses during storage and transport, however, is the pressure from the product loaded on top. Figure 2 illustrates how the damage by pressure can be reduced by the installation of partition floors or by using self supporting crates (Schoneveld, 1992). Clearly, the pressure on the lowest product is least for solution G (small self supporting crates).
Vegetables can be stored in a cool, dark, well ventilated place. Leaf vegetables must be wetted occasionally, to avoid drying out. The best keeping is in a cool room, but this method is too expensive for the individual farmer and for most dealers. Some vegetables keep well at low temperatures of 1-2°C (Allium crops, cabbage, radish), but most other products will suffer damage at those temperatures. For example capsicum pepper stores better at 5-7°C, cucumber at 10-12°C. Onion, shallot and garlic also store well in the lowlands in well ventilated sheds; a temperature above 27°C impedes early sprouting. Leafy vegetables may be packed with shredded ice to keep them fresh during long distance transport. Fresh exports require sophisticated post harvest facilities and transport infrastructures to deliver fast and timely, as well as top quality produce.
Utilization and processing
The utilization of vegetable products is changing constantly. The development of new types of food is in general leading to higher levels of consumption. In South East Asia it is customary to consume vegetables as fresh as possible. Many housewives buy fresh vegetables once or even twice a day. Yet processed vegetables are becoming more popular for reasons of convenience.
Processing techniques are of the utmost importance in the vegetable sector, because of the perishable nature of the product. Apart from adding value, processing enables the fresh market to be relieved when prices are low due to a glut in production, and also avoids wasting produce which is not marketable because of its small size or less attractive appearance.
Vegetable products are mainly processed by drying or dehydration, pickling, canning, and freezing. Drying or dehydration is one of the oldest preserving methods; the principle consists of reducing the moisture content below that at which microorganisms grow and reproduce. It is usually accomplished through heat (e.g. sunshine) and ventilation; for aromatic vegetables dehumidifiers are more suitable, in order not to lose the volatile oils. The drying of green leaves (Corchorus olitorius L., Sesamum radiatum Thonn. ex Hornem.) and fruits (okra, capsicum pepper, local eggplant, pumpkin) and their preservation as powder is common practice in Africa. Some of the food value is lost in the process, but drying merits more investigation as it is a simple technique that can be widely used throughout the tropics. Pickling is preservation in brine or vinegar, with or without bacterial fermentation. There are many traditional methods for preparing salted and pickled vegetables in South East Asia.
Mixed vegetable and fruit juices are becoming increasingly popular.
Canning fruits and vegetables is becoming an established practice in South East Asia, but preservation by freezing is still in its infancy.
Genetic resources and breeding
Knowledge of the use of wild plants is disappearing rapidly. It is only natural that the importance of the wild flora as a direct food source is decreasing, but it is inadmissible for genetic resources to be destroyed before the true value has been assessed objectively.
The development of modern horticulture has led to a huge reduction in the number of vegetable species. It has also resulted in a narrowing of the genetic base of the remaining species, because a large number of local, unselected cultivars have been replaced by fewer highly selected planting materials. Plant breeders have to rely on genetic resources, which can be found in the primary and secondary centres of diversity or in artificial germplasm collections.
The establishment of the International Board of Plant Genetic Resources (IBPGR) in 1974 greatly increased the awareness of the importance of crop germplasm and led to the establishment of numerous collections of species endangered by genetic erosion. Based on a study of tropical vegetables and their genetic resources (Grubben, 1977), in 1979 IBPGR prioritized eight vegetable genera or groups for immediate action, i.e. Abelmoschus, Allium, Amaranthus, Capsicum, Cruciferae, Cucurbitaceae, Lycopersicon, and Solanum melongena (van Sloten, 1980).
Information on existing collections was compiled in a Directory of Germplasm Collections. 4. Vegetables (Bettencourt & Konopka, 1990).
Selection and breeding have an important role to play in the improvement of vegetable crops. Named, well defined cultivars have so far only been developed in an estimated 60 species out of the 225 primary use vegetables described in this book, and a large part of these originate from outside South East Asia. The development and release of cultivars in most South East Asian countries is still the task of public research and extension agencies, although the involvement of the private sector is increasing.
The establishment of the Asian Vegetable Research and Development Center (AVRDC) in 1971 has given a strong impetus to the development of advanced breeding programmes in South East Asia on a number of vegetable crops such as tomato and Chinese cabbage. Its training programmes have also strengthened the national programmes in other crops.
There are two aspects to the breeding philosophy in South East Asia. The first, easily overlooked or neglected, is to select or breed for low input farms where standard cultivars are required that respond to low levels of fertilizer, are adapted to a wide range of environmental conditions, and are tolerant of common diseases and pests. This implies collecting, evaluating and maintaining the germplasm of a wide range of crops, doing research and gathering information on crop characteristics, and finally, selecting suitable cultivars by conventional selection techniques. These tasks are best performed by government research and extension agencies with an overall responsibility for the sector; they should supply local private seed companies with breeding material or selections to be multiplied into commercial, good quality seed for the farmer.
The other aspect is the development of high yielding cultivars for commercial farms, which give maximum response to optimal input. It is at this level that good commercial opportunities exist for the private sector.
This volume is proof of the great wealth and diversity of vegetables in South East Asia. The development of the horticultural sector is first and foremost a matter of allocation of resources. However, new revolutionary solutions have to be found to achieve sustainable production systems. As the train of horticultural development gathers speed, efforts should be made not to repeat the mistakes of the industrialized world, i.e. environment-unfriendly production methods and a considerable loss of genetic diversity.
Notwithstanding the significance of the vegetable sector in the agricultural economy, the diffuse distribution and species diversity have made it hard to develop a compelling rationale for allocating appropriate resources for vegetable crop research.
Most countries in South East Asia have facilities for vegetable research (LEHRI, Indonesia; MARDI, Malaysia; IPB, the Philippines; Institute of Horticulture, Thailand; Institute of Agricultural Science, Vietnam), but in general the allocation of resources does not reflect the economic (and nutritional) importance of the sector. In setting priorities within the sector, the "exotic" highland species (cabbage, potato, etc.) have received much attention, and the indigenous vegetables have scored low, but they have not yet lost the battle. Numerous new initiatives have recently been taken to promote lowland vegetable research, e.g. at LEHRI, Indonesia.
At the international level, the Asian Vegetable Research and Development Center (AVRDC) has done pioneering research on Chinese cabbage and tomato. In its strategic plan for the 1990s (AVDRC, 1991), it has clearly opted to give first priority to the lowland humid and subhumid tropics, to concentrate on small scale commercial production, and to expand its commodity coverage to capsicum peppers, eggplant, and the important Allium crops (onion, shallot, garlic). It plans to put more emphasis on a decentralized organizational set up with regional research networks, and by so doing is pursuing the same line as the Consultative Group on International Agricultural Research (CGIAR), which is studying new ways of promoting tropical vegetable research, possibly through a new coordinating body (such as IBPGR) with the task of stimulating the development of national research systems (Winrock International, 1986).
National policies tend to emphasize the development of exports rather than domestic consumption, but a well supported domestic market is the best possible basis for export. The rapid expansion of the supermarket system of selling fresh vegetables, with its insistence on quality, will in time stimulate the adoption of improved marketing methods. The prospects for fresh exports to large urban centres (Singapore) and nearby industrialized countries (Taiwan, Japan) are certainly good, but more is to be expected from exports of preserved and processed products. However, this export sector should develop as a by product of processing industries aimed at the large domestic markets. Vegetables in general have a positive income elasticity and with increasing economic prosperity, the production of market vegetables will increase.
The rapid expansion of commercial vegetable production creates a market for high quality seed. Growers are changing their attitude from considering vegetable seeds as a cheap internal input to the conviction that it pays to start a crop with healthy market seed of an improved cultivar purchased from a professional seed producer.
In some cases it may be justified that the public sector (National Agricultural Research Systems) produces market seed itself. However, international experience has proven that farmers are generally better off when the public sector takes care of the more fundamental part of research in support of the private seed sector. The public sector should be responsible for independent testing of the value of new and existing cultivars, the release policy for new cultivars, and the control of seed quality.
The size of the national seed market determines whether the seed can be produced in a country. A sound government policy should stimulate breeding activities and seed production in the vegetable production areas of the country in the interest of farmers and consumers. Apart from a few exceptions (e.g. white cabbage), it is technically and economically feasible to produce all vegetable seed in the South East Asian region.
G.J.H. Grubben, J.S. Siemonsma & Kasem Piluek<|endoftext|>
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Home | Courses | Back to Chapter | Any Question? |
# Physics - 11 Complete Syllabus Notes
### Chapter 2 - Vector
Scalar Quantities::
The physical quantities having only one magnitude are called scalar quantities.
eg. distance,speed etc.
Vector Quantities:
The physical quantities having magnitude and direction and obeying the law of vector addition are called vector quntities.
eg: displacement, velocity, etc.
A vector is represented graphically by straight line with arrow head. The length of the straight line represent magnitude and arrow head represent direction.
Terms regarding in vector:
i) Equal Vector:
Those vector are said to be equal vector if they have same magnitude and direction.
ii) parallel vector:
Vector acting in same direction with equal or unequal magnitude are called parallel vector.
iii)Anti-Parallel Vector:
Vector acting in opposite direction are called Anti- Parallel vector.
iv)Colinear Vector:
Vector acting in same line are called collinear vector.
v) Coplaner Vector:
Vector acting in same plane is called coplaner vector.
vi) Unit vector:
A vector having unit magnitude is called unit vector.
In certesian coordinate system x-axis, y-axis and z-axis are represented by i^, j^ and k^ respectively where i ^, j^ and k^ are unit vectors.
Statement: If two vectors acting at a point are represents at both in magnitude and direction by two adjacent side of parallelogram. Then the diagonal from that point represent their resultant both in magnitude and direction.
Explanation:
Let two vector $\overrightarrow{P}$ and $\overrightarrow{Q}$ are represented at both in magnitude and direction by two adjecent size $\overrightarrow{OA}$ and $\overrightarrow{OB}$ of a parallelogram OACB as shown in figure ii) then from parallelogram law of vector addition the diagonal $\overrightarrow{OC}$ represent their resultant $\overrightarrow{R}$ .
Let $\theta$ be the angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$ and $\beta$ be the angle between $\overrightarrow{R}$ and $\overrightarrow{P}$.
Magnitude of Resultant:
From C draw perpendicular, CD and OA produced.
In $\Delta$ ODC,
OC 2 = OD 2 + CD 2
OC 2 = (OA + AD) 2 + CD 2
R 2 = (P + Q COS $\theta$) 2 + (Q Sin $\theta$) 2
On solving with R,
we get,
$R=\sqrt{p^{2}&space;+&space;2PQcos\Theta&space;+Q^{2}}&space;......i)$
Direction Of Resultant:
In $\Delta$ ODC,
$tan\beta$ = $\frac{CD}{OD}$
$tan\beta$ = $\frac{CD}{OA + AD}$
$tan\beta&space;=&space;\frac{Qsin\Theta&space;}{p&space;+&space;Q&space;cos\Theta}$
$\therefore&space;\beta&space;=&space;Tan^{-1}[\frac{Qsin\Theta&space;}{P&space;+&space;Q&space;cos\Theta&space;}]....ii)$
eqn i) represent the magnitude of resultant and eqn ii) represent the direction of resultant with $\overrightarrow{P}$.
Special Cases:
i) WHEN $\theta$ = 00
Magnitude of Resultant:
$R=\sqrt{p^{2}+2pqcos0^{0}+q^{2}}$
$R=\sqrt{p^{2}+2pq+q^{2}}$
$R=\sqrt{(p+q)^{2}}$
$R=(p+q)$
$R=(p+q)=Maximum&space;Value$
Direction of Resultant:
$\beta&space;=&space;Tan^{-1}\left&space;(&space;\frac{Qsin\theta}{P&space;+&space;Qcos0^{0}}&space;\right&space;)$
$\therefore&space;\beta&space;=0$
i.e. the direction of resultant is in the direction of given vector (either P or Q vector)
ii) WHEN $\theta$ = 900
Magnitude of Resultant:
$R=\sqrt{p^{2}+2pqcos90^{0}+q^{2}}$
$R=\sqrt{(p^{2}+q^{2})}$
Direction of Resultant:
$\beta&space;=&space;Tan^{-1}\left&space;(&space;\frac{Qsin90^{0}}{P&space;+&space;Qcos90^{0}}&space;\right&space;)$
$\therefore&space;\beta&space;=Tan^{-1}&space;\frac{Q}{P}&space;&space;&space;with&space;\overrightarrow{P}$
iii) WHEN $\theta$ = 1800
Magnitude of Resultant:
$R=\sqrt{p^{2}+2pqcos180^{0}+q^{2}}$
$R=\sqrt{(p-q)^{2}}$
$R=P-Q$ or, $R=P-Q$
Direction of Resultant:
The direction of the resultant is in the direction of greater vector.
Statement: If two vector are represented both in magnitude and direction by two sides of a triangle taken in same order than the closing side ( third side ) of triangle taken in opposite order represents theirresultant both in magnitude and direction..
Explanation:
Let two vector $\overrightarrow{P}$ and $\overrightarrow{Q}$ are represented at both in magnitude and direction by two side $\overrightarrow{OA}$ and $\overrightarrow{AC}$ of a triangle OAC as shown in figure ii) then from trianglem law of vector addition the closing side $\overrightarrow{OC}$ represent their resultant $\overrightarrow{R}$ .
Let $\theta$ be the angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$ and $\beta$ be the angle between $\overrightarrow{R}$ and $\overrightarrow{P}$.
Magnitude of Resultant:
From C draw perpendicular, CD and OA produced.
In $\Delta$ ODC,
OC 2 = OD 2 + CD 2
OC 2 = (OA + AD) 2 + CD 2
R 2 = (P + Q COS $\theta$) 2 + (Q Sin $\theta$) 2
On solving with R,
we get,
$R=\sqrt{p^{2}&space;+&space;2PQcos\Theta&space;+Q^{2}}&space;......i)$
Direction Of Resultant:
In $\Delta$ ODC,
$tan\beta$ = $\frac{CD}{OD}$
$tan\beta$ = $\frac{CD}{OA + AD}$
$tan\beta&space;=&space;\frac{Qsin\Theta&space;}{p&space;+&space;Q&space;cos\Theta}$
$\therefore&space;\beta&space;=&space;Tan^{-1}[\frac{Qsin\Theta&space;}{P&space;+&space;Q&space;cos\Theta&space;}]....ii)$
eqn i) represent the magnitude of resultant and eqn ii) represent the direction of resultant with $\overrightarrow{P}$.
In case of any problem ask me in qustions section!!!!<|endoftext|>
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# How do you solve [(2x), (2x+3y)]=[(y), (12)]?
Aug 5, 2017
$x = \frac{3}{2} \text{ and } y = 3$
#### Explanation:
Here we have the system
$\left[\begin{matrix}2 x \\ 2 x + 3 y\end{matrix}\right] = \left[\begin{matrix}y \\ 12\end{matrix}\right] {\to}_{R 2 - R 1}$
We can subtract the first equation from the second equation to cancel out the $2 x$ factor
$\left[\begin{matrix}0 \\ 0 + 3 y\end{matrix}\right] = \left[\begin{matrix}0 \\ 12 - y\end{matrix}\right]$
Which then leaves us with the equation
$3 y = 12 - y$
$\iff$ Add $y$ to both sides
$4 y = 12$
$\iff$ Divide both sides by $4$
$y = 3$
Then, since we know that $y = 2 x$ from the first equation, we can plug in
$2 x = 3$
$\iff$ Divide both sides by $2$
$x = \frac{3}{2}$
We can check out answer by plugging in $\left(\frac{3}{2} , 3\right)$ into the second equation
$2 \left(\frac{3}{2}\right) + 3 \left(3\right) = 12$
$\iff$
$3 + 9 = 12$
$\iff$
$12 = 12$<|endoftext|>
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In 1944, at the Rockefeller Institute, New York City, NY, USA, a Canadian born Bacteriologist by the name of Oswald Theodore Avery discovered evidence that DNA was in fact genetic material.
Avery and his colleagues, M. McCarty and C. M. MacLeod, conducted experiments that determined the “Transforming Principle” where heritable characteristics of one generation of bacteria was passed on to the next generation was the nucleus acid called Deoxyribonucleic Acid or for short and simple DNA.
Oswald Theodore Avery
Rockefeller Institute, New York City, NY, USA
For More Information:
- Oswald Avery – Wikipedia
Oswald Theodore Avery Jr. ForMemRS (October 21, 1877 – February 20, 1955) was a Canadian-American physician and medical researcher. The major part of his career was spent at the Rockefeller University Hospital in New York City.
- Oswald Theodore Avery :: DNA from the Beginning
Oswald Avery was born in 1877 in Halifax, Nova Scotia. His father was a Baptist minister, and when Oswald was ten, his father became the pastor at the Mariners’ Temple in New York’s Lower East Side.
- Oswald Avery – Geneticist, Scientist – Biography.com
Oswald Avery discovered cell transformation. He recognized that DNA carries a cell’s genetic material and can be altered.
- DNA – Wikipedia
Deoxyribonucleic acid (Listeni/diˈɒksiˌraɪboʊnjʊˌkliːɪk, -ˌkleɪɪk/; DNA) is a molecule that carries the genetic instructions used in the growth, development, functioning and reproduction of all known living organisms and many viruses.
- Deoxyribonucleic Acid (DNA) Fact Sheet – National Human Genome
We all know that elephants only give birth to little elephants, giraffes to giraffes, dogs to dogs and so on for every type of living creature. But why is this so?<|endoftext|>
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## Intermediate Algebra (6th Edition)
Published by Pearson
# Chapter 2 - Cumulative Review - Page 115: 46
x=2
#### Work Step by Step
We are given that $|x+3|=|7−x|$ So $x+3=7−x$ or $x+3=−(7−x)$ We can solve both of these equations for x. For $x+3=7−x$, subtract 3 from both sides of the equation. $x=4−x$ Add x to both sides of the equation. $2x=4$ Divide both sides of the equation by 2. $x=2$ For $x+3=-(7−x)$, use the distributive property to simplify the right side of the equation. $x+3=x-7$ Subtract x from both sides of the equation. $3=−7$ This is false, so the only solution is x=2.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.<|endoftext|>
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During the Scottish Enlightenment, economists and philosophers debated the legitimacy of the death tax. John Locke believed it was proper for government to protect the rights and property of the individual and wrote that it was natural and good for parents to pass wealth on to their children (but not others).
In the 18th century, William Blackstone disagreed, arguing that a person's claim to wealth ended when life ended, and the state should regulate any inheritance. Utilitarian Jeremy Bentham went even further, arguing for strong regulation of inheritance for the sake of the public good. On the other hand, Adam Smith favored the rights of the individual and free market for the ultimate good of society.
The American founders were very familiar with this philosophical struggle. At first, U.S. law aligned with Adam Smith's view of limited regulation, and the tax was used only occasionally. The Stamp Tax of 1797 was used to fund a naval war with France. The tax required people to purchase federal stamps on inventories of the deceased person's effects, on receipts for legacies (tangible property and goods) or shares of the estate, and on probates of will and letters of administration. Estates under $50 were exempt, and larger estates were taxed on a graduated scale. The tax exempted widows, children and grandchildren [source: Luckey].
In 1802, Congress repealed the tax after the need for revenue ended. Congress considered passing the death tax again during the War of 1812, but the war ended before it passed.
A federal death tax next appeared during the Civil War, but with three significant changes. First, the Revenue Act of 1862 taxed the receivers of the inheritance directly, rather than tax the legal instruments used to convey the inheritance by required stamps.
Second, the Internal Revenue Act of 1864 extended the scope of the tax to include real property (i.e., land and buildings) rather than just personal property such as legacies and distributive shares of businesses.
Third, the amendment contained what we would recognize as a gift tax because it taxed the transfer of real property if sold for less than fair market value during life. This last provision was an attempt by Congress to prevent people from dodging the death tax by transferring assets before death. Congress repealed this Civil War-era death tax in 1870.<|endoftext|>
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September 14, 2012 report
Britain to use spent nuclear fuel for batteries to power deep space craft
The idea would follow designs already used by the United States to power its Cassini and Voyager space probes and now in use by the Mars rover, Curiosity. Nuclear material gives off heat for many years, which can be used directly to keep a craft warm, or be converted into electricity for use by electronic components. The team has reportedly already harvested some amount of americium-241 from the plutonium waste left over from the production of nuclear weapons. The Sellafield facility reprocesses or separates plutonium, uranium and other fissionable materials from spent nuclear fuel, some of which is used for other purposes such as creating new fuel for nuclear reactors. It's also the site of what will be a new nuclear power station due to begin operation in 2025.
The ESA is keen to find a suitable replacement for plutonium-238, as it's only currently available from the United States and Russia, and believes americium-241, harvested from already existing plutonium waste would make a good choice. Each nuclear battery would only need about 5 kg of the material, which would mean Britain could supply all that would be needed (the Sellafield facility is believed to house some 100 tonnes of waste plutonium) by the ESA for the foreseeable future. Batteries made using it could be used to support missions to other planets and other exploratory projects.
It's also been noted that such batteries could be used for other purposes as well, such as in long term undersea probes, or in buoys outfitted with sensors to monitor sea conditions and other countries such as China and India have already expressed interest in using them for various projects. Thus, the market for long duration nuclear batteries might be expanding, which would make harvesting americium-241 not only cost efficient but perhaps at some point, profitable.
© 2012 Phys.org<|endoftext|>
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# Basic Trig – Sine and Inverse Sine
If you have never had a trig class, you can do some pretty useful things with just a few trig functions. Bob Welds discusses the sine of an angle and how to use it. He also tells how to find an angle from two known sides.
Video Transcript:
Hi I’m Bob Welds and this is a first look a trigonometry.
Today we are going to pick up where the Pythagorean theorem left off. If you don’t know how the Pythagorean theorem works then you probably want to go learn about that first. Also, we’ll use a little bit of algebra but i don’t think it will be a problem for you if you already know some basic algebra. To keep it simple, we’re only going to talk about a thing called the sine of an angle today. Later we’ll see other trig functions. But really, if you understand how the sine works, you’ll get the others pretty easily.
Ok. Let’s look at this right triangle. You see that it has a 30 degree angle in one corner. The side opposite of the 30 degree angle is 1 meter long, and the hypotenuse is 2 meters long. (remember that the hypotenuse is the longest side of a right triangle).
let’s compare the opposite side to the hypotenuse by dividing one by two and see what we get….1 divided by two equals .5 — a half. um. so .. that was easy. What’s the big deal? The deal is that we just found the “sine” of thirty degrees. The sine of thirty degrees is one half. we write it like this:
Here, let’s do it again. this right triangle has a height of 4 meters and a hypotenuse of eight meters. The sine of 30 deg is four divided by eight. 4 divided by eight reduces to one half, again, the sine of thirty degrees is one half, or point five. You see it doesn’t matter how big the triangle is; 30 degrees always has a sine of point five. Any 30 degree right triangle has a hypotenuse that is exactly twice as long as the opposite side. Here, you can google it…type sin 30 degrees and see what you get. be sure to type degrees or you’ll get something else.
The sine of an angle tells us the answer to the division problem we get when we put the opposite side (that is, the side across from the angle) over the hypotenuse. So the sine of the angle is “ratio” of the opposite side to the hypotenuse. the opposite over the hypotenuse. In this case, we put one over two and see that it is one half or point five.
You might be asking what good it is to know that ratio is since we already know how to divide one by two or four by eight… well let’s take a closer look. Here is another triangle, but with the same 30 degree angle in the corner. This time, we know the length of the hypotenuse and angle, but we don’t know the opposite side. Look what we could do if we knew the sine of the angle…instead of saying “sine of 30 degrees” we could say point five. Because .5 is the sine of thirty degrees. ….and instead of saying hypotenuse, we could say fourteen, because that’s how long this hypotenuse is. Now instead of using that silly question mark, we could call the length of the opposite side: “a”. And voila! we have an algebra problem instead of a trigonometry problem.
Here is how we could solve the algebra part….multiply both sides by 14… on the left, we see 14 times .5 equals seven. and on the right side .the 14’s cancel each other out…and look! A equals seven. That is something we could not have found if we didn’t know this little bit of trigonometry.
Lets see how useful the sine of an angle can be. We know the sine of thirty is, but with our calculator we can find the sine of ANY angle. Lets try to find the length of the hypotenuse of this triangle. The angle is 25 degrees and the side opposite of the angle is 1.69 meters long. First write down what we know…The sine of 25 degrees is equal to the opposite side–1.690 over the Hypotenuse. Now, we need the sine of 25 degrees…This time, I’ll use my calculator instead of google. First, we’ll turn it to scientific mode…..then I’ll be sure it knows to use degrees….now I’ll type 25 and hit the sin button. Different calculators might have you hit the sine key first, so check your answers to be sure you are doing what your calculator expects….0.422 618 261 and it keeps going. We only need the first few decimal places of that. So instead of saying “sine of 25 degrees” we will say 0.4226. Here, I rewrite our equation.
Now it’s an algebra problem. We need to isolate thy hypotenuse to see what it’s value is. I’m going to call it “C” to keep things neat and clean. I’ll multiply both sides by c….the c’s cancel out on the right…..now i’ll divide both sides by .4226. this leaves C on the left and a little division problem on the right. we use our calculator to type 1.690 divide by .4226 and we get something just a smidgen over 3.999 — a number very close to four meters.
Now we’ve seen that we can use the sine of an angle to find the sides of the triangle, and you can use the opposite side and the hypotenuse to find the sine of the angle. What if you knew the sine of an angle, not the angle itself? There’s a trick for that. Let me show you. Here we have another right triangle. The side opposite of our angle is 2.5 meters long, and the hypotenuse is 3.889 inches long. You may be able to see right away that we could find the sine of the angle like we did a few minutes ago. here, remember? You see we just put 2.5 over 3.889. Since a fraction is just a division problem, we’ll divide 2.5 by 3.889. That gives us .64283. So just like before, we know what the sign of the angle is. Only now we don’t know what the angle itself is. To find the angle when we know the sine of the angle, we just run the sine function in reverse. That’s called taking the ARC Sine or the INVERSE SINE. You will see it on your calculator as a sine with a -1 exponent, or you may see it spelled out as “A”sin or “arc sine” or even inverse sine. A lot of the time it’s the same button as the sine button, but you push another button to enable this feature. I’ll do it with this little calculator so you can see. I’ll type .64284 and push this up arrow button. You’ll see that my sine key turned into an inverse sine key.
Now when i press the inverse sine button i get an angle! Isn’t that teh cats pajamas? My angle is just a tiny bit more than forty degrees. Ill round it off to the nearest tenth of a degree. Here’s the original problem you can remember what we were solving. We knew the side opposite of the angle was 2.5 meters long, and we knew the hypotenuse was 3.889 meters long. We wanted to know the angle. First we found the sine of the angle, opposite over hypotenuse (2.5 over 3.889). We divided and found the sine of the angle was .64283. Then we used our new trick, the inverse sine to find teh angle was very close to forty degrees. I just think that’s great. That’s going to be very useful.
Ok, let’s work some problems. Im going to give you a problem where you find the missing opposite side first. When you see sparky’s paws, Pause teh video and try to fill in the blanks in the equation. You don’t need to solve it yet, just fill the blanks in with what you see on the diagram….[pause]. Ok, here are the blanks filled in. The sine of 30 degrees is equal to “a” over 15. “a” is the opposite side, and 15 is the hypotenuse. teh sine of an angle is the opposite over the hypotenuse. Now lets see if we can turn the trig problem into an ordinary algebra problem. Let’s see instead of saying sine of thirty degrees, what could we say? use a calculator to find the sine of 30 degrees, even if you know what it is. This way you can be sure your calculator is set up correctly.
I’ll pause while you calculate. ok. i know you probably remembered that the sine of thirty degrees was .5, but hopefully you tried it on your calculator. It is really important that you know how to get the sine of an angle. Now we just have a little algebra problem on our hands. See if you can solve for “a.” a is equal to 7.5. If you were able to solve it by looking at it, let me encourage you to go through the algebra steps to get good at them. Here, we multiply both sides by 15 to get rid of that fraction…. Then we see that 15*.5 is equal to 7.5. that’s our answer.
Let’s do one more. Let’s do one where we find the angle by using the opposite side and the hypotenuse…. let’s remember that if we know the opposite side and the hypotenuse, we can find the SINE of the angle. Let’s do that first. Pause the video and see if you can fill in the blanks. Then find the sine of the unknown angle by dividing. That’s right, 1 on top, 4.810 on the bottom. That means the sine of this angle is .2079 . Now for the second part, lets use the arc sine to find what the angle is….remember we know the sine is .2079 Olk, pause the video and be sure you know how to find the angle using the sine of the angle. Fill in the blanks and do the calculation on your calculator. OK, I got an answer that was ridiculously close to 12. I’m going to round it off to the nearest tenth of a degree.
Well that’s our first look at trigonometry. I hope you can see how useful it is to find angle from those lengths and lengths from the angles. There is a LOT more to cover, but if you understand how the sine and inverse sine work you are on you way to using some very useful tools. I’m bob welds, and these are weldnotes!
### 1 Response
1. Robert Vaillancourt says:
I’ve watched four of the video’s and they answered the questions I had. Easy listening Thank you.<|endoftext|>
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Fractions and Decimals: Teaching and Learning Activities
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Document Sample
Fractions: Teaching and Learning
Activities
Penny Lane
Conceptual understanding vs procedural knowledge
The most important thing we have to bear in mind when teaching fractions is that
our students need to develop conceptual understanding of fractions, not
procedural knowledge. That is, we have to ensure students really understand
fraction concepts, rather than simply follow procedures for working with fractions.
Classroom snapshots
Let’s take a look at what some of our students think about fractions.
Year 6 students were shown a shoe and asked how many pairs of
shoes they could see. Of 30 students asked, four gave the answer
half a pair. Twenty five other students said none. One student said
What a cool shoe; where did you get it?
Year 4 students were shown this circle, and asked what
fractions they could see.
The whole class agreed with what one student said:
You have to have equal parts to have fractions, so there
aren’t any fractions in this circle.
1
Year 1 students were shown a third of a circle, and asked what it
was.
It’s a fat quarter.
If you put four together you’ll get a fat circle.
Year 4 students were shown a circle divided into thirds.
They’re quarters.
That’s three quarters.
Year 4/5 students were shown a third of a circle.
It’s a big quarter.
It’s three quarters.
Year 4 students were shown a quarter of a circle divided into three
equal parts and asked what one of the parts was.
Three quarters.
Year 4 students were asked to suggest some fraction words.
Half
Quarters
… a long silence…
Fifths?
Year 4 students were asked what fifths are.
It’s like when they draw a circle divided into six parts and they tell
you to colour in five of the parts.
2
Two Year 3 students with a considerable amount of procedural
knowledge and very quick to respond with number facts when
called on to do so, were asked to divide a circle into five equal
parts.
They drew a line dividing the circle in half.
Then they drew two more diagonals to divide the
circle into six equal parts. They counted the parts
and erased all their lines.
Next they divided their circle in half with a line from
top to bottom, added two more diagonals, counted
the six parts and erased their lines.
It’s impossible.
After being unsuccessful on the above task, the same two Year 3
students asked the teacher to give them some tables questions.
They were asked what was 6 x 2.
12
Then they were asked what was 6 x 2½.
12½
3
A Year 6 student was asked how much pie each person would get
if two pies were shared among three people. He drew this.
They’d get two-sixths each, because it’s two parts out of six.
Year 2/3 students were asked to share two pizzas among three
people. They drew this.
They get two thirds each.
anchovies, and each person wanted one slice of each pizza.
They drew this.
They get two sixths each. They get two parts out of six.
The teacher asked them did they get the same amount of pizza as
before.
Yes, two thirds is the same as two sixths.
How can that be?
It’s two sixths because it’s two parts out of six, and it’s two thirds
because there are two slices and the slices are thirds.
4
Perhaps teachers are doing too much scaffolding, too much modelling, too much
explaining.
We present each concept neatly and develop it sequentially according to the
knowledge we have about fractions, rather than the knowledge students have
about fractions; we model concepts and language that can lead to confusion,
such as ‘one part out of four’; we explain that fractions are related to equal parts
and we demonstrate equal parts with models and diagrams that make the equal
parts concrete, without understanding the complexity of this concept (it is not so
much the concept of concrete equal parts that is at the heart of understanding
unit fractions, it is the size of one part in relation to the whole); we focus on
specific fractions rather than on general fraction concepts; we make the
connections among fractions for students, for example when we begin with
halves and proceed to quarters then eighths, rather than allowing students to
make connections; we ask students to colour in three parts out of five parts that
have already been drawn for them, rather than supporting them to explore and
represent such fractions in their own ways; we base our teaching on specific
syllabus content items rather than on what is already in students’ minds.
We can give students a range of challenging tasks that elicit their prior
knowledge and give them opportunities to create new knowledge, note their
unscaffolded responses, and base further questions and tasks on these. In this
way we will build on what students already know and can express to work
towards the syllabus outcomes.
Students should be challenged and supported to devise their own strategies for
working with fractions, as this benefits the development of fraction concepts and
reveals to their teachers their strong understandings, limited understandings and
misunderstandings, and their effective use, limited use and misuse of language
in the context of fractions.
5
What content?
We need to support students to build general fraction concepts as well as
concepts of specific fractions, and sometimes we need to go beyond the syllabus
content for a particular stage in order to build strong concepts. These general
fraction concepts include the following.
Objects, collections and measurements can be divided into parts.
Unit fractions are named according to the number of parts of that size that make
up the whole.
There are relationships among different size parts of an object, collection or
measurement.
Non-unit fractions are named according to their relationship to unit fractions.
Specific fractions can be expressed in more than one way because of the
relationships among the different size parts.
We must keep returning to these general fraction concepts to ensure students
understand the concepts related to specific fractions. So, for example, if we are
building the concept of equivalent fractions by focusing on halves, quarters and
eighths, we must challenge students to transfer this concept to fractions
generally, rather than move onto equivalence among another set of specific
familiar fractions such as thirds, sixths and twelfths.
Fraction models
Students need to investigate a range of fraction models. Circles are often
preferable to rectangles for working with fractions of whole objects because once
a rectangle is cut it looks like two new wholes, whereas when a sector is cut from
a circle the original circle is still apparent. A circle is also a familiar shape in real
fraction contexts such as dividing pizzas, pies and cakes.
6
When using rectangles, it is a good idea to have a ‘base’ rectangle the same size
as the rectangle that is to be folded or cut, so students always have an image of
the whole to hand.
When using circles and naming them as pizzas, be aware that bought pizzas
have the added complexity of being delivered already cut into slices, usually
eight. This feature can be exploited at times, or students can be asked to imagine
the pizza before it is cut.
Students’ own problems
At all stages, students can pose fraction problems, including oral and written
word problems, and problems for investigation. In this way, students explore their
own interests, develop their understandings, and reveal their levels of
understanding of fraction concepts and language.
Also, students are highly interested in working on problems posed by their peers,
and like to challenge their classmates with increasingly more sophisticated
problems, thus building their own understanding and skills.
Suggested teaching and learning activities
Following are several suggested teaching and learning activities, many of which
have been trialled with students K-6 in the St George District.
This symbol appears in lessons that have been trialled and the
‘likely student responses’ are actual responses from those lessons.
The syllabus Fractions and Decimals content covered in each lesson is specified.
Relevant outcomes from other strands are recorded; fractions tasks have a lot of
potential to integrate with other strands.
7
Early Stage 1
Outcome: Describes halves, encountered in everyday contexts,
as two equal parts of an object
8
Sharing a birthday cake
Content
K&S: Recording fractions of objects using drawings
WM: Explain the reason for dividing an object in a particular way
Materials
Paper and pencils
Ask students to draw a birthday cake cut into slices and describe why they are
dividing it the way they are.
Likely student responses include the following.
Three people are eating my cake. That’s ten pieces.
The pieces are the same size.
9
What is a whole?
Content
WM: Using fraction language in everyday situations
Materials
None
Likely students responses include the following.
A big round circle you dig with a spade
Wombats can dig a hole
Ask: What is ‘a whole cake’?
Likely student responses include the following.
A round shape
You’ve got all of it
A cake that hasn’t been eaten
A cake that’s not a half cake
Ask: What does ‘our whole class’ mean?
Likely student responses include the following.
Everyone
All of us
Students can participate in ‘collecting’ holes and wholes to develop an
understanding of the meanings of, and the difference between, the two words.
10
What do you do if 2 people want to share a cake?
Content
K&S: Sharing an object by dividing it into two equal parts
WM: Describe how to make equal parts
Materials
Paper and pencils
The following tasks focus on sharing food items that are different shapes.
Ask: If you have a cupcake and you want to share it with a friend, how could you
do it?
Students can describe what they could do.
Students can draw what they could do.
Ask: If you have a jelly snake and you want to share it with a friend, how could
you do it?
Students can describe what they could do.
Students can draw what they could do.
Ask: If you have a slice of bread and you want to share it with a friend, how could
you do it?
Students can describe what they could do.
Students can draw what they could do.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
11
What is a half?
Content
WM: Using fraction language in everyday situations
Materials
None
Constructing definitions is an important mathematical activity. Students explore
both the concept of a half and the associated language.
Likely student responses include the following.
It’s not a full thing, it’s a part of it.
It’s like a birthday cake with a cut down the middle.
12
Has this shape been divided into halves?
Content
K&S: Recognising that halves are two equal parts
K&S: Recognising when two parts are not halves of the one whole
Materials
Circles and rectangles with lines dividing them into two parts, some with equal parts and some
with unequal parts (see below)
Ask: Are these shapes divided into halves?
Students are likely to say the first and third rectangles are divided
into halves, and the middle one is not, explaining that the line must
be in the middle to make the two sides the same size.
Students are likely to say the first circle is divided into halves, and
the middle one is not, explaining that the line must be in the middle
to make the two sides the same size. They are likely to say the third
circle is not divided into halves because the line has to be straight
down.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
13
How many circles?
Content
K&S: Recognising that halves are two equal parts
Materials
12 semicircles, all the same size
With the students sitting in a circle on the floor, place the semicircles in a pile on
the middle, and ask: What are these?
Likely student responses include the following.
They’re like circles.
They’re half circles.
They’re circles cut in half.
They’re semicircles.
Ask: Can we use these shapes to make circles?
Then ask: How many circles do you think we can make with these shapes?
Because the semicircles are in a pile, this task requires students to make
estimates of the answer. Ask some students to explain how they worked out their
estimates.
Spread the semicircles out and ask the students again: How many circles can we
make? Ask some students to explain their strategies.
On another occasion, ask students to work out how many circles they could
make if they had say 14 half circles, and ask them to show on a sheet of paper
how they worked out their answer.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
14
How much is left?
Content
K&S: Recording fractions of objects using drawings
WM: Explain the reason for dividing an object in a particular way
Materials
Paper and pencils
Ask: If I have a cupcake and I eat half of it, can you draw a picture showing how
much of the cupcake is left?
When students have drawn their pictures, ask: Why did you draw it this way?
Likely student responses include the following.
The straight line is where you cut the cupcake and the curvy line is
the edge of the cake.
It’s half the cake.
It’s a semicircle.
On other occasions the students could be asked to draw other objects that can
be represented with half circles such as half a pie or a half moon, or they could
be asked to draw what is left after half a pizza has been eaten.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
15
Drawing half circles
Content
K&S: Recording fractions of objects using drawings
WM: Explain the reason for dividing an object in a particular way
Materials
Paper and pencils
This is a follow-up task for students who did not draw semicircles in response to
the previous task, How much is left? See the work samples from Ilina, Sophie
and Rana. In some cases, the students might not be able to conceptualise the
solution (eg possibly Rana), and in other cases the students may be limited by
their inability to draw a semicircle (eg possibly Ilina).
Ask the students to draw circles. Can they do this successfully?
Give each student a circle cut from paper and ask them to show where they
would draw a line to divide their circle in half, and explain why they would draw it
there. If they are successful, ask them to draw the line.
Next they can cut along the line. Ask them to describe the shapes they now
have.
Finally, get them to draw one of the shapes.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
16
Investigating folding and cutting shapes in half
Content
K&S: Sharing an object by dividing it into two equal parts
K&S: Recognising that halves are two equal parts
Materials
Paper shapes – rectangles (oblongs and squares), circles, diamonds; scissors
Set students the task of investigating what happens when you fold and cut a
shape in half.
If they don’t include folding and cutting a rectangle along a diagonal, lead them to
doing so by giving them a square and asking them if there is a way to fold it other
than from one edge to the opposite edge. Then give them other rectangles to try.
They might now fold these along a diagonal or from one corner to the opposite
corner, both of which produce interesting halves.
Folding allows students to see the shape divided while remaining whole, and
cutting allows them the opportunity to deconstruct the shape, match the two
halves by overlapping, and reconstruct the original shape. It is particularly useful
to be able to cut out and match the two halves in the case of an oblong halved
along a diagonal because the two halves don’t match when folded.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
17
Investigating folding and cutting triangles in half
Content
K&S: Sharing an object by dividing it into two equal parts
K&S: Recognising that halves are two equal parts
K&S: Recognising when two parts are not halves of the one whole
Materials
Paper triangles (equilateral, isosceles and scalene), scissors
Set students the task of investigating what happens when they try to fold and cut
a triangle in half.
This activity will help students extend their understanding of what is a triangle,
because they will be working with different types of triangle.
They should find that only equilateral and isosceles triangles can be folded and
cut into matching halves. Ask them to talk about the shape of each half.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
18
What is the shape?
Content
K&S: Recognising that halves are two equal parts
Materials
Shapes (oblongs, squares, circles, triangles) cut from paper and folded in half
This activity helps students visualise shapes and sizes.
Show each folded shape in turn and ask: If this is half a shape, what is the whole
shape?
When students have given their suggestions and explanations, open up the
shapes.
Other outcomes covered
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
19
Covering half the area of a rectangle
Content
K&S: Recognising that halves are two equal parts
Materials
Large rectangles drawn on paper; assorted coloured cardboard rectangles, each one being half
the area of one of the large drawn rectangles
Give students a large rectangle drawn on paper, and ask them to find a
cardboard rectangle that covers half its area.
Other outcomes covered
MES1.2: Describes area using everyday language and compares areas using direct comparison
SGES1.2 Manipulates, sorts and describes representations of two-dimensional shapes using
everyday language
20
Filling and half-filling containers
Content
K&S: Recognising that halves are two equal parts
Materials
Clear plastic containers such as cylindrical jugs; plastic cups; water
Ask students to estimate how many cupfuls of water it will take to fill a specific
container. Then ask them to estimate how many cupfuls will half-fill the same
container. They should explain their strategies for making their estimates.
They can investigate filling and half-filling containers with water and record (in
drawing and writing) what they find out.
They can discuss what happens when they half-fill a cylindrical jug and a tapered
jug.
Other outcomes covered
MES1.3: Compares the capacities of containers and the volumes of objects or substances using
direct comparison
21
Sharing a collection of objects
Content
K&S: Recognising that halves are two equal parts
Materials
Collections of up to 20 objects (eg oranges, marbles, blocks, counters)
Give students a collection of up to 20 objects and ask: If you share these so you
and a friend get half each, how would you do it? Allow them to use the objects to
demonstrate their strategies.
This task gives students the opportunity to transfer their understanding of the
concept of ‘half’ to a collection of objects. They can also be given sharing tasks
to work out mentally, such as the following.
Show students a small number of objects (up to 10) and ask: How many objects
you would get if you were given half of these? They should explain their
strategies.
Show up to 30 counters in a pile and say: Look and think about how many
counters you would get if you were given half of these. Again, they should
explain their strategies. Estimation tasks such as this help build students’
reasoning skills and concepts of number relationships.
Other outcomes covered
NES1.3: Groups, shares and counts collections of objects, describes using everyday language
and records using informal methods
22
Stage 1
Outcome: Describes and models halves and quarters, of objects
and collections, occurring in everyday situations
23
Which piece would you like?
Content
WM: Explain why the parts are equal
Materials
Two identical oblong chocolate slices; a knife; two identical oblong pieces of cardboard that
measure 4 counters by 6 counters
Show students the two chocolate slices and establish they are the same size.
Cut one slice lengthwise and the other one crosswise and put one of each on a
plate.
Ask: Which one would you prefer to have or doesn’t it matter? Why?
Likely students responses include the following.
I want this one because it’s longer.
I want this one because it’s fatter.
Show the two cardboard oblongs, establish they are identical, then cut one
lengthwise and the other crosswise. Ask: Which one is bigger, or are they the
same size?
Pick up a crosswise half and ask: How many counters do you think it will take to
cover this shape? Place counters on the shape until it is covered. Repeat with a
lengthwise half. Guide students to discuss why 12 counters cover each of the
halves. Likely students responses include the following.
They’re the same size. Hey, they must be because they’re both
halves.
One’s fatter and one’s longer but they’re both halves.
Other outcomes covered
MS1.2: Estimates, measures, compares and records areas using informal units
24
What is a half?
Content
WM: Explain why the parts are equal
Materials
None
Likely student responses include the following.
It’s something that’s got a line down the middle and there’s a part
on this side of the line (gesturing to the right) and a part on this side
of the line (gesturing to the left) and both parts are the same size.
25
Black and white
Content
WM: Explain why the parts are equal
Materials
Cards with rectangles which are half black and half white (see below)
Show each card in turn and ask: How much of this shape is black and how much
is white?
This task should help students understand that ‘a half’ doesn’t have to be just
one region of a whole, nor is it a standard rectangular shape.
Students can be set the task of making a rectangle and colouring it half one
colour and half another colour, in an interesting way. They should justify their
pattern, explaining how they know it is half one colour and half the second colour.
At another time, they can be given a square of paper marked out in squares (say,
16 or 36 squares) and asked to colour the paper to make a design that is half
one colour and half another colour, using the square grid as a guide. They could
be asked to make the design symmetrical, as another challenge.
Other outcomes covered
SGS1.2 Manipulates, sorts, represents, describes and explores various two-dimensional shapes
26
Comparing lengths
Content
WM: Visualise fractions that are equal parts of a whole
Materials
2 lines drawn on cardboard – one line 6 counters long and the other 12 counters long
With students sitting in a circle, put out a cardboard sheet with the shorter line
drawn on it and ask: Look and think… how many counters will fit along this line?
Once the students have made their estimates, begin putting counters along the
line, and allow the students to change their estimates as the counters are placed.
Pause after three counters are on the line and ask if they can now work out how
many will fit along the line. Likely responses include the following.
Three more, because the bit of line that’s left is the same as the
covered bit.
Six, because you’ve covered half the line.
Cover the line. Next, put out the longer line and ask students to think about how
many counters will fit along it. Likely responses include the following.
10, because it’s longer than the other line.
12, because it’s double the other line.
12, because the other line is half the long line.
Other outcomes covered
MS1.1: Estimates, measures, compares and records lengths and distances using informal units,
metres and centimetres
27
Cutting holes
Content
WM: Visualise fractions that are equal parts of a whole
Materials
Sheets of paper; scissors
Show the students that you are folding a sheet of paper, then cut a rectangular
hole on the fold.
Ask: What shape will the hole be when I unfold the sheet of paper?
Ask students to fold a sheet of paper and cut a hole that will be a specific shape
(eg circle, square, triangle) when they unfold the paper. Ask them to explain why
they think their hole will be the shape specified.
Likely student responses include the following.
Mine will be a circle because I’ve cut out a curve.
It will be a circle because I cut a semicircle.
Mine is a square because I cut out half a square.
I think mine will be a triangle because I cut out a triangle…it’s not,
it’s a diamond.
(The triangle is quite a challenge.)
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
28
Is it a half?
Content
K&S: Describing parts of an object or collection as ‘about a half’, ‘more than a half’ or ‘less than a
half’
Materials
Counters
Cards showing black and white circles
Put about 20 blue counters and about 20 yellow counters in a clear plastic jar or
in a pile on the floor. Ask: Are about half the counters blue, or more than half, or
fewer than half? Why do you think that?
Vary the proportions of the two colours and ask the same questions.
Show cards such as the following and ask similar questions.
29
Sharing Problems
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Problems written on sheets of paper and put up on display (Masters follow)
Share 9 marbles between 2 people.
Share 9 cupcakes between 2 people.
If 9 people are driving to the beach in 2 cars, how many people should travel in each car?
These are all problems involving division of 9 by 2, but each one is answered differently. The first
answer is 4 marbles each with one left over; the second answer is 4½; the third answer is 5 in
one car and 4 in the other.
Another set of problems could be as follows.
Share 21 marbles among 4 people.
Share 21 jelly snakes among 4 people.
If 21 people have 4 tables to sit at, how many people should sit at each table?
Display the problems on a classroom wall. When students have worked out
explain what they did.
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
30
Dividing a cake into equal slices
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Paper and pencils
Form the students into different size groups of between 3 and 6, and ask each
group to draw a round birthday cake (top view) and divide it into enough slices for
each person in the group to have the same size slice, with no cake left over.
Likely student responses include the following.
It’s quarters
It’s four halves
It’s one four half
We started with a dot in the middle and
we drew the lines. We just looked and
thought where the lines would go.
We drew the line to cut it in half then we
worked out there would be three slices
in each half of the cake.
They don’t look right.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
31
Dividing a pie into three equal slices
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Paper and pencils
It is important that students explore fractions other than halves and quarters in
order to build the understanding that halves and quarters are not the only
fractions. If we limit students’ experiences to halves and quarters, some students
might think that any fraction that is not a half is called a quarter.
Ask students to draw a circle on a sheet of paper, imagine it is the top view of a
pie, and draw lines to divide the pie into 3 equal slices with none left over.
Encourage them to work out where their lines will go before they draw them; they
could use pencils or popsticks to show where the lines could go.
Likely student responses are shown in the photographs overleaf.
The students can also explore pies that are in the shape of a square, an oblong
and an equilateral triangle.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
32
Three Year 1 boys worked together to divide a ‘pie’ into three equal slices. The
following four photos show the progress of their thinking.
33
How old is the person having the birthday?
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
‘Birthday cake slices’ (one being a cardboard 3D model of a slice of cake with a candle in it; the
others being cardboard segments of circles, with a mark on each showing where a candle would
go); paper and pencils
Show the 3D model of the birthday cake slice and explain that the mark on the
cake is where a candle was, and the candles were equally spaced around the
cake. Ask: How old is the person having the birthday? How do you know?
Give pairs of students 2D ‘slices’ and ask them to work out the age of the person
having the birthday. Have paper and pencils available, because students often
use the strategy of drawing around the slice enough times to create a cake.
This person is 24. We did 4 slices and that’s 12 candles because
it’s 3, 6, 9, 12, and it’s half the cake so we doubled it.
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
34
In 20 seconds
Content
WM: Use fraction language in a variety of everyday contexts
Materials
None
Tell the students: In 20 seconds I can write the word ‘happy’ 8 times. Ask: How
many times would I write it in 10 seconds? Why? Do they use the word ‘half’ in
their explanations?
Ask: How many times can I write ‘happy’ in 15 seconds?
The students can time themselves writing words or numbers, or doing other
actions such as clapping. They can pose problems for their classmates based on
their results.
Other outcomes covered
MS1.5: Compares the duration of events using informal methods and reads clocks on the half-
hour
35
Fat Cat and Little Kitten Problems
Content
K&S: Modelling and describing a half or a quarter of a collection of objects
Materials
Problems, perhaps recorded in booklets, based on two characters, a fat cat and a little kitten. The
fat cat eats twice as much as the little kitten.
My little kitten ate 3 pieces of fish. My fat cat ate twice as many pieces as my little kitten. How
many pieces of fish did my fat cat eat?
My fat cat ate 10 pieces of meat. My little kitten ate half as many pieces as my fat cat. How many
pieces of meat did my little kitten eat?
My fat cat ate 5 fish. My little kitten ate half as many fish as my fat cat. How many fish did my little
kitten eat?
My little kitten ate 1½ cans of cat food. My fat cat ate twice as many cans as my little kitten. How
many cans of cat food did my fat cat eat?
My fat cat ate 11 sardines. My little kitten ate half as many sardines as my fat cat. How sardines
did my little kitten eat?
Read the problems to the students and ask them to explain their strategies for
They could work with materials, use drawings or work mentally to work out the
Students could write their own Fat Cat and Little Kitten problems.
36
Measuring
Content
WM: Use fraction language in a variety of everyday contexts
Materials
Large paperclips, classroom objects
Have a pile of paperclips on show. Hold up something like a pencil that is about
3½ paperclips long and ask students how many paperclips they think would fit
along it. Line up paperclips alongside the pencil and ask how many paperclips
long is the pencil.
Note that some students might say it is 4½ paperclips long, counting the fourth
Students can be set tasks involving half-unit measurements, such as
constructing a cylinder that is 5½ paperclips long.
Other outcomes covered
MS1.1: Estimates, measures, compares and records lengths and distances using informal units,
metres and centimetres
37
Halfway and quarter way
Content
WM: Use fraction language in a variety of everyday contexts
Materials
Pictures of paths, lines, hills
Show a picture of a hill, a path or a line and ask questions such as the following.
Where would a person be if they walked halfway along this path?
Halfway up this hill?
Run your finger along the line and stop when you get to the halfway point.
Where would a person be if they walked a quarter of the way along this path?
A quarter of the way up this hill?
Run your finger along the line and stop when you get a quarter of the way.
Other outcomes covered
MS1.1: Estimates, measures, compares and records lengths and distances using informal units,
metres and centimetres
38
A one-handed clock
Content
WM: Use fraction language in a variety of everyday contexts
Materials
A model of a clock made with only the hour hand
Show the one-handed clock with the hand pointing to the four, ask the students
to work out what the time would be, and ask them to explain their thinking.
Likely student responses include the following.
It’s four o’clock because the hand is pointing to the four.
Move the hand, with the students watching its clockwise movement, and stop it
when it points to another number. Ask the students to work out what the time
would be. Repeat several times.
Move the hand and stop it midway between two numbers and ask the students to
work out the time and explain their thinking.
Likely student responses include the following.
It’s half-past seven because it’s halfway between the seven and the
eight.
Half-past seven because it’s halfway past the seven.
It’s seven thirty because it’s halfway past the seven.
Other outcomes covered
MS1.5: Compares the duration of events using informal methods and reads clocks on the half-
hour
39
How many circles?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
24 quarter circles, all the same size
With the students sitting in a circle on the floor, place the quarter circles in a pile
on the middle, and ask: What are these?
Likely student responses include the following.
They’re shapes.
They’re cake shapes.
They’re semicircles.
They’re kind of like a puzzle; you put shapes together.
They’re quarter circles.
Ask: Can we use these shapes to make circles?
Then ask: How many circles do you think we can make with these shapes?
Because the quarter circles are in a pile, this task requires students to make
estimates of the answer. Ask some students to explain how they worked out their
estimates.
Count the quarter circles, then put them away and ask the students to work out
how many circles they could make with 24 quarter circles. Ask them to show on
paper how they worked it out. Likely students responses are shown overleaf.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
40
Likely student responses to the task of working out how many circles can be
41
Has this shape been divided into quarters?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Circles and rectangles with lines dividing them into four parts, some with equal parts and some
with unequal parts (see below)
Ask: Are these shapes divided into quarters?
Students often say the third rectangle above is divided into quarters without
questioning the fact that the four triangles are not identical. While each triangle is
actually a quarter of the area of the rectangle, this needs to be explored.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
42
Are these quarters?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
An oblong divided by two diagonals
Show students an oblong divided by two diagonals.
Ask: Has this rectangle been divided into quarters? How do you know?
Discuss the two different triangles they can see.
Give pairs of students photocopies of the rectangle and ask them to work out if
the two different triangles have the same area.
Other outcomes covered
MS1.2: Estimates, measures, compares and records areas using informal units
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
43
What is a quarter?
Content
K&S: Modelling and describing a half or a quarter of a whole object
K&S: Modelling and describing a half or a quarter of a collection of objects
Materials
None
Constructing definitions is an important mathematical activity. Students explore
both the concept of a quarter and the associated language.
Work with students to produce a definition of a quarter that reflects their current
level of understanding and use of language, write it on a poster and display it on
a classroom wall. Return to it from time to time to refine it.
44
What shape can a quarter of a square be?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Paper and pencils
Students can investigate the different ways that a square can be divided into
quarters.
If they produce a limited range of responses, ask them to recall the different ways
a square can be divided into halves, and base their divisions into quarters on
those. Then they might produce solutions such as the following.
This is an appropriate context to introduce the terms trapezium and quadrilateral.
Students could investigate the following question: Does every quarter of a square
have a square corner?
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
45
Has this rectangle been divided into quarters?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Card with the diagram below. The rectangle should be made so that each quarter can be
measured by counters eg the card could be 12 counters by 8 counters, so that the longer
quarters measure 3 by 8 counters and the wider quarters measure 6 by 4 counters.
Show the following card and ask: Has this rectangle been divided into quarters?
This task focuses on the concepts of matching and non-matching quarters.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
46
Halving regular polygons
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Templates of regular polygons (pentagon, hexagon, heptagon, octagon, nonagon, decagon) for
students to trace around
Students can investigate how regular polygons can be halved. They should find
that regular polygons with an even number of sides can be halved by a line
drawn from corner to corner or from the a point on one side to a point on the
opposite side. Regular polygons with an odd number of sides can only be halved
by a line joining a corner to the midpoint of the opposite side.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
47
Quartering regular polygons
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Templates of regular polygons (pentagon, hexagon, heptagon, octagon, nonagon, decagon) for
students to trace around
Students can investigate if regular polygons can be divided into matching
quarters.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
48
Quarters game
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Dice marked with ¼, 2/4, ¾, 4/4 and two blank faces; paper and pencils
Students play in pairs, taking turns to throw the dice and attempting to be the first
to complete five circles. They draw what they throw, so if they throw ¾, they draw
three-quarters of a circle.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
49
Quarter hours on a one-handed clock
Content
WM: Use fraction language in a variety of everyday contexts
Materials
A model of a clock made with only the hour hand
Show the one-handed clock and move the hand until it is a quarter of the way
past a number. Ask the students what time they think it is, and why they think so.
Likely student responses include the following.
It’s not half-past four because it’s not halfway after it.
It’s quarter-past I think.
It might be ten minutes past four or 20 minutes past four or
something like that.
Is it four and a quarter?
The students should discuss the different responses.
Other outcomes covered
MS1.5: Compares the duration of events using informal methods and reads clocks on the
half-hour
50
Collecting data
Content
K&S: Modelling and describing a half or a quarter of a collection of objects
Materials
Paper and pencils
Form students into groups of 8 or 12. Direct them to find a food that only half the
group likes and a food that only a quarter of the group likes, and to show the
information in graphs.
Other outcomes covered
DS1.1: Gathers and organises data, displays data using column and picture graphs, and
interprets the results
51
How much of the shape is black?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Cards showing circles or rectangles partly coloured black (see below)
Show a card and ask: How much of this shape is black? Students should explain
their thinking.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
52
How many circles?
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
18 thirds of circles, all the same size
With the students sitting in a circle on the floor, place the circle thirds in a pile in
the middle, and ask: What are these?
Likely student responses include the following.
They’re quarters.
They’re big quarters.
Ask: Can we use these shapes to make circles?
Then ask: How many circles do you think we can make with these shapes?
Because the circle thirds are in a pile, this task requires students to make
estimates of the answer. Ask some students to explain how they worked out their
estimates.
Count the thirds, then ask the students if they can work out how many circles can
be made, and explain their strategies.
Other outcomes covered
SGS1.2: Manipulates, sorts, represents, describes and explores various two-dimensional shapes
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
53
Which is bigger – a quarter or a third?
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Several quarters and thirds of circles of different sizes; use different coloured cardboard for
different size circles from which to make the circle sectors, so that, say, a green third and a green
quarter are from the same circle and a red third an a red quarter are from a larger circle, thus
allowing comparisons among different size quarters or between a third and a quarter from the
same circle
Ask students: Which is bigger – a quarter or a third? They should give reasons
Put out the quarters and thirds of circles and ask them to discuss the sizes of the
pieces.
Likely student responses include the following.
Thirds can be bigger and quarters can be bigger.
It depends on how big the circle is.
A red third is bigger than a red quarter and a green third is bigger
than a green quarter.
Ask questions such as: Would you rather have a third of this finger bun or a
quarter of the bun? Why? When would I cut it into thirds and when would I cut it
into quarters? What if I shared it among five people; would the pieces be larger
or smaller?
54
Sharing a blackbird pie
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Paper and pencils
Students often find it easier to divide an object (such as a pie) into fractions than
a collection, so this activity is valuable because it links finding fractions of a
collection to fractions of an object.
Students begin by dividing a pie into halves and quarters. Discuss the blackbird
pie in ‘Sing a Song of Sixpence’ which has 24 blackbirds.
Ask: If there are 24 blackbirds in a pie, how many would be in each half of the
pie, if the blackbirds are shared equally between the two halves?
How many blackbirds would there be in a quarter of the pie, if the blackbirds are
shared equally among the four quarters?
Students can investigate how many different ways the blackbird pie can be
divided so each slice has the same number of blackbirds.
Likely student responses are overleaf.
This type of activity helps build the understanding that halves and quarters are
not the only fractions.
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
55
Likely student responses to the task of sharing a blackbird pie.
56
Sharing a plum pie
Content
K&S: Describing equal parts of a whole object or collection of objects
Materials
Paper and pencils
This activity is similar to ‘Sharing a blackbird pie’. It allows a focus on any
number as the number of plums in the plum pie can be varied.
Students can find halves and quarters of pies with 20 plums (or any other
number divisible by 4), with 18 plums (or any other number which will require 2
plums being halved), with 21 plums (or any other number requiring a plum to be
quartered), and perhaps with 23 plums.
Students can be set the task of investigating how many ways they can divide a
pie with, say, 30 plums, with a whole number of plums in each slice. This type of
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
57
What fraction is red?
Content
K&S: Modelling and describing a half or a quarter of a collection of objects
Materials
Counters
Give each pair of students eight counters - four red and four of a second colour.
Add eight more counters of the second colour to each collection so each pair of
students has four red counters and twelve counters of a second colour.
Ask students to explain the strategies they used.
58
How many oranges?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
Some oranges cut into halves; picture of several half oranges; card with ‘½’ written several times
Show students the oranges already cut into halves and ask: How many oranges
are there?
They should discuss their answers (because some students are likely to count
each half as a whole orange) and their strategies.
Show the picture and ask the same question.
Show the card with ½ written several times and ask: What would the total be if
we added all these halves? Some students may think the question is ‘How many
halves are there?’ so it is a good idea to ask students to explain what the
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
59
How many half oranges?
Content
K&S: Modelling and describing a half or a quarter of a whole object
Materials
None
Ask students questions such as: How many half oranges would there be if I cut
six oranges into halves?
This type of question can be recorded as 6 – ½, which reads as ‘6, how many
halves?’
Other outcomes covered
NS1.3: Uses a range of mental strategies and concrete materials for multiplication and division
60
Stage 2
NS2.4 Models, compares and represents commonly used
fractions and decimals, adds and subtracts decimals to two
decimal places, and interprets everyday percentages
61
Dividing a circle into equal parts
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
Large circle drawn on butchers’ paper; lengths of tape secured at the centre of the circle
Have students in a circle on the floor or around a table, and put out a large circle
drawn on paper, with its centre marked. Secure some lengths of tape at the
centre of the circle and ask the class: Which students should hold the end of the
four lengths of tape to divide the circle into equal parts?
Encourage the students to discuss what is happening.
Change the number of lengths of tape and repeat.
62
A Year 4 class used tape to divide a circle into equal parts. The students stood in
a circle around the table and worked out who should hold the end of each length
of tape in order to make the parts equal.
63
Dividing a cherry cake into equal slices
Content
K&S: Modelling, comparing and representing fractions
Materials
A circle drawn on paper with 30 ‘cherries’ (red counters) placed around the perimeter;
photocopies of the ‘cherry cake’ (Master follows)
Show students the model of the cherry cake and ask them to work out the
number of cherries on the edge. Ask: How could we use the cherries to help us
divide the cake in half?
Give pairs of students photocopies of the cherry cake and ask them to divide the
cake into more than two equal slices, with none of the cake left over. Do they use
the cherries as a guide?
Some work samples follow to show likely student responses.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
64
These students investigated the different ways they could divide the cake into
equal slices, with a whole number of cherries on each slice. This type of work
builds the concept of factors.
65
Dividing a cake into fifths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
A circle drawn on paper with 30 ‘cherries’ (red counters) placed around the perimeter;
photocopies of the ‘cherry cake’; paper and pencils
Ask students to draw a circle representing the top view of a cake and ask them to
divide the cake into five equal slices, with no cake left over. Ask students to
discuss their strategies for dividing their cakes.
Give each student a photocopy of a cherry cake and ask them if the cherries on
the edge of the pie would help them divide the cake into five equal slices. Again,
ask students to discuss their strategies for dividing the cakes.
66
Strategies for dividing cakes into equal slices
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
Card showing a circle divided into halves
When students discuss their strategies for dividing cakes in activities such as
‘dividing a cake into fifths’, they need their unsuccessful strategies to be
challenged.
Some students sometimes begin by drawing a line through the middle of the
circle, thus dividing it in half. They should discuss why this was unsuccessful for
dividing the cake into fifths. Show a drawing of a cake divided in half, and ask: If I
start by cutting the cake into halves, how many slices do you think I will be able
to divide the cake into if I make some more cuts?
Some students do not visualise, or cannot represent, the slices as lines radiating
from the centre of the circle.
They need to be asked to describe what they can see as a real cake is being cut,
and when lines are drawn from the centre of a circle to the perimeter.
67
Dividing a circle into thirds
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
Cards showing circles divided into three parts with a ‘Y’,
one with equal parts and some with unequal parts eg
When one student divided a ‘cake’ into fifths, he began by dividing it into thirds,
then he added two more radii. He said I did a Y first. He thought he had divided
the cake into five equal slices. There are three aspects here (equal parts, from
thirds to fifths, the ‘Y’) that need challenging at an individual level and at a whole
class level.
The class can discuss strategies for checking that the parts are equal. Possible
student responses include the following.
You could cut out one of the parts out and put it on top of the other
parts to see if they are the same size.
You could cut out all the parts and put them on top of each other.
You can just tell by looking carefully.
The students could work in pairs to discuss whether or not a circle that has been
divided into thirds could then be divided into fifths.
It’s not possible, because if you start with thirds and then divide
the thirds all in half, you’d get sixths.
Students often discuss the fact that a circle divided into thirds looks like a peace
sign, a Mercedes logo or a Y inside a circle. They need to be challenged to build
the understanding that the three lines are equally spaced around the circle. Show
them some circles with different ‘Y’ shapes drawn inside and ask them if each is
divided into thirds.
68
Dividing a cake into quarters
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
A circle drawn on paper with 30 ‘cherries’ (red counters) placed around the perimeter;
photocopies of the ‘cherry cake’; paper and pencils
Give each student a ‘cherry cake’ and ask them to divide it into quarters.
As they work, ask them how many cherries there will be on each quarter.
If they have unequal numbers of cherries on their slices, ask them if the cake is
divided into fair shares. Then ask them to try again to divide the cake into
quarters.
69
What shape is it?
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
Materials
About 15 cardboard isosceles triangles with sides 5cm, 5cm and 6cm and the angles being 72
degrees and 54 degrees (two) – mark the 72 degree angle in some way, eg with a coloured dot
Paper and pencils
Give each pair of students a cardboard triangle and ask: If this is a fifth of a
shape, what could the shape be?
The mark on the 72 degree angle will help students orient the triangle if they try
rotating, flipping or sliding it.
A related activity would be to give each pair of students an equilateral triangle
and ask: If this is a sixth of a shape, what could the shape be?
Other outcomes covered
SGS2.2a: Manipulates, compares, sketches and name two-dimensional shapes and describes
their features
70
How many marbles?
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100 by
extending the knowledge and skills covered to fifths, tenths and hundredths
WM: Pose questions about a collection of items
Materials
None
Ask students questions such as: If 10 marbles is one-fifth of my collection of
marbles, how many marbles do I have?
If 21 marbles is three-fifths of my collection, how many marbles do I have?
(This question is appropriate for students who are very familiar with the multiples
of 7.)
Students can pose problems of this type for their peers.
These types of questions can be rephrased as clues to the number of items in a
collection when students are briefly shown the collection and asked to estimate,
then calculate, how many there are. For example, show briefly a bag of 40
marbles or a card with 40 marbles pictured and ask: How many marbles do you
think I have in this bag? (Allow students to make an estimate.) I’ll give you a clue:
five marbles would be an eighth of my collection.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
71
Equivalent fractions
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by finding
equivalence among halves, quarters and eighths
Materials
Diagrams of a circle divided into quarters, a circle divided into eighths
Hold up the card showing a circle divided into quarters and ask: What can you
see?
After students have described the circle as being divided into quarters, ask: If I
cut the circle in half, what would you see? Cut the circle in half and hold up one
half.
Likely student responses include the following.
I can see a half of a circle.
A semicircle.
Two quarters.
Do the same with the circle divided into eighths.
Summarise what students are saying: So half a circle is the same as two
quarters of a circle and the same as four eighths of a circle. Is that right? … We
say that one-half and two-quarters and four-eighths are equivalent fractions.
What do you think ‘equivalent’ means? … Can you find any other fractions that
are equivalent to one-half?
72
Equivalent fractions - fifths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Diagram of a circle divided into fifths
Hold up the card showing a circle divided into fifths and ask: What can you see?
After students have described the circle as being divided into fifths, ask: If I cut
the circle in half, what would you see? Cut the circle in half and hold up one half.
Likely student responses include the following.
Half a circle.
Two fifths and another fifth cut in half.
Two and a half fifths?
Direct students to work in pairs to discuss what fractions equivalent to a half they
can see or imagine in the half circle.
Likely student responses include the following.
You can see there would be five tenths, because you can imagine a
line down the middle of each fifth to match this bit (pointing to the
tenth where the fifth has been cut in half).
Two and a half fifths is the same as five tenths.
73
Equivalent fractions game
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by finding
equivalence among halves, quarters and eighths
Materials
A game: Pizzas, using the following materials: 2 cardboard circles cut into halves, 2 cut into
1 1 2 1 2 3
quarters and 2 cut into eighths; a dice marked 2, 4, 4, 8, 8, 8
This is a game for two players, or teams of players. The players take turns to
throw the dice and collect the ‘pizza slices’ (circle sectors) matching the fraction
thrown. The first player or team to make two pizzas wins. Pizzas can be made
with non-matching slices.
A variation on the game is to complete two pizzas, each one having identical
slices.
Both variations of the game involve students in exchanging slices, so help
develop equivalent fraction concepts.
Similar games can be made with fifths, tenths and twentieths, and with thirds,
sixths and twelfths.
74
Unequal fractions
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by finding
equivalence among halves, quarters and eighths
Materials
Diagram of a circle divided as follows: (It is divided into a half, a quarter, a sixth
and a twelfth.)
About 15 photocopies of the diagram
Show students the circle divided into unequal fractions and ask: Is this circle
divided into fractions?
Likely student responses include the following.
No, because fractions have to be the same size.
You know, like quarters are four parts the same size, so this can’t
be fractions.
No, they’re all different.
You have to have equal parts to have fractions, so there aren’t any
fractions in this circle.
Give each pair of students a copy of the circle and say: Have a careful look at the
circle. You might recognise some fractions there. Talk about what you can see.
Likely student responses include the following.
We can see a half over here, and a quarter.
They’re different sizes but they’re still fractions.
We know this is a half and this is a quarter and we think these ones
are a sixth and a twelfth.
75
Folding a paper strip
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
A4 paper cut into strips about 4cm wide
Give each student some paper strips and ask them to work out how they can fold
them to make halves, quarters and eighths. They should discuss the strategies
they are using.
Some students will be able to fold into halves but not into the other fractions. If
they do understand that there are four quarters in a whole and eight eighths, the
problem is with the folding and visualising. Allow these students to cut a strip into
halves, then ask them how they could make quarters and allow them to cut
again. If they are successful, help them line up the four quarters to reconstruct
the strip and ask them to describe where the cuts are. Give them another strip
and ask them to fold it into quarters, explaining at each step what they are
planning to do.
If there are students who do not understand there are four quarters in a whole
and eight eighths, they can be asked to ‘fold the strip into four equal parts’ and, if
they are successful with this task, into ‘eight equal parts’.
76
Cutting a paper strip into three equal pieces
Content
K&S: Modelling, comparing and representing fractions
Materials
A4 paper cut into strips about 4cm wide; scissors
Hold up one strip, fold it in half end to end, hold a pair of scissors as though
about to cut it in the centre between the ends and the fold, and ask: What would I
have if I cut this strip across here and opened it up?
Likely student responses include the following.
You’d have a long piece and two short pieces, because where the
its folded you open it up and it makes a long piece.
You’d have a half and two quarters.
Cut the strip and open it up to show the results. Then fold a second strip the
same way and ask students where it should be cut to get three strips the same
length. Give each student a strip of paper and some scissors, direct them to fold
the paper strip and not to open it again until they had made their cut. As they
work, ask them to explain their strategies for working out where to cut the strip.
They can measure, but without opening up the strip.
Likely student responses are overleaf.
77
```
78
Cutting a paper strip into fifths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
A4 paper cut into strips about 4cm wide; scissors
This is the same as the previous task, but students are set the task of working
out where to cut the folded strip to divide the strip into fifths.
Likely student responses follow.
79
80
Measuring area
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
Cardboard rectangles whose areas can be measured with counters, including half-counters (7 x
1½, 3 x 3½, 2½ x 4, 2½ x 2½); counters
Show students a rectangle (that is nine counters long by two counters wide) and
ask them to estimate how many counters it would take to cover it. Ask them to
demonstrate how they would work it out. Show another rectangle (6 counters
long by 3 counters wide) and repeat. Ask them if there are any other rectangles
that would hold 18 counters. They might suggest a rectangle measuring 18
counters by one counter. Ask if there are any others. They are likely to say no.
Give students the cardboard rectangles and direct them to work out how many
counters would fit on each one.
Ask them to show on paper how they worked it out.
Likely student responses are overleaf.
At the end of the lesson, return to the question of whether there are any other
rectangles that would fit 18 counters; some students might suggest a rectangle
that measures 36 counters by half a counter.
81
A student works out how many counters fit on this rectangle.
82
83
As an extension task, a pair of students was asked to draw a rectangle that
would fit 20¼ counters.
84
Half-filling a jug
Content
K&S: Modelling, comparing and representing fractions
Materials
Clear cylindrical jug, clear tapered jug the same height
Show the students the two jugs and ask them to describe what is different about
them. They should notice that one is a cylinder and the other is almost a cone.
Set the students the task of drawing the two jugs and showing them half-filled
with water.
Likely student responses include the following.
Most students are likely to draw the top of the water halfway up the vertical
height of the tapered jug, so draw this on the board and ask the students if it
looks correct. They will be likely to say it does. Leave the diagram there.
With the students watching, fill the cylindrical jug with cupfuls of water, then ask
the students to work out how to half-fill the jug. They might suggest pouring out
half the number of cupfuls.
85
Repeat with the tapered jug, and encourage the students to discuss why the jug
doesn’t look half-full when half the water has been poured out.
to suggest where the water will come up to when the jug is half-full.
Other outcomes covered
MS2.3: Estimates, measures, compares and records volumes and capacities using litres,
milliliters and cubic centimetres
86
Dividing rectangles into quarters
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
Paper rectangles; 3 cardboard rectangles that measure 12 by 8 counters, divided as follows:
Give each student some paper rectangles and ask them to divide them into
quarters in different ways. They can use both sides of the paper. Ask them to
record on each quarter how many counters they estimate would fit on it. Remind
them they can think of using half-counters so the whole surface is covered.
Likely student responses include the following.
11 22
16 16 12 12 12 12 11
22 22
11
16 16
22
11
22 21
26 26 27 27 27 27 22
21 21
22
26 26
21
22
Some students think there will be an odd number of counters in the quarters of
the last rectangle because ‘one counter will fit in the pointy bit’ of the triangle,
indicating the angle at the midpoint of the rectangle.
20 30
15 20 20 15 25 30 24
20 20
22
15 25
30
23
87
At the conclusion of the activity, when students are gathered together and
discussing what they did, put out the three cardboard rectangles divided into
quarters with the undivided sides showing and establish with the students that
they are all the same size. Turn them over and ask the students what they notice.
They should realise that the rectangles are divided into quarters in different ways.
Ask them to estimate how many counters will fit on each quarter. Then begin to
cover them with counters. The students will find that each quarter will fit the same
number of counters, and should discuss reasons for this.
In the photo below, note that instead of individual counters being used, there are
strips of photocopied counters, which makes the task quicker and supports the
development of multiplication and division concepts.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
88
How much of this pie has been eaten?
Content
K&S: Modelling, comparing and representing fractions
Materials
Pictures of divided ‘pies’ with one slice missing, for example:
Pictures of undivided ‘pies’ with one piece missing, for example:
Hold up each divided ‘pie’ in turn and ask: How much of this pie has been
eaten? Students are likely to say in each case that one piece has been eaten,
except when shown a pie with half missing when they will say that half has been
eaten.
Spread out the pies to which the students responded ‘one piece has been eaten’
and ask: Have the people who have eaten the slices of pie had the same amount
from each pie? Is there a way we could say how much of each pie has been
eaten that will show how big each slice is? What did you say when I showed you
this pie (show the pie with half missing)?
These questions should generate discussion of fractions generally.
Likely student responses include the following.
The bits that are missing are fractions.
You know how many slices there are so you can tell what the
fraction is.
The slices in each pie are the same size so it’s fractions.
Show the undivided pies with a piece missing to generate further discussion.
Likely student responses include the following.
You can sort of tell what the fraction is because you can imagine
where the other slices would be.
89
Sorting circle fractions
Content
K&S: Modelling, comparing and representing fractions
Materials
Several quarters, fifths and thirds of circles of different sizes
Ask students to sort the shapes and explain how they have sorted them.
Ask questions such as: Which is larger – a quarter or a third? A quarter or a fifth?
A fifth or a sixth? The last question will challenge students to extend their thinking
to fractions other the ones they can see.
90
Visualising thirds
Content
K&S: Modelling, comparing and representing fractions
Materials
3 sectors of circles, one being a third of a circle and the others being fractions that look similar to
a third (eg five-twelfths and seven-twentyfourths)
Hold up the three sectors and ask: Which one of these is a third of a circle?
This is a useful task because some students may think of a third of a circle in
terms of its shape rather than in terms of its size in relation to the circle.
It is also useful because it helps students build familiarity with thirds, which will
lead to visualising and conceptualising the fraction two-thirds.
As an extension task, the other fractions could be displayed in the classroom and
students set the challenge of working out what fractions of a circle they are.
91
Visualising tenths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Various sectors of a circle, including a tenth
Display the sectors and ask: Which one of these is a third of a circle? Why do
you think it is that one?
Give each group of students one of the sectors and set them the task of showing
on paper what fraction of a circle it is. They will probably draw around the sector
to make a circle, or half a circle.
Ask questions such as: Which is larger – a tenth or an eighth? Why? Which
fraction is twice as big as a tenth? What fraction do you think would be half as
large as a tenth? (Do not have a twentieth available – this question is to extend
their thinking beyond the fractions available.)
92
Investigating shapes that can be divided into
tenths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Templates of various shapes including circles, rectangles, pentagons, pentagrams and decagons,
and others that cannot easily be divided into tenths
Set students the task of investigating which two-dimensional shapes can be
divided into tenths.
Other outcomes covered
SGS2.2a: Manipulates, compares, sketches and names two-dimensional shapes and describes
their features
93
What fraction is black?
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
WM: Pose questions
Materials
Cards showing rectangles divided into tenths, twentieths, thirtieths and so on, with some of the
squares black and some white, as follow:
The last three cards have two sides,
with the grid not shown on one of the
sides.
Show one card at a time (showing the side without the grid) and ask: What
fraction of this rectangle is black?
Students should explain their thinking. Turn the cards over to show the grids to
help students check their ideas.
of five-tenths and one-half.
Set students the task of drawing their own shapes, colouring part and asking
their peers fraction questions about them.
94
Chocolate hundredths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
A ‘giant block of chocolate’ – a brown cardboard square, say 20cm by 20cm, marked into 100
squares in a 10 by 10 grid; several brown cardboard parts of a ‘block of chocolate’ such as half a
block, 99-hundredths and 30-hundredths
Show students the ‘block of chocolate’ with the grid showing, without calling it
that, and ask them what it is. Several students are likely to suggest it looks like a
block of chocolate.
Display the block of chocolate so the grid is not seen, hold up the half-block, and
say: This is what is left after some of the chocolate has been eaten. How much is
left? Students usually say that half the block is left. Turn the block of chocolate so
the grid can be seen and use it to check how much is left.
Show the 99-hundredths-block and ask: How much of this block has been eaten?
Students are likely to say that one piece has been eaten. Ask them: What
fraction of the block of chocolate has been eaten?
Show other part-blocks, asking what fraction of the block has been eaten or how
much is left, turning over the block so the grid is seen and using it to check the
students’ responses.
Likely student responses include the following.
(Being shown a five piece by five piece section of a block)
A quarter of the block.
Twenty five hundredths, because it’s five rows of five I think.
Teacher: So do we all agree it’s a quarter of the block?
(She turns it over to show the grid.) And is it twenty five
hundredths?
Twenty five hundredths is the same as a quarter.
95
96
100 echidnas
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
100 five-cent coins or large replicas of them
Put out the hundred coins or coin replicas and ask students to estimate how
many there are. Begin to count them into groups of ten and let the students
change their estimates as you do so.
Ask the students how they could be arranged in a pattern to see easily that there
are 100 coins.
Arrange them in a ten by ten array. If using enlarged photocopies of coin
pictures, perhaps glue them to a sheet of cardboard for display; to make it easier,
make photocopies of a line of ten coins, thereby being able to glue ten coins at a
time.
Ask: How much money is here altogether? How did you work it out?
When it is established that there is \$5, ask: What is a tenth of \$5? How do you
know? What is a hundredth of \$5? How do you know? How many 50c items
could you buy for \$5? How many 5c items could you buy for \$5?
The same activities can be done with ten-cent coins.
Then students could be asked: What if the coins were 20c coins, how much
money would there be altogether? What if they were 50c coins?
Later questions could be of the type: What is a tenth of \$8? What is a hundredth
of \$8?
97
Measurement relationships
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Metric measurement equipment
Set students in small groups the task of investigating the relationships between
centimetres and metres, including making models to show how long a tenth of a
metre is and how long a hundredth of a metre is. They can use materials such as
paper streamers, tape, metre rulers and cardboard strips. One group could be
set the task of making a visual representation for display.
Repeat this task for area, asking students to investigate and model a tenth of a
square metre and a hundredth of a square metre. They need a large space for
this, so could go outside and use materials such as string to peg out a square
metre. One group could be set the task of making a visual representation for
display.
Repeat the task for volume and capacity, using materials such as water, cubic
centimetre blocks, and cardboard to make models, to investigate tenths and
hundredths of a litre and relationships between MAB blocks measuring one cubic
centimetre and 1 000 cubic centimetres.
Other outcomes covered
MS2.1: Estimates, measures, compares and records lengths, distances and perimeters in metres,
centimetres and millimeters
MES2.2: Estimates, measures, compares and records the areas of surfaces in square
centimetres and square metres
MS2.3: Estimates, measures, compares and records volumes and capacities using litres,
millilitres and cubic centimetres
98
Measurement tenths
Content
K&S: Modelling, comparing and representing fractions with denominators 5, 10 and 100
Materials
Metric measurement equipment
Set students in pairs to investigate the relationship between tenths of a litre and
tenths of a kilogram. (There are 100g in a tenth of a kilogram and 100mL in a
tenth of a litre.)
Some students will benefit from using measurement equipment to investigate.
For example, they might pour a litre of water into ten cups, putting the same
amount in each cup, then working out, with support, how much is in each cup
and noticing it is about 100mL in each case. They might measure a kilogram in
smaller masses, say 10g masses, and divide the smaller masses into ten equal
groups to work out how much is in each tenth.
Other students will be able to work without equipment, simply using their
knowledge of the number of millilitres in a litre and the number of grams in a
kilogram.
Other outcomes covered
MS2.3: Estimates, measures, compares and records volumes and capacities using litres,
millilitres and cubic centimetres
MS2.4: Estimates, measures, compares and records masses using kilograms and grams
99
Farms
Content
K&S: Modelling, comparing and representing fractions
Materials
‘Farms’, A5 sheets of card divided into sections, for example:
Hold up one card at a time and explain that the card is a farm, with the lines
being fences breaking the farms up into fields. Ask: How many cows will there be
in each field if there are 36 cows altogether and they need to have the same
amount of grass each?
Likely student responses include the following.
You can work out halves first because one fence goes
across in the middle, so that’s eighteen up the top,
then it’s half again so it’s nine in the quarter one, then
you can work out how many go in the smaller bits, it’s
three and three and three.
Well, see how’s there’s kind of like six the same size
but a line’s missing, so you divide six into 36 and then
it’s six in each one and twelve in the big one.
You can do thirds and that’s twelve in the first field,
and six in the two fields at the end because six is half
of twelve, and six in the top middle field because it’s
the same as the one next to it, and three in the bottom
middle fields because three and three is six.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
100
101
Party
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils
Record the following sequence of numbers on the board and ask the students
what they notice: 1, 5, 9, 13, 17, 21 …
Likely student responses include the following.
It’s plus one to the multiples of four.
You started at one and then counted by fours after that.
Say: Imagine a party with four people sharing 1 pizza, 5 cupcakes, 9 sausages
and 13 jelly snakes. How much would each person have if they shared
everything fairly and there was nothing left over?
Likely student responses are overleaf.
Students should discuss what they have found out when they have completed
Likely student responses include the following.
The numbers are all in the sequence you wrote on the board.
Every answer is something and a quarter.
Because you have to divide the last thing into quarters, that’s why.
Related tasks are to share 1, 4, 7, 10 and 13 items (multiples of 3, plus 1) among
3 people, and so on.
102
103
Football fractions
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
Paper and pencils
Set each student the task of writing ‘a fraction problem’. Possibly the only
fractions represented in their problems will be a half, a quarter and three-
quarters, and this can result in a discussion of why this is so.
usually mention half-hour shows on television, clock times, pizza slices, cutting
toast. Most of the fractions they mention will be unit fractions, particularly a half
and a quarter.
Set the students to work in groups to draw a rugby league field, and ask them
what fractions they used when drawing. They will probably mention the half way
line. Ask them if there are any other ‘football fractions’, for example in the field
positions. They should be able to name the ‘half’, ‘three-quarters’ and ‘five-
eighth’. Ask them to talk in their groups about why these positions have these
names, and if it has anything to do with the field layout. They can use their
drawings of the field to think about and demonstrate what they know.
104
Mixing cordial
Content
K&S: Modelling, comparing and representing fractions
Materials
Clear cylindrical jug
Mark the jug into sixths up the side.
Pour concentrated cordial into the jug, up to the first mark. Add water up to the
fifth mark.
Ask: What fraction of the mixture is cordial and what fraction is water?
The amount of mixture can be measured in millilitres, and the amounts of cordial
and water worked out from that measurement.
Other outcomes covered
MS2.3: Estimates, measures, compares and records volumes and capacities using litres,
milliliters and cubic centimetres
105
Pizza sharing problems
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils
Pose the following problems and allow students to work with pencil and paper.
Ask the students: If you shared a pizza among four people, how much pizza
would each person get?
Likely student responses include the following.
Two pieces each because there are eight pieces.
A quarter.
Ask: Imagine I’ve made a pizza. I’m in the kitchen, about to slice the pizza. If I
share it among five people, how much would each person get?
Likely student responses include the following.
A fifth.
A sixth, if you’re having some as well as the other five people.
If you cut it into ten slices, they can have two slices each.
Ask: What if the pizza has to be shared between two people, but one of the
people is a very hungry adult and the other is a very young child?
Likely student responses include the following.
Cut it into thirds, two-thirds for the adult and one-third for the child.
Cut it into one-quarter for the child and three-quarters for the adult.
Ask: How could you share a pizza among two hungry adults and two very young
children?
Likely student responses include the following.
One-third for each adult and one-sixth for each child.
Three-eighths for each adult and one-eighth for each child.
106
Two students’ responses to the question: How could you share a pizza among
two hungry adults and two very young children?
107
Fraction sequences
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils
Begin by having in mind the number 2 048, and saying to the students: I’m
thinking of a four-digit number. Its digits are all even and they are all different. 6
is not one of the digits. The final digit is double the third digit, and the first two
digits add to 2. What number am I thinking of? The ‘clues’ can be written and put
on display.
When they have worked out the number, ask them to begin with 2 048, halve it,
halve the new number and keep repeating the process. They should generate
the sequence 2 048, 1 024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1 … Some
students may continue halving and add some fractions to the sequence. If not,
ask the students to keep on going once they reach 1. They will notice some
interesting patterns in the sequence and also learn about fraction relationships as
they investigate how to halve fractions.
Students can be set the task of creating other fraction sequences as an on-going
focus.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
PAS2.1: Generates, describes and records number patterns using a variety of strategies and
completes simple number sentences by calculating missing values
108
Yes/No game
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
None
Draw a vertical line on the board, creating two spaces, and write ‘Yes’ at the top
of one and ‘No’ at the top of the other.
Tell the students you are collecting certain numbers and they have to work out
which ones you are collecting. They take turns to suggest numbers, perhaps
within a specified range such as from 0 to 100, and you will record them in the
‘Yes’ space if they are being collected and in the ‘No’ space if they are not.
Numbers being collected might be even numbers, or multiples of 3, or two-digit
numbers, or single-digit odd numbers, or two-digit numbers ending in 6 and so
on.
In this case, specify the range from 1 to 20, and have in mind to collect mixed
numbers. The students will probably suggest all the whole numbers from 1 to 20
before someone realises mixed numbers as a possibility.
109
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8
Materials
Cards with three or four mixed numbers, two of which add to 10
For example:
8½ 6½ 5½
2½
1½ 4½ 2½
Hold up each card in turn and ask the students to work out which pair of numbers
add to 10, and explain how they know they are correct.
When students are familiar with the task and confident with mixed number
combinations of 10, ask them to work out the total of the numbers on the card.
What strategies do they use? Do they use their knowledge of the combinations of
mixed numbers of 10?
Likely student responses include the following.
8½ plus 1½ is 10, plus 2½ is 12½.
8½
2½ 8 and 2 is 10, plus 1 is 11, plus two halves is
1½ 12 and then one more half is 12½.
110
Multiplying mixed numbers
Content
K&S: Modelling, comparing and representing fractions
Materials
Cards with a mixed number repeated several times
For example:
3½ 3½
3½
3½
3½ 3½
3½
3½ 3½
3½
Hold up a card briefly and ask: If you added all the numbers, what would the total
be?
Students should explain and discuss their strategies.
Hold up the card again and allow students to work out the total. Again, they
should explain and discuss their strategies.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
111
Investigating multiplying mixed numbers
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils; coloured paper squares about 3cm squared
Begin to write the following sequence on the board, and ask students to
comment on what you are doing: 1½, 2½, 3½, 4½, 5½ …
Set them the task of investigating multiplying the numbers in the sequence by 5,
and ask them to use arrays to work out the sequence of answers. They can draw
arrays or use the paper squares to construct them, cutting them in half when
necessary and gluing them in place.
Ask the students to explain why they get the sequence of answers that they do.
Ask questions such as: What will be the hundredth number in the sequence of
Other investigations of this type can be undertaken, such as investigating
multiplying 2½ by the numbers in the sequence of even numbers, investigating
multiplying numbers in the sequence 1½, 2½, 3½, 4½, 5½ … by 10.
Other outcomes covered
NS2.3: Uses mental and informal written strategies for multiplication and division
PAS2.1: Generates, describes and records number patterns using a variety of strategies and
completes simple number sentences by calculating missing values
112
Number lines
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by placing
halves, quarters and eighths on a number line
Materials
Paper and pencils
Set each student the task of making their own number line, beginning and ending
with numbers of their choice, and making regular markings on their line.
Likely student responses include the following. They usually work
with numbers with which they feel confident. Some of them will
fractions and/or include mixed numbers. Occasionally a student
will include decimals, such as a girl whose number line (1.5, 2, 2.5,
3 and so on), she explained, was like a number line showing depths
at her local swimming pool.
10 20 30 40 50 60 70 80 90 100
1 1½ 2 2½ 3 3½ 4 4½ 5 5½
No matter which numbers a student has placed on their number line, ask each
them to add one more number somewhere on their line between the first and last
numbers they already have. That is, they cannot simply extend their line to
continue the counting sequence they already have. So the student who
constructed the second number line above would have to add a number like 1¼
or 4¾ .
113
Fraction number lines
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by placing
halves, quarters and eighths on a number line
Materials
Paper and pencils
Draw a number line on the board, showing the numbers from 0 and 10 at each
end, and leaving large spaces between them. Ask: Are there any other numbers
we could place on this number line?
Students should be able to add some fractions and mixed numbers to the
number line.
Set students the task of investigating showing fractions on number lines. They
should discuss their own number lines in comparison with those made by others.
They are likely to draw lines showing mixed numbers rather than fractions on a
line from 0 to 1.
Later, set them the task of investigating showing fractions on number lines with
the stipulation that the number lines begin with 0 and end with 1.
114
Equivalent fraction number lines
Content
K&S: Modelling, comparing and representing fractions with denominators 2, 4 and 8 by placing
halves, quarters and eighths on a number line
Materials
Paper and pencils
Set students in pairs the task of making a set of number lines that show
equivalence among halves, quarters and eighths. Tell them one line should show
halves, one should show quarters and one eighths.
They should realise they will need to make their number lines the same length. It
is preferable they are not told this, nor given photocopied number lines of equal
length, as they then won’t have the opportunity to think the task through for
themselves and in so doing come to understand an important aspect of
equivalence.
Some students could be challenged to extend the task to other fractions.
115
Stage 3
NS3.4 Compares, orders and calculates with decimals, simple
fractions and simple percentages
116
Thirds
There are several activities outlined in Stage 2 that are useful in developing
concepts of thirds.
Dividing a circle into equal parts
Dividing a cherry cake into equal slices
Dividing a circle into thirds
How many marbles?
Equivalent fractions games
How much of this pie has been eaten?
Sorting circle fractions
Visualising thirds
Farms
Party
Mixing cordial
Pizza sharing problems
Fraction sequences
Yes/No game
Multiplying mixed numbers
Investigating multiplying mixed numbers
Number lines
Fraction number lines
117
From thirds to other fractions
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
Photocopies of a circle with lines dividing it into thirds
Give each pair of students photocopies of the circle and ask them to investigate
what other fractions the circle could be divided into, using the lines already there.
Likely student responses include the following.
You could draw lines halfway through each third and make sixths.
And twelfths if you do the same again.
Or you could do ninths, you would draw two lines in all the thirds.
Any multiple of three.
118
Cutting a paper strip into thirds
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
A4 paper cut into strips about 4cm wide; scissors
Hold up one strip, fold it in half end to end, hold a pair of scissors as though
about to cut it in the centre between the ends and the fold, and ask: What would I
have if I cut this strip across here and opened it up?
Likely student responses include the following.
You’d have a long piece and two short pieces, because where the
its folded you open it up and it makes a long piece.
You’d have a half and two quarters.
Cut the strip and open it up to show the results. Then fold a second strip the
same way and ask students where it should be cut to cut it into thirds. Give each
student a strip of paper and some scissors, direct them to fold the paper strip and
not to open it again until they had made their cut. As they work, ask them to
explain their strategies for working out where to cut the strip. They can measure,
but without opening up the strip.
119
Cutting a paper strip into thirds, sixths and
twelfths
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
A4 paper cut into strips about 4cm wide; scissors
Set students the task of investigating what fractions result when they fold a strip
of paper lengthwise, then lengthwise again, and, leaving it folded, making a cut
one-third from its end. The ends are different (one having two folds and the other
having one fold) so students should work with two strips, cutting one third in from
the end with one fold on one strip, and one-third in from the end with two folds on
the other strip.
They will find thirds, sixths and twelfths.
They can produce reports that show in diagrams what they found out.
120
Equivalent fraction number lines
Content
K&S: Placing thirds, sixths or twelfths on a number line between 0 and 1 to develop equivalence
WM:Eplain or demonstrate why two fractions are or are not equivalent
Materials
Paper and pencils
Set students the task of making a set of number lines that show equivalence
among thirds, sixths and twelfths. Tell them one line should show thirds, one
should show sixths and one twelfths.
They should realise they will need to make their number lines the same length. It
is preferable they are not told this, nor given photocopied number lines of equal
length, as they then won’t have the opportunity to think the task through for
themselves and in so doing come to understand an important aspect of
equivalence.
Some students could be challenged to extend the task to other fractions.
121
Investigating shapes that can be divided into
thirds, sixths or twelfths
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
Templates of various shapes, including regular dodecagons. It will be useful for this investigation
Set students the task of investigating which two-dimensional shapes can be
divided into identical thirds.
Repeat for which two-dimensional shapes can be divided into identical sixths.
Repeat for which two-dimensional shapes can be divided into identical twelfths.
Other outcomes covered
SGS3.2a: Manipulates, classifies and draws two-dimensional shapes and describes side and
angle properties
122
Picnic
Content
K&S: Modelling, comparing and representing fractions
Materials
Paper and pencils
Record the following sequence of numbers on the board and ask the students
what they notice: 1, 4, 7, 10, 13, 16 …
Likely student responses include the following.
19, 22, 25 … Just add threes.
You started at one and then plussed three every time.
Every number is a multiple of three plus one.
Say: Imagine a picnic with three people to share the food. How would they share
one bread stick, four sausages, seven square slices of cheese, ten slices of
tomato and thirteen carrot sticks so they all had the same amount to eat and
there was nothing left over?
Students should explain their strategies for sharing the items and discuss their
results. Why did they get the results they did?
A later task would be to share 2, 5, 8, 11 and 14 items (multiples of 3, plus 2)
among 3 people, and compare the results with the results from this task.
123
When would you need to find a third?
Content
K&S: Modelling thirds, sixths and twelfths of a whole object or collection of objects
Materials
None
Ask students: When would you need to find a third of a collection of objects?
Likely student responses include the following.
If you have some marbles and your mum or dad makes you share
them with your brothers and if you’ve got two brothers.
When you and two friends want to share some food.
Ask: Which numbers of things are easy to find a third of?
Students should work in pairs on this question. Note that for some students it is
‘easy’ to find thirds of almost any whole number, whereas others will come up
with the answer that the multiples of three are ‘easy’ to find thirds of.
The same questions can be asked about sixths and twelfths, and later students
can be asked: Can you find a number that it is it easy to find thirds and sixths and
twelfths of? What other numbers is it easy to find thirds and sixths and twelfths
of?
124
Investigating shapes that can be divided into
sevenths
Content
Materials
Templates of various shapes, including regular heptagons. It will be useful for this investigation if
there are rectangles with linear dimensions that are multiples of 7.
Students have explored relationships among halves, quarters and eighths,
among fifths, tenths and hundredths, and among thirds, sixths and twelfths, so it
is useful for them to be challenged to transfer their conceptual understandings to
unfamiliar fractions.
Set students the task of investigating which two-dimensional shapes can be
divided into identical sevenths.
Ask: What other fractions can these shapes be divided into?
Other outcomes covered
SGS3.2a: Manipulates, classifies and draws two-dimensional shapes and describes side and
angle properties
125
Sevenths
Content
WM: Pose and solve problems
Materials
Paper and pencils
Ask: If you had enough food to last you a week, how much of the food would you
eat each day?
Ask: What if you had 28 slices of bread, how many would you eat each day of the
week if you had the same amount each day and had none left over at the end of
the week? 42 strawberries? 8 chocolate bars? 23 slices of cheese?
Students could pose some questions like this for their peers.
of the money would you need to keep for the weekend? If the amount you had
was \$35, how much money would you spend on the weekend?
126
Lots of quarters
Content
K&S: Expressing mixed numerals as improper fractions
Materials
Three cardboard circles (not divided); cardboard quarter-circles
Perhaps begin by reciting the following rhyme, Quarters, written by some Stage 1
students.
One quarter
Two quarters
Three quarters
Four.
And that’s where we stop
Because there aren’t any more.
Ask: Is it correct that there aren’t any more than four quarters?
Show five cardboard quarter-circles and ask: How many quarters can you see?
Show a whole circle (not divided) and ask: How many quarters can you see?
Show three whole circles and ask: How many quarters can you see?
Ask: How could you write twelve quarters? Is twelve quarters the same as four?
12 20
Could we write 4 = 3 ? How would we finish this one: 4 = ?
10
And this one: 4 = ?
Set students in pairs the task of investigating other fractions like twelve-quarters.
Tell them that fractions like this are called improper fractions, and ask them to
suggest why they might be called this.
Perhaps students could suggest an alternative last two lines to the rhyme above.
127
Fraction posters
Content
K&S: Expressing mixed numerals as improper fractions
K&S: Developing a mental strategy for finding equivalent fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Poster-size paper and pencils
In the centre of a poster-size sheet of paper record the number 3 and record two
improper fractions equivalent to 3 as follows.
12
4
3
9
3
Ask students to comment on what they see and ask them to suggest some other
improper fractions equivalent to 3 that could be written on the poster.
Set students in groups the task of making a similar poster with a number of their
choice, other than 3, in the centre.
They should discuss their strategies for working out the improper fractions.
128
A dozen eggs
Content
K&S: Developing a mental strategy for finding equivalent fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Egg-cartons, one marked by different colours in halves, one in quarters, one in thirds, one in
sixths and one in twelfths; cardboard ‘eggs’; poster-size paper and pencils
Have the egg-cartons on display and ask students to talk about what they see.
Lead them to notice that an egg-carton can be divided into different fractions.
Ask: How many eggs are in a dozen? In a half-dozen? In a quarter of a dozen?
In a third of a dozen? In a sixth of a dozen?
Ask questions such as: How many eggs are there in three-quarters of a dozen?
How many eggs are there in five-twelfths of a dozen?
Ask: How many eggs are there in six-twelfths of the dozen? What other fractions
of the dozen are 6 eggs?
The students can help make a poster for fractions equivalent to a half, beginning
with the fractions found with the egg-cartons, as follows. They can add other
equivalent fractions.
6
12
1
2
2
4 3
6
129
More fraction posters
Content
K&S: Developing a mental strategy for finding equivalent fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Poster-size paper and pencils
Students can work in groups to make posters for fractions equivalent to the
following familiar fractions: a quarter, a third, two-thirds and three-quarters.
130
Puzzling posters
Content
K&S: Developing a mental strategy for finding equivalent fractions
K&S: Reducing a fraction to its lowest equivalent form
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Poster-size paper and pencils; equivalent fractions posters made previously
Provide posters such as the following and set students the task of adding some
more equivalent fractions, including the one that belongs in the centre. Explain
that there is only one possible fraction that can be placed in the centre and ask
them to work out what it should be, including by referring to the posters they have
3 16
30 32
5
50 7
70 20
40
8
7 48
35
15
75 4 5
20 30
Students can make some of their own ‘puzzling posters’ for their peers to
complete.
131
Folding and cutting quarters
Content
K&S: Reducing a fraction to its lowest equivalent form
WM: Explain or demonstrate why two fractions are or are not equivalent
WM: Pose and solve problems
Materials
A4 paper cut into strips about 4cm wide; scissors
Ask students to fold a strip of paper lengthwise, then lengthwise again, and,
leaving it folded, making four small cuts at the edge of the strip at the one-
quarter, half and three-quarters spots.
Tell them to open up their strips and work out what fractions the small cuts and
the folds divide the strip into (sixteenths).
Set them the task of investigating what fractions result when they re-fold the
paper and make a cut one-quarter from an end. The ends are different (one
having two folds and the other having one fold) so students should work with two
strips, cutting one quarter in from the end with one fold on one strip, and one-
quarter in from the end with two folds on the other strip.
Ask students to name each fraction that results in its lowest form.
Students can pose their own paper folding and cutting problems.
132
Subtraction problems
Content
K&S: Using written, diagram and mental strategies to subtract a unit fraction from any whole
number
WM: Pose and solve problems
Materials
Paper and pencils
Pose problems such as the following.
If I had a bottle of drink and poured a third of it into a glass, how much would be
left in the bottle? To extend some students: If I poured half of what was left into
another glass, how much would be left in the bottle?
If I bought three chocolate bars and I ate half of one of them, how many bars
would I have left?
If I had eight socks and I lost one sock, how many pairs of socks would I still
have?
Show a picture of a shoe and ask: One shoe is missing; how many pairs of shoes
are left?
Students can pose similar problems
133
What can you see?
Content
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
A circles divided into sixths; photocopies of circles divided into fifths
Show the circle divided into sixths, cut off half so that students can see three-
sixths, and ask: What do you have if you cut off half of this circle?
They are most likely to tell you there is half left, or three-sixths.
Show a circle divided into fifths and repeat as above. Students might say there
are left, but it is likely they will not be sure they can call it ‘two-and-a-half fifths’.
Set them the task of investigating in pairs what fraction it could be called, other
than a half or two-and-a-half fifths.
Likely student responses are as follows, including overleaf.
134
135
Equivalent fraction problems
Content
WM: Explain or demonstrate why two fractions are or are not equivalent
WM: Pose and solve problems
Materials
Egg-cartons, one marked by different colours in halves, one in quarters, one in thirds, one in
sixths and one in twelfths; cardboard ‘eggs’; poster-size paper and pencils
Have the egg-carton and ‘eggs’ on display and pose problems such as the
following.
How many eggs would be left in a carton if three-twelfths of them have been
eaten? How do you know?
How many eggs would be left in a carton if a quarter of them have been eaten?
How do you know?
Why are the answers to the two questions the same?
And:
How many eggs have been eaten if four-sixths of the eggs are left?
How many eggs have been eaten if two-thirds of the eggs are left?
Students can pose similar problems.
136
Reducing a fraction to its lowest equivalent form
Content
K&S: Reducing a fraction to its lowest equivalent form
Materials
None
Ask questions such as the following.
Why do we hear the word ‘half’ in everyday talk, rather than four-eighths or ten-
twentieths or six-twelfths?
If I have 30 counters, 15 red and 15 blue, is it sensible to say ‘fifteen-thirtieths of
my counters are red’ or is it sensible to say ‘half my counters are red’?
How do you know the most sensible way to say a fraction?
Students can be set the task of investigating fractions that can be reduced to
one-quarter, or fractions that can be reduced to two-thirds. At the conclusion of
the investigation, students can explain their strategies for reducing any fraction to
its simplest form.
137
Problems involving fractions with the same
denominator
Content
K&S: Adding and subtracting fractions with the same denominator
WM: Pose and solve problems
Materials
Paper and pencils
Pose problems such as the following.
If I had a pizza, and I ate two-eighths and my sister ate three-eighths, how much
would be left for our brother?
If my three friends each had a pizza, and each friend gave me two-eighths of
their pizza, would that be fair?
If my sister and brother each had a pie, and my sister gave me a third of hers
and my brother gave me a third of his, would that be fair?
The students can discuss how they add and subtract fractions with the same
denominator.
Students can pose their own problems like the ones above. They might use
fractions with different denominators, so this will naturally lead to discussing and
investigating how they might handle such problems.
138
Adding and subtracting fractions with related
denominators
Content
K&S: Adding and subtracting simple fractions where one denominator is a multiple of another
WM: Pose and solve problems
Materials
Paper and pencils
Following on from students posing their own fraction problems, which might
involve a mix of fractions with different denominators, ask them to discuss which
groups of fractions are easy to add.
Set students in pairs to investigate adding and subtracting a mix of fractions.
Some could investigate a mix of halves, quarters and eighths, others a mix of
fifths and tenths, and others a mix of thirds, sixths and twelfths.
The students should gather together and share their findings about the different
groups of fractions, perhaps using diagrams and models to explain their thinking.
To transfer their thinking beyond the groups of fractions investigated, ask: What
fractions would be easy to add and subtract along with sevenths?
139
Which is closest to a half?
Content
K&S: Comparing and ordering fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Paper and pencils
Set students in pairs to work out which of the following fractions is closest to a
half: five-eighths, four-sixths, seven-twelfths.
They should explain their strategies as they work. They might use diagrams,
models and number lines.
After the students have completed the task, gather them together to share their
findings and the different strategies they used. If they did not use number lines,
suggest to them that using number lines is a strategy worth exploring.
Ask them to investigate how to use number lines to compare fractions and find
equivalent fractions.
You could consider having students help make a class set of number lines, all
the same length but showing different fractions.
140
Which fraction is the largest?
Content
K&S: Comparing and ordering fractions
WM: Explain or demonstrate why two fractions are or are not equivalent
Materials
Paper and pencils
Write the following fractions on the board and ask students to work out the
pattern: 2 3 4 5
3, 5, 7, 9
(Each denominator is twice the numerator, minus one; or each numerator is half
the denominator, plus a half.)
Ask them to think about which of the fractions is the largest, and set them the
Other outcomes covered
PAS3.1a: Records, analyses and describes geometric and number patterns that involve one
operation using tables and words
141
Problems involving unit fractions
Content
K&S: Calculating unit fractions of a collection
Materials
Paper and pencils
Pose problems such as the following.
I made some cakes and iced them in three different colours. One third of the
cakes have white icing. One-sixth have pink icing. 12 cakes have yellow icing.
How many cakes did I make?
How many pairs of footwear do I have if one-third of the pairs are sandals, one-
sixth of the pairs are thongs, one-twelfth of the pairs are slippers, and I have 5
pairs of shoes?
Some students will be able to investigate how to find non-unit fractions of a
collection, such as two-thirds of a collection. They can be set problems such as
the following.
I have a collection of large and small marbles. One-third of my marbles are red.
Five-sixths of my marbles are large. I have three small marbles. How many red
marbles do I have?
142
Multiplying fractions by whole numbers
Content
K&S: Multiplying simple fractions by whole numbers
Materials
Paper and pencils
Pose problems such as the following.
Eight children shared some jelly snakes. If each child got two and a half jelly
Six people shared some cakes. They ate one and a third cakes each, so how
many cakes did they have altogether?
Eight children shared some sausages. They were able to have two and a quarter
sausages each, so how many had they started with?
A bathroom floor was covered with large square tiles. There were eight rows of
tiles with nine and a half tiles in each row. How many tiles were used altogether?
143
Multiplying mixed numbers
Content
Materials
Squares of coloured paper; large sheets of paper; glue
Ask students to investigate what happens when they make arrays to work out the
answers to 1½ x 1½, 2½ x 2½, 3½ x 3½, and so on.
Likely student responses are as shown in the photograph below.
144
Adding and subtracting fractions with unrelated
denominators
Content
Materials
Paper and pencils
This is not a topic listed in the syllabus for Stage 3, but is an interesting topic to
extend students, and to observe how students apply their understanding of
fraction concepts.
Pose problems such as the following.
If I ate half a pie and my brother ate one-third of it, how much would be left?
145
146
Related docs
Other docs by HC120302125729
1086349211 Valmistuneet 04
FAIR EMPLOYMENT TRIBUNAL - DOC 1<|endoftext|>
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What is 'Look-Back Race'?
This is the first 'Question' mini-skill. Questioning, and all that goes with it, is vital to good comprehension. We introduce the question mega-skill as follows:
"Answering questions, which shows we can understand what we are reading, is really important. But also thinking up our own questions and discussing questions with our friends help us to make meaning. They are important too!"
So the recognition of both answering questions, but also asking questions of texts, is emphasised to learners.
Look-back works on answering questions effectively. The aim of the look-back race mini-skill is to introduce, hone and consolidate skimming and scanning, identifying key words in a question, and then finding these key words in a text along with the answer. This technique is often used to answer literal questions. As The Master says:
Don’t forget this is all about looking back in the text for answers. These are those literal questions, they answers are literally... right there!
Developing the ability to move effectively between question and text is incredibly useful for learners. Look-back is a strategy that empowers them to do this, and by structuring the activity as a game (the race element), learners enjoy and want to repeat the exercise. They aim to increase their speed and accuracy.
Assessment and look-back
Interestingly, there has been some debate around whether look-back should be allowed as a strategy when conducting comprehension assessments. One idea is that the option to look back may prevent us from truly understanding whether a learner comprehends the text. However, another view is that removing the text when answering assessment questions significantly increases reliance on memory - which is different, but related to, reading. Do you have any thoughts on this?
The vast majority of assessments allow learners to revisit the text that is being interrogated by the questions, so looking-back is essential in, for example, SATs style exams.
Look-back race in the Red Stone of Calcutta
The following image is an extract from The Red Stone of Calcutta.
As you can see, the initial task for the learner is to select 1 or more key words from the question. Excellent practice!
Once the key words have been selected, the learner then moves to the episode they are working on (the text on the left), using the key words to pick out the answer. As usual, learners are able to hear individual words, or sentences before making their selection.
The race element is represented by the timer, a relaxed, fun way for the learner to monitor their speed.
What do you think?
We'd love to hear your thoughts about look-back as a comprehension strategy, and ways that you've integrated this practice into your classroom. Have you seen anything special?
The ReadingWise Demo
Short, friendly and informative - you might like to arrange a demo with our team at a time to suit you.So far we've reached many thousands of learners and would love to explore working with you if it might help more children read.<|endoftext|>
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Dr Arun Mitra
Climate Change implies significant variations in temperature, wind patterns and precipitation that may occur in cycles over decades, hundreds, thousands and millions of years; sometimes these changes may be random occurrences. These result in rain or hail, snowfall or extreme weather events like thunderstorms, cyclones, tornados etc. We have witnessed such events in much greater frequency in recent years.
According to Thomas Stocker, Co-Chair of IPCC Working Group I, “in the 20th century, scientists began to detect that the Earth was warming up abnormally—In the first decade of the new millennium the common people began to feel the effects. Since the 1950s atmosphere and oceans have warmed, the amount of snow and ice has diminished, sea level has risen and the concentration of carbon dioxide has increased to a level unprecedented in at least the last 800,000 years”. This is attributed to be the result of human activity.
Currently Global Warming is being caused by Greenhouse Gases (GHGs) resulting from mainly industrial development and urbanization, agriculture and changes in land-use patterns. Added to this is excess of cutting of trees resulting in soil erosion with much serious impact on the mountain ranges and coastal regions. Different gasses have different Global Warming Potential (GWP) which means how much damage they cause to the environment. Carbon dioxide (CO2) has GWP of 1; Methane – GWP of 21 and Nitrous oxide – GWP of 310.
The effects have been felt in India as well. There was country-wde drought in 2002. There was severe drought in 2009 as well. The year 2010 was one of warmest. In the year 2013 there were extreme rainfall events in Uttarakhand. We have seen mountains becoming barren due to large scale cutting of trees. This is resulting in hills becoming weak.
Climate and weather have always had a powerful impact on human health and well-being. Global climate change is a newer challenge to ongoing efforts to protect human health. In the past few years there has been increased incidence of mosquitoes breeding Malaria, Dengue and Yellow Fever. Increase in temperature by 2-3º C would increase the number of people who, in climatic terms, are at risk of malaria by around 3-5 %, which means several hundred million people globally. According to World Health Report 2002, climate change was estimated to be responsible in 2000 for approximately 2.4% of worldwide diarrhea cases, and 6% of malaria in some middle-income countries.
Changes in atmosphere, coupled with vehicular and industrial emissions, lead to smog and poisonous gases which cause difficulty for those with cardio vascular disease, respiratory disorders as asthma, emphysema, chronic bronchitis and allergy problems. Similarly water pollution related diseases are on the increase. There is poor quality of drinking water because water resources are threatened by drought, leading to bacterial, viral, Protozoal and Parasitic diseases.
As a result of rising sea levels and flooding of coastal areas, there occurs increase in population density due to migration of population to safe areas. People have to live in make shift camps in poor, unhygienic living conditions, which cause several infectious diseases. Children lose their school. There may be violence for want of food and other basic amenities. This may lead to psycho trauma and post traumatic stress disorder.
Extreme changes in climate may occur in the event of nuclear fallout. A study conducted by Dr Ira Helfand, MD, and Alan Robock et al on the climatic consequences of a regional nuclear war shows that even a “limited” nuclear conflict, involving as few as 100 Hiroshima-sized bombs, would have global implications with significant effects on weather patterns throughout the world. Debris injected into the atmosphere from the explosions and resulting fires would produce an average surface cooling of -1.25ºC that would last for several years. Because of fall in temperature there would occur crop failures. This would result in serious food shortage. This could put the lives of over 2 billion people at risk.
Human-induced depletion of stratospheric ozone is another issue affecting human health. Stratospheric ozone absorbs much of the incoming solar ultraviolet radiation (UVR), especially the biologically more damaging, shorter-wavelength, UVR. The solar ultraviolet radiation may cause diseases of skin like Malignant Melanoma, Non-Melanocytic Skin Cancer – Basal Cell Carcinoma, Squamous Cell Carcinoma, Sunburn, Chronic Sun Damage, Photo dermatoses etc. They may affect our eyes in the form of acute photo keratitis and photo conjunctivitis, climatic droplet keratopathy, pterygium, cancer of the cornea and conjunctiva, cataract, Uveal melanoma, acute solar retinopathy and macular degeneration
There is also negative impact on immunity. There may occur suppression of cell mediated immunity, Increased susceptibility to infection, impairment of prophylactic immunization and activation of latent virus infection. Climate changes also cause mutations leading to development of new types of viruses. These produce new types of diseases.
The climate change leads to altered general well-being, disturbed sleep/wake cycles, Seasonal affective disorder and disturbance in mood. The number of heat stroke related diseases is on rise during summer. Similarly due to erratic winter there is detrimental effect on human health.
The health effects of climate change also depend on other relevant factors like age and gender, socio economic condition, geographic locations — already cold/warm areas/temperate regions, population density, sanitation and healthcare, nutrition, preexisting diseases, public healthcare system, literacy, infrastructure. People who are socially, economically, culturally, politically, institutionally, or otherwise marginalized are especially vulnerable to climate change.
There is urgent need to take steps to prevent climate change to save health. These may include policy making and strategies to reduce the risks. There is need to develop better infrastructure to combat the negative effect of climate change. A holistic approach is needed to health care in the form of better nutrition, job opportunities, housing, shelter, clean water etc. Also important are changes in life style, equity, seriousness to the problem, international agreements and people’s campaigns.
“We have the means to limit climate change,” said R. K. Pachauri – former Chair of the IPCC. “The solutions are many and allow for continued economic and human development. All we need is the will to change, which we trust will be motivated by knowledge and an understanding of the science of climate change.” (IPA Service)<|endoftext|>
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We have to add only the numerators and write the resultant as the sum for the common denominator.
### Examples:
(i) 2 ⁄ 7 + 4 ⁄ 7
Solution:
Since the denominators are same we have to add only the numerators.
= (2+4)/7
= 6/7
(ii) 3/11 + 5/11 + 2/11
Solution:
As the denominators are same we have to add only the numerators.
=(3+5+2)/11
= 10/11
Solution:
As the denominators are same we have to add only the numerators.
= (2+5)/9
= 7/9
Solution:
As the denominators are same we have to add only the numerators.
=(2+2)/5
=4/5
Solution:
As the denominators are same we have to add only the numerators.
= (1+2)/3
= 3/3
=1
Addition of like fractions are explained with squares below.
Solution:
As the denominators are same we have to add only the numerators.
= (2+7)/5
= 9/5
Solution:
As the denominators are same we have to add only the numerators.
= (1+5)/6
= 6/6
We can simplify this using 6 times table.
So we will get 1.
Solution:
As the denominators are same we have to add only the numerators.
= (3+5)/11
= 8/11
We cannot simplify this by any table.
So we will get 8/11.
Solution:
As the denominators are same we have to add only the numerators.
= (5+6)/11
= 11/11
We can simplify this using 11 times table.
So we will get 1.
Solution:
As the denominators are same we have to add only the numerators.
= (5+6)/11
= 11/11
We can simplify this using 11 times table.
So the final answer is 1.
Solution:
As the denominators are same we have to add only the numerators.
= (3+6)/16
= 9/16
We can not simplify this using any table.
So we will get 9/16
We had seen examples for adding like fractions. Students can practice the problems and try to solve the practice problems given below. Parents and teachers can guide the students to do the practice problems using the same method given in the examples. If you have any doubt you can contact us through mail, we will help you to clear the doubts.
Practice problems:
Related topics:<|endoftext|>
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## Calculus Tutorial
#### Intro
Before diving into the world of limits, derivatives, and integrals, it is essential to learn some basic trigonometric facts.
Over the years, math teachers have created several mnemonics to help students remembers trigonometric properties. The first is “SOH CAH TOA” (read: sew, kah, tow-ah). This strange sounding series of acronyms stands for “sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.” This means that when taking the function sine of an angle (denoted ), it is equal to the side opposite of the angle divided by the hypotenuse of the triangle. (E.g., in the triangle above, since is the side opposite to the angle and is the hypotenuse of the triangle.) Further, when taking the cosine of an angle (denoted ), it is equal to the side adjacent to divided by the hypotenuse. (E.g., since is on the side adjacent to and is the hypotenuse.) Finally, the tangent of an angle (denoted ) is defined to be the side opposite of the angle divided by the side adjacent to the angle . (E.g. since is the opposite side to and is the adjacent side to .)
To apply trigonometric equations to Calculus, one may need to take into account angles which are outside the bounds of a traditional right triangle. The phrase “All Students Take Calculus” (ASTC) stands for “All, Sine, Tangent, Cosine.” This references which trigonometric functions are positive in the four quadrants of the -plane. In the Quadrant I (upper right), all trigonometric functions are positive. In Quadrant II (upper left), only sine of an angle is positive, while the other two functions are negative. Quadrant III (lower left) has positive values for the tangent of an angle and negative values for sine and cosine of an angle. Finally, in Quadrant IV (lower right), only the cosine of an angle is positive while sine and tangent are negative.
#### Sample Problem
Let be an angle such that and . What is ?
#### Solution
Since is negative and is positive, our angle must lie in quadrant IV. Further, since cosine is adjacent over hypotenuse and , this means that the adjacent side (side in the image) has length and the hypotenuse (side in the image) has length .
By the Pythagorean Theorem, so plugging in our values, we have:
Therefore, we know the length of all four sides and we can find . By TOA, we know that tangent of is the opposite side over the adjacent side . Further, we know that we are working in the fourth quadrant, so tangent is negative. Putting this together, we see that .<|endoftext|>
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Lesson 12 Functions
Episode 2
Reasoning Resources: Table from Worksheet 1 on board or OHT or pupils' copies, Worksheet 2
Whole class preparation
Demonstrating ways of working with arrows and loops.
.
With pupils’ help, extend the table on Worksheet 1 up to input 15. Using clear linking lines and loops, show a couple of examples of doubling the input, then looking at what happens to the output. Also show a tripling strategy by arrowed loops from number to number.
Ask pupils to describe in their own words what is happening to the input, then the output, in each case, Press pupils who use the words ‘the same’ to explain more precisely that the same action has to be performed on the output as on the input. Ask: If the arrow this way means doubling, what would it mean if the direction of the arrow were the other way?
Looking at ratio. Unlikely to be assimilated fully by the pupils until they carry it out for themselves.
Pair and small group work
Extracting examples from the table of values allows a look at proportion problems in a fresh way. Give out Worksheet 2. Pupils now identify similar patterns in their own tables. Tell them that their task is to find examples and pairs of values in their table for question 1.This will help them to answer in words question 2 on the worksheet.
Recognising the two relationships: across and down. Recognising the inverse relationships.
Working with different values across and down.
An alternative way of solving proportion problems to the procedure of ‘cross- multiply and divide!' It links intuition and sense of size of numbers to formal work.
Whole class sharing and discussion
Collect different phrases from pupils’ answers to question 2, writing them on the board. Take two pairs of paired values from the table separately and show that there are two relations: one across, the X 5, and the other down, the X 2. If any of the four numbers in this double relationship is missing, we can find it. Ask pupils to explain how. Show with another example that this also applies to tripling.
Extension
Some classes could consider other proportion questions, first with easy then with harder relationships. Pupils should recognise that they can use either relationship, normally the easiest one, and that some relationships may not be in whole number multiplication.
Note that often some pupils lapse into additive relationship while in all cases the ‘across-relation is ratio, or multiplicative. The same relation must be kept for the second pair of values in each four.To ensure the focus on the ratio relation, more than one pair can be given with another with a missing value, e.g.<|endoftext|>
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One day after China announced it grew the first plants on the Moon, the fledgling plants have been pronounced dead. Rest in peace, lunar sprouts.
On Tuesday, China’s space program said that cotton seeds had germinated in a biosphere carried to the Moon by the nation’s Chang’e-4 lunar lander.
By Wednesday, mission leads had broken the news that the plants perished as the lunar night fell over the probe’s landing site.
The Sunday arrival of the lunar night, which lasts 14 days, deprived the plants of sunlight. During a lunar night, temperatures can plummet as low as −170°C (−274°F).
Meanwhile, daytime temperatures on the Moon can reach a sweltering 127°C (260°F). These massive fluctuations are one of the main obstacles encountered by lunar explorers.
The remaining seeds and fruit fly eggs contained in the mission’s biosphere are not likely to be viable after two weeks of light deprivation and freezing temperatures.
According to China’s National Space Administration, they will decompose and remain sealed to avoid contaminating the lunar surface.
Chang’e-4 went into sleep mode on Sunday to prepare for the harsh night. The lander will rely on a radioisotope heat unit (RHU) to stay warm until sunlight returns in late January.
The mission’s rover Yutu 2, which rolled off a ramp to the lunar surface on January 3, is also dependent on an RHU during the cold spell.
Chang’e-4 is the first spacecraft ever to land on the far side of the Moon, which is commonly mistaken for the “dark” side of the Moon.
Though the far side is always angled away from Earth, it is not always angled away from the Sun. Both lunar faces experience roughly 14 days of daylight and 14 days of darkness in a regular lunar cycle.
Chang-e-4’s biosphere may have only survived for a brief week, but it still made history as the first garden planted at the surface of an alien world.
Please like, share and tweet this article.
Pass it on: New Scientist<|endoftext|>
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Two ring tailed lemurs and one brown collared lemur were born in late March and are making their public debut. Both species live in a naturalistic habitat depicting the Malagasy Spiny Forest along with critically endangered radiated tortoises and several bird species including vasa parrots, red fodies, grey-headed lovebirds, and ground doves.
Guests hoping to catch a glimpse of the new additions will have to observe closely as young lemurs cling to their mothers and nestle in their fur.
The Bronx Zoo has had tremendous success breeding lemurs as part of Species Survival Plans, cooperative breeding programs designed to enhance the genetic viability of animal populations in zoos accredited by the Association of Zoos and Aquariums.
The Wildlife Conservation Society works to save lemurs and their disappearing habitat in the African island nation of Madagascar - the only place in the world where lemurs are found in the wild. Brown collared lemurs are native to the tropical forests of southeastern Madagascar. Ring tailed lemurs are native to the forests and bush in the south and southwestern portions of the island. Their habitats are being destroyed by human activity including charcoal production and slash-and-burn agriculture.
Ring-tailed lemurs are very social and live in large matriarchal groups that often contain several breeding females. They are capable climbers, but spend much of their time on the ground. Newborns will ride on their mothers' chest and back for the first few weeks and will begin move around on their own within two-to-four weeks, but still stay close to their mother.
Collared lemurs use their long tails to balance when leaping through the forest canopy. They live in groups of males and females but are not matriarchal like many other lemur species. The young ride on their mother's back hiding in her fur for the first few months of their lives.
All lemur species are in trouble due to devastating loss of suitable habitat. The International Union for Conservation of Nature classifies both ring-tailed lemurs and brown collared lemurs as endangered in the wild.
There are six species of lemurs on exhibit in the Madagascar! exhibit: ring-tailed lemurs, brown collared lemurs, red ruffed lemurs, gray mouse lemurs, crowned lemurs, and Coquerel's sifaka. Black and white ruffed lemurs are on exhibit at the Central Park Zoo.
Opened in 2008, Madagascar! educates zoo visitors about the country's incredible biodiversity and how the Wildlife Conservation Society and partners are working to save its wildlife and wild places.<|endoftext|>
| 3.75 |
203 |
Returns the next character from the standard input (stdin).
It is equivalent to calling getc with stdin as argument.
On success, the character read is returned (promoted to an int value).
The return type is int to accommodate for the special value EOF, which indicates failure:
If the standard input was at the end-of-file, the function returns EOF and sets the eof indicator (feof) of stdin.
If some other reading error happens, the function also returns EOF, but sets its error indicator (ferror) instead.
1 2 3 4 5 6 7 8 9 10 11 12 13
/* getchar example : typewriter */
int main ()
puts ("Enter text. Include a dot ('.') in a sentence to exit:");
} while (c != '.');
A simple typewriter. Every sentence is echoed once ENTER has been pressed until a dot (.) is included in the text.<|endoftext|>
| 4.0625 |
535 |
Lesson 12ª
How do we Rationalize a Denominator Which is Composed by both Positive and Negative Signs?
Imagine we must rationalize:
To rationalize the denominator of a fraction composed by a binomial, we need to multiply the numerator and denominator of the fraction times the conjugate of the denominator.
What is a binomial conjugate?
A binomial conjugate is another binomial similar to the first but the sign of the second term is the opposite than it previously had:
Examples:
The conjugate of
The conjugate of
The conjugate of
Remember: the sum of two numbers by their difference is equal to the difference of their squares.
The square of any quantity in a square root results in removing the radical and keeping the radicant:
If the numerator and denominator of are multiplied by their conjugate, which is, I would get the difference of their squares in the denominator, 3 – 2 = 1:
Remember that if I multiply or divide both terms in a fraction by the same number, the value of the fraction doesn't vary.
10.83 Calculate:
Solution:
We multiply the numerator and denominator by .
We see that in the numerator we have the square of the sum of two numbers:
.
We know that the square of the sum of two numbers is equal to:
the square of the first: ,
plus two times the first by the second: ,
plus the square of the second: ,
which is what you have next:
10.84 Calculate:
Solution:
We see numbers without square roots in the numerator and denominator. The procedure is the same because, in order to eliminate square roots in the denominator, we need to have square differences and the square of a whole number is another (higher) whole number.
We multiply both members in the fraction times the conjugate of the denominator, which is: .
In the numerator, we have the square of the difference of two numbers; which is the same as the square of the first (1) minus 2 times the first times the second
times the square of the second :
Finally, we multiply times -1 eliminating the denominator,
10.85 Calculate:
Solution:
.
10.86 Calculate:
and also .
10.87 Calculate:
Solution:
We multiply each term in the multiplier by the terms in the multiplicand. Remember that to be able to multiply square roots, they must have the same index. Then, we multiply the sub-radical quantities: keeping the same root.
To multiply a number times a square root; consider that: .
Solving the exercise:
10.88 Calculate:<|endoftext|>
| 4.8125 |
955 |
A blood test, or the results of a blood test, that measures levels of lipids, or fats, including cholesterol and triglycerides. Factors such as your age, sex, and genetics influence your lipid profile. Certain aspects of your lifestyle, including your diet, level of physical activity, level of diabetes control, and smoking status, also affect your lipid profile. And some medical conditions can raise or lower cholesterol and triglyceride levels.
A lipid profile is a direct measure of three blood components: cholesterol, triglycerides, and high-density lipoproteins (HDLs). Cholesterol is a vital substance that your body uses to produce such things as digestion-aiding material, hormones, and cell membranes. It is both produced by the body and absorbed from some of the foods you eat. Cholesterol and triglycerides are transported in the blood by combinations of lipids and proteins called lipoproteins. HDLs, the so-called “good” or “healthy” cholesterol, are lipoproteins made mostly of protein and little cholesterol. HDLs can help to clear cholesterol deposits in blood vessels left by another blood component called low-density lipoproteins, or LDLs.
LDL levels may be calculated from the three directly measured lipids or may be more accurately measured by a direct test. LDLs and very-low-density lipoproteins (VLDLs) are the so-called “bad” cholesterols. Unlike HDLs, LDLs and VLDLs are high-cholesterol particles. While cholesterol is necessary for various bodily functions, too much cholesterol is harmful, since excess cholesterol can be deposited in blood vessel walls. These fat deposits can lead to atherosclerosis, or hardening of the arteries, and cardiovascular disease, the number one killer in the United States. High levels of triglycerides are also associated with an increased risk of heart disease.
People with Type 1 diabetes whose blood glucose levels are controlled tend to have lipid levels similar to people who don’t have diabetes and may even have higher levels of HDL, possibly because of the effects of insulin. People who have Type 2 diabetes, however, tend to have high levels of triglycerides and low HDL levels. And while people with Type 2 diabetes tend to have similar LDL levels as people who don’t have diabetes, their LDL particles seem to be smaller and more prone to causing damage. Premenopausal women who do not have diabetes tend to have better cholesterol levels than men, but this protective effect seems to be negated in women with diabetes. Both sexes show an increase in lipoprotein levels as they age.
The American Diabetes Association (ADA) recommends that people with diabetes who have lipid levels that need correction have a lipid profile taken at least annually. Because alcohol and food intake before the test can cause temporary elevations of cholesterol and triglycerides, it is necessary to fast for 9–12 hours before the test and to abstain from alcohol for 24 hours before.
According to the ADA, low-risk cholesterol levels in adults with diabetes are LDL levels below 100 mg/dl, HDL levels above 40 mg/dl (above 50 mg/dl for women), and triglycerides below 150 mg/dl. If all your levels fall in these low-risk categories, you and your doctor may decide that you only need to have a lipid profile done every two years. People with diabetes who have cholesterol levels out of the low-risk ranges may be given a repeat test to verify the results.
A child with diabetes older than two years should be tested at the time of diagnosis if there is a family history of high cholesterol or heart disease or if family history is unknown. If there is no family history of high cholesterol or heart disease, the child should be tested at puberty. Children 12 and older who are diagnosed with diabetes should have a lipid profile done at the time of diagnosis and again after blood glucose levels have been brought under control. Subsequent checks of children need only be done every two years if levels are acceptable.
Lowering your risk for cardiovascular diseases by raising your HDL level, lowering your triglyceride level, and lowering your LDL level can be accomplished by keeping your blood glucose level as close to the normal range as possible, exercising regularly, stopping smoking, losing weight if you are overweight, maintaining a diet low in saturated fat and cholesterol, and consuming more soluble fiber (the type of fiber found in barley, beans, and oats, for example). Read food labels carefully to keep track of your intake of saturated fat, cholesterol, and fiber, as well as carbohydrate. Some people may require drugs in addition to lifestyle changes to improve their lipid profile.<|endoftext|>
| 3.78125 |
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