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.