What learning math sometimes looks like – rectangle edition

Recently my younger son and I have been study some introductory geometry in our Prealgebra book, and several of the questions in this chapter have sparked great conversations. Today I saw a fascinating question about a rectangle and couldn’t wait to hear his ideas. So, off we went.

We talk through the problem for a bit at the start – this is a challenging problem just to understand! Since he’s a little stuck, he decides to try out a few test cases and draws a 2×1 rectangle. This simple rectangle immediately gives him some new ideas – great start!

It is nice to hear all of his ideas, and especially “I think I see something . . . . ”


I broke the talk into two pieces when it seemed that his idea at the end of the last video didn’t quite work. The trouble he’s run into is that when the width increases by two units the area changes in a way that is different that when the length increased by one unit.

One bit of luck is that he’s working with some small numbers and that probably helped him notice that the change was twice what he was expecting. Upon noticing that fact, he’s able to write down the dimensions of the rectangle. The last part for him is just checking. At the end I show him a little bit of geometry that was hiding in the problem that helps us simplify the calculations a little.


When I think of what learning math looks like, a lot of the ideas in these two videos are what I think of. When I think of kids who are not enjoying learning math – or worse – I’d love to walk through a problem like this, or even just show them the ideas here so that they can see what learning math looks like. Questions, ideas, trying out a few things here and there, trying to find what works and what doesn’t, and maybe even finding the solution to a pretty challenging problem.

Fun morning!


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