Moving on from trial and error

My son wanted to talk through an interesting problem from his math club homework this morning:

Mary notices that the number of miles on her car’s odometer is (i) a multiple of 1,000 and (ii) is also a perfect cube. She does some calculations and determines that she has to travel another 4,921 miles until the odometer will be at a perfect cube again. What mileage is her odometer showing now?

My son solved the problem through trial and error, so we spent a little time today talking about an alternate way to solve the problem. I’m sad that the conversation is a little rushed, but that happens sometimes.

Here’s how we got started with the conversation today. He introduces his trial and error method and we talk a little bit about last digits of perfect cubes just to clarify a small point in his thinking.

So, with the little last digit point behind us, we returned to talking about the problem. He notices that 1,000 can’t be the answer because the next cube is 1,331 and that’s not 4,921 away from 1,000. He also thinks that he can eliminate 8,000, but isn’t sure.

His idea about 8,000 inspired a short conversation about how you could walk through a list more efficiently than checking every number.

Rather than having him multiply out a bunch of numbers, I decided to show him an algebraic solution to the problem. This is where I really wish we weren’t running low on time – I would have liked to have talked through the idea here in much more depth.

So, definitely not our best day in terms of talking about ideas, but I did want to help him see how to move beyond trial and error solutions.