I have a few lessons that, over the years, I have worked hard on to improve, and that, as a result, I felt not-embarrassed to share. (Is that the best teachers ever feel - not embarrassed? Aren't we ever pleased with our work? But I digress.) This year, I am using geogebra right out of the gate, and I am amazed to see how much of a difference it has made, even on those not-embarrassing lessons.
Here's an activity I did last week, along with one I plan to do this week that builds on it. I wanted them to work together to develop a procedure for finding the rule of an absolute value function, so I made up some geogebra worksheets. In each example, they are given some points, and they have to input the rule that fits all the points. And of course, change the function's colour to match the points, since I am a geek for colour-coding. I started out easy, by giving them a few examples where they know the vertex and one other point:
Note that geogebra doesn't give them the rule, it just graphs the rule that they type into the input bar at the bottom, so they have to put the vertex into the rule properly, and they have to figure out the value of parameter a. The hardest part for them seemed to be getting used to using geogebra, but in no time, I had all kinds of ggb files uploaded to their dropboxes:
Note that for the black one, all I gave them was the vertex. "But that's not enough information!" some objected. "Well," I said, "it might not be enough for you all to get the exact same answer, but it is enough to get AN answer..." just to warm them up to the idea of minimum conditions.
The next worksheet was a bit more challenging:
Again, I shut myself up, and just said, you know everything you need to do this, work together, check out each others' ideas...In the red function, some tried making A the vertex, which geogebra soon showed them couldn't be. The tool did the intervention, not me.
Most figured out the red and turquoise functions pretty quickly. I saw some nice discussions happening in the breakout rooms. Some found h, the x-coordinate of the vertex, by using midpoint, others found slope of one side of the function.
But the purple example was REALLY cool to watch. It took a while for them to absorb the significance of not having that vertex. You don't realize what you have until it's gone! I did have to give a few hints, like which one of those points is the vertex ("oh! none!"), would you expect the vertex to be above those three points or below ("oh! below! ok I got it!") A couple of kids jumped right into the algebra - found the rules of the two linear halves of the function, then solved the system to find their meeting point. But others just used the line drawing feature in ggb to DRAW the answer. They drew two lines with equal and opposite slopes, located their meeting point on the graph, then boom, they had the graph of the function, which then revealed the rule, instead of vice-versa:
Now I figured my job was to show that these two methods of finding the answer were related. That each drawing step corresponds to an algebra step. AND, something I previously only dreamed of getting through to my students, that if it's possible to draw one and only one function to fit the points, that must mean there is enough info to find the rule algebraically. So I spent some time next class comparing and aligning the steps:
I think, I hope, that this validated the kids who drew the answer, and opened the minds of those who algebra-ed the answer, to see that even the most abstract of ideas has a foundation in the concrete world. Geogebra is 100% responsible for however much of that idea made its way through their neurons.
Next job for geogebra:
This coming week, we're doing the square root function. When we get to finding the rule, I will re-use the first worksheet, by merely replacing the words "absolute value" with "square root", and see what happens. I am hoping like crazy that someone will say, "Hey isn't that the exact same worksheet?" "Same points," I will say, "but now they belong to another function." And I will have shown them that these points didn't BELONG to any one particular function to begin with, we are the ones deciding that - bending the points to our will, if you will. And who knows, maybe someone will learn something that I hadn't anticipated, or even known myself!
And my next not-embarrassing lesson to be geogebra-ed:
I call it "The Big Picture", which is coming up in a few weeks. It brings together a few different functions. Enough said. The problem is, for every idea I have about how I will do that, 5 more seem to bubble up. It feels like I'm standing on an iceberg. It's a bit scary but, as my yoga instructor says, "Bring it on."
By the way, I would love to share my ggb files here, but I'm not sure how to do that. They are embedded, but does that mean they can be downloaded? Don't know. I know I can upload to the geogebra wiki, which I will, but as soon as I can figure out how to do that here, I will. Any advice would be GREATLY appreciated.