I've been working with my Freshmen on linear patterns for the last couple of weeks. I've been really amazed by how well they've taken to them, and how quickly they were able to start figuring out the function rules. I've been using lesson ideas from The Pattern and Function Connection by Fulton and Lombard.
First, I gave students matchstick patterns and boxes of toothpicks, and had them build the patterns. This was important, because often students would draw the picture incorrectly (not seeing where toothpicks were repeated, for example) and would not quite see how the pattern was growing.
Then, I gave them other types of patterns like this one, and asked them to describe how they were growing, and to determine the number of items in the 30th step and in the n-th step. I think that building a concrete connection between the variable "n" and the "step number" will be invaluable as we move into solving decontextualized equations. I was really excited when the first student figured out that this pattern could be thought of as a center square surrounded by "groups of n squares" and explained it to the rest of the class.
After this, we moved into organizing the steps of the pattern into a t-table, and then plotting them on a set of axes. We defined "n" as the step number, and "f(n)" as the function value, meaning the "number of items at step n". I worked through one example with them, but then I found that most students took off on their own and blazed through the whole set of patterns I gave them. During the following lesson, we repeated this work, but I introduced the concept of "rate of change" and "starting point" and we talked about how these values are seen in the different representations of the pattern.
I really liked the following diagrams from the book:
Today, I gave them some function rules, and they had to then work backwards to generate a pattern that could fit that rule. They struggled with this quite a bit at first, and it showed me that, although they are doing really well, they are far from having fully internalized the connections between the representations.
On Monday, I will give students a pattern, and tell them how many pieces are in the last step. From there, they will have to figure out what the step number is. I think this will be a good scaffold for understanding how to solve equations. For now, I am going to treat it like a problem solving activity and am interested to see what they will come up with.
Below are some of the handouts I made to supplement the excellent materials in the Fulton book.
Pattern Project handouts (doc) / (pdf)
Working Backwards handouts (doc) / (pdf)