We're studying quadratics....and even the name usually strikes fear in the heart of the most competent adult. I didn't want it to be that way for my math students. I wrote a good lesson plan and then I let students help me modify it---they "taught" me how to teach them better thru the interaction and feedback we gave to each other during the learning process.
I create a scaffolding technique, but students helped me add, delete and amend it until it worked for the way they think.
Even if you don't teach math, this works...because I've done things like it in science and social studies. I guess I just had forgotten about using it. So the trick is to create a means for scaffolding and then let students help you fine-tune it...meeting their needs. In addition, you'll learn so much 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 300BC. I think that sort of impresses students that they are following in the steps of ancience Greeks. We were also able to pair this ancient technique with modern-day technology using Geometer's Sketchpad and Algebra-Tiles sketch that was available in the software library.
From there we started moving into a more symbolically 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.....
If I divided my SmartBoard into two areas: one side represented what 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, they helped me refine my thinking list.
- Look for GCF
- Look for a letter that could be factored
- Find both factors(it's a mulitplication 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 look at the process list. 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 that published this formula way back in the late 1500s. Here's the second thing I learned...they 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
I can imagine them in high school...thinking all this was foolishness. And it will be then. But right now, where they are developmentally in building up their quadratic muscles, it is perfect.