For Algebra II/Trig I decided that I would emphasize the relationships between graphs and equations throughout.

Even though there is no unit, today, as we are just starting some basic trig, the classes are taking a “graphing” quiz in which I ask them a few questions about fitting some curves to points, about graphing, and about transforming an arbitrary graph.  (“Here’s the picture that goes with f(x). Sketch the picture that goes with f(2x).” That sort of thing)

Yesterday I did some prep for that. But I started them off with two problems that they found highly challenging. Great discussions. Good buzz. See what you think:

1.  Sketch xy ≥ 0

2. Sketch $y^2 \geq x^2$

None of the kids had ever worked on a problem like either of these, and there were a ton of places to slip up. See if you can anticipate some. But the discussion was rich, and many seemed to have a sense of accomplishment afterwards.

1. March 24, 2010 am31 8:57 am 8:57 am

Isn’t #1 the whole first and third quadrants + the axes? It’s only challenging if you try to get procedural on it. I like it.

• March 29, 2010 pm31 2:39 pm 2:39 pm

So, the range of response was tremendous. And I had them interacting, so they engaged with some thought process that was not mine nor their own.

In each class, at least one kid just quick-graphed and quick-shaded. (Doesn’t less than 0 mean negative?)

In each class a handful pulled out their TIs and … and … how do I explain the strange things they produced??

I had several kids in one class who used the second problem to divide the plane with y = x and y = -x, and then proceed to misinterpret the axes as borders as well. I walked by to watch some interesting graphs, slowly shading the plane, one-eighth at a time.

In all, the variety was fascinating, the discussion worthwhile. It was the best 15 – 20 minutes of classtime I have thrown away in a while.

2. March 24, 2010 am31 9:15 am 9:15 am

I like what you said, Kate: “It’s only challenging if you try to get procedural on it.”

Of course, most students think procedural is all there is.

This feels like a Zen thing somehow. (I’ll try to keep thinking about this, so I can say something clearer than that…)

3. March 24, 2010 am31 10:24 am 10:24 am

Great problems! When I first looked at them I tried not “get procedural on them,” but afterwards I realized that there is one “procedure” that I think makes this problem simpler: Graph the boundary first.

Do you think it is easier for students to graph $xy=0$ than $xy>0$? Or easier to graph $y^2 = x^2$ than $y^2 \geq x^2$?

• March 29, 2010 pm31 2:28 pm 2:28 pm

Excellent question. I sweated this myself.

After watching the kids, I think equality would have been easier for the second, and a pure inequality for the first. My leqs made each problem a shade more challenging, or provided some minor misdirection.

4. March 24, 2010 pm31 12:49 pm 12:49 pm

Aaron, at the very least, I think students are more used to testing points on the equalities than the inequalities (which can help getting to grips with either problem).

5. March 24, 2010 pm31 1:10 pm 1:10 pm

My “procedure” was to think about y^2 = x^2 being equivalent to abs(y) = abs(x). That got me thinking about how y^3 >= x^3 would look…

• March 29, 2010 pm31 2:28 pm 2:28 pm

I wish you had been in class that day…

6. April 5, 2010 pm30 6:56 pm 6:56 pm