A major expectation for our grade 9 applied class is to “*connect various representations of a linear relation, and solve problems using the representations.*” Early in the spiralled grade 9 course I bring in Fawn’s Visual Patterns website as warm ups. We routinely continue the patterns, create tables, equations, and graphs to show the representations. Students also create their own patterns.

More and more I notice that grade 9 applied students don’t see what I see when looking at patterns (which is definitely not a bad thing). I love hearing all about how students see the patterns. However, I always see the patterns as growing/shrinking…..what I mean is that I see that one shape morphing into a bigger/smaller version. What I’ve heard from some students though is that they see each figure as a separate object, separate things that looks slightly different. I wanted to explore if students seeing the patterns morph instead of seeing them as separate objects could help them with seeing connections among the different forms of the relation.

To start the class I showed this video:

I asked: What do you notice?

Students described the pattern to each other while sitting in pairs. We decided that if the first set of shapes represented figure 1….then every figure after that showed two more shapes being added in. I asked them to go ahead and find out how many shapes were in figure 108.

I gave out the following set of instructions:

- Create your own animated pattern video
- Create a tough pattern for your classmates to discover. Ex: Show how the pattern changes in other ways than figure 1 then figure 2 then figure 3. Maybe show how your pattern changes from figure 1 to figure 3 then figure 5.
- Create a question for your fellow classmates to solve about your pattern.
- Display your video around the room for a gallery walk. In your display hide the table and equation and answer to your question.

They went to work on building & shooting their patterns. Having them skip figure numbers made them really think about how to create their patterns and how the equations related. Since they were invested in their own patterns they worked hard at creating the tables and equations.

After they created their video they were to create a display for a gallery walk. The gallery walk gave us a purpose to practice finding rates of change, determining equations, generating equations and solving problems. We wanted to see the creative patterns from our classmates and see if we could solve each others problems. Like a challenge! Each display showed the video and then under flap of paper was an answer to a problem with a table and equation. Students left their display and visited each others displays with a recording sheet.

We spent two class days working on building the videos/patterns and the gallery walk. There are a variety of stop animation apps on the app store. My students used various different ones. Some students used iMovie.

I felt students were stronger on knowing why we need to find the rate of change for our equations and not just take the first difference value. The one-two combo of actually building the patterns and then making them move through animation built a deeper understanding of the representations than just completing a worksheet!!

This is amazing! We’re working on patterning and algebra right now and can’t wait to do this with my 7’s!!

Hi Jon,

Thanks for sharing an excellent lesson and your insights. I wonder if you have tried having students come up with recursive equations for their patterns before explicit? I ask because many students are able to describe the recursive pattern before they are able to define an explicit rule (see https://mathbitsnotebook.com/Algebra2/Sequences/SSRefreshRecursive.html “a recursive formula is easier to create than an explicit formula. The common difference/ratio is usually easily seen, which is then used to quickly create the recursive formula.” and https://problemproblems.files.wordpress.com/2016/02/on-visual-patterns.pdf “The journey from recursive to functional thinking can be rocky. It’s hard for a lot of kids.”)

I am excited about the animations because I think this helps with the stumbling block you identified (each figure is different), but it still lends itself to recursive thinking (the pattern is adding two blocks each time, or a_n = a_n-1 + 2).

Best,

Evan

Thanks Evan, yes students always identify the pattern in a recursive way before explicit. I wonder how many students would understand writing equations in the recursive way and not get more confused by the subscripts?