Exploring Rectangles

Friday is a special day in our math classes at the Main Line Classical Academy.  We read and discuss mathematical stories and we do exploration projects.  Here is the project that we did with the 2nd-4th grades last Friday.

It began with one of my favorite questions to discuss with kids: What is a rectangle?  Some of the kids in each class had participated in previous discussions with me on this topic, but this was close to 2 years ago and so probably had very little effect on the outcome.

Here is what the boards looked like after the 2nd grade and the 3rd/4th grade discussions respectively:

The kids used a lot of hand motions in their initial descriptions, but I told them to pretend that we were talking on the phone and I couldn’t see them.  They would also sometimes come up with very long and convoluted explanations, which I also refused to write on the board.  After each initial set of properties, I’d try to draw a shape on the board that fit them all but was not a rectangle or did not fit some of them and was a rectangle (some of the shapes unfortunately did not make it into the pictures).  The kids had a lot of laughs when I would draw a silly shape and ask them “is this a rectangle?”  In the end though, I believe that we settled on a set of properties that succinctly characterized rectangles.

The second part of the class consisted of making all possible rectangles out of a given number of squares.  The kids had to make them out of snap cubes and then draw them on graph paper.  The second graders all got 12 snap cubes while the 3rd/4th graders initially got 12 and then each their own different number between 18 and 32.

I was very surprised that no one tried to draw the same rectangle in different orientations.  Some kids did, however, try to make and draw rectangles with holes in them.  A few of the second graders initially had trouble because the squares on the graph paper were smaller than the snap cubes, so tracing the structure did not work.  However, after a brief discussion, they were all able to make the one-to-one correspondence between the cubes and the squares.

Here are some pictures of the process:

In the end, we discussed with both groups how to make sure that we have made all the possible rectangles.  One of the older kids pointed out the connection with factors/divisors of a number.  None of the kids had formally studied area or multiplication (although most know what those are to various degrees), but those will both be big topics in the 3rd/4th grade class this year.  I think that this served as a good indirect introduction to them.