The fun of learning to count things

Had a really nice conversation with my older son yesterday about this old AMC 10 problem:

Problem 14 from the 2005 AMC 10a

Here’s the problem: “How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?”

I decided to turn the problem into a short project with both kids this morning and got a really nice surprise when my younger son approached the problem in a completely different way than my older son did. Sometimes these little counting problems help you see neat connections that you never noticed before.

Here’s a quick introduction to the problem and my younger son’s idea (btw, I’m not letting my older son touch the marker because he has pink eye!) :

 

and here’s my older son’s approach:

 

Definitely fun to see these two different approaches and a, I think anyway, a pretty nice example of the interesting connections you find when you are learning about counting. Excited to see where this counting project takes us this summer!