If I wait until I have an hour to craft the perfect blog post, this post will never happen. And, it certainly can't help anybody if it's sitting in my drafts folder. And, you can't tell me how to make this lesson better if it's sitting in my drafts folder. So, this post is going to be quick. If you have questions, leave them in the comments, and I'll try to answer them!

Completing the square. AKA my least favorite way of solving a quadratic equation. I would skip teaching it if I could. If I'm dealing with a quadratic, I'm going to either factor it if it's factorable, solve it using a graphing calculator if one is handy, or turn to the quadratic formula. The majority of my students prefer the graphing calculator route, as well. But, there is a high likelihood that my students will see a question on their EOI at the end of the year that asks them what number must be added to both sides of the equation in order to complete the square. So, I spend a day on completing the square.

This student obviously did not pay attention on that day. I guess he did complete the square, but...

To illustrate completing the square, I got out a set of algebra tiles. I only have one set of algebra tiles, so I used these under the document camera. I began by putting out the blue x squared tile and two green x tiles. Class, how many yellow tiles are needed to complete the square? One.

What if I have four or six green tiles?

We kept adding green tiles and determining how many more yellow tiles we would need to add. Some students could visualize what we were doing. Others acted like this was the hardest concept in the world.

As we experimented, I had several students collect data in class. If we have 2 green tiles, we need 1 yellow tile. If we have 4 green tiles, we need 4 yellow tiles. As I started to run out of tiles, I asked the students to begin generalizing. How many yellow tiles would I need if I had 26 green tiles?

Only after discovering the formula for determining what to add to each side of the equation to complete the square did I pass out our notes to fill in.

Here's the notes and the facing page for reference.

I'm not completely happy with this lesson, but that's normal. Every year I strive to teach things better. I learn by posting my stuff on the Internet for others to modify, tweak, and critique.

Want to download the files? Click here!

thanks a lot........I learnt a lot from your blog for my teaching & learning Maths II exam which is going to be tomorrow...I have got many great ideas....keep it up.....:)

ReplyDeleteYou're very welcome!

ReplyDeleteI am getting ready to introduce completing the square in the next couple weeks. Thank you for this. It will make it easier to teach with this visualization.

ReplyDeleteHi, I was wondering if I could ask you a question about the example. I noticed that -12x seems to vanish into thin air! Could you please tell me what happens to it? Thanks.

ReplyDeleteThe trinomial x^2-12x+36 factors into (x-6)(x-6) which can be rewritten as

Delete(x-6)^2

Thanks for explaining!

Delete