# A nicely timed problem from Megan Hayes-Golding

I started a new section in our geometry book with my older son today and spent the morning talking about sums of interior and exterior angles. He had an appointment for an orthopedist to look at his broken hand from a sledding accident, so I wanted to keep things fairly light for the rest of the day. This problem I saw on twitter hit the spot exactly:

My son’s initial reaction was great – make everything totally symmetric and see what happens. Love it:

But, that doesn’t solve the problem – we don’t know that all of the sides of that interior pentagon are equal. What next?

It turned out that this was a nice example of how simplifying a problem can help you find a solution since basically the same solution to the simple problem works in general. He had a little hiccup at the end with the final subtraction, but all in all a nice solution to a neat exercise. Love it when I see great problems on Twitter đź™‚

# Going through David Wees’s Snap Cube exercise with my older son

Last evening I went through David Wees’s snap cube exercise with my younger son. That post is here:

Going through David Wees’s Snap Cube Exercise with my younger son

After publishing my younger son’s project, it was my older son’s turn. Unfortunately he broke his hand sledding and can’t put the snap cubes together too one with one hand super well, so my younger son helped out in the background. I think that my younger son looking on proved to be a happy accident since he got to see an approach to this lesson that was pretty different from how he approached it.

My son chose n = 1, 3, and 4 as the numbers that he wanted to use to explore the first expression. Just as my younger son did, he (eventually) represented the numbers in this expression as rectangles. We decided to draw the 5×7 rectangle rather than putting together 35 snap cubes for the last number.

At the end of this movie something was bothering my younger son – not sure what it was, but he was ultimately satisfied that the numbers we’d written down were correct.

One big difference in my older son’s approach to the second expression compared to my younger son’s approach was that he represented the $n^2$ term using squares. For the first two numbers, n = 1, and n = 3, he was able to see how the shapes he made from the second expression could be transformed into the shapes from the first expression. For n = 4, though, he couldn’t quite figure out how to make it work. He could see that the algebra made the two expressions the same, though, so we left it at that for now.

For the last expression, representing the figure as a square less one cube really helped him see the relationship between expression 3 and expression 1. This is the part that I was happy that my younger son got to see.

We finished up this section talking through the algebra. He saw right away that you got the same quadratic when you expanded the first and third expressions. I also wanted him to notice that you could transform expression 3 into expression 1 using a difference of squares, though I think my approach to “helping” him see this relationship was actually not all that helpful.

Overall I enjoyed this exercise. I think that each of the kids was able to take something away from it. Interestingly both of the kids seemed to gravitate to the algebraic connection, though my older son representing the $n^2$ terms as squares did help him with the geometry. I’ll be interested to see if seeing my older son’s approach helped my younger son see