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.

How to Drive to Sonic

Recently I noticed my students having a few misconceptions about the different strategies for solving quadratic equations:

A student assumed that a non-factorable quadratic had no solution.

Several were surprised when quadratic formula and completing the square yielded the same solution.

I came up with this little analogy to help clear things up, and it ended up working pretty well.

I started out by talking about how students might drive to Sonic, a favorite fast food place that is 10 miles away from our tiny town. Students named several paths and we talked about their pros and cons. The interstate is fast, but it does not provide a direct path to Sonic. Most students would choose to take Old 40 highway, but it is well known for road construction and can be blocked for days/weeks/months at a time. Someone even pointed out that there is a dirt path through a corn field that many of us have used to bypass the road construction. Perfect.

Solving a quadratic equation is very much the same.

On the left you have your quadratic equation. You want to get over to the solution, on the right. Factoring, complete the square, and quadratic formula give you three potential paths to get there.

Sometimes, when you take the factoring route, you find that your equation can't be factored. When this happens, its like a road closed sign on the path to your solution. The solution is still there, but you'll have to take a different route to get to it.

Other times, when you're completing the square, you run into a bunch of fractions. We aren't afraid of fractions here, but they can turn an otherwise straight path to the solution into a long and winding road. If we're looking for the most efficient way to get there, we want to avoid this situation.

Then there's the quadratic formula. It may not always be the most efficient, but it always works. It is particularly useful for avoiding the road block and the winding road. 

To follow up this discussion, I asked students to be in charge of road signs for these three paths. Your job is to put up some signs instructing people on which road to take. What do the signs say? Here is what one group created:

 My favorite quadratic formula sign said "when you don't want to think about which road to take". They're not wrong.

For a follow up activity, this card sort would be perfect.

Visionary

I always think its silly when people start a blog post by talking about how long its been since they've blogged, but . . . It's been over a year and that's probably worth mentioning. If any dear readers are still here, thanks for not giving up on me.

I'm starting the third week of a brand new school year. There have been some growing pains, but I will talk about those another day. I want to talk about one of our back-to-school inservice meetings. Our principals shared about the book Soup by Jon Gordon. At one point, they would describe a characteristic of a good teacher, and ask our groups of four to name one of our colleagues who most represents that quality.

One characteristic that was mentioned was "visionary". We were asked to name a colleague who is always evolving and learning new things, striving to get better. I was touched that many of my colleagues named me. This is the kind of teacher that I want to be and that I try to be, but I haven't felt particularly visionary in the past year(s). Even this year is off to a rough and chaotic start.

In all fairness, I've had a lot of life happen outside of the classroom. Our now two-year-old son has had multiple surgeries to repair his cleft lip and palate. Then last spring our family grew by one more through what can only be described as a surprise adoption. The rare and special nature of how our family has grown is not lost on us, and we are thankful.

Being parents of three (particularly the two littles) is exhausting. Some days are all about survival. I'm going to cut myself some slack for my absence from blogging, but still . . .

I've been thinking about how this is the place where my vision began . . . The blogosphere is where I began to be more adventurous as a teacher . . . trying new things . . . writing about them . . . learning from others. I thought if I came back here it might trigger what I've been missing.

So here I am . . . inspired by my colleagues' perspective of me, inspired by a gentle nudge from @druinok a few months ago, and inspired by watching TMC17 from afar. I could spend a few more hours re-reading and editing this post, but instead I'll #pushsend.