Act 1
Today’s 3 act math activity was in the style of Fawn Nguyen‘s Visual Patterns. I presented the class with this:and asked what Mathematical questions could we ask & solve? Students responded via Pear Deck:I want my students to be thinking about good questions to match a given scenario as this is what our MFM2P board-wide summative task will require of them.

So I then asked them to estimate the number of cubes in step #62 (being careful to pick a high enough number that they won’t want to count up like they did last time):

Act 2

I asked them to solve at their boards with their groups. All of them had a table of values created and had started to determine the first differences for the most part. One group also had the 2nd differences listed. Most were stuck as to how to continue. I overheard the following:

“We can’t [make a table] all the way to 62 …”

“There’s got to be a better way to do this”

One student realized they could multiply the term number by itself & multiply by 2 (effectively $2x^2$). But they were only counting blue blocks (when they should have been counting red + blue). I told her to keep that in mind but that they needed the red blocks too.

At this point I asked them to take a seat at their desks & I did a quick “teacher-centred moment”. We reviewed the linear & quadratic regressions with Desmos that we learned & practiced last week. We also talked about 1st & 2nd differences and how they allow us to determine whether or not a relation is linear, quadratic or neither. It looked like this:

Then I sent them back to their boards. A few groups took their chromebook with them & got to work with Desmos. Here’s a group that worked their way through the quadratic regression w/ Desmos w/ some prompting from me along the way:

Another group had a member that recognized a pattern:

I asked them to explain how they got 124 at the bottom of their table. They said they counted up knowing it was increasing by 2 each time. I asked them if they could figure out that number just based on the step number without counting up. I pointed out the pattern 1 x 2, 3 x 6, 4 x 8 … and they realized that it was twice the step number (which I scribed on their board for them as they explained it to me).

That group then decided to go around and explain their pattern to the other groups so you can bits & pieces of their calculations (without the pattern or thinking behind it) on these groups’ boards:

This last group didn’t quite get to the end point that we wanted:

Act 3

The actual answer was 7691 blocks, which the groups had figured out. I called students back to their seats even though the last group didn’t yet have an answer. I reviewed the different solutions on the first and second boards pointing to things as I went, clarifying their written work. I showed how the pattern the 2nd group came up with could be written as n x (n x 2) + 3. I also wrote on the original image to show them how it could be $n^2+n^2+3$ or $2n^2+3$.

They reflected on their original guesses with a little help from my prompts about expectations of linear VS quadratic:

And then I asked them to consolidate their learning with some practice using this handout for homework tonight.

The whole set of materials is here.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON