The Mathematical Association of America recently posted this wonderful video from James Tanton about a problem on the 2003 AMC 10a. It is a fantasatic example of what solving math problems looks like:
By lucky coincidence we had talked through this problem just a few days ago. My older son works on challenge problems from the AMC 10 occasionally and this one gave him quite a bit of trouble. Since I’ve been talking about different bases with my younger son, we all ended up talking through this problem together after my older son finished.
After seeing Tanton’s presentation I thought it would be fun to revisit the problem with the kids tonight. Since we talked through the problem recently I had each of them talk about how they would solve the problem. Importantly, this isn’t the first time they are seeing the problem, they are remembering a solution from last week and then putting it into their own words.
My older son went first. He remembers the idea that the answer will be between the lowest base 11 number and the largest base 9 number. This part shows that the fact that we’ve already gone through this problem is a little unlucky, because he knows the right path and charges down it. He makes a little bit of an arithmetic mistake at the start, but finds his way back onto the path.
My younger son also remembers the problem a little bit. I’ve not talked much about probability with him, so I spend a little extra time at the beginning making sure that the probability part of the problem doesn’t confuse him. After we get past that part he talks through the base number arithmetic. I think that he did not remember the problem as much as my older son did, but he does manage to work through the computations in a really nice way. At the end we got to talk just a little bit about fractions.
After they completed their talks I had them watch the James Tanton video. Their first reaction was awesome – “wow, he can write in the air!” When the video finished I asked them what they thought about Tanton’s approach. I got a laugh when my older son said that one thing that Tanton did differently was “show his work.” Ha – I’ll just go ahead and assume that lesson is now learned . . . .