## Counting Rectangles

This week we continued playing with rectangles, but this time we were counting them.  This can be more tricky than it sounds.  I started the lesson by drawing a single rectangle on the board.  The kids all agreed that there was one rectangle (although some were trying to be clever by saying that the board itself was a rectangle).  Then I drew a straight line down the middle:

How many rectangles are on the board now?  The kids’ answers included all numbers from 0 to 3 – all of them can be easily explained.

0 – This answer was given by the kids who still hadn’t accepted that a square is a rectangle, and they forgot about the original rectangle.

1 – Same as for 0 but they counted the original rectangle.

2 – Counted the two smaller but not the larger rectangle.

3 – Well that’s how many there actually are 🙂

With the older group, as is often the case, they initially said the apparent answer 2, but then realized that it was actually 3 without me having to explain anything.  These are some of my favorite moments in class.

We then moved on to slightly more complicated problems of this type.  I showed them that they can give each region a name and then count the combinations of letters that account for rectangles:

It was interesting to see that some kids found that this helped them count the rectangles while others were confused by the letters.  Here is the worksheet we gave them and one kid’s solutions (we had them count the rectangles made up of different numbers of regions separately and then add up the results):

Overall, the kids caught on pretty quickly and I was impressed by how well they did.  In fact, they were so into it that when at the end of the lesson we suggested we play a game and they finish the worksheets later, almost all of them demanded to continue working on the problems.

And here they are, thoroughly engaged. 🙂