My students are using GeoGebra again today and yes, it is still just as amazing as I previously thought. Today both my Pre-Calculus and Calculus classes are using it. The Pre-Calculus activity was awesome and it generated lots of discussion and exploration today.
Before I tell you about it though, I think you should watch this TED talk. It sets the stage for the kind of thinking I was expecting from my kids today. Really, go watch it. Its only 12 minutes long and well worth your time.
I can wait.
Cooperation, technology and my Pre-Calculus Students
I gave my students the Introduction to GeoGebra on GeoGebra today. It was a great example of how I want my students to work in cooperation with the technology to make meaning out of the mathematics. Wanna know the coolest part? I built that activity last week and didn’t the see TED talk (you watched it right?) until after my first period completed the activity. I felt quite validated as you can imagine.
In the first activity, the big “thinking” exercise was only possible because the technology provided an interface the students could use to explore, generate and test hypotheses and justify their thinking.
For a problem like this one:
students were asked to use GeoGebra to explain why it is true. They do not yet have the algebraic expertise to see this yet without the aid of the technology, but they can approach this equivalence statement through algebraic or graphical means with the aid of GeoGebra.
I want my students to explore mathematics and connections. I am very worried that Pre-Calculus will be a series of disconnected topics. I’ve written about this already. GeoGebra is giving me one way to build little sandboxes for my students to go play in.
That’s actually exactly how I think about: Sandboxes. First, I write a lesson like this thinking about the big target I want my students to hit. The target for today was to help students begin to think about polynomials as products of linear factors, or at least as products of other, “smaller” polynomials.
I put this picture on the board today while students were working:
and I explained to my students how I see polynomials. I see them as little blocks that you can connect, disconnect and rebuild in different ways… Just like legos!
Today’s activity was me saying to my students:
“Go play in this sandbox. The sand today will be polynomials and monomials and you will be putting them together and taking them apart. I want you to look for patterns and GeoGebra will help you do that. What I most want is for you to analyze the results. Tell me why these things you see are happening. Now, go play.”
How it went
Today they only had time to play and write down their observations. Tomorrow we will be discussing “why” and “how you know” kinds of questions. This discussion will lead to me reminding them of a particularly useful tool when breaking apart polynomials into their factors: polynomial long division. They have used this tool before. I want to make sure that they see how it is useful in our context, so that they can make decisions about how and when to use it.