When a student asked why the binary number card trick worked, I decided to help them figure it out. Besides wanting to know why it worked, the kids wanted to know why we could only choose the numbers 1 to 50. These explorations are what makes mathematics exciting. Why do we study mathematics? Not to prepare us for the next course, but to discover and explore why the world works as it does.
We began by making a table of the addends of the numbers from 1 to 50. The only addends we could use were 1, 2, 4, 8, 16, and 32, since we only had one of each:Many students quickly saw patterns in this table: there is only one sum that begins with 1, two with 2, four with 4, etc. Also, 1 was only used once, before it was “changed” to a 2, then a 2 was used twice before it “changed” to a 4, etc.
Some students had difficulty seeing this, so I devised a “tally chart” so that we could see these patterns more clearly:
Very quickly, all students saw the patterns of the zeros and ones.
I then defined some terms for them so that we could speak the same language (if I ever do this again, I may delay this step). I showed them how we had shown that 11(base 2) = 3(base 10), and that this is called the binary number system. It is a way to count using only ones and zeros. We then talked about how the card trick worked because you can write any number with only one of each card that has a power of two on it. Many students saw that in order to choose numbers higher than 63, we would need to add one more card to our deck. We never did discuss why we only choose numbers from 1 to 50 instead of from 1 to 63.
Because of the many Christmas activities in the school, this was as far as I was able to go with one class. One class I never even saw during the whole week so I asked their homeroom teacher to show them the card trick and cut out their cards, which he did. In the future, I would add the NCTM Illumination activity so that students would make their own cards. I would also develop the connections to the base 2 number system and our base 10 system. We might even try other number systems (would a card trick work with base three numbers, even though you would need two cards for each place value?) Would there ever be enough time to do all this?
A Merry Binary Christmas
Binary Card Trick, Part 2
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