Answers.Body: <p>I think the biggest piece of confusion comes from the words representing quantities; in particular, the word TEN.</p>
  
  <p>If you disassociate TEN from the quantity that represents the number of your fingers, and instead frame TEN as word noting when it's time to "start over" with a new quantity set, you are more than halfway towards your goal.</p>
  
  <p>And before you get to binary, and to reinforce that "TEN" is just a word representing a quantity, I'd instead focus on teaching Octal.  Once your students can get used to a number system where you skip the "words" EIGHT and NINE, your students will be more able to wrap their brains around binary.</p>
  
  <p>You might introduce the Octal system by way of frogs since frogs have 8 toes total on their front legs. That's to say what would the number system look like if it were created by frogs (8 instead of 10 digits)?</p>
  
  <p>An interesting dynamic also for frogs is that there are 10 toes on their back legs.  Perhaps an interesting story could be derived from this where frogs find themselves having to switch between the Octal and Decimal system.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T20:55:00.780Z
Answers.Text: It is said that if you ask a horse to count to 10 by stamping a foot, you hear "stamp, stamp, stamp, stamp".
Answers.UserDisplayName: Buffy
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T21:11:11.257Z
Answers.Text: Rabbits count One Two Three Many (from Watership Down).
Answers.UserDisplayName: Scott Rowe
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-06-24T23:59:26.753Z
Answers.Text: Ten is a certain number of things. So is 10--the same number of things. Where you write "TEN" here you should write "ONE THEN ZERO (WHEN IT MIGHT NOT MEAN TEN)".
Answers.UserDisplayName: philipxy
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T20:51:04.317Z
Answers.LastActivityDate: 2018-11-05T20:51:04.317Z
Answers.UserDisplayName: flaZer
Answers.Body: <p>For German speaking folks take a look at</p>
  <p><a href="https://kinder.wdr.de/tv/die-sendung-mit-der-maus/video-bis--zaehlen-102.html" rel="nofollow noreferrer">https://kinder.wdr.de/tv/die-sendung-mit-der-maus/video-bis--zaehlen-102.html</a></p>
  <p>where fingers are used.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-07T02:57:10.653Z
Answers.Text: Now that I have unmarked this as spam, it's worth noting that this link appears to have gone dead.  Would you mind writing in a simple description?
Answers.UserDisplayName: Ben I.
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-10T21:09:54.773Z
Answers.Text: thanks for the note, I updated the link.
Answers.UserDisplayName: OBu
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-10T21:26:17.557Z
Answers.Text: Sorry again about that. I felt awful!
Answers.UserDisplayName: Ben I.
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-10T21:52:00.283Z
Answers.Text: Even still, it would be good if you [added some information to this answer](https://meta.stackexchange.com/q/225370/280611).
Answers.UserDisplayName: Ben I.
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-02-08T08:40:10.907Z
Answers.Text: This works. I used this to teach binary counting to kids in the target age group back in the early 2000s. Their favourite number of course was the $132_{10}$ in binary.
Answers.UserDisplayName: Kai
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-03T21:06:40.970Z
Answers.LastActivityDate: 2021-01-10T21:09:33.020Z
Answers.LastEditDate: 2021-01-10T21:09:33.020Z
Answers.UserDisplayName: OBu
Answers.Body: <p>I would start by reviewing how the mathematics curricula explains the numbers to these children.</p>
  
  <p>I have been surprised in the past when looking at education books of young children that when teaching numbers, <em>they explained bases</em>! There is a lot of counting numbers as <em>X hundreds, Y tens Z units</em>, using colored stickers for each position, etc.</p>
  
  <p>Then you are never again asked to write 123 as 100 + 20 + 3.</p>
  
  <p>However, when having to learn binary or hexadecimal, that's exactly what you need to do, realize that 123d is just 1×10² + 2×10¹ + 3×10⁰ and 123h will be 1×16² + 2×16¹ + 3×16⁰, and so for any base.</p>
  
  <p>So, start by checking how they are|will already learn the numbers and see how to make binary encoding fit into them. If a yellow dot will mean tens, use yellow for 2, not for 8 (maybe with an extra symbol to mean binary), etc.</p>
  
  <p>The curricula will vary a lot between places and editions, and this may not be applicable to your specific case, but I think it's important to adapt to how they are learning the numbers (or will learn them later) so that they can learn the concept as the one thing it really is, instead of being taught by different people as separate and unrelated things.</p>
  
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Answers.CreationDate: 2018-11-05T15:28:34.397Z
Answers.LastActivityDate: 2018-11-05T15:28:34.397Z
Answers.UserDisplayName: Ángel
Answers.Body: <p>I've used Garlikov's Socratic approach to teaching binary in the classroom to rave reviews:</p>
  <p><a href="http://www.garlikov.com/Soc_Meth.html" rel="nofollow noreferrer">http://www.garlikov.com/Soc_Meth.html</a></p>
  <p>I think not just the format but also the particular questions he asks and how he frames the discussion make his use of this technique effective.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-16T21:00:39.883Z
Answers.Text: Hi, and welcome to [cseducators.se]!  This answer has a few problems: first, it's a link-only answer, and since we accumulate a lot of links, we ask that people give at least *some* sort of summary in case the link goes dead. Secondly, (and more importantly) it's not specifically an answer to the question being asked.  You can teach virtually anything using the socratic method (to better or worse effect).  There is nothing about binary or binary encoding within your link.
Answers.UserDisplayName: Ben I.
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-17T12:35:19.057Z
Answers.Text: @BenI. I will update with more detail. That link contains a discussion of binary and binary encoding that takes up the vast, vast majority of the page. Are you sure you followed the correct link?
Answers.UserDisplayName: Jason Hemann
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-17T13:42:19.503Z
Answers.Text: Ah, I see... I didn't scroll down far enough. On mobile, it appeared to be a very long wall of text about the socratic method, so I didn't go past the first 4 screens.  You're right that there is binary in there.
Answers.UserDisplayName: Ben I.
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-17T19:37:03.293Z
Answers.Text: Yes. Although he mentions binary early on in the post, you're right it doesn't become exclusively focused on teaching binary until you scroll a half page (on my screen). I should have said it was specifically about teaching binary in my original post, though.
Answers.UserDisplayName: Jason Hemann
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-01-14T23:04:40.007Z
Answers.LastActivityDate: 2021-01-17T12:36:45.830Z
Answers.LastEditDate: 2021-01-17T12:36:45.830Z
Answers.UserDisplayName: Jason Hemann
Answers.Body: <p>James Tanton makes it very clear how binary numbers emerge: Exploding Dots</p>
  <p><a href="https://www.youtube.com/watch?v=bvglHQk-Ygs&amp;list=PLoBaXWXBi2d6JJ-HpXk38SmkCFrGX9iJD&amp;ab_channel=JamesTanton" rel="nofollow noreferrer">https://www.youtube.com/watch?v=bvglHQk-Ygs&amp;list=PLoBaXWXBi2d6JJ-HpXk38SmkCFrGX9iJD&amp;ab_channel=JamesTanton</a></p>
  
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Answers.CreationDate: 2020-11-26T13:27:30.660Z
Answers.LastActivityDate: 2020-11-26T13:27:30.660Z
Answers.UserDisplayName: danbst
Answers.Body: <p>Why don't just consequently use <span class="math-container">$11_{2}$</span> and <span class="math-container">$11_{10}$</span> and teach it the same way it's done at universities? I don't see how this would be any more difficult than starting confusing things with symbols that just add complexity on top of a difficult thing.</p>
  <p>The most confusing thing in school for me was when teachers tried to simplicate stuff ...</p>
  <p>If it was me, I'd teach them the whole abstract thing and leave simplified constructs unless your students really need it.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-09-03T01:14:35.507Z
Answers.Text: Hello, and welcome to [cseducators.se]! I just removed the bit at the end.  Please note the most recent incarnation of the [Be Nice](https://cseducators.stackexchange.com/conduct) policy.
Answers.UserDisplayName: Ben I.
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-09-04T11:17:33.307Z
Answers.Text: Thanks for the correction, failed to notice the ambiguity.
Answers.UserDisplayName: kaiya
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Answers.CreationDate: 2020-09-02T19:50:52.797Z
Answers.LastActivityDate: 2020-09-03T01:11:22.417Z
Answers.LastEditDate: 2020-09-03T01:11:22.417Z
Answers.UserDisplayName: kaiya
Answers.Body: <blockquote>
  <p>I'm teaching some introduction lessons to a classroom of 6-8 year olds about basic 21st century skills. A fundamental part of this is how computers represent numbers, words and images, which all starts with understanding binary representation of a number.</p>
  <p>...</p>
  <p>So I'm thinking of using different symbols. Like ● for 'this bit is on', and ⊗ for 'this bit is off' ...</p>
  <p>-- Roel</p>
  </blockquote>
  <p>My son just turned 8 and I've run into this same problem many times. I've found it an interesting and rewarding challenge to try to explain concepts without getting too deep into technical details.</p>
  <p>For my son, (and even many adults) I've found it extremely valuable to <strong>not</strong> talk much (if at all) about binary, place value, and number bases (at least at first) as bytes are what everyone uses and talks about near daily. I've also found it good to avoid talking about specific numerical quantities and focusing instead on what those quantities represent, such as letters, brightnesses, etc. In some conversations I also found it useful to explain that bytes in a computer are taking up physical space inside the memory of the machine, just as letters on paper take up space.</p>
  <h2>Teaching Binary Numbers:</h2>
  <blockquote>
  <p>Sometimes light bulb on/off symbols are used ...</p>
  <p>-- totera</p>
  <p>An example from Basic Computing book is coming to my mind here: Suppose you have some torches or light bulbs and want to convey a message using the combination of those bulbs/torches/LEDs.</p>
  <p>-- Failed Scientist</p>
  <p>... I have never not confused anyone with notation.</p>
  <p>-- Roel</p>
  </blockquote>
  <p>When I tried to teach my son binary in order to explain how numbers worked in a computer I found that what worked best was to keep things simple and only build from ideas he already knew and nothing else. The usual light switch (or LED) analogy seemed to help some, but working with our hands was best. I considered using blocks, drawing pictures, and so on, but ended up not needing to.</p>
  <p>Simple counting, doubling, and adding were things he already knew. Place value, not so much, but it wasn't hard for my son to get the general idea that each of his fingers could represent a number twice as much as the one before it, and you add them all together after you hold them up. Each finger can either be up or down; added or not.</p>
  <p>He was then very impressed with how we could count to 31 on just one hand, and 1023 using both hands. This understanding actually came <strong>after</strong> explaining what bytes were and getting him to understand them.</p>
  <h2>Teaching How Computers Store Letters, Words, and Images</h2>
  <p>The really important part was to explain that a &quot;byte&quot; was a number in the computer, and that it could represent different things, like one letter, or how bright a color was. Explaining that a byte was 8 bits came later when he began learning what bits were.</p>
  <p>As for words, I told him that you could lay out the letters of the alphabet in a row and count them, and that's how computers know what letters are, because the people who build computers decided which numbers would be which letters. I was then able to explain from there how many bytes it would take to make various kinds of documents around our house.</p>
  <p>Explaining pictures also started with a recap on bytes. I explained that 1 byte told the computer how much of a color was in a picture. The computer would read that number and see how much red was in the picture. I next Googled &quot;color mix venn diagram&quot; so we could have a conversation about how pictures are only made of red, green, and blue, and thus 3 bytes are needed to talk about color. From there I was able to explain that those 3 bytes don't actually describe the entire picture, just one very tiny dot, and that you then need <strong>a lot of bytes</strong> to draw one image.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-08T02:16:15.687Z
Answers.Text: I think this might be my favorite answer here. Nicely done!
Answers.UserDisplayName: Ben I.
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-08T04:55:01.787Z
Answers.Text: @BenI. Thank you very much! To be honest, I probably spent way too much time writing this up today. I'm glad it was appreciated. :-)
Answers.UserDisplayName: RoboticForest
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-07T21:49:46.067Z
Answers.LastActivityDate: 2018-11-07T21:49:46.067Z
Answers.LastEditDate: 2020-06-18T08:36:57.023Z
Answers.UserDisplayName: RoboticForest
Answers.Body: <p>At that age, I would skip binary notation entirely and focus on <strong>combinatorics</strong> problems (simple ones) whose answers are most clearly understood as combinations of powers of two.</p>
  <p>Any combinatorics (counting) problem where the full set of answers can be <em>systematically written down by hand,</em> and then counted, is the most appropriate.  Or if it can be physically constructed, that's even better than being written down.</p>
  <p>For instance, someone else mentioned colored beads.  That's great.  So get some actual beads, red and green or whatever, just two colors, and sit the student down and ask, &quot;How many <em>different</em> patterns of three beads in a row can you make with these?&quot;  Of course they'll eventually come up with the answer &quot;8,&quot; but it may take them a while to get there.</p>
  <hr />
  <p>When you consider it in depth, the fundamental reason why binary notation is useful <em>is</em> because it naturally represents the answers to combinatorics problems involving two possible states.  If our computer systems were to have three possible states rather than two, then ternary notation would be a more natural system for representing computer data.</p>
  <p>From that standpoint, introducing combinatorics <em>first</em> before discussing notation is simply a single instance of a more general teaching philosophy I follow: in particular, that the formulas or notations must come <em>after</em> the underlying <em>reasons</em> why they're important to know.</p>
  <p>This principle can be sloppily summarized as &quot;motivating examples&quot; but that loses a lot in translation; really what I'm saying is to <a href="http://www.appliedscholastics.org/l-ron-hubbard/teaching.html" rel="noreferrer">&quot;Form patterns of computation in the individual with regard only to their usefulness.&quot;</a>  Binary notation <em>is</em> useful, but it's more important to lead the student to realize that for himself or herself, than it is to teach it as a rote pattern.</p>
  <hr />
  <p>The keynote with regard to conveying that &quot;1001101&quot; can be used to represent the same quantity as &quot;77&quot; is to start with the <em>meanings</em> of those symbols, and worry about the actual symbols used later on.  This could all be done in one short tutoring session so long as the student's mind is coaxed into action rather than passively recording the information that you're &quot;teaching.&quot;</p>
  <p>I would <strong>have the student choose his or her own symbols to use.</strong>  They're completely unimportant, so don't let the student get hung up on them.  I might start out by using stars and diamonds, just to make the point.  Those are obviously inconvenient to draw.  So then we might switch to circles and squares at the student's suggestion.  And then squares and circles, just to switch things up.  And then I could switch us to minus and plus for a minute or two, and then tell the student:</p>
  <blockquote>
  <p>So you can see it doesn't really matter what pictures we use or what we draw, what matters is that we have two of them, and that we've decided which of the two comes first.  You can use anything you like and you'll still wind up with the same answer, 8, when you list the possibilities out and count them up.  But the pictures that we <em>usually</em> use, so other people will understand our list also, are just &quot;0&quot; and &quot;1&quot;.  And 0 comes before 1, so then our list would look like:</p>
  <p>000<br />
  001<br />
  010<br />
  011<br />
  100<br />
  101<br />
  110<br />
  111</p>
  </blockquote>
  <p>Then you can number the list &quot;1&quot; through &quot;8&quot; and the connection should come pretty quickly, at least with an 8 year old.  (A 6 year old I would do the same thing but not try to teach actual binary numbers.)</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T15:42:00.563Z
Answers.Text: Interestingly, tertiary computers *were* built. They *did have* advantages, but the problem was that because of the way they were built, using the exact same hardware and removing a piece gave a quaternary computer...aka binary computer with double the bit resolution.
Answers.UserDisplayName: Draco18s no longer trusts SE
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-02T22:05:28.717Z
Answers.LastActivityDate: 2018-11-02T22:05:28.717Z
Answers.LastEditDate: 2020-06-18T08:36:57.023Z
Answers.UserDisplayName: Wildcard
Answers.Body: <p>I was going to start by asking what was wrong with <span class="math-container">$11_2$</span> and <span class="math-container">$11_{10}$</span>, and then I noticed the ages you were referring to.  Six year olds!  </p>
  
  <p>First, a bit of frame-challenging advice: I'm not sure that counting in binary is what I would focus on at that age at all.  You might have better luck with finite state machines, which are great CS concepts that you can play lots of physical games with.  You can put states on the floor and have kids jump around, or you can have kids pass picture-notes to one another based on state and observe the results.  You can even have them try to draw out simple machines by watching behaviors of other kids acting out your "script".</p>
  
  <p>However, if the goal <em>really is</em> binary counting at 6-8 years old, I would start with symbols like "•" and "◦", or even start with symbols like </p>
  
  <pre><code>□□    □    
  □□    □    □    L    |
  </code></pre>
  
  <p>... using "count the lines" as a property, and then extend that to dots, so </p>
  
  <pre><code>□□    □    
  □□    □    □    L    |
  •     ◦    •    ◦    •
  </code></pre>
  
  <p>... becomes ...</p>
  
  <pre><code>□□         
  □□         □         |
  •          •         •
  </code></pre>
  
  <p>... becomes 21.  And only when they're familiar with the dots, at the very end of the lesson, I'd get into something <em>very silly</em> that people do with this system!  They use 1s and 0s for •s and ◦s!  If the kids laugh, then your joke landed, and the kids will be beginning to see.</p>
  
  <p>Then see if they can connect 10101 with </p>
  
  <pre><code>□□         
  □□         □         |
  </code></pre>
  
  <p>Good luck!</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-02T16:56:33.340Z
Answers.Text: I always wonder what number-base should the subscripts be written in.
Answers.UserDisplayName: ctrl-alt-delor
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-03T03:59:07.190Z
Answers.Text: The subscripts shouldn't be numeric. They should be *b* and *d*. Binary and decimal.  Or else the word $1011_\text{two}$, $123_\text{ten}$.  Of course, if they are numeric, they must use some common understood base (like ten); if they use the indicated number's **own** base, it will always uselessly be "10".
Answers.UserDisplayName: Kaz
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-03T15:38:11.333Z
Answers.Text: @kaz if the preferred system is a single letter, how would you deal with duodecimal? I could accept using the full word, but there is no advantage there to the digits, as the full word is also in decimal.
Answers.UserDisplayName: Ben I.
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-04T12:39:47.377Z
Answers.Text: I wasn't sure about whether this would be suitable for this age either, but I'm working off a curriculum set by the Dutch association for primary education who put this material with this age bracket, so I'm just going with it. The more I get into it though, the more I start to see how this is basically a launching pad for tying together other things - like how do you represent 'strings' when all you have to work with are numbers (and base 2 numbers for that matter), or even higher level concepts like 'video' or anything else that can be represented by computers, really.
Answers.UserDisplayName: Roel
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T10:27:41.133Z
Answers.Text: @Roel That representation issue was a massive elephant in the room to me for a very long time, well past the point that I was writing commercial code. You should work it into a new question...
Answers.UserDisplayName: Sean Houlihane
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T15:21:27.363Z
Answers.Text: @Ben I. I simply wouldn't deal with duodecimal; I'm not aware of it being widely used. I mean, of course kids have to learn about arbitrary bases, but maybe at the stage where they are confused by the interpretation of digits, the common ones used in computing are good enough.
Answers.UserDisplayName: Kaz
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T15:27:31.777Z
Answers.Text: ^  And of course sexagesimal, ubiquitous in time representation.
Answers.UserDisplayName: Kaz
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-02T13:48:49.403Z
Answers.LastActivityDate: 2018-11-02T13:48:49.403Z
Answers.UserDisplayName: Ben I.
Answers.Body: <p>We made bracelets in APCS Principles (9th - 12th grade). I bought a bunch of blue and white (school colors) beads. Blue was on, white was off. They then made bracelets of their names or initials using 5 bits. A was 1 (w-w-w-w-b), B was 2 (w-w-w-b-w), etc. </p>
  
  <p>Kids seemed to enjoy it, and I caught a few wearing their bracelets walking around the halls days later. </p>
  
  <p>My own kids, 6 and 9, came in after school and saw the beads and wanted to make their own bracelets. By the time they were finished the 9 year old had made 6 or 8 bracelets for different friends and was going to explain how they worked the next day at school.</p>
  
  <p>Only thing I'm planning on changing for next year is adding a different bead color to represent the start / end of the string.  </p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-04T12:36:01.723Z
Answers.Text: I'm working off a curriculum, proposed by what is essentially 'the Dutch association for primary education', and they have an exercise in making bracelets. I was a bit hesitant about that though, representing essentially 'something' and 'nothing' with two color, but when I think about it, it's no worse than using the symbols I described. So in your experience, children were able to make the link between the concept of an 'on bit' and 'off bit' and the colors quite readily? 9th-12th grade, that is ages 8-11, right?
Answers.UserDisplayName: Roel
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-04T12:40:17.280Z
Answers.Text: 9-12th is students 14-18 years old. The activity came from CSUnplugged, and was meant for younger students, but worked pretty well with the older students too.
Answers.UserDisplayName: Ryan Nutt
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T20:59:04.647Z
Answers.Text: I've made bead necklaces using Morse code. It was easy to read them, just sound it out in my head.
Answers.UserDisplayName: Scott Rowe
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-02T17:33:35.583Z
Answers.LastActivityDate: 2018-11-02T17:33:35.583Z
Answers.UserDisplayName: Ryan Nutt
Answers.Body: <p>Use money.  Tell your students to image a country has bills that are in denominations of powers of 2:  1, 2, 4, 8, 16....  Now tell them this:  For a given number, count out that money with as few bills as possible.  You use a "greedy algorithm:"</p>
  
  <ol>
  <li>Use the largest bill possible.</li>
  <li>Repeat this until you have counted out the amount.</li>
  </ol>
  
  <p>Questions to ask:</p>
  
  <ul>
  <li>Did you ever use more than one bill from any denomination?</li>
  </ul>
  
  <p>They should observe this is never necessary.</p>
  
  <p>Now write a 1 for the largest bill.  If the next denomination down is not present, put a 0; otherwise, put a 1.  Repeat until you descend through the denominations.  You have the binary expansion of the integer you specified.  </p>
  
  <p>This is concrete and very easy for students to grasp.  This "greedy algorithm" works for any base: Use the largest bill you can can count out as many as you can at each stage.  </p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-04T12:29:15.207Z
Answers.Text: Thanks, yes this seems like a good way to go in terms of giving someone an algorithm to do conversions. I'm not convinced though that it's a clear way to explain the concept of there being different number bases at all, since I would still be talking in terms of 'count out 6' or 'count out 13'. But still, I am going to try to work this into my preparation.
Answers.UserDisplayName: Roel
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Answers.CreationDate: 2018-11-02T14:41:51.463Z
Answers.LastActivityDate: 2018-11-02T14:41:51.463Z
Answers.UserDisplayName: ncmathsadist
Answers.Body: <p>Just to calibrate expectations: at the lower end of that age range, the kids will have only just started learning about the place-value system in decimal. I wouldn't try to leverage knowledge of things like "carrying" in decimal over to binary.</p>
  
  <p>My own kids are at boundaries of your age range, and in first and second grade they use things like "tens frames" and "hundreds frames". You might consider making twos/fours/eights frames or perhaps better: blocks, <span class="math-container">$1\times 2$</span> joined block pairs, <span class="math-container">$2\times 2$</span> squares of joined blocks, and <span class="math-container">$2\times 2\times 2$</span> cubes of joined blocks. Then show them (or let them explore) how you only ever need at most one of each block type to make a number. Binary then just consists of a table of the number of each block type that you used. </p>
  
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Answers.CreationDate: 2018-11-04T12:31:45.323Z
Answers.Text: My children are 5 and 7, so they will be my guinea pigs :) Part of the challenge is that this is for a Montessori school with so-called 'vertical groups', i.e. children of ages 6-8 are in one classroom (so are children 4-5, and 9-11). So I might need to differentiate the material a bit for various sub-groups. I don't quite understand what you mean by 'tens frames', can you elaborate?
Answers.UserDisplayName: Roel
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Answers.CreationDate: 2018-11-04T12:59:41.790Z
Answers.Text: Tens frames are size 10 grids (usually $2\times 5$) which the student fills in to visualize counting up to 10.  The following PDF is a good description and has a link to youtube which is even more thorough: https://www.ru.ac.za/media/rhodesuniversity/content/sanc/documents/SANC%2010-frames%20reading.pdf
Answers.UserDisplayName: Adam
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Answers.CreationDate: 2018-11-04T13:41:00.187Z
Answers.Text: Oh I see, it's an interesting concept, and one I think would adapt well to counting in various bases. I will talk about this with the teachers, see how it fits within the overall pedagogical math framework they use; I don't want to get in the way of how they teach basic counting, of course. Thanks!
Answers.UserDisplayName: Roel
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Answers.CreationDate: 2018-11-02T14:25:49.577Z
Answers.LastActivityDate: 2018-11-02T14:25:49.577Z
Answers.UserDisplayName: Adam
Answers.Body: <p>The way I got my kids interested in binary representation was to discuss up to what number they could count on one hand. Learning they could count up to 31 and then understanding why, made the trick. You can then impress them yet again when you show them that with ten fingers then can count to 1023. </p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T21:07:50.723Z
Answers.Text: Just be careful showing some of those numbers.
Answers.UserDisplayName: Scott Rowe
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T21:54:46.840Z
Answers.Text: Indeed, there was some giggling with "four".
Answers.UserDisplayName: Martin Argerami
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Answers.CreationDate: 2018-11-04T16:48:08.587Z
Answers.LastActivityDate: 2018-11-04T20:52:26.610Z
Answers.LastEditDate: 2018-11-04T20:52:26.610Z
Answers.UserDisplayName: Martin Argerami
Answers.Body: <p>Please see the note in the last few paragraphs for an important caveat. </p>
  
  <p>Before you get too deep into this, you should look at the work of Piaget. For example: <a href="http://alumni.media.mit.edu/~stefanm/society/som_final.html" rel="noreferrer">http://alumni.media.mit.edu/~stefanm/society/som_final.html</a>. Note that children of that age have almost no ability to do abstraction and won't for several years. Whatever you do has to be very concrete. </p>
  
  <p>For a wider discussion, see: <a href="https://www.verywellmind.com/piagets-stages-of-cognitive-development-2795457" rel="noreferrer">https://www.verywellmind.com/piagets-stages-of-cognitive-development-2795457</a>.</p>
  
  <p>In particular, while symbols can be meaningful to them, they are still likely pre-operational. </p>
  
  <p>You  might also think about contacting a local Montessori school to see what goes on there with students of that age. </p>
  
  <hr>
  
  <p>Note, of course, that the reason that they can't grok using old symbols for new ideas is, precisely, the lack of abstract thinking ability. Catch them again around age 12 or so. </p>
  
  <p>So, thinking a bit more on this topic, let me suggest that introducing new symbols might be counterproductive. They may get the idea firmly embedded in their minds that you are teaching them things that are important and should be retained. Then, in junior high or so, they will have to <em>unlearn</em> what you so carefully taught them. In my view, it is a serious error to teach people things they later need to unlearn/forget/ignore/reject. </p>
  
  <p>I once had a college student who couldn't grasp the idea of operator precedence. It turns out he was taught the wrong rules in HS and internalized them deeply. He couldn't understand why his Java programs didn't work correctly. His earlier teacher was highly respected so he fought me rather than learn the correct rules. He though I was lying to him that Java performed multiplications before additions rather than strict left to right evaluation of all operators. </p>
  
  <p>I'm starting to think this may be a minefield best left unexplored. </p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-04T12:47:08.847Z
Answers.Text: Actually, this is at a Montessori school. I'm working with a few of the teachers who want to incorporate some of this into their curriculum, although me, I'm just a dad who is doing one or two short introduction sessions, as a way to explore where to go from there. Montessori actually has a system with beads (and 'sticks' which are 10 beads, and 'plates' which are 10 sticks) to teach about the importance of the place of a digit in a number. I'm hoping to find a way to tie in with that.
Answers.UserDisplayName: Roel
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-02T19:47:31.570Z
Answers.LastActivityDate: 2018-11-02T23:09:33.670Z
Answers.LastEditDate: 2018-11-02T23:09:33.670Z
Answers.UserDisplayName: Buffy
Answers.Body: <p>Sometimes light bulb on/off symbols are used, and in fact if you show a real circuit with a battery, a switch and a light bulb before you start teaching binary numbers you could explain why the computer needs just two symbols to count.</p>
  
  <p>Anyway, as a personal recollection, I learned binary on my own thanks to an 80's version of this toy: <a href="https://www.youtube.com/watch?v=EM_-APWFX2s" rel="noreferrer">https://www.youtube.com/watch?v=EM_-APWFX2s</a></p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-04T12:49:56.753Z
Answers.Text: That is a really cool toy, I now have to order this :) Thanks.
Answers.UserDisplayName: Roel
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-04T18:52:32.140Z
Answers.Text: By the way, "rami" is Italian for "branches"
Answers.UserDisplayName: totera
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Answers.CreationDate: 2018-11-02T21:30:42.000Z
Answers.LastActivityDate: 2018-11-02T21:30:42.000Z
Answers.UserDisplayName: totera
Answers.Body: <p>An example from Basic Computing book is coming to my mind here: Suppose you have some torches or light bulbs and want to convey a message using the combination of those bulbs/torches/LEDs.</p>
  
  <p>I can understand that this example can still be bit too much for 6-8 years old, but you can make it a fun activity for them by giving them torches and give them some examples to work with. </p>
  
  <p>Maybe you can use some cartoon characters pairs: Tom or Jerry, Buggs Bunny or Daffy Duck, etc. <em>(unsure whether kids watch these classic cartoons anymore, so you can find a more relatable cartoon pair)</em> or maybe some example from basic technology <em>(Our 6-8yr old these days are pretty good at basic knowledge of mobile phones, TVs, etc.)</em></p>
  
  <p>Having said this all, I confess its not easy to convey it all to kids who have just begun the decimal system. The first time ever I heard of binary (it was part of text) was at the age of 12 and even then, I couldn't grasp it too quickly. I feel for the kids if they are being asked to perform the binary arithmetic as well.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-04T21:25:52.937Z
Answers.Text: It looks like you chose torches out of 'em all... I think you're going to be very responsible in front of kids. :-P
Answers.UserDisplayName: EKons
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T14:48:37.637Z
Answers.Text: @EKons Lol I haven't ever taught the primary school kids but LEDs are safe ;)
Answers.UserDisplayName: Failed Scientist
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T15:00:16.870Z
Answers.Text: No, the joke is that I meant real torches, i.e. fire-powered...
Answers.UserDisplayName: EKons
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T23:26:13.297Z
Answers.Text: @EKons, in British English, "torch" simply means "flashlight."
Answers.UserDisplayName: Wildcard
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-06T03:46:57.410Z
Answers.Text: @EKons Yeah I did get you there. Reminds me of a student who ignited fire crackers in evening school back in my college days
Answers.UserDisplayName: Failed Scientist
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-03T04:03:50.347Z
Answers.LastActivityDate: 2018-11-03T04:03:50.347Z
Answers.UserDisplayName: Failed Scientist
Answers.Body: <p>The largest problem is probably trying to sell binary code as a number system, particularly in this age group. As some other answers suggested the combinatorics aproach is probably the way to go. I would even stay away from 0s and 1s because that already has another meaning to them.</p>
  
  <p>I also think that the beads is a good starter to first give them a grasp of how the number of possibilities in which to order them.<br>
  My idea would be a game where 5 kids are sitting in a row. This is basically the second phase after the beads. Each of the 5 kids has a number assigned in the order 16, 8, 4, 2, 1. Then the class is asked for a number between 0 and 31. If it is 21 the procedure goes like this. The first kid just has to check if his number fits into the target number. In this case it does and he stands up, the next one (or the class) has to now check the difference 21-16=5. Again check if the nummer fits, which it doesn't, and the kids remains seated and might say "done". The next on fits again and get's up, leaving a remainder of 1, kid no. 4 remains seated and kid no. 5 gets up for that remaining 1. This could be written to the board as 21 = +o+o+ or filled an empty circles or the class can be asked which symbols they would prefer. <strong>u</strong> and <strong>d</strong> might also be intuitive for <strong>up</strong> and <strong>down</strong>.<br>
  There can be a few examples, eventually well picked neighbouring numbers to show a flip of 2 adjacent positions.<br>
  If it goes well up to that point you can even count up, but as a teacher you should be familiar with that, so that it goes right and the system is applied correctly. To explain what I mean it should go like this. Right kid gets up "one", first kid down and second up "two", first kid up "three", first and second down and third up "four" ... Imagine the power of for example 8 when 3 kids sit down and one gets up.<br>
  To finally close the bridge to a general application of this you can explain that the same can be done for the alphabet and instead of counting up numbers you can also count up letters.<br>
  It is possible that 31 is a bit high for 5 year olds for the calculation part.
  The good part is the physical involvement, which I think is a useful tool to enter their brains.</p>
  
  <p>What also just came into my mind is Morse Code, which is also somehow binary in using short and long sounds. But I really have no idea if kids today know anything about it and if this would be a useful tool. Maybe a question to the class might give some feedback in that regard. It could also be mentioned on the side in a few explaining sentences.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T10:31:34.727Z
Answers.Text: I wonder about extending this to show how 5 beads and 5 boxes are enough to represent any letter of the alphabet (or a variation on that), but I have no idea how accessible this would be to the audience.
Answers.UserDisplayName: Sean Houlihane
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T01:03:45.437Z
Answers.LastActivityDate: 2018-11-05T01:09:08.523Z
Answers.LastEditDate: 2018-11-05T01:09:08.523Z
Answers.UserDisplayName: Javatasse
Answers.Body: <p>Your students already know what 1 and 0 mean, and those meanings are applicable to binary arithmetic. With different glyphs, your students will have to mentally decode those glyphs into 1s and 0s in order to understand what you are saying, so it will be harder to understand than it should be.</p>
  
  <p>Also, ⊗ looks a lot like ⊕, which could get confusing later if you want to discuss Boolean logic.</p>
  
  <p>I suggest representing the radix with parenthesis and subscript eg <span class="math-container">$(10101010)_2$</span>. Or use the binary literal format of a modern programming language eg <code>0b10101010</code>. These make the radix unambiguous without confusing your students.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T01:57:15.623Z
Answers.Text: What is the difference between the symbols in Boolean Logic? I've been using them synonymously, perhaps in error!
Answers.UserDisplayName: Ben I.
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-11-05T09:53:07.300Z
Answers.Text: @BenI. ⊕ means "exclusive OR" or XOR. ⊗ means "tensor product", but nothing related to Boolean logic AFAIK. Here's a list https://en.wikipedia.org/wiki/List_of_mathematical_symbols
Answers.UserDisplayName: James Hollis
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Answers.CreationDate: 2018-11-05T01:37:04.527Z
Answers.LastActivityDate: 2018-11-05T01:37:04.527Z
Answers.UserDisplayName: James Hollis
Answers.Body: <p>I just taught my 8yo daugter to count in binary, in her head, &amp; very quickly.<br></p>
  <p>First we count in base-10:<br>
  <em><strong>&quot;Count from 0 to 100&quot;</strong></em><br>
  0,1,2,3,4,5,6,7,8,9 ... 98,99,100.<br></p>
  <p>Then base-8<br>
  <em><strong>&quot;Count from 0 to 100 and skip every number with an 8 or 9&quot;</strong></em><br>
  0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,21 .... 75,76,77,100</p>
  <p>Then base-5<br>
  <em><strong>&quot;Count from 0 to 100 and skip every number with a 5,6,7,8 or 9&quot;</strong></em><br>
  0,1,2,3,4,10,11,12,13,14,20,21 ... 43,44,100<br></p>
  <p>Then base-2 or binary.<br>
  <em><strong>&quot;Count from 0 to 1000 and skip every number with a 2,3,4,5,6,7,8 or 9&quot;</strong></em><br>
  0,1,10,11,100,101 ... 1000<br></p>
  <p>Then to understand the value of each binary numbers, write down the sequence in a table 0 is 0, 1 is 1, 10 is 2, 11 is 3 etc<br></p>
  <p>She quickly realised she was 1000 years old.<br></p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-01-25T14:57:36.357Z
Answers.Text: This is a really clever approach! I'm going to try it with my kid, but I'm also going to try it with my students.  The "how to count" part has always been a bit hand-wavy at the beginning of my binary unit.  I kinda show them and hope that they catch the pattern.  This is a much clearer approach.
Answers.UserDisplayName: Ben I.
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Answers.CreationDate: 2023-01-25T11:43:41.673Z
Answers.LastActivityDate: 2023-01-25T11:43:41.673Z
Answers.UserDisplayName: MathDad
Answers.Body: <p>I once saw how my brother had figured out binary numbers (many, many years ago) and it seemed just so intuitive! And I figured out an even better way of showing it.</p>
  
  <p>You've no doubt seen mechanical counters. Like this:</p>
  
  <p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Gebuehrenzaehler57.JPG/240px-Gebuehrenzaehler57.JPG" alt="Mechanical counter"></p>
  
  <p>In the counter you have multiple rolls with digits on them. When the first roll completes a full circle, the second one moves ahead by one. Etc.</p>
  
  <p>Binary is the same, except instead of 10 digits on each roll, you have just two. But counting still works! There are even some videos on YouTube where someone has made a mechanical binary counter.</p>
  
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Answers.CreationDate: 2018-11-05T19:15:25.627Z
Answers.LastActivityDate: 2018-11-05T19:15:25.627Z
Answers.UserDisplayName: Vilx-
Body: <p>I'm teaching some introduction lessons to a classroom of 6-8 year olds about basic 21st century skills. A fundamental part of this is how computers represent numbers, words and images, which all starts with understanding binary representation of a number.</p>
  <p>When explaining this in the past, I have never not confused anyone with notation. For example trying to explain the difference between '11' (binary) and '11' (decimal) is hopelessly confusing because it looks like the exact same thing, and saying 'one one' almost always leads someone to think 'well one plus one is two, what is this dude going on about it being 3'.</p>
  <p>I think the problem is that we use the same symbol for decimal and binary notation here, while there is not strictly a reason for it (it's convenient if you already know how it all works, but not before that).</p>
  <p>So I'm thinking of using different symbols. Like ● for 'this bit is on', and ⊗ for 'this bit is off', so that I would write e.g.</p>
  <pre><code>3 = ●●
  </code></pre>
  <p>and</p>
  <pre><code>5 = ●⊗●
  </code></pre>
  <p>Any ideas on this? Would this help? Are there already other symbols being used for this purpose, symbols that don't rely on the glyphs we use for 0-9?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-02T13:49:08.900Z
Comments.Text: Excellent first question here.  Welcome to [cseducators.se]!
Comments.UserDisplayName: Ben I.
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-02T16:52:48.683Z
Comments.Text: number is not the foundation of all of the other stuff. It is just on interpretation of a sequence of bits. It is no more real than any other. Therefore consider starting at some other place.
Comments.UserDisplayName: ctrl-alt-delor
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-02T16:54:21.990Z
Comments.Text: This may be of use https://csunplugged.org/en/topics/binary-numbers/ or https://classic.csunplugged.org/binary-numbers/
Comments.UserDisplayName: ctrl-alt-delor
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-02T16:55:16.420Z
Comments.Text: What other number bases do we use, in everyday life?
Comments.UserDisplayName: ctrl-alt-delor
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-02T21:34:06.707Z
Comments.Text: I'm not convinced that understanding base two numbers is a "basic 21st century skill."  It is an _essential_ thing to know if you want to understand how electronic circuits can perform mathematical and logical functions, but you can write an awful lot of computer software these days without even having a strong understanding of base 10, let alone other place-value number systems.
Comments.UserDisplayName: Solomon Slow
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-03T02:16:08.337Z
Comments.Text: Different symbols may work well, but it also might help to use a verbose notation like `0b11` = `0d3`. By making use of prefixes it could make it easier to grasp that those are the same value just written in different ways.
Comments.UserDisplayName: isaacparrot
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-03T03:55:49.127Z
Comments.Text: If you talk about 11 and 11 with grownups who are math literate, there will still be confusion. You need an indication of the base if you're mixing bases: a little subscript *b* or *d* or something: $11_b$ versus $11_d$.
Comments.UserDisplayName: Kaz
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-04T00:38:04.973Z
Comments.Text: [Vaguely relevant joke](https://m.imgur.com/II5W6Pl)
Comments.UserDisplayName: Darren H
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-04T12:23:25.363Z
Comments.Text: @ctrl-alt-delor Sure, it's true that all computers just work on some sort of signals that represent some state, and there is no inherent meaning. I can also say that computers don't need electricity, because it's possible to build a computer that doesn't use electricity. Apart from such pedantry though, a common and useful 'baseline abstraction' is the one where bits are aggregated to 8-bit bytes, where strings are sequences of encoded numbers and bitmaps are arrays or matrices of 'pixels', each of which is 3 or 4 numbers. But that wasn't what my question was about.
Comments.UserDisplayName: Roel
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-04T12:26:14.880Z
Comments.Text: @Solomon Slow The point is not to give a 'programming' class; in fact, the explicit goal is for it to *not* be a programming class. We have plenty of people who know how to write software without understanding numbers, and some of that software is even useful; but in primary education, we're concerned with teaching children foundations and concepts, not with whipping them into CRUD app slingers asap. But again, this is a wholly different discussion, and one that is as old as CS departments themselves are, for that matter.
Comments.UserDisplayName: Roel
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-04T12:27:28.243Z
Comments.Text: @Kaz fair enough, and I think I should indeed do so when talking about this with adults. Thanks, I hadn't really internalized that this would make a difference.
Comments.UserDisplayName: Roel
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-04T12:42:51.733Z
Comments.Text: Additional thought: how many working programmers need to write 'sort' algorithms today? Yet one of the most basic 'CS unplugged' (i..e, without using any electronics) exercises is to get a group to devise an algorithm to sort that group from shortest to tallest. It's not about job skills, it's about conceptual understandings how things work. Another: how often do people today open up their computers? Yet letting them unscrew a case and look inside is very illuminating, when you can point to various components that together make up a von Neumann architecture (even if you never use that term).
Comments.UserDisplayName: Roel
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-04T12:59:26.247Z
Comments.Text: @Roel algorithms are important (even if you never implement a sort). Other ways to see the bits (not as numbers), is fundamental (not implementation). See work of Ada of Lovelace for examples. Bits can represent images, audio, text …, and you don't have to go through number to explain it. See also Euclid, for an example of an explanation of geometry without reference to number. (Numbers are over used in the teaching of maths in schools.)
Comments.UserDisplayName: ctrl-alt-delor
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2018-11-05T01:15:28.813Z
Comments.Text: You might find inspiration from "[exploding dots](https://www.explodingdots.org/)", whose lower levels are suitable for fairly young students. Briefly, a row of boxes with 0 or 1 dot in them, and when two or more dots are in a box, they "explode" for the carried bit.
Comments.UserDisplayName: Mark S.
Tags: <introductory-lesson>
Title: Teaching binary to 6-8 year olds
ThreadText: # Teaching binary to 6-8 year olds
  
  (Asked by: username_0 on 2018-11-02)
  
  I'm teaching some introduction lessons to a classroom of 6-8 year olds about basic 21st century skills. A fundamental part of this is how computers represent numbers, words and images, which all starts with understanding binary representation of a number.
  
  When explaining this in the past, I have never not confused anyone with notation. For example trying to explain the difference between '11' (binary) and '11' (decimal) is hopelessly confusing because it looks like the exact same thing, and saying 'one one' almost always leads someone to think 'well one plus one is two, what is this dude going on about it being 3'.
  
  I think the problem is that we use the same symbol for decimal and binary notation here, while there is not strictly a reason for it (it's convenient if you already know how it all works, but not before that).
  
  So I'm thinking of using different symbols. Like ● for 'this bit is on', and ⊗ for 'this bit is off', so that I would write e.g.
  
  ```
  3 = ●●
  
  ```
  
  and
  
  ```
  5 = ●⊗●
  
  ```
  
  Any ideas on this? Would this help? Are there already other symbols being used for this purpose, symbols that don't rely on the glyphs we use for 0-9?
  
  **Tags:** introductory-lesson
  
  > **Comments:**
  > **username_1**: Excellent first question here.  Welcome to [cseducators.se]!  
  > **username_2**: number is not the foundation of all of the other stuff. It is just on interpretation of a sequence of bits. It is no more real than any other. Therefore consider starting at some other place.  
  > **username_2**: This may be of use https://csunplugged.org/en/topics/binary-numbers/ or https://classic.csunplugged.org/binary-numbers/  
  > **username_2**: What other number bases do we use, in everyday life?  
  > **username_3**: I'm not convinced that understanding base two numbers is a "basic 21st century skill."  It is an _essential_ thing to know if you want to understand how electronic circuits can perform mathematical and logical functions, but you can write an awful lot of computer software these days without even having a strong understanding of base 10, let alone other place-value number systems.  
  > **username_4**: Different symbols may work well, but it also might help to use a verbose notation like `0b11` = `0d3`. By making use of prefixes it could make it easier to grasp that those are the same value just written in different ways.  
  > **username_5**: If you talk about 11 and 11 with grownups who are math literate, there will still be confusion. You need an indication of the base if you're mixing bases: a little subscript *b* or *d* or something: $11_b$ versus $11_d$.  
  > **username_6**: [Vaguely relevant joke](https://m.imgur.com/II5W6Pl)  
  > **username_0**: @username_2 Sure, it's true that all computers just work on some sort of signals that represent some state, and there is no inherent meaning. I can also say that computers don't need electricity, because it's possible to build a computer that doesn't use electricity. Apart from such pedantry though, a common and useful 'baseline abstraction' is the one where bits are aggregated to 8-bit bytes, where strings are sequences of encoded numbers and bitmaps are arrays or matrices of 'pixels', each of which is 3 or 4 numbers. But that wasn't what my question was about.  
  > **username_0**: @username_3 The point is not to give a 'programming' class; in fact, the explicit goal is for it to *not* be a programming class. We have plenty of people who know how to write software without understanding numbers, and some of that software is even useful; but in primary education, we're concerned with teaching children foundations and concepts, not with whipping them into CRUD app slingers asap. But again, this is a wholly different discussion, and one that is as old as CS departments themselves are, for that matter.  
  > **username_0**: @username_5 fair enough, and I think I should indeed do so when talking about this with adults. Thanks, I hadn't really internalized that this would make a difference.  
  > **username_0**: Additional thought: how many working programmers need to write 'sort' algorithms today? Yet one of the most basic 'CS unplugged' (i..e, without using any electronics) exercises is to get a group to devise an algorithm to sort that group from shortest to tallest. It's not about job skills, it's about conceptual understandings how things work. Another: how often do people today open up their computers? Yet letting them unscrew a case and look inside is very illuminating, when you can point to various components that together make up a von Neumann architecture (even if you never use that term).  
  > **username_2**: @username_0 algorithms are important (even if you never implement a sort). Other ways to see the bits (not as numbers), is fundamental (not implementation). See work of Ada of Lovelace for examples. Bits can represent images, audio, text …, and you don't have to go through number to explain it. See also Euclid, for an example of an explanation of geometry without reference to number. (Numbers are over used in the teaching of maths in schools.)  
  > **username_7**: You might find inspiration from "[exploding dots](https://www.explodingdots.org/)", whose lower levels are suitable for fairly young students. Briefly, a row of boxes with 0 or 1 dot in them, and when two or more dots are in a box, they "explode" for the carried bit.  
  
  ---
  
  ## 20 answers
  
  ### Answer by username_23 on 2018-11-02. Score: 40
  
  Use money. Tell your students to image a country has bills that are in denominations of powers of 2: 1, 2, 4, 8, 16.... Now tell them this: For a given number, count out that money with as few bills as possible. You use a "greedy algorithm:"
  
  1. Use the largest bill possible.
  2. Repeat this until you have counted out the amount.
  
  Questions to ask:
  
  * Did you ever use more than one bill from any denomination?
  
  They should observe this is never necessary.
  
  Now write a 1 for the largest bill. If the next denomination down is not present, put a 0; otherwise, put a 1. Repeat until you descend through the denominations. You have the binary expansion of the integer you specified.
  
  This is concrete and very easy for students to grasp. This "greedy algorithm" works for any base: Use the largest bill you can can count out as many as you can at each stage.
  
  > **Comments:**
  > **username_0**: Thanks, yes this seems like a good way to go in terms of giving someone an algorithm to do conversions. I'm not convinced though that it's a clear way to explain the concept of there being different number bases at all, since I would still be talking in terms of 'count out 6' or 'count out 13'. But still, I am going to try to work this into my preparation.  
  
  ---
  
  ### Answer by username_24 on 2018-11-02. Score: 14
  
  Just to calibrate expectations: at the lower end of that age range, the kids will have only just started learning about the place-value system in decimal. I wouldn't try to leverage knowledge of things like "carrying" in decimal over to binary.
  
  My own kids are at boundaries of your age range, and in first and second grade they use things like "tens frames" and "hundreds frames". You might consider making twos/fours/eights frames or perhaps better: blocks, $1\times 2$ joined block pairs, $2\times 2$ squares of joined blocks, and $2\times 2\times 2$ cubes of joined blocks. Then show them (or let them explore) how you only ever need at most one of each block type to make a number. Binary then just consists of a table of the number of each block type that you used.
  
  > **Comments:**
  > **username_0**: My children are 5 and 7, so they will be my guinea pigs :) Part of the challenge is that this is for a Montessori school with so-called 'vertical groups', i.e. children of ages 6-8 are in one classroom (so are children 4-5, and 9-11). So I might need to differentiate the material a bit for various sub-groups. I don't quite understand what you mean by 'tens frames', can you elaborate?  
  > **username_24**: Tens frames are size 10 grids (usually $2\times 5$) which the student fills in to visualize counting up to 10.  The following PDF is a good description and has a link to youtube which is even more thorough: https://www.ru.ac.za/media/rhodesuniversity/content/sanc/documents/SANC%2010-frames%20reading.pdf  
  > **username_0**: Oh I see, it's an interesting concept, and one I think would adapt well to counting in various bases. I will talk about this with the teachers, see how it fits within the overall pedagogical math framework they use; I don't want to get in the way of how they teach basic counting, of course. Thanks!  
  
  ---
  
  ### Answer by username_19 on 2018-11-02. Score: 11
  
  At that age, I would skip binary notation entirely and focus on **combinatorics** problems (simple ones) whose answers are most clearly understood as combinations of powers of two.
  
  Any combinatorics (counting) problem where the full set of answers can be *systematically written down by hand,* and then counted, is the most appropriate. Or if it can be physically constructed, that's even better than being written down.
  
  For instance, someone else mentioned colored beads. That's great. So get some actual beads, red and green or whatever, just two colors, and sit the student down and ask, "How many *different* patterns of three beads in a row can you make with these?" Of course they'll eventually come up with the answer "8," but it may take them a while to get there.
  
  ---
  
  When you consider it in depth, the fundamental reason why binary notation is useful *is* because it naturally represents the answers to combinatorics problems involving two possible states. If our computer systems were to have three possible states rather than two, then ternary notation would be a more natural system for representing computer data.
  
  From that standpoint, introducing combinatorics *first* before discussing notation is simply a single instance of a more general teaching philosophy I follow: in particular, that the formulas or notations must come *after* the underlying *reasons* why they're important to know.
  
  This principle can be sloppily summarized as "motivating examples" but that loses a lot in translation; really what I'm saying is to ["Form patterns of computation in the individual with regard only to their usefulness."](http://www.appliedscholastics.org/l-ron-hubbard/teaching.html) Binary notation *is* useful, but it's more important to lead the student to realize that for himself or herself, than it is to teach it as a rote pattern.
  
  ---
  
  The keynote with regard to conveying that "1001101" can be used to represent the same quantity as "77" is to start with the *meanings* of those symbols, and worry about the actual symbols used later on. This could all be done in one short tutoring session so long as the student's mind is coaxed into action rather than passively recording the information that you're "teaching."
  
  I would **have the student choose his or her own symbols to use.** They're completely unimportant, so don't let the student get hung up on them. I might start out by using stars and diamonds, just to make the point. Those are obviously inconvenient to draw. So then we might switch to circles and squares at the student's suggestion. And then squares and circles, just to switch things up. And then I could switch us to minus and plus for a minute or two, and then tell the student:
  
  > So you can see it doesn't really matter what pictures we use or what we draw, what matters is that we have two of them, and that we've decided which of the two comes first. You can use anything you like and you'll still wind up with the same answer, 8, when you list the possibilities out and count them up. But the pictures that we *usually* use, so other people will understand our list also, are just "0" and "1". And 0 comes before 1, so then our list would look like:
  >
  > 000  
  > 001  
  > 010  
  > 011  
  > 100  
  > 101  
  > 110  
  > 111
  
  Then you can number the list "1" through "8" and the connection should come pretty quickly, at least with an 8 year old. (A 6 year old I would do the same thing but not try to teach actual binary numbers.)
  
  > **Comments:**
  > **username_20**: Interestingly, tertiary computers *were* built. They *did have* advantages, but the problem was that because of the way they were built, using the exact same hardware and removing a piece gave a quaternary computer...aka binary computer with double the bit resolution.  
  
  ---
  
  ### Answer by username_1 on 2018-11-02. Score: 10
  
  I was going to start by asking what was wrong with $11\_2$ and $11\_{10}$, and then I noticed the ages you were referring to. Six year olds!
  
  First, a bit of frame-challenging advice: I'm not sure that counting in binary is what I would focus on at that age at all. You might have better luck with finite state machines, which are great CS concepts that you can play lots of physical games with. You can put states on the floor and have kids jump around, or you can have kids pass picture-notes to one another based on state and observe the results. You can even have them try to draw out simple machines by watching behaviors of other kids acting out your "script".
  
  However, if the goal *really is* binary counting at 6-8 years old, I would start with symbols like "•" and "◦", or even start with symbols like
  
  ```
  □□    □    
  □□    □    □    L    |
  
  ```
  
  ... using "count the lines" as a property, and then extend that to dots, so
  
  ```
  □□    □    
  □□    □    □    L    |
  •     ◦    •    ◦    •
  
  ```
  
  ... becomes ...
  
  ```
  □□         
  □□         □         |
  •          •         •
  
  ```
  
  ... becomes 21. And only when they're familiar with the dots, at the very end of the lesson, I'd get into something *very silly* that people do with this system! They use 1s and 0s for •s and ◦s! If the kids laugh, then your joke landed, and the kids will be beginning to see.
  
  Then see if they can connect 10101 with
  
  ```
  □□         
  □□         □         |
  
  ```
  
  Good luck!
  
  > **Comments:**
  > **username_2**: I always wonder what number-base should the subscripts be written in.  
  > **username_5**: The subscripts shouldn't be numeric. They should be *b* and *d*. Binary and decimal.  Or else the word $1011_\text{two}$, $123_\text{ten}$.  Of course, if they are numeric, they must use some common understood base (like ten); if they use the indicated number's **own** base, it will always uselessly be "10".  
  > **username_1**: @kaz if the preferred system is a single letter, how would you deal with duodecimal? I could accept using the full word, but there is no advantage there to the digits, as the full word is also in decimal.  
  > **username_0**: I wasn't sure about whether this would be suitable for this age either, but I'm working off a curriculum set by the Dutch association for primary education who put this material with this age bracket, so I'm just going with it. The more I get into it though, the more I start to see how this is basically a launching pad for tying together other things - like how do you represent 'strings' when all you have to work with are numbers (and base 2 numbers for that matter), or even higher level concepts like 'video' or anything else that can be represented by computers, really.  
  > **username_21**: @username_0 That representation issue was a massive elephant in the room to me for a very long time, well past the point that I was writing commercial code. You should work it into a new question...  
  > **username_5**: @username_1. I simply wouldn't deal with duodecimal; I'm not aware of it being widely used. I mean, of course kids have to learn about arbitrary bases, but maybe at the stage where they are confused by the interpretation of digits, the common ones used in computing are good enough.  
  > **username_5**: ^  And of course sexagesimal, ubiquitous in time representation.  
  
  ---
  
  ### Answer by username_22 on 2018-11-02. Score: 9
  
  We made bracelets in APCS Principles (9th - 12th grade). I bought a bunch of blue and white (school colors) beads. Blue was on, white was off. They then made bracelets of their names or initials using 5 bits. A was 1 (w-w-w-w-b), B was 2 (w-w-w-b-w), etc.
  
  Kids seemed to enjoy it, and I caught a few wearing their bracelets walking around the halls days later.
  
  My own kids, 6 and 9, came in after school and saw the beads and wanted to make their own bracelets. By the time they were finished the 9 year old had made 6 or 8 bracelets for different friends and was going to explain how they worked the next day at school.
  
  Only thing I'm planning on changing for next year is adding a different bead color to represent the start / end of the string.
  
  > **Comments:**
  > **username_0**: I'm working off a curriculum, proposed by what is essentially 'the Dutch association for primary education', and they have an exercise in making bracelets. I was a bit hesitant about that though, representing essentially 'something' and 'nothing' with two color, but when I think about it, it's no worse than using the symbols I described. So in your experience, children were able to make the link between the concept of an 'on bit' and 'off bit' and the colors quite readily? 9th-12th grade, that is ages 8-11, right?  
  > **username_22**: 9-12th is students 14-18 years old. The activity came from CSUnplugged, and was meant for younger students, but worked pretty well with the older students too.  
  > **username_10**: I've made bead necklaces using Morse code. It was easy to read them, just sound it out in my head.  
  
  ---
  
  ### Answer by username_25 on 2018-11-04. Score: 7
  
  The way I got my kids interested in binary representation was to discuss up to what number they could count on one hand. Learning they could count up to 31 and then understanding why, made the trick. You can then impress them yet again when you show them that with ten fingers then can count to 1023.
  
  > **Comments:**
  > **username_10**: Just be careful showing some of those numbers.  
  > **username_25**: Indeed, there was some giggling with "four".  
  
  ---
  
  ### Answer by username_26 on 2018-11-02. Score: 7
  
  Sometimes light bulb on/off symbols are used, and in fact if you show a real circuit with a battery, a switch and a light bulb before you start teaching binary numbers you could explain why the computer needs just two symbols to count.
  
  Anyway, as a personal recollection, I learned binary on my own thanks to an 80's version of this toy: <https://www.youtube.com/watch?v=EM_-APWFX2s>
  
  > **Comments:**
  > **username_0**: That is a really cool toy, I now have to order this :) Thanks.  
  > **username_26**: By the way, "rami" is Italian for "branches"  
  
  ---
  
  ### Answer by username_9 on 2018-11-02. Score: 6
  
  Please see the note in the last few paragraphs for an important caveat.
  
  Before you get too deep into this, you should look at the work of Piaget. For example: <http://alumni.media.mit.edu/~stefanm/society/som_final.html>. Note that children of that age have almost no ability to do abstraction and won't for several years. Whatever you do has to be very concrete.
  
  For a wider discussion, see: <https://www.verywellmind.com/piagets-stages-of-cognitive-development-2795457>.
  
  In particular, while symbols can be meaningful to them, they are still likely pre-operational.
  
  You might also think about contacting a local Montessori school to see what goes on there with students of that age.
  
  ---
  
  Note, of course, that the reason that they can't grok using old symbols for new ideas is, precisely, the lack of abstract thinking ability. Catch them again around age 12 or so.
  
  So, thinking a bit more on this topic, let me suggest that introducing new symbols might be counterproductive. They may get the idea firmly embedded in their minds that you are teaching them things that are important and should be retained. Then, in junior high or so, they will have to *unlearn* what you so carefully taught them. In my view, it is a serious error to teach people things they later need to unlearn/forget/ignore/reject.
  
  I once had a college student who couldn't grasp the idea of operator precedence. It turns out he was taught the wrong rules in HS and internalized them deeply. He couldn't understand why his Java programs didn't work correctly. His earlier teacher was highly respected so he fought me rather than learn the correct rules. He though I was lying to him that Java performed multiplications before additions rather than strict left to right evaluation of all operators.
  
  I'm starting to think this may be a minefield best left unexplored.
  
  > **Comments:**
  > **username_0**: Actually, this is at a Montessori school. I'm working with a few of the teachers who want to incorporate some of this into their curriculum, although me, I'm just a dad who is doing one or two short introduction sessions, as a way to explore where to go from there. Montessori actually has a system with beads (and 'sticks' which are 10 beads, and 'plates' which are 10 sticks) to teach about the importance of the place of a digit in a number. I'm hoping to find a way to tie in with that.  
  
  ---
  
  ### Answer by username_27 on 2018-11-03. Score: 3
  
  An example from Basic Computing book is coming to my mind here: Suppose you have some torches or light bulbs and want to convey a message using the combination of those bulbs/torches/LEDs.
  
  I can understand that this example can still be bit too much for 6-8 years old, but you can make it a fun activity for them by giving them torches and give them some examples to work with.
  
  Maybe you can use some cartoon characters pairs: Tom or Jerry, Buggs Bunny or Daffy Duck, etc. *(unsure whether kids watch these classic cartoons anymore, so you can find a more relatable cartoon pair)* or maybe some example from basic technology *(Our 6-8yr old these days are pretty good at basic knowledge of mobile phones, TVs, etc.)*
  
  Having said this all, I confess its not easy to convey it all to kids who have just begun the decimal system. The first time ever I heard of binary (it was part of text) was at the age of 12 and even then, I couldn't grasp it too quickly. I feel for the kids if they are being asked to perform the binary arithmetic as well.
  
  > **Comments:**
  > **username_28**: It looks like you chose torches out of 'em all... I think you're going to be very responsible in front of kids. :-P  
  > **username_27**: @username_28 Lol I haven't ever taught the primary school kids but LEDs are safe ;)  
  > **username_28**: No, the joke is that I meant real torches, i.e. fire-powered...  
  > **username_19**: @username_28, in British English, "torch" simply means "flashlight."  
  > **username_27**: @username_28 Yeah I did get you there. Reminds me of a student who ignited fire crackers in evening school back in my college days  
  
  ---
  
  ### Answer by username_29 on 2018-11-05. Score: 3
  
  The largest problem is probably trying to sell binary code as a number system, particularly in this age group. As some other answers suggested the combinatorics aproach is probably the way to go. I would even stay away from 0s and 1s because that already has another meaning to them.
  
  I also think that the beads is a good starter to first give them a grasp of how the number of possibilities in which to order them.  
  My idea would be a game where 5 kids are sitting in a row. This is basically the second phase after the beads. Each of the 5 kids has a number assigned in the order 16, 8, 4, 2, 1. Then the class is asked for a number between 0 and 31. If it is 21 the procedure goes like this. The first kid just has to check if his number fits into the target number. In this case it does and he stands up, the next one (or the class) has to now check the difference 21-16=5. Again check if the nummer fits, which it doesn't, and the kids remains seated and might say "done". The next on fits again and get's up, leaving a remainder of 1, kid no. 4 remains seated and kid no. 5 gets up for that remaining 1. This could be written to the board as 21 = +o+o+ or filled an empty circles or the class can be asked which symbols they would prefer. **u** and **d** might also be intuitive for **up** and **down**.  
  There can be a few examples, eventually well picked neighbouring numbers to show a flip of 2 adjacent positions.  
  If it goes well up to that point you can even count up, but as a teacher you should be familiar with that, so that it goes right and the system is applied correctly. To explain what I mean it should go like this. Right kid gets up "one", first kid down and second up "two", first kid up "three", first and second down and third up "four" ... Imagine the power of for example 8 when 3 kids sit down and one gets up.  
  To finally close the bridge to a general application of this you can explain that the same can be done for the alphabet and instead of counting up numbers you can also count up letters.  
  It is possible that 31 is a bit high for 5 year olds for the calculation part.
  The good part is the physical involvement, which I think is a useful tool to enter their brains.
  
  What also just came into my mind is Morse Code, which is also somehow binary in using short and long sounds. But I really have no idea if kids today know anything about it and if this would be a useful tool. Maybe a question to the class might give some feedback in that regard. It could also be mentioned on the side in a few explaining sentences.
  
  > **Comments:**
  > **username_21**: I wonder about extending this to show how 5 beads and 5 boxes are enough to represent any letter of the alphabet (or a variation on that), but I have no idea how accessible this would be to the audience.  
  
  ---
  
  ### Answer by username_30 on 2018-11-05. Score: 2
  
  Your students already know what 1 and 0 mean, and those meanings are applicable to binary arithmetic. With different glyphs, your students will have to mentally decode those glyphs into 1s and 0s in order to understand what you are saying, so it will be harder to understand than it should be.
  
  Also, ⊗ looks a lot like ⊕, which could get confusing later if you want to discuss Boolean logic.
  
  I suggest representing the radix with parenthesis and subscript eg $(10101010)\_2$. Or use the binary literal format of a modern programming language eg `0b10101010`. These make the radix unambiguous without confusing your students.
  
  > **Comments:**
  > **username_1**: What is the difference between the symbols in Boolean Logic? I've been using them synonymously, perhaps in error!  
  > **username_30**: @BenI. ⊕ means "exclusive OR" or XOR. ⊗ means "tensor product", but nothing related to Boolean logic AFAIK. Here's a list https://en.wikipedia.org/wiki/List_of_mathematical_symbols  
  
  ---
  
  ### Answer by username_31 on 2023-01-25. Score: 2
  
  I just taught my 8yo daugter to count in binary, in her head, & very quickly.
  
  First we count in base-10:  
  ***"Count from 0 to 100"***  
  0,1,2,3,4,5,6,7,8,9 ... 98,99,100.
  
  Then base-8  
  ***"Count from 0 to 100 and skip every number with an 8 or 9"***  
  0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,21 .... 75,76,77,100
  
  Then base-5  
  ***"Count from 0 to 100 and skip every number with a 5,6,7,8 or 9"***  
  0,1,2,3,4,10,11,12,13,14,20,21 ... 43,44,100
  
  Then base-2 or binary.  
  ***"Count from 0 to 1000 and skip every number with a 2,3,4,5,6,7,8 or 9"***  
  0,1,10,11,100,101 ... 1000
  
  Then to understand the value of each binary numbers, write down the sequence in a table 0 is 0, 1 is 1, 10 is 2, 11 is 3 etc
  
  She quickly realised she was 1000 years old.
  
  > **Comments:**
  > **username_1**: This is a really clever approach! I'm going to try it with my kid, but I'm also going to try it with my students.  The "how to count" part has always been a bit hand-wavy at the beginning of my binary unit.  I kinda show them and hope that they catch the pattern.  This is a much clearer approach.  
  
  ---
  
  ### Answer by username_32 on 2018-11-05. Score: 2
  
  I once saw how my brother had figured out binary numbers (many, many years ago) and it seemed just so intuitive! And I figured out an even better way of showing it.
  
  You've no doubt seen mechanical counters. Like this:
  
  ![Mechanical counter](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Gebuehrenzaehler57.JPG/240px-Gebuehrenzaehler57.JPG)
  
  In the counter you have multiple rolls with digits on them. When the first roll completes a full circle, the second one moves ahead by one. Etc.
  
  Binary is the same, except instead of 10 digits on each roll, you have just two. But counting still works! There are even some videos on YouTube where someone has made a mechanical binary counter.
  
  
  ---
  
  ### Answer by username_8 on 2018-11-05. Score: 1
  
  I think the biggest piece of confusion comes from the words representing quantities; in particular, the word TEN.
  
  If you disassociate TEN from the quantity that represents the number of your fingers, and instead frame TEN as word noting when it's time to "start over" with a new quantity set, you are more than halfway towards your goal.
  
  And before you get to binary, and to reinforce that "TEN" is just a word representing a quantity, I'd instead focus on teaching Octal. Once your students can get used to a number system where you skip the "words" EIGHT and NINE, your students will be more able to wrap their brains around binary.
  
  You might introduce the Octal system by way of frogs since frogs have 8 toes total on their front legs. That's to say what would the number system look like if it were created by frogs (8 instead of 10 digits)?
  
  An interesting dynamic also for frogs is that there are 10 toes on their back legs. Perhaps an interesting story could be derived from this where frogs find themselves having to switch between the Octal and Decimal system.
  
  > **Comments:**
  > **username_9**: It is said that if you ask a horse to count to 10 by stamping a foot, you hear "stamp, stamp, stamp, stamp".  
  > **username_10**: Rabbits count One Two Three Many (from Watership Down).  
  > **username_11**: Ten is a certain number of things. So is 10--the same number of things. Where you write "TEN" here you should write "ONE THEN ZERO (WHEN IT MIGHT NOT MEAN TEN)".  
  
  ---
  
  ### Answer by username_12 on 2018-11-03. Score: 1
  
  For German speaking folks take a look at
  
  <https://kinder.wdr.de/tv/die-sendung-mit-der-maus/video-bis--zaehlen-102.html>
  
  where fingers are used.
  
  > **Comments:**
  > **username_1**: Now that I have unmarked this as spam, it's worth noting that this link appears to have gone dead.  Would you mind writing in a simple description?  
  > **username_12**: thanks for the note, I updated the link.  
  > **username_1**: Sorry again about that. I felt awful!  
  > **username_1**: Even still, it would be good if you [added some information to this answer](https://meta.stackexchange.com/q/225370/280611).  
  > **username_13**: This works. I used this to teach binary counting to kids in the target age group back in the early 2000s. Their favourite number of course was the $132_{10}$ in binary.  
  
  ---
  
  ### Answer by username_14 on 2018-11-05. Score: 1
  
  I would start by reviewing how the mathematics curricula explains the numbers to these children.
  
  I have been surprised in the past when looking at education books of young children that when teaching numbers, *they explained bases*! There is a lot of counting numbers as *X hundreds, Y tens Z units*, using colored stickers for each position, etc.
  
  Then you are never again asked to write 123 as 100 + 20 + 3.
  
  However, when having to learn binary or hexadecimal, that's exactly what you need to do, realize that 123d is just 1×10² + 2×10¹ + 3×10⁰ and 123h will be 1×16² + 2×16¹ + 3×16⁰, and so for any base.
  
  So, start by checking how they are|will already learn the numbers and see how to make binary encoding fit into them. If a yellow dot will mean tens, use yellow for 2, not for 8 (maybe with an extra symbol to mean binary), etc.
  
  The curricula will vary a lot between places and editions, and this may not be applicable to your specific case, but I think it's important to adapt to how they are learning the numbers (or will learn them later) so that they can learn the concept as the one thing it really is, instead of being taught by different people as separate and unrelated things.
  
  
  ---
  
  ### Answer by username_18 on 2018-11-07. Score: 1
  
  > I'm teaching some introduction lessons to a classroom of 6-8 year olds about basic 21st century skills. A fundamental part of this is how computers represent numbers, words and images, which all starts with understanding binary representation of a number.
  >
  > ...
  >
  > So I'm thinking of using different symbols. Like ● for 'this bit is on', and ⊗ for 'this bit is off' ...
  >
  > -- username_0
  
  My son just turned 8 and I've run into this same problem many times. I've found it an interesting and rewarding challenge to try to explain concepts without getting too deep into technical details.
  
  For my son, (and even many adults) I've found it extremely valuable to **not** talk much (if at all) about binary, place value, and number bases (at least at first) as bytes are what everyone uses and talks about near daily. I've also found it good to avoid talking about specific numerical quantities and focusing instead on what those quantities represent, such as letters, brightnesses, etc. In some conversations I also found it useful to explain that bytes in a computer are taking up physical space inside the memory of the machine, just as letters on paper take up space.
  
  ## Teaching Binary Numbers:
  
  > Sometimes light bulb on/off symbols are used ...
  >
  > -- username_26
  >
  > An example from Basic Computing book is coming to my mind here: Suppose you have some torches or light bulbs and want to convey a message using the combination of those bulbs/torches/LEDs.
  >
  > -- username_27
  >
  > ... I have never not confused anyone with notation.
  >
  > -- username_0
  
  When I tried to teach my son binary in order to explain how numbers worked in a computer I found that what worked best was to keep things simple and only build from ideas he already knew and nothing else. The usual light switch (or LED) analogy seemed to help some, but working with our hands was best. I considered using blocks, drawing pictures, and so on, but ended up not needing to.
  
  Simple counting, doubling, and adding were things he already knew. Place value, not so much, but it wasn't hard for my son to get the general idea that each of his fingers could represent a number twice as much as the one before it, and you add them all together after you hold them up. Each finger can either be up or down; added or not.
  
  He was then very impressed with how we could count to 31 on just one hand, and 1023 using both hands. This understanding actually came **after** explaining what bytes were and getting him to understand them.
  
  ## Teaching How Computers Store Letters, Words, and Images
  
  The really important part was to explain that a "byte" was a number in the computer, and that it could represent different things, like one letter, or how bright a color was. Explaining that a byte was 8 bits came later when he began learning what bits were.
  
  As for words, I told him that you could lay out the letters of the alphabet in a row and count them, and that's how computers know what letters are, because the people who build computers decided which numbers would be which letters. I was then able to explain from there how many bytes it would take to make various kinds of documents around our house.
  
  Explaining pictures also started with a recap on bytes. I explained that 1 byte told the computer how much of a color was in a picture. The computer would read that number and see how much red was in the picture. I next Googled "color mix venn diagram" so we could have a conversation about how pictures are only made of red, green, and blue, and thus 3 bytes are needed to talk about color. From there I was able to explain that those 3 bytes don't actually describe the entire picture, just one very tiny dot, and that you then need **a lot of bytes** to draw one image.
  
  > **Comments:**
  > **username_1**: I think this might be my favorite answer here. Nicely done!  
  > **username_18**: @BenI. Thank you very much! To be honest, I probably spent way too much time writing this up today. I'm glad it was appreciated. :-)  
  
  ---
  
  ### Answer by username_15 on 2021-01-14. Score: 0
  
  I've used Garlikov's Socratic approach to teaching binary in the classroom to rave reviews:
  
  <http://www.garlikov.com/Soc_Meth.html>
  
  I think not just the format but also the particular questions he asks and how he frames the discussion make his use of this technique effective.
  
  > **Comments:**
  > **username_1**: Hi, and welcome to [cseducators.se]!  This answer has a few problems: first, it's a link-only answer, and since we accumulate a lot of links, we ask that people give at least *some* sort of summary in case the link goes dead. Secondly, (and more importantly) it's not specifically an answer to the question being asked.  You can teach virtually anything using the socratic method (to better or worse effect).  There is nothing about binary or binary encoding within your link.  
  > **username_15**: @BenI. I will update with more detail. That link contains a discussion of binary and binary encoding that takes up the vast, vast majority of the page. Are you sure you followed the correct link?  
  > **username_1**: Ah, I see... I didn't scroll down far enough. On mobile, it appeared to be a very long wall of text about the socratic method, so I didn't go past the first 4 screens.  You're right that there is binary in there.  
  > **username_15**: Yes. Although he mentions binary early on in the post, you're right it doesn't become exclusively focused on teaching binary until you scroll a half page (on my screen). I should have said it was specifically about teaching binary in my original post, though.  
  
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  ### Answer by username_16 on 2020-11-26. Score: 0
  
  username_30 Tanton makes it very clear how binary numbers emerge: Exploding Dots
  
  <https://www.youtube.com/watch?v=bvglHQk-Ygs&list=PLoBaXWXBi2d6JJ-HpXk38SmkCFrGX9iJD&ab_channel=JamesTanton>
  
  
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  ### Answer by username_17 on 2020-09-02. Score: 0
  
  Why don't just consequently use $11\_{2}$ and $11\_{10}$ and teach it the same way it's done at universities? I don't see how this would be any more difficult than starting confusing things with symbols that just add complexity on top of a difficult thing.
  
  The most confusing thing in school for me was when teachers tried to simplicate stuff ...
  
  If it was me, I'd teach them the whole abstract thing and leave simplified constructs unless your students really need it.
  
  > **Comments:**
  > **username_1**: Hello, and welcome to [cseducators.se]! I just removed the bit at the end.  Please note the most recent incarnation of the [Be Nice](https://cseducators.stackexchange.com/conduct) policy.  
  > **username_17**: Thanks for the correction, username_27 to notice the ambiguity.  
  
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