# How to teach binary numbers to 5th graders?

I already tried the direct approach, starting with "this is how it works". That turned out ok but took too long and was boring for all of us.

My second attempt was using the twofingered alien. This worked better but needs improvement. The problem was that most kids in 5th grade are not aware of how the 10 based number system works. They just use it. I ended up having to explain the 10 based system in the middle of my binary alien story. That kind of interrupted the flow. But explaining the 10 based system before starting binary takes away most of the "suprises" of the aliens counting with just two fingers.

edit: I should probably add that I'm a math teacher in Germany and that binary numbers are on the agenda at this grade.

• My kids are already using the base 10 system quite well. The problem is they're not aware of how it works. The "surprise" is that it's possible to count with just two digits instead of 10. Using the alien story gives them more to think about and lets them discover a lot on their own, making it more memorable. I wanted them to see the analogy between the two systems. – Esmaya Sep 10 '14 at 22:14
• Perhaps the very purpose of teaching binary numbers is to highlight the structure of the base 10 system by comparison? In that case, there may be no way around talking about the nature of the base 10 system first. – DavidButlerUofA Sep 10 '14 at 23:01
• To make binary fun, one must also study hexidecimal. The digit grouping trick is really cool. – James S. Cook Sep 11 '14 at 4:58
• Iain Banks's "The Business" (en.wikipedia.org/wiki/The_Business_%28novel%29) has a plot point that turns on the use of binary numbers, which is motivated by the question "How can you count to 1000 on your fingers?" – LSpice Sep 11 '14 at 21:01
• I agree with @DavidButlerUofA that learning binary can be incredibly useful in trying to solidify an understanding of base 10. With my 10th grade geometry students, we did a day on binary and by the end they not only understood base 2 but also were excited about how much more they understood base 10 and were able to make connections to ancient numbering systems (base 60, base 20 etc) – celeriko Nov 13 '14 at 17:39

A sketch of one idea. I think it's probably better spread over a couple of days.

Day one:

Start them counting, from zero, out loud to you. Write the numbers on the board as they go. Zero (0), one (1), two (2), ... , ten (10). Stop here. Prompt a discussion about what happened - how is the most recent number different than all of the previous numbers?

I expect (admittedly I don't have a lot of experience with this age group) that they'll be able to come up with the fact that it's now a two digit number. Press them to explain why we weren't able to express ten as a single digit. With some guidance, I think it's possible for them to come up with something that approximates 'because we don't have any more numbers'. Go into that - explain that the digits 0-9 are symbolic in nature, and that we use them to represent quantities. Since we have ten of them, we're able to express ten different values with single digits, and when we want to express other values we have to use the digits in combination.

Start over now, except ask the class to imagine that there were only 5 numeric symbols in the alphabet (0, 1, 2, 3, 4) and that 5 through 9 never existed. Again, count up from zero to four on the board, but put it to the class how five might be represented. This may be a struggle since lots of them will want very badly to write '5' - potential for some humor here though as you can remind them that '5' doesn't exist.

Hopefully you're able to get the suggestion of '10' (5) out of the class, and then you're able to proceed with '11' (6), '12' (7), on up to '14' (9). If you're able to get them to '20' for ten, then you're set - they get it.

Day two:

Run through the base 5 counting exercise again. 1,2,3,4,10,11,12,13,14,20,21, etc. If things go well enough here then it's time to get into generalized discussion about how this sort of counting works in comparison with 'normal' counting. Ask if there's anything special about counting with 10 digits, or with 5 digits, or with any particular number of digits. I expect that they'll say no.

Move to binary now - tell the kids that there are only two number symbols (0 and 1), and get them to count aloud and instruct you on how to represent the numbers. They'll probably get through 0, 1, 10, 11 without much trouble, but you can help them move from here to 100 if they're stuck by asking about the transitions from 9 to 10 and 99 to 100 in 'normal' numbers. If things are going OK at this point, it might be appropriate to talk about computers, yadda yadda.

Thoughts...

My original thought after reading this question was that binary for elementary students is an awful idea, and representative of the bad habit of 'back-porting' desired skills into inappropriate age groups. (We need computer technicians? Computers use binary! Teaching binary to kids will help!).

But done carefully, it can potentially speak to one of the great joys in math - the whole idea is that you can make suppositions and then wonder about their implications. Suppose 5-9 don't exist? Absurd, but OK. Now how does counting work again?? Uh, did I understand counting in the first place? It's confusing in the way that mathematicians (of any age) love to be confused.

I think it's best to take the intermediary step here (base 5) because it's substantially less of a jump than going straight to binary. In base five, things proceed in the natural fashion most of the time - when incrementing the ones digit - and only the 'rollover' conditions are different. Binary rolls over with every other increment, and requires a carrying rollover very early on, which is really a difficult thing. By learning base 5 first, you make it easier for them to understand the fundamental validity of different numbering systems, which makes it easier to process the ultimate extreme of using binary.

• Very nice idea, I'll be sure to try that. – Esmaya Sep 11 '14 at 15:51
• Base 3, 4, and 8 are good possibilities too. If your aliens have 4 or 8 fingers, you can work your story around this still. – Sue VanHattum Sep 11 '14 at 17:52
• If gas meters are still a thing, I think that they are a great way to visualize what happened between 9 and 10. – Roland Dec 27 '15 at 18:27

When teaching binary to any age group I always start by taking a set of kitchen weights into the classroom. 2lb (32oz) 1lb (16oz) 8oz 4oz 2oz 1oz. Then make a table asking what weights yo1u would use to make up certain values. If a weight is used you show it with a "1" if it is not used you show it with a "0".

                                    32    16    8    4    2    1


What weights would you use for 1oz 1 2oz 1 0 3oz 1 1 4oz 1 0 0 5oz 1 0 1 6oz 1 1 0 7oz 1 1 1 8oz 1 0 0 0 9oz 1 0 0

• This is a great idea! – A.Ellett Jul 21 '17 at 17:16

I've never done this, but maybe something like the following will work. Since they don't really understand base $10$ (indeed, one can consider that being the end goal of doing the binary stuff), you don't want to get all tangled up in binary position notation and the like.

Start by looking at the numbers you get when you begin with $1$ and successively double:

$$1, \;\; 2, \;\; 4, \;\; 8, \;\; 16, \;\; 32, \;\; 64, \;\; 128, \;\; 256, \;\; 512, \;\; \ldots$$

Leave this list on the blackboard or in some other place where the students can see it in what follows.

Then tell the students (in an age-appropriate manner) that even though this process removes relatively more and more numbers from a listing of all positive integers, the numbers that are left have a really neat property. Namely, every positive integer can be written as a sum of these numbers where each of these numbers is either used just once or not at all. Moreover, there is only one way to do this for each positive integer.

For example, $13 = 8 + 4 + 1$ and $23 = 16 + 4 + 2 + 1.$

Then pick a number, say $53,$ and show students how they can find the powers of $2$ that work. First, find the largest power of $2$ that doesn't "overshoot" the integer. In this case it's $32.$ Now subtract $32$ from $53$ to see what's left: $53 - 32 = 21.$ Now find the largest power of $2$ that doesn't "overshoot" the part that's left. This will be $16.$ Next, subtract $16$ from $21$ to get $5,$ and so on in the manner we all know about. At the end, tell the students that we'll record the result using a shorthand code. In the case of $53,$ the shorthand code is $110101$ because (reading the $0$'s and $1$'s from right to left) we used $32,$ we used $16,$ we didn't use $8,$ we used $4,$ we didn't use $2,$ and we used $1.$

You can mention that the shorthand code always starts with a $1$ (because there will always be a largest power of $2$ we can use), but after that anything can happen.

Perhaps then give students two or three examples to try among themselves (small groups ought to be great for this). After most students seem to understand how to carry this out, you can then consider the reverse problem where someone gives you a code for the number and you figure out what the number is, such as $10010$ being $2 + 16 = 18$ and $110011$ being $1 + 2 + 16 + 32 = 51.$ Point out that when deciphering the code for a number, it's easiest to work through the digits from right to left (opposite the direction we used when finding the codes).

So far there is nothing about positional notation or how this is base $2$ and we're used to working with base $10,$ but at least this will get across some aspects of binary notation without delving into abstract positional notation system stuff (which will likely fly right over their heads if you start there).

You might also be able to work in the game of $20$ questions, explaining (carefully!) how the number of digits in the code tells you how many yes/no questions are needed.

A colleague of mine has pretty good success with a sitting and standing activity. The basic idea is this:

Line up 3 to 5 chairs in front of the class with a person sitting in each chair. The rules are: You sit down or stand up (i.e. change position) if the person on your left sits down and the person all the way to the left can sit and stand freely. The goal is to find the number of moves the person all the way to the left has to make to get the person all the way to the right to stand up.

• Good idea as an exercise, to see if they understood the counting. And as energizer. – Esmaya Sep 11 '14 at 15:52
• So the leftmost person causes a cascade of changes one at a time until the last person moves? This might take a while. Another version of the rule is for all the players to sit or stand simultaneously based on how the people to the left are currently arranged: the leftmost person always switches; everyone else switches when all the people to the left are currently standing. – DavidButlerUofA Nov 12 '14 at 21:10
• If you have all your players seated so they are facing to the right relative to the rest of the class (like these letter p's: PPPPP). Then the rule can be to sit or stand at the next change when everyone in front of you is standing currently. – DavidButlerUofA Nov 12 '14 at 21:33

I often invent a land where the money comes in units of 1, 2, 4, 8, 16, ... and then ask how to pay for certain amounts, using exact change, with the fewest possible coins. It's nice that the greedy algorithm of always using the largest possible coin always works!

Then I make the analogy starting with \$101 meaning one hundred-dollar bill, no tens, and one one to lead them toward using binary notation to represent the way that they're paying the amount of money that they need to pay.

For completeness I'll add a reference to Rick Garlikov's use of the Socratic method to teach binary arithmetic to a third grade class. It took 75 questions in the instance he describes. A complete transcript and a summary of his thoughts on the process are at The Socratic Method: Teaching by Asking Instead of by Telling.

Esmaya, I have done math circles around base 8 and base 3. I made a little story for base 8 called Eight Fingers. Your students might have fun illustrating it. (The 'code' at the end is actually binary. If you have time for that sitting and standing activity, they would go together well.)

I have a suggestion for getting a feel for how the number system works, which might even convince them that it is a useful thing to know how to do.

Step 1: A method of counting

Take a pile of small objects (beads, lollies, blocks, pieces of spirali pasta, Go counters -- whatever takes your fancy -- though anything that naturally joins together might have a slight advantage). Then say you are going to count how many there are by a new and creative method.

First you'll group them into pairs. There may or may not be one left over but that's ok. Next you'll group your pairs into pairs, making groups of four. There may or may not be a pair left over but that's ok. Now you'll group your groups of four into pairs, making groups of 8.Now group the groups of 8 into pairs, making groups of 16.Now group the groups of 16 into pairs, making groups of 32.If you haven't chosen too many things in your pile it means that now you'll have to stop because there are no pairs of groups. Of course if there are more you'll just keep going on pairing up the biggest groups.

Now how many have we got? Well let's start with the biggest group. We have one group of 32, one group of 8, one group of 2 and a single object. So that's 43 things. As a way of counting it was quite efficient because we weren't actually counting at any point, only pairing things up.

At this stage you can give everyone a pile of things and they can do the pairing up themselves and figure out how many objects they have.

[You can also imagine this being done with the students themselves -- get them to pair up, then pair up the pairs etc. Of course some of them might be a bit miffed at being the ones left out at the end.]

Step 2: A new way to represent numbers

You should be able to say at this point that in fact every number can be grouped like this because they can see that all their piles have been effectively split into pairs progressively. So that means it's a neat way of representing what number you have. But there's an even neater way to do it and here it is.

List the sizes of the piles on the board/document camera/screen: 32, 16, 8, 4, 2, 1 Now underneath say you'll write how many of each pile you have. In our example you have 1 0 1 0 1 1.

Ask them to do this for their own piles and get some or all of them to write their numbers and their list of piles on the board.

At this stage they should be able to see that every number has a representation as a string of 0's and 1's. Ask them if it's possible to get anything other than 0 or 1 (the answer is no, but you want them to be sure).

Then you can say that this is the binary representation of a number. Each digit is how many groups of that size you get when you successively pair things up.

[It may be worth doing this process for other sized groupings, like grouping into 10's, which they may be familiar with from earlier years.]

Now you can give them various binary representations and ask them to tell you what number they represent.

Now pair up your students and get them to add their numbers together and find out what the new binary representation is. See if they can do this without having to start from scratch.

Hopefully they hit upon the idea that they can join any groups that are the same size into bigger groups, and continue this process until there are no more. They'll probably do this organically with no structure, and after discussion you can practice by always starting with the smallest piles.

Then you can see how this works on paper with the binary representation.

I think this method could work quite well to get them started on binary numbers, though including some other activities involving counting in binary with the fingers of your hand (as others have described) wouldn't go astray.

Give them some motivation to learn it.

Three years ago I was meeting 3 fifth-graders once a week for half an hour of math. Mostly we did puzzles and games. One time I gave them the rules of Nim; we played it in various combinations, and (naturally) I could always win. I even played all 3 of them simultaneously and won all the games. So by then they were asking how I did it.

Some later week (reminding them about Nim) we discussed using bases other than 10, and converting back and forth.

Then a third week I told them the strategy for Nim, based on writing the numbers in base 2.

• If I may add: What better motivation is there than knowing that every MP3 file, every YouTube video, is delivered to their screens as a stream of binary numbers... – Joseph O'Rourke Nov 14 '14 at 0:29

It may be helpful when introducing binary numbers to begin with showing the difference between counting and arithmetic in binary and binary codes:

http://en.wikipedia.org/wiki/Binary_code

This may help clarify the complex issue of symbols and what these symbols are used for. By the 5th grade students have a fair amount of experience with the decimal system. Decimal codes are less common than binary codes and so when one moves to showing students how one can count and do arithmetic in other systems than decimal it may be useful that binary codes be part of what one does. Furthermore there are lots of lovely applications of binary codes (Huffman codes, for example).

I'm posting this answer to complement @JoshuaZucker's one, whose idea is basically the same I'm introducing. Instead of using money, use weights and... the magic cards (they'll love them).

Suppose we have a set of weights. One weight of 1kg, one of 2kg, one of 4kg, one of 8kg and one of 16 kg. Then, I propose the students to make a table specifying which weights are required to weigh 1kg, 2kg, 3kg, 4kg, 5kg, ... 31kg. For example, to weigh 7kg you need the weights of 4kg and 3kg. And so on. Actually, when they'll finish, they'll have a complete binary representation of numbers from 1 to 31.

Just for amusement (motivation) and reinforce it, they can prepare the so called magic cards. You could even start the lesson showing the power of this cards. Show the cards with the PC and tell a student to think a number. Then tell him/her what number it is.

Magic cards are just tables in which you put all the weights you can weigh with each one of your original weight set. The first card includes all numbers that require the 1kg weight. The second, that require the 2kg weight. The third, that require the 4kg weight. And so on.

Once the students are done with this activity, the best option is to continue as @DavidButlerUofA explains, just to make them realize how counting systems work.

Then tell them how would they count when having eight fingers instead of ten!

I haven't tried this in a class but it worked in one-on-one conversation with kids at basically all ages (well, ability to count is needed but that's basically it). It also has the benefit, that you do not need to speak about positional notation or the 10 based system at all for the beginning.

State:

I can count to 1023 with my fingers.

To show how, hold your closed fists, showing the backs of your hands. Show your right thumb and say "one". Fold the thumb in and the index out and say "two". Let the index out and also show the thumb and say "three", and so on.

At some point some kids get where this is going and figure out how the 1023 comes into play. Some other kids need some more time but in my experience, basically anybody can figure out what the system is by doing it himself. Of course, many kids will find it funny to show the number 4 or (even better) 132. You can also ask how far two kids can count if they use all four hands…

From that point on you can take different. You could go into the direction that the fingers really do not matter here and use another binary notation. You could also investigate numbers that the single fingers represent. No matter how you do, you'll end up explaining the binary system.

I am also in Germany and confirm that this is a topic in 5th grade and also that the books I've seen make no reference to "counting with your fingers to 1023" at all.

• When you say "four" aren't you going to be sticking out your middle finger at the students? At least in the US that is not such a nice gesture... – KCd Dec 16 '16 at 6:13

I already fully understand how base 2 and base 10 work so I assume it's just because I have the natural talent to do so so I'm not quite sure how to teach grade 5s base 2 in such a way that they can understand. I have an idea of how to inductively define the binary notation of every natural number counting zero as a natural number which is a simpler but equivalent definition to the ones given in C Wren, Davi L Renfro, and Pablo B's answer.

First, we can show that every natural number can be expressed uniquely as 2 times a natural number plus 1 or 0. First we say that we define 0 to be the notation for 0 and 1 to be the notation for 1. Next we say 2 can be expressed as (2 $$\times$$ 1) + 0 but we already defined the notation for 1 so we define the notation for 2 to be the notation for 1 followed by the digit 0 so its notation is 11. Now 3 can be expressed as (2 $$\times$$ 1) + 1 but we already defined the notation for 1 so its notation is the notation for 1 followed by the digit 1. 4 can be expressed as (2 $$\times$$ 2) + 0 but we already defined a notation for 2 so its notation is the notation for 2 followed by the digit 0 which is 100. We can show that when ever you express a nonzero natural number as 2 times a natural number plus 1 or 0, the second of the 3 terms in the expression is a smaller number than the number you're trying to figure out how to express that way. We can show that for any natural number, once you have defined a notation for all the smaller natural numbers, you can define a notation for that natural number so by induction, you can define a notation for all natural numbers. You can also show that conversely, all nonzero length strings of the digits 1 and 0 that don't start with 0 and the string 0 and no other finite length strings of the digits 1 and 0 represent a natural number and each string that represents a natural number represents only one natural number.

If you're given a string of 1's and 0's whose first digit is 1 and you're asked to calculate the notation for its successor, you can tell already that the string represents exactly one natural number so you don't need to take on blind faith that no string represents more than one natural number and they gave you a string that represents a natural number. If the last digit is 0, all you have to do is change it to a 1. If the last digit is 1 and the second last digit is 0, all you have to do is change the last digit to 0 and the second last digit to 1. It's easy to show that the rule is to change the last 0 digit to 1 and all later digits to 0 if not all digits are 1 and the string is not 0 and when all the digits are 1, you change them all to 0 and stick a 1 onto the beginning. However, it might be better to leave the students to figure out on their own the notation for every natural number from 0 to 31 as a homework problem after they're learned the simple inductive definition for the notation of every natural number because that will get them trying to figure out how to determine the notation for the next number and that might make them smarter later than they would be if they just got told how to do it.