Henri Picciotto’s factoring activity

[sorry for the tired write up – I’m a little sick today]

Saw this neat tweet from Henri Picciotto last night:

At the end of the blog post is this activity:

Geometry and Graphing Connection by Henri Picciotto

and the last page of this activity really caught my attention. I’d been doing a bit of work on factoring and completing the square with my older son anyway, so I thought I’d give it a try with him tonight.

One warning is that I’m pretty sick and out of gas today. I wanted to see how he’d react to using factoring ideas with geometry, but I didn’t have the energy to pursue all of the ideas. That said, I’m actually very happy with how he reacted to the ideas – especially with little help from me since I was fully in the useless zone tonight. We’ll definitely be revisiting this activity.

I started off with a quick introduction to how the blocks could be used to represent the various pieces of a quadratic polynomial. He seemed to catch on fairly quickly using the example x^2 + 3x + 2:

At the end of the last video he’d picked a 2nd example quadratic with negative coefficients. I wanted to do an example with negative coefficients, but I wanted to wait to do that example later in the project.

Here he pics the example x^2 + 5x + 6 which makes another rectangle. He’s able to see how the blocks show that this quadratic factors into (x + 2)(x + 3).

The next two examples showed what this method would look like for a perfect square, and how it could be used for completing the square.

Now we returned to a polynomial with negative coefficients. The geometric ideas here were a little harder for him to see. I need to find a better way to communicate the ideas for this part so that he sees the -2 as -3 + 1:

My only idea for helping him see the geometry in the case with negative coefficients was to use an example that was a little easier. That example was x^2 - 2x + 1. He did seem to see the geometry here, which made me happy. I was especially happy that he saw how a block was taken away twice.

So, I think this is a really great project from Henri. I’m really excited to try some more with it this weekend.

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