In an effort to provide a challenge worthy of the students who always get EVERYTHING, I put the following problem on their last review packet with the intention that not everyone would get it, but that it would give the super smarties something to chew on.
Consider line AB that goes through an x-intercept of -8 and a y-intercept of 3. This line goes through the altitude in triangle DEF from vertex D. Find D, E, and F, and convince me that your points satisfy these conditions.
I gave my students the key to every other problem in the packet, so they could study for their test, but not this one. I was gone to a workshop while they took their test, and today when I came back, I had some students present their answers. No one got it (yet), but it provided some great conversation about what they did and how we knew it didn't work.
I was interested to see that some students just used a slope approach, and tried to find D, E, and F using slopes of things they knew. Mixed success. Some students forgot how to find points on a line, so we had a bit of review. I still didn't tell them the way (a way) to solve this, and mentioned that somehow, before the end of the 6 weeks, they have to each convince me that they can do it (for a grade). It's interesting having students come up to the document camera and present. We get to have a conversation on how to do so and how to talk to an audience. I also get to pick their brains about their understanding and also poll the class about their understanding (even though they're not the ones in front). ... We'll see what they come up with next.
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