## Spiralling MFM2P Update

Another month flies by in my spiralled MFM2P! I don’t know how it’s possible for me to be both bubbling over with energy and totally exhausted at the same time, but seems like that’s teaching for me!

Here’s what October looked like in my class:

Some things that happened this month that I didn’t expect:

- My students had a rough time when they came across a problem they didn’t know how to solve, so we spent some time working on creating a classroom culture of problem solving (read about it here)
- Doing Peter Liljedahl’s taxman problem was a great choice – thanks Mr. Hogg for the recommendation! It actually ended up working out better than expected as it reviewed factors, which was helpful when we started factoring about a week later
- My students really struggled with similar triangles. Consequently, I ended up teaching similar triangles three times. Based on formative assessments and my own observations, it seems like they got it, but I haven’t marked their quizzes yet so I’m going to wait and see how those went before I say the third time’s the charm.
- In our last lesson of the triangles and trigonometry strand, I did Jon Orr’s new 3-act math, Eye-to-Eye. Students struggled with this and a few groups had to try a few different strategies before they figured out what they needed to do. Unfortunately we ran out of time and not every group got to the point where they found the answer correctly, but I was so impressed with my students for continuing to work on the problem and not giving up! I was so proud of them!

## Planning lessons with the #mtbos on Twitter

A couple of weeks ago I came across this question on an old test:

I debated showing my students a similar question with different numbers to help “prepare” them for a question like this on an assessment, but I wanted something that involved more thinking. I tried making the question better myself, but I was feeling a bit stuck. So I decided to throw the question up on Twitter and tag some experienced and creative teachers to see if they had any better ideas:

I got some great responses, including connections to optimization, but Jon Orr‘s was my favourite:

That would have been my warm up, but Mr. Hogg asked a question that made me take a second look at the original situation, and I realized that the details of the original wording and numbers chosen, which I hadn’t been too careful about when I started mucking around, were actually pretty important:

Reading the question more carefully, I noticed two very important things:

One was that the question was looking for a **binomial** expression (expanded and simplified) to represent the remaining area. The other was that the inside shape was not a 3 x 4 rectangle, but a square, which makes this situation literally a *difference* of *squares*.

I’m not planning to come back to factoring until December in MFM2P, but I needed to make a lesson plan for one of the modules in my math honour specialist course, so I decided to run with Jon’s idea and use it as an introduction to Difference of Squares. The full lesson plan can be found here, but here are the highlights:

## Difference of Squares Lesson

Warm up: this one from Which One Doesn’t Belong:

If when you take it up the students don’t point out that 9, 16 and 25 are all perfect squares and 43 is not, lead them to the idea and help them with the vocabulary of perfect squares and square roots.

Action:

Have students draw a big square on their paper, then draw a smaller square in the top right corner, like so:

Label the dimensions of the small square 3m x 3m.

Explain that this is your backyard and the 3 x 3 square in the corner is a shed. The shaded region is covered in grass.

**Ask: **what dimensions of your backyard are possible? What would be the area of the grass? Send students to the VNPS to work out different dimensions with the restriction that the backyard must be a square (not a rectangle). Have students calculate the area of the grass (outside – inside).

Bring students back to their seats. Say that now, let the dimensions of the backyard be x by x. Send students back to the VNPS to create an expression for the* shaded area *(grass) in *factored form*.

For students who struggle to factor x² – 9, lead them to rewrite the expression as x² + 0x – 9. (This lesson assumes students have already learned factoring simple trinomials.)

Once most groups have factored the expression, explain briefly that a quadratic expression of the form x² – a² is called a Difference of Squares.

Send students back to the VNPS with the following change: what if the shed were 4m by 4m instead? What would the area of the grassy region be? What would the area of the grassy region be in factored form? Once they find the factored form, have the students do it again for a shed of 5m by 5m, and again for 6m by 6m. Once they begin to grasp the pattern, have students create a general expression for the area of the shaded region in factored form for a side length of a. Their final result should be:

For students who finish quickly, provide more challenging questions, such as those that require common factoring first (eg. 3x² – 27). For an academic class (MPM2D), you can extend into more complex differences of squares of the form a² – b² (such as 4x² – 1).

Consolidate:

Here’s a template students can fill out to summarize what they learned if you do that in your class:

Many thanks to Jon and everyone else who contributed to helping create this lesson. What a shame to (almost!) waste such a rich problem on an assessment. 😉

I’m so excited for this lesson! Check my Twitter in a few weeks to see how it goes!

Follow my adventures in MFM2P through my spiralled Long Range Plans here.