Last week I introduced my son’s third grade class to binary numbers. I wanted to build on my prior visit, where I introduced them to the powers of two. By teaching them binary, I showed them that place value is not limited to base ten, and that there is a difference between numbers and numerals.
My presentation was based on base-ten-block-like imagery, since I knew the students were comfortable expressing numbers with base ten blocks. I thought extending the block model to other bases would work well. I think it did.
Before my presentation, I put twenty-seven tape flags on the whiteboard, in an unorganized fashion like this:
(I would have preferred to use magnets instead of tape flags, since they would have been easier to move and align; but I didn’t have twenty-seven identical magnets.)
I started my presentation by telling the class that I would teach them about something called binary numbers, but that first I would review the numbers they already know — decimal numbers (I took a moment to explain that this was not the same as “decimals”). The first thing we did was count the tape flags, and as we counted together I rearranged them into a line:
I asked them how they would write that number. One student came up and wrote “27,” which is the first answer I expected. Other suggestions were Roman numerals (“XXVII”) and “twenty-seven,” also as I anticipated. One student suggested writing it in Japanese (I was expecting a foreign language, but Spanish: “veintisiete”). Some students suggested arithmetic expressions, like 20 + 7. One unexpected answer was from a girl who wrote it on the board in base ten blocks, which is how I was planning to rearrange the tape flags next!
I suggested tally marks as another alternative, and wrote twenty-seven in tally marks on the board.
I singled-out the answer “27” and said it is written in place value. I reviewed how the places were powers of ten. Then, as the class counted along to twenty-seven, I rearranged the flags into base ten block powers of ten groups, under headings labeled “tens” and “ones”:
We counted the powers of ten and wrote the totals in the blanks I drew below each grouping of blocks; we came up with the numeral “27”: two tens and seven ones.
I told the class that place value is not limited to base ten. I said, for example, you could write any number in base five, or quinary. (I wanted to take an intermediate step to binary, which is the simplest base, having only a maximum of one instance of each power.) I had them compute the powers of five from one to 625, and I explained that these are the places in quinary. I told them we would group the flags into powers of five. I wrote three headings on the board: “twenty-fives,” “fives,” and “ones.”
I asked “are there any twenty-fives in twenty-seven” and they said “yes.” We then counted out twenty-five flags, which I removed from the decimal grouping we’d just done. I built a block as we went, under the twenty-five label. Next I asked if there were any more twenty-fives in the flags that remained, and they quickly said “no.” They could also see there were no fives, and that there were only two ones left, which I moved under the ones label.
We counted the powers of five and wrote them under each grouping of blocks, coming up with the numeral “102”: one twenty-five, zero fives, and two ones. Some kids wanted to pronounce this as “one-hundred and two”, but I told them you pronounce it as “twenty-seven,” or “one-zero-two base five.”
Now I said let’s look at another example of place value: base two, or binary. I said it is based on powers of two. We computed the powers of two from one to thirty-two (my son was rattling them off to 4096 before I could cut him off :)), which they remembered from my last visit.
We proceeded as above, except we pulled out the powers of two (from the flags in the quinary grouping): first we looked for sixteens, then eights, then fours, then twos, and then ones.
We counted the powers of two and wrote them under each label, coming up with the numeral “11011”: one sixteen, one eight, zero fours, one two, and one one.
Number of Digits
When I was done with the tape flag examples, I took a moment to explain that base ten has ten digits, base five has five digits, and base two has two digits. As an example, I said that in base ten you could never have a 10 in any place, because that would be the same as a 1 in the next higher place. Similarly for base two, a 2 in a place would equal the next higher power of two, which also would be the same as a 1 in the next higher place.
I told the class that you could write any whole number in any base. One kid asked if I could do it in a base that was greater than ten (I forget which base he used as an example). I said any number could be the base, but you’d have to have enough symbols. I briefly explained why you wouldn’t want a multi-digit number in a place (it would make the numeral ambiguous). I mentioned base sixteen, and said it uses the letters A through F for the values ten through fifteen. (I did not intend to get into hexadecimal, but hey, I wanted to answer the question!)
Students as Binary Numbers
At the front of the classroom, just below the whiteboard, I arranged five chairs, facing the class. I wrote the names of the binary places above the chairs, left to right from the class’s point of view: sixteens, eights, fours, twos, ones. I got five volunteers to come up, and said that I would turn them into a binary number. I said if I told them to sit in their chair, they would count as a 0; if I told them to stand in front of their chair, they would count as a 1.
For my first example, I put the students in the pattern 11011, which the class correctly read as twenty-seven (they added the place values above the chairs of the standing students — that or they read the numerals I had left on the board under the tape flags :)). I did a few other examples like this, which amounted to binary to decimal conversion. They got them all right.
Next I did what amounted to decimal to binary conversion, asking the class how to arrange the volunteers to represent a given number. For example, when I said “nine,” they called out instructions to make the volunteers stand and sit to make the pattern 1001. They got all of these examples correct as well.
The above discussion took about twenty-five minutes, so with the extra five minutes I squeezed in a demonstration of a binary counter. I took a new set of five volunteers and had the class direct them through the sequence zero to thirty-one. We got through the count, but I think a few students got lost as some of the faster adders called out instructions. In any case, there were definitely some who understood the process, enough to know that when I asked them to display thirty-two, they said we would need another volunteer.
If I had more time, I would have done the count a second time, with the volunteers driving the counting; I came up with this scheme after I left the class:
- All volunteers start out sitting, representing zero.
- Whenever I say “count”
- The ones place volunteer does the opposite of what she is currently doing: if she’s sitting, she stands; if she’s standing, she sits.
- For everyone not in the ones place, if the kid to your left sits, you do the opposite of what you’re currently doing.
I think this would have made the counting easier and more fun.
(Update 11/7/12: I gave this presentation again recently — to fifth graders — using the new counting scheme. It did not go over like I imagined. The kids were confused about when to stand and sit, and weren’t having fun. In the future, I’d omit binary counting; in hindsight, it seems too “computery” for this context.)
I mentioned briefly that there is an equivalent of decimals in binary numbers. Instead of the tenths, hundredths, etc. places there are the halves, quarters, eighths, etc. places.
I think most of the kids understood the presentation; certainly, they were all engaged. I’d like to think it gave them a better understanding of decimal, even if they didn’t understand the details of binary. I told them “you may not understand this now, but when you see it again someday, you’ll remember back to this day in third grade and it will come to you.” Someone then asked what grade they teach this in. I said it’s not really part of any particular math class (as far as I know) but that they would be taught it in a high-school computer class if they took one.
I used number words when I wanted to avoid writing decimal numerals; for example, when describing a number or when labeling places. Unfortunately, number words have decimal place value built-in, but that’s the closest I know how to get to a base-independent description of a number. That said, I don’t think the class recognized this, so I don’t think it caused any confusion.
I didn’t explain why we broke the numbers down by starting with the largest powers and working down. If I had more time, maybe I would have let them discover the algorithm themselves.
I use the term “number” when I really mean “numeral”, as in “binary number” or “decimal number.” This terminology is unfortunate, but it is standard.
- Rick Garlikov’s use of the Socratic method to teach binary numbers to third graders.
I used a different approach, but a lot of the same concepts are involved. Rick’s method centered on binary counting, which lead to a discussion of powers and places. My method started with powers and places, and lead to binary numerals and then binary counting. Rick discussed other bases after discussing binary, whereas I discussed them before. Also, he discussed binary arithmetic, but I did not.
One thing I liked about my approach is that I built in the concept of base conversion, showing the equivalence of whole numbers written in any base. I also liked the way I exhibited the concept of “number vs. numeration.”
- Computer Science Unplugged Page on Binary Numbers.
This page contains videos on binary counting, which inspired my own binary counting demonstration.
- My article “How I Taught My Mother Binary Numbers.”
I taught my mother a little differently (at least in my second attempt), mainly because I think most adults don’t think explicitly about place value.
Please Try This
I’d love to know if this method works for you; if you try it, please let me know!