Geometry and counting

We started a new chapter in our Introduction to Counting and Probability book today. The chapter covers “basic counting techniques” and is filled with wonderful counting problems. We worked through one of the examples from section 2.2 – “Casework” – for our project today.

The problem is simple to state, but coming up with all of the cases requires some careful thinking:

How many squares of any size can be formed by connecting the dots in a 5×5 grid?

I thought using snap cubes for the grid would help the boys understand the problem (though I’m not sure if it helped that much). Here’s what the table looked like at the end of the project:

Counting Pic

We started by identifying the cases. First the cases where the sides of the squares are parallel to the edges of the grid:


Next the cases where the sides of the squares aren’t parallel to the edges of the grid. The first two parts here were actauly one continuous discussion. I thought breaking the discussion into two pieces made sense for this writeup. Finding all of the different squares in the 2nd part is challenging.


Next we moved on to counting the number of different types of squares. The first thing we did was count the squares whose sides were parallel to the grid. It took a minute for the boys to understand the geometry in this situation, but once they made sense of the geometry for the first case, all four cases went fairly smoothly:


Now things get a little tricky! Counting the number of squares whose sides are not parallel to the grid. There are two different types of squares here – the “\sqrt{2}” squares and the “\sqrt{5}” squares, and there’s an interesting difference in the way these figures interact with the grid.

First the \sqrt{2} squares:


Next the \sqrt{5} squares. Counting these cases helps you to understand a bit about rotations, and we used some extra snap cubes to illustrate the rotations for the camera:


So, a really neat counting problem. It is fun to see the connections between geometry and counting in this problem. Another fun piece of this project is that there’s not a lot of math background needed to tackle this problem. Hopefully that makes it a great introductory counting problem for kids.

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