Tuesday, October 3, 2017

Inside Out Quadratic Formula

I noticed that my students, while using the quadratic formula, typically had trouble in one of two places. They either find the wrong value for b^2 - 4ac, or they have trouble simplifying the radical.

I decided to try having my first-timers evaluate the formula from the inside-out. Maybe if we put our focus on the most-likely-to-mess-up parts of the formula minus the noise of the whole thing, then we would be more successful and consistently correct.

Here's what I mean:

The solution looks like this: First, we find b^2 - 4ac.

Next, we simplify the radical. My students are awesome at this when it is an isolated operation, but often struggle when its part of a larger problem.

Then we put it all together.

At this point you may be done, or maybe you need to reduce.

Time will tell how much of a difference this approach makes (if any), but I have noticed a few other benefits:

We don't have a learning target for the discriminant in our curriculum, but if we did this would be an easy lead in. That number we found first? It has a name, and it can tell you what kind of solution you are going to have before you do anything else.

Also, I kind of like how the solution seems more efficient . . . it feels less noisy and crowded compared to writing out all the parts again and again as you simplify the answer.


  1. I don't quite see it: What's different from the standard approach? Not computing the whole expression but instead take auxiliary calculations?

    I guess yeah, for first-timers that's probably more organized, so they are less prone to errors.

  2. Yep that's it. It was a new approach for me to have students calculate the individual pieces one at a time before the formula as a whole, but I would agree that it isn't revolutionary.

  3. I like the equation split into two fractions. Seems to work better for my students. Also, facilitates the understanding of a line of symmetry plus or minus equal amounts, giving the two roots

    1. Yes! I've noted these two points in my post (see comment below). I find "distributing the denominator", which splits the expression into 2 fractions, is the most reliable for students to avoid mistakes. And the line of symmetry is a nice "surprise" that relates the quadratic equation's solutions to the graph.

  4. I think this is a great approach that enables success with the arithmetic. I've mentioned your post in my blog post "Searching for Structure" https://karendcampe.wordpress.com/2017/10/13/searching-for-structure/ THANK YOU!