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 (2

*n*- 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 6*n*-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 4*n*+ 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!

I really like how you spent so much time on visual patterns. I think this will really help your students understand quadratics. Once you get into the 'nuts and bolts' you'll be able to relate things back to these patterns.

ReplyDeleteThanks for reading, Dave. I agree - I think that my students will have a really solid grounding as we move forward with quadratics.

DeleteI like the progression of patterns. Do you have the patterns that you used that you could post here?

ReplyDeleteThe link is on the day 2 post: http://marybourassa.blogspot.ca/2015/09/mpm2d-day-2-testing-our-models-starting.html

DeleteAnd you can find lots more at visualpatterns.org

Hi Mary,

ReplyDeleteI borrowed your visual pattern handout and modified it for my split 2P/2D French Immersion class. I am attempting spiralling for the first time with this class to try to reconcile the two curricula! We have done a few visual patterns as part of our daily warm-ups, but only simple linear patterns so far.

With the quadratic ones on this handout, some students made tables and found differences, saw that they were quadratic, and knew their equation would have to contain an x^2, and then guessed and checked from there to find the rule.

Most students felt stuck, so I nudged them toward the idea of the finding expressions for the length and width. As with your students, this definitely set off some lightbulbs. It was so nice to hear the discussions in the room as one student would figure it out and explain to his/her group how he/she found the rule. One student actually said: "Miss, this length and width thing is like magic!"

Thank you for openly sharing your resources and daily plans. Your 2P posts from last year and your 2D posts so far have been a huge help for me in spiralling my split class.

Thank you for taking the time to share your experience and for the kind words. A 2P/2D split class must be quite challenging to teach - I'm glad you are finding some helpful resources and that your students are enjoying it!

DeleteI just wanted to thank you for your dedication to teaching. I was fortunate enough to hear you speak about your teaching last year and was truly inspired. This is my first time teaching a 2D course in it's entirety and your blog has been my go-to. I used patterns to algebra in grade 9 and it's amazing how quickly my students were able to adapt to patterns that can be modelled by a quadratic equation. They love the challenge of trying to figure out the pattern!

ReplyDeleteThanks for the kind words. Patterns are awesome - it's so cool to see how students develop the ability to see patterns in multiple ways. Glad things are going well for you!

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