MPM1D1 - Day 35 Quadratic Pattern & More Volume

Before we started today we talked about the definition of a prism. It was interesting to hear what students thought the definition was. Eventually we settled on a definition (that came from one of the students). We then talked about how we could find the volume of any prism.

We branched out a bit today and tried this visual pattern:

Groups began right away and created a table immediately. They found the first differences and realized that the pattern was not linear. Some groups showed that the second differences were the same. Others noticed it but didn't include a column for the second differences. Once groups realized that the pattern wasn't linear, they didn't know how to proceed. They realized that finding the slope wasn't an option. One group did find the y-intercept. After giving some time to struggle I stopped the class and told them that they needed to find another method. I told them that they should look at how the length and width grow from one step to the next. This was enough to get the groups to come up with an expression for both the length and width. They knew that to find the number of helmets they would need to multiply the length by the width but they struggled with how to multiply two algebraic expressions. That makes sense given that we haven't really done that yet. As a class we put the pieces together and came up with a possible solution.

Without saying, the warm-up took quite some time but I think it was time well spent. Once we were done we went back to the Big Nickel. We looked at the question of what happens to the volume if you double the height. We then explored, as a class, what happens if you double the width as well. What if we double all three dimensions? It seemed to make sense to them. I was hoping to get to the rest of the extensions of the nickel task but I also wanted to compare the volume of a cone to a cylinder. I held up the cylinder and cone and asked how many cones would fit into the cylinder (the areas of the bases are the same).

After taking some guesses I filled the cone with water then dumped it into the cylinder. I asked if anyone wanted to change their guess and a few students did. I added another one and asked if anyone wanted to change their guess. As I dumped the third one in many students were convinced that it wouldn't all fit but sure enough it did. We talked about how this relates to the formula. It seemed like lots of light bulbs went on at that point. I gave them a couple of pages to practice (here and here) and the period was over.