5th Grade /
Unit 6: Multiplication and Division of Decimals
Multiplication and Division of Decimals
Students use their knowledge of multiplication and division with whole numbers and with fractions to multiply and divide with decimals, and apply this understanding to the context of measurement conversion.
Unit Practice
In Unit 6, students use their procedural knowledge of multiplication and division with whole numbers, combined with their newly acquired understanding of multiplication and division with fractions, to multiply and divide with decimals, reasoning about the placement of the decimal point. They then apply this to the context of word problems, including those involving measurement conversion.
In Grade 4, students were first introduced to decimal notation for fractions and reasoned about their size (4.NF.5—7). Then, in the first unit in Grade 5, students developed a deeper understanding of decimals as an extension of our place value system, understanding that the relationships of adjacent units apply to decimal numbers, as well (5.NBT.1), and using that understanding to compare, round, and represent decimals in various forms (5.NBT.2—4). Next, students learned to multiply and divide with whole numbers in Unit 2 (5.NBT.5—6), skills upon which decimal computations will rely. In Unit 3, students explored the other two operations with decimals not addressed in this unit: addition and subtraction (5.NBT.7). In Unit 5, students learned to multiply and divide with fractions, including relating fractions to the operation of division; multiplying a fraction by a fraction, including mixed numbers; and dividing a unit fraction by a whole number and vice versa (5.NF.3—7), which will help them make sense of analogous cases of decimal multiplication and division. Thus, this unit is dependent on a lot of prior learning, both in Grade 4 and Grade 5.
This unit starts with multiplying a decimal by a single-digit whole number, then multiplying a decimal by a multi-digit whole number, and finally multiplying a decimal by another decimal. Then, students progress to dividing a decimal by a single-digit whole number, then dividing a decimal by a two-digit whole number, and finally solving cases involving decimal divisors. Throughout these topics, students use the same methods to compute decimal products and quotients as they did for whole-number products and quotients, but they must reason about the placement of the decimal point. It is only in the last lesson of each topic that students generalize the pattern of the placement of the decimal point. The various lines of reasoning, and their advantages and disadvantages, can be read on pages 19 and 20 of the NBT Progression linked in the "Unit-Specific Intellectual Preparation" section. Students also solve myriad word problems as well as write and solve expressions involving decimals as a way to support the major work (5.OA.1, 5.OA.2). Finally, the unit closes with students learning to convert among different-sized customary measurement units within a given measurement system and solve word problems that use those conversions (5.MD.1), which extends the work from Grade 4 of converting from a larger unit of measurement to a smaller one in Grade 4 (4.MD.1—2). As noted in the Progressions, "this is an excellent opportunity to reinforce notions of place value for whole numbers and decimals, and the connection between fractions and decimals (e.g., meters can be expressed as 2.5 meters or 250 centimeters)" (GM Progression, p. 26), as well as computations with these types of numbers (5.NBT.7, 5.NF), thus connecting the work of unit conversion with major work of the grade.
Reasoning about the placement of the decimal point affords students many opportunities to engage in mathematical practice, such as constructing viable arguments and critiquing the reasoning of others (MP.3) and looking for and expressing regularity in repeated reasoning (MP.8). For example, "students can summarize the results of their reasoning as specific numerical patterns and then as one general overall pattern such as 'the number of decimal places in the product is the sum of the number of decimal places in each factor'" (NBT Progression, p. 20).
In Grade 6, students will become fluent with all decimals computations that they've developed in Grade 5 (6.NS.3). In Grade 7, students will also learn that every fraction can be represented with a decimal that either terminates or repeats. Then in Grade 8, students learn that terminating and repeating decimals are rational numbers and that there are numbers that are irrational whose decimal expansion does not repeat. Then, students use the work they start in this unit in Grade 8 in the context of scientific notation. Thus, this unit has many interesting connections and applications for many years to come.
Pacing: 27 instructional days (24 lessons, 2 flex days, 1 assessment day)
For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 5th Grade Scope and Sequence Recommended Adjustments.
Expanded Assessment Package
Problem Sets for Each Lesson
This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.
Download Unit Assessment
Download Unit Assessment Answer Key
Download Student Self-Assessment
Intellectual Prep
Intellectual Prep for All Units
Read and annotate "Unit Summary" and "Essential Understandings" portion of the unit plan.
Do all the Target Tasks and annotate them with the "Unit Summary" and "Essential Understandings" in mind.
Take the unit assessment.
Unit-specific Intellectual Prep
Read pp. 19—20 of Number and Operations in Base Ten, K-5 of the Progressions for the Common Core State Standards in Mathematics.
Familiarize yourself with the reference sheet that students will be given on their end-of-year assessment based on the state in which you live, so that you know what information is given to students and what should be internalized. This unit has been designed based on the Massachusetts Comprehensive Assessment System Grade 5 Mathematics Reference Sheet.
Essential Understandings
General methods used for computing products and quotients of whole numbers extend to products and quotients of decimals, with the additional issue of placing a decimal point in the solution. There are several lines of reasoning that students can use to explain the placement of the decimal point in products and quotients of decimals.
Students may use estimation to assess the reasonableness of their solution. As with whole-number computation, some computational estimates can be better than others, depending on what numbers are chosen to use in place of the actual values. Using estimation to place the decimal point, however, isn't always a reliable strategy, especially in cases they will see in later grades such as $${0.0043 \times 0.00078}$$.
When multiplying, it is more efficient to decompose the value with fewer digits. In the standard algorithm, this means writing the number with fewer digits on the bottom. With decimals, the number with fewer digits does not always imply the number with the smallest value (e.g., $${17.15 \times 42}$$).
conversion factor
5th Grade Vocabulary Glossary
Unit Materials, Representations and Tools
Area models
Standard algorithm for multiplication
Standard algorithm for division
Tape diagrams
Grade 5 MCAS Reference Sheet (or the conversion rates reference sheet of your choosing)
Additional Unit Practice
With Fishtank Plus you can access our Daily Word Problem Practice and our content-aligned Fluency Activities created to help students strengthen their application and fluency skills.
Topic A: Multiplying Decimals
Multiply a single-digit whole number by a decimal in cases that involve basic facts. Estimate the product of a single-digit whole number with a decimal by rounding numbers to their largest place value.
Multiply a single-digit whole number by a decimal.
Construct viable arguments and critique the reasoning of others regarding the placement of the decimal point in computations that involve multiplying a single-digit whole number by a decimal.
Multiply a multi-digit whole number by a decimal.
Multiply a decimal by a decimal in cases that involve basic facts. Estimate the product of two decimals by rounding numbers to their largest place value.
Multiply a decimal by a decimal.
Multiply with decimals in all cases, reasoning about the placement of the decimal point.
Topic B: Dividing Decimals
Divide a decimal by a single-digit whole number in cases that involve basic facts. Estimate quotients with single-digit divisors by rounding numbers to compatible numbers.
Divide a decimal by a single-digit whole number.
Divide a decimal by a single-digit whole number that requires decomposition in the smallest place value.
Divide a decimal by a two-digit whole number.
Divide a decimal by a two-digit whole number that requires decomposition in the smallest place value.
Divide a whole number or a decimal by 1 tenth or 1 hundredth.
Divide a whole number or a decimal by a decimal in cases that involve basic facts. Estimate quotients with decimal divisors by rounding numbers to compatible numbers.
Divide a whole number or a decimal by a decimal.
Divide with decimals in all cases, reasoning about the placement of the decimal point.
Topic C: Decimal Expressions and Word Problems
Solve real-world problems that require interpretation of the remainder involving all possible cases, including further decomposition into a decimal.
Solve real-world problems involving multiplication and division with decimals.
Write and evaluate numerical expressions involving multiplication and division with decimals.
Topic D: Measurement Conversion and Word Problems
5.MD.A.1
Express measurements in a whole number of larger units or mixed units in terms of a smaller unit.
Express measurements in a fractional or decimal number of larger units in terms of a smaller unit.
Express measurements in a whole number of smaller units in terms of a larger unit.
Express measurements in a decimal or fractional number of smaller units in terms of a larger unit.
Solve real-world problems involving measurement conversions.
Key: Major Cluster Supporting Cluster Additional Cluster
5.MD.A.1 — Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
5.NBT.B.7 — Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
5.OA.A.1 — Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
5.OA.A.2 — Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
4.MD.A.1 — Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …
4.MD.A.2 — Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
5.NBT.A.1 — Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
5.NBT.A.2 — Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
5.NBT.B.5 — Fluently multiply multi-digit whole numbers using the standard algorithm.
5.NBT.B.6 — Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Number and Operations—Fractions
5.NF.B.3
5.NF.B.3 — Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
5.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.7 — Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade
Future Standards
Expressions and Equations
6.EE.A.2
6.EE.A.2 — Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.3 — Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.A.4 — Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
6.NS.B.3
6.NS.B.3 — Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
6.NS.B.4 — Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1—100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
Standards for Mathematical Practice
CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.
Multiplication and Division of Fractions
Patterns and the Coordinate Plane