The Question Reveals . . .

I saw this quote somewhere a while ago, I don't remember where:

"It is better to solve one problem five ways than to solve five problems one way."

Recently, for bell work, I asked my students to give a response to that quote. I was kind of proud of myself, because it might be the first time my bell work has been something other than a problem to solve. The lesson for the day was solving quadratic inequalities both algebraically and graphically (they have done both, but I wanted students to see them side-by-side), so it seemed to fit.

The responses were limited to just a few thoughts:

You can use one method to check another method.
Sometimes one of the methods may not work.
You might forget one method so you could use another.

I am not sure what I was looking for. Maybe I was hoping that someone would think about how solving a problem multiple ways helps you better understand the concepts and how they all fit together?

What I think is interesting, is how much these answers reveal about where my students are at in their understanding of math. To my students, math is still a bunch of procedures to remember and repeat.

And here's the thing:  I think I am still mostly teaching that way.

Sorting Out Quadratic Methods

Now that my students can solve quadratics in five different ways, I wanted them to weigh the pros and cons of each method. I wanted them to be able to look at a quadratic equation and choose an efficient method for solving. Maybe it is just me, but watching someone pull out the quadratic formula when the equation can be factored kinda makes me cringe. I also wanted to review all the methods at the same time.

First, I gave them this sheet. It has the bell work and the practice problems.

For bell work, students worked out an example of each method in the first column as a review. Then we had a class discussion about the strengths and weaknesses of each method. We talked about how factoring may be the shortest method, but you can only use it if the quadratic isn't prime. And so on. We also talked about why you might choose one method over another. (Like how complete the square is so much nicer when the coefficient of x-squared is one and the coefficient of x is even.)

Next, I gave them this set of 16 cards* and a piece of card stock divided into four sections. I told them to try to put exactly four quadratic equations in each section. They needed to choose carefully, because next they will use that method to solve that problem. I just walked around and coached them a bit as they worked, and mostly I heard some good discussions going on. A few groups struggled, mostly because they had trouble figuring out how to tell if the equation could be factored or not. That is one problem I was hoping to correct with this activity, and it was pretty easy to identify who needed some help with that.

Then, students worked out the practice problems using the method they chose.

The next day I used a similar set of four quadratic equations for a quiz. Students could solve using any method they chose, but they could get bonus points for using each method only once. Students did fabulously with this. Yay!

*I also put the 16 problems on the practice sheet. Whenever I do an activity like this, I try to create the sheet so that someone who was absent could so something similar outside of class.