Last week I wrote about some great shapes that one of the kids in our neighborhood made just playing around with our Zometool set:

Today we used the shape below as a basis for our Family Math talk:

To start off I put the original shape in front of the kids and asked them to come up with a question that we could ask about it. My older son’s question was about the area of the shape and my younger son’s question was about the number of struts in the shape.

We approached the question about the number of struts after we recreated the top of the shape. My younger son’s two approaches to counting the struts shows how great the Zometool set is for facilitating math conversations with kids. Though the question itself is not particularly complicated, it gets my son thinking about (i) systematic ways of counting and (ii) the beginnings of the inclusion / exclusion principle as his second way of counting over counts some of the struts.

Now we came to the question about the area of the ring. My older son also found two ways of approaching this problem. Both ways use the area formula for an equilateral triangle that he’s recently learned from our *Introduction to Geometry* book, though you could have this conversation without knowing that formula by simply assuming that the area of these small triangles was 1. The math ideas in this conversation are ideas like (i) breaking a difficult problem up into pieces that you already know how to do, (ii) some basic ideas about symmetry, and (iii) by coincidence, a little bit of the inclusion / exclusion idea from the last question.

Last up was my question – what is the ratio of the area of the big hexagon to the area of the little hexagon in the picture?

Each kid gave an answer to this question using a different idea. At first, my older son calculated the areas the same way that he did in the last video. At the end he was able to answer the question just by counting the smaller equilateral triangles that lived inside of both shapes.

My younger son used ideas of scaling to discuss the relationship between the two areas.

So, a super fun project inspired by a little creative playing around with the Zometool set. Pretty amazing to see the possibilities for exciting math conversations that come from the shapes kids create!