**The Problem**:

**Find the x-intercepts of y = x**

^{2}+ 10x + 16.**Me**: How many different ways could we do this?

**Class**: Quadratic Formula, completing the square and factoring.

**Student 1**: Mr. Cox, why does y = 0? Isn't this a function where y can equal a bunch of things?

**Student 2**: Yeah, but we want y = 0 because that's when it crosses the x-axis.

**Student 1**: Ok, so we are just focusing on one possible value of y.

**Student 3**: So then, if we let y = 0, then we are finding the x values that make y = 0, right? But what if we want to know when y = 1? Can we do that too?

**Student 4**: I guess we should just let y = 1, but we'd have to subtract it from both sides so we get:

0 = x

^{2}+ 10x + 15. So now we are finding the x-intercept again.

**Me**: Is this an x-intercept? What does y equal?

**Student 4**: Oh, no, y = 1.

**Student 5**: Mr. Cox, can we see what this looks like in GeoGebra or something?

We graphed it and then moved a point around on the parabola to verify our results and looked at how the points where y =1 maintained the same symmetry with the x-intercepts. Then, a kid pipes up:

Are there graphs that have more than two x-intercepts?

**Me**: Hmm. Maybe. I'll tell you what...you guys give me a function that has x intercepts at -4, 0 and 3. You have 15 minutes. Go!

About 5 minutes later, J.V. walks up with this:

**y = (x + 4)(x - 0)(x - 3)**

**y = x**

^{3}+ x^{2}- 12x**J.V**.: Remember the other day when we had to come up with parabolas with x-intercepts? I figured I could just work backwards like we did then.

I'm amazed at how often these kids will take the lesson into places I wouldn't have thought to go. It's just a matter of letting go a little. And the more I look for open ended opportunities, the return is exponential.

## 3 comments:

That's awesome! Good stuff

Okay, fess up Mr. Cox: were you a ghost author on "The Art of Problem Posing"? There's just a really good vibe going on in your class. Kudos.

Paul Hawking

Blog:

The Challenge of Teaching Math

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Pulled form the comments feeds (2-28-11)

http://challenge-of-teaching-math.blogspot.com/2011/02/pulled-from-comments-feeds-2-28-11.html

hello!!

really good work!

How old are these pupils?

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