We’re studying quadratics in my 8th grade class. Even the name can strike fear in the heart of the most competent adult. I didn’t want it to be that way for my math kids.

I wrote a good lesson plan and then I let students help me modify it. Essentially, they “taught” me how to teach them better through the interaction and feedback we gave to each other during the learning process.

**Here’s some of what it looked like**

I create a scaffolding technique, but students helped me add, delete and amend it until it works for the way they think. In essence, we built the scaffold together.

Even if you don’t teach math, this strategy is a good one. I’ve done things like it in science and social studies, but it’s been awhile. I guess I’d just forgotten about using it.

The trick is to create a means for scaffolding and then let students help fine-tune it. As a bonus (trust me here), while you’re figuring out the scaffolding, you’ll learn a lot about the topic that would never occur to you otherwise.

We started off slowly using the geometric area model. Our textbook tells us that Greek mathematicians used this method as long ago as 300 BC. I think that sort of impresses students — the idea that they are following in the footsteps of ancient Greeks. We were also able to pair this ancient technique with modern-day technology using Geometer’s Sketchpad and an Algebra-Tiles sketch that was available in the software library.

From there we started moving into a more symbolic version of finding the “x” solutions, figuring out what this means in real life and how to even use an old kindergarten Valentine making technique. It wasn’t easy but they hung in there.

**Here’s what I learned**

I divided my SmartBoard into two areas: one side represented the problem we were studying and the other side represented the thinking someone would need to do in order to solve the problem. This helped tremendously and as we worked problem after problem, students helped me refine my thinking list.

- Look for GCF
- Look for a letter that could be factored
- Find both factors (it’s a multiplication expression after all): usually we write something like this to remind us —-> a product=factor * factor
- Solve for zero—finding both “x”s
- Set those “x”s equal to the x-intercepts
- If you need to find a min or max, find the line of symmetry — we would say “this is like when you folded the paper in two and cut out your Valentine heart” (and we always did the hand motions!)
- Use the line to find the “y” of that max or min

So all of this is on one side and then they use the other side to solve the equation. It’s scaffolding, and it helps them ingrain the process in their brain.

I’m not sure they realized how much they helped me think about their thinking, but student feedback helped me zero-in on what they needed me to “think aloud” for them. Throughout the unit, you would see students able to stop and look at the process list and go on. They could perform this procedure independently.

To enrich the lesson, we did a one-day mini-lesson that showed them the quadratic formula. It ties what the ancient Greeks did to another pretty old mathematician….Francois Viete. He was the French mathematician who published this formula way back in the late 1500s.

**Here’s the second thing I learned.** My students couldn’t use this formula on problems unless they were in the standard form of the quadratic. Again, I could scaffold this by simply writing the standard form and then helping them use the process.

I can imagine them in high school thinking all of this was foolishness. And it will seem that way then. But right now, where they are developmentally in building up their quadratic muscles, it is perfect.

*This post, written by Marsha Ratzel, appeared originally on this blog in May 2013. *

#### Marsha Ratzel

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