I’ve had multiplication on my mind lately and I think it’s because I have been planning a lesson on expansion for my grade 8 students. I needed them to learn that a*(b+c) = ab + bc, but I also wanted to know why that is true. After much deliberation, and some very helpful conversations with a colleague, this is what I decided to do…
I used this slideshow to guide students through an activity in which they used rectangles to model different multiplication problems. They started with a 5 * 4 rectangle, which they immediately recognized as 20. When I asked them to write an algebraic expression to represent the area of the rectangle, they quickly wrote it as 5 * 4. I had students working in their table groups and each group had a small whiteboard, so they could write their answer and show me from anywhere in the room.
The next step started with the same rectangle, with an extra 2 units added to the length. They quickly recognized that the area was 28. When asked to write it as an expression, most groups wrote 4 * 7, but when I pushed them to write each term separately, they were able to write it as 4*(5 + 2). I also had them write the area of each part of the rectangle, so that they could see that 4*(5 + 2) = 20 + 8.
Once they had seen a few rectangles with specific dimensions, they went back to the original 4 * 5 rectangle and increased it’s length by an unknown amount, x. Having worked through earlier examples, they could see that the rectangle represented 4*(5 + x). By writing expressions for the area of each section, they could also see that the same expression could be written as 20 + 4x.
We went through a number of examples, and eventually students worked on problems that required them to use expansion in order to solve for the variable.
I often struggle to keep this group on-task when I see right before lunch (as I did today), so I was surprised by their level of engagement in this activity. It seemed to strike a good balance of success and challenge. The early questions were simple and student felt successful in being able to answer them quickly. They were then motivated to persist with the tougher questions that pushed their thinking in new directions.
When the activity was done and students started some practice problems, I was pleased to see that many were sketching out rectangles to model the problems. Instead of just doing a*(b+c) = ab + bc because I had told them it worked, they were showing that it worked in every problem. Eventually, they will skip to the short cut and expand it the same way that I was taught to expand when I was in high school, but instead of doing it because their teacher told them to, they will do it because they know what the rule means and why it works.