We jumped right in to our visual patterns, starting with two that we had already looked at. I wanted my students to start seeing them in a different way.
Instead of looking at how many blocks were being added each time, we discovered how the shapes related to the step number. They could see that the "extra" block on each side were constant, as was the width of the rectangle. Finding a pattern, based on the step number, for the length of the rectangle was more challenging. I wrote the values in a table to help them come up with a relationship. I had seen it as double the step number minus 1, which is also how at least one of my students saw it. But another said that it was the step number added to the previous step number. We figured out how to write that and simplified to show that both ways of seeing it produced the same result (2n - 1). It was great to have more than one way of seeing this. I asked how many blocks were in the rectangle at step 3. Although some counted the blocks to get 15, others said that they could multiply the length and width. We did this for our length and width at step n, simplified with a little help from the distributive property, and arrived at 6n -1, which was our result from yesterday.
Next, we took another look at the volume pattern from yesterday.
Again, we found the pattern without having to go back to step 0, instead relating what we could see to the step number.
Then, my favourite pattern and several ways of finding its rule.
There is so much richness in this pattern. The first three ways really reinforced the idea of relating the number of blocks to the step number. To work out the rule based on the area of the large square and the area of the inside square, we needed to step into the world of quadratics. My students have never really worked with quadratics before - the word itself was new to them. I loved that one student had created a table of values AND had calculated the first and second differences. We figured out a way of writing the area of the outside square for step n, but in order to show that the rule did still work out to 4n + 4, we needed to expand. They had never done this before, but I tied it back to the distributive property which seemed to make sense to them (I normally use some kind of area model to introduce multiplying binomials, but did not want to add that to the mix today).
And we kept going:
Then I let them work on the next few in their groups. It was so cool to have the same conversation over and over about the length of the rectangle for pattern 8.
They would tell me that you couldn't get it by adding something to the step number and as soon as I suggested multiplying the step number by something, the wheels started turning and almost immediately the light bulb went on. Visibly! Their faces lit and and they said an enthusiastic "Oh!". And I walked away.
I asked them to try to finish the visual patterns over the weekend (although #11 is hard) and gave them this linear review, as well. This is a modification of something I had created for my grade 10 applied class a few years ago. There is a worked example and then a practice question for each skill. As we haven't really done any formal taking of notes, I thought this would be a good approach.
I was planning on doing Solve Me mobiles next class, but now I think I will stick with quadratics for a while since we landed there today. My planning is always a work in progress!