One of our PoWs this week involved finding three consecutive numbers that added to 63. The four main approaches were:

1. Divide 63 by 3 and use that as the middle number
2. Guess and check, generally starting in the teens or low 20s
3. Write an equation: x + (x + 1) + (x + 2) = 63*
4. Think of 63 and 60 + 3 and find three consecutive digits that add to 3 (o, 1, 2) and then divide the 60 into three 20s, yielding 20+0, 20+1, 20+2. (This was the rarest strategy).

I’m always trying to define for myself what the heck Mathematical Practice #7, Look for and Make Use of Structure, means in actual kids’ thinking. I think strategy #4 is a particularly good example of a middle- or elementary-school version of making use of structure, taking advantage of place value.

I wonder what would have happened if the three consecutive numbers had added to 60 or 61.

I wonder if the fact that the structure they found doesn’t make every type of problem easier invalidates it as a strategy somehow?

Hmm….

Update: an upper-elementary student used a strategy that combined algebraic representation and looking for structure:

Y + (Y+1) + (Y+2) = 63.
Y + (Y+1-1) + (Y+2-2) = 63 – 1 – 2
Y + Y + Y = 60
3Y = 60
Y = 60 ÷ 3 = 20