Ace of Bases

Jose Vilson Education, Jose

Ace of Base

This week, I’ve taken Greg Tang’s advice from the NCTM Conference and started working with the kids on different bases. And by different bases, I mean different ways of looking at the number systems we use. One of the biggest reasons why kids don’t get math in general is because the numbers themselves don’t make sense to them. For example, a child who looks at 1,234 can tell me that it’s more than 1000, but when it comes to dividing, they can’t tell me how many 100s or even why the 3 isn’t just a 3 but a 30. Using Greg Tang’s advice then (to work in different number systems and hope they can develop rules that are applicable to the base 10 number system that they’re already familiar with).

At first, the whole idea of trying binary and tertiary number systems was ludicrous. Why would I want to teach them something about these number systems when they hardly get their own? Fair enough. Once I flipped it on them and told them it was a game, and the rules were that you could only use the digits lower than the base number, they ate it up. A hook. So for base two, they said,

0, 1, 2, … 10, 11, 12 … 20, … 100, 101, 110, 111, 1000 … (eventually they got it right)

So we finally crossed that counting barrier. Now that we were actually in the shore of where I wanted to get them, I tried seeing if they could do it for base 3. Same results.

0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120 … (they were much better at this one after seeing that last one)

Then I asked them to help me make some rules, and all three classes basically saw the same patterns (after some good questions on my part, honestly):

1. We start with the number 0.

2. We work with all the digits less than the base number.

3. Once we get to the base number in that row, we use the next row, and start from 0 again in that first row.

So far so good. Today, we tried to turn numbers in base 2 to their corresponding number in base 10. Again, I give them a oversimplified version of the place value system we already established and maybe they’ll come to some conclusions about how to arrive at our conversions. I made little boxes around the number and put the number representing the place value over it (not the name, but the actual number) just to make it easier for them to see.


110 in base 2 equals 6 in base 10. Why? Because 1 is in the 4s place, the next 1 is in the 2s place, and 0 is in the 1’s place, so we have 1 4, 1 2, and 0 1s i.e. we have a total of 6.

Then one of my kids yells out,

“So what you’re saying is that we multiply the place value by what’s in the box?”

SUCCESS! Hopefully he saw the sign. (har har har). Yes, I’m a math teacher. I’m allowed a corny joke here and there.

jose, who loves teaching math when the kids are actually learning something …