We were essentially asked to come up with an algebraic equation for this pattern. I started by writing a table of values, but it didn't help me too much right away. I could see that the pattern wasn't linear (ie - increasing by the same amount of squares every time), but I was finding it really hard to find the equation. I looked around me and saw everyone else writing furiously, so I knew I had to do something. I figured that since the first term had 2 squares only and that every subsequent term had 2 squares on either side of it, that I could start with a "+2" at the end. Now I just had to figure out an equation for the middle of each figure. I ended up using area.
An easy way to see it is that you can see that figure 2 has an area of 1x3, figure 3 has an area of 2x4 and so on. This works out to (n-1)(n+1). So, the final equation is (n-1)(n+1) + 2.
The interesting thing is, not everybody saw it as an area with two squares on either side. Some saw it very differently. In fact, here are some examples of the different ways people saw this pattern:
Now, they all work out to the same simplified answer of n^2 + 1, but it just goes to show you how different people look at patterns in different ways. It's another reminder that not everyone sees math the way I do!
As I keep going on this journey to becoming a math teacher, it just keeps getting confirmed to me that I'll need to approach my teaching from many different angles. I think I really just always assumed that there is one "best" way to teach a concept. In a way that's true...there is one best way for me. And there will be one best way for every single person in my class. The problem is, that best way will not be the same for everyone. And that's something I'm going to remember.
I liked your comment - "I looked around me and saw everyone else writing furiously, so I knew I had to do something." When I'm stuck on something, but everyone else is writing, I sometimes just write random stuff down so people will think I know what I'm doing! But sometimes it does help me get started. I wonder if that kind of situation is a deterrent for some students though (increases anxiety?).
ReplyDeleteYour last paragraph resonates with me - I need to learn to become more flexible as a teacher. I think it's great that Liisa keeps doing tasks like these in class. That concept (that teaching needs to be approached from many different angles) is difficult for some people to grasp, I think - and needs to be reinforced.