One thing that I try to emphasize to my students is that mathematicians don’t spend their time (usually) doing calculations. They spend a lot of time trying to make sense of scenarios, and trying to answer bigger questions by breaking them down, and they spend a lot of time being curious. One of the most important skills of a mathematician, to me, is to try to find a good question. Last week, we had one of my favorite geometry lessons of the year, in which we just spend the entire class playing and posing questions.
I asked students to spend the first 10 minutes of class just constructing isosceles triangles, and whatever may go along with them. At this point in the year, we have covered constructions of perpendicular bisectors, midpoints, angle bisectors, and we just discovered a variety of triangle centers: orthocenters, centroids, incenters, and more. Students are encouraged to use either pencil and paper, GeoGebra, or both. After playing around and having several sketches and constructions completed, I encourage them to spend 5 minutes looking at their own drawings as well as those of the other three group members, and do a little noticing and wondering. While they do this, I model this process as well, both with a doc camera and with GeoGebra on my SmartBoard.
I’m always impressed with the variety of questions that students come up with, and this year was no different:
Check out how creative these students are! How curious! How thoughtful! How perceptive! I encouraged them to try to find questions that they don’t know the answers to, and then challenged them to find questions that they think I don’t know the answers to.
I’d also like to share something that I found, when playing around with GeoGebra. I constructed an isosceles triangle based on a circle, and then constructed the orthocenter, and then animated one of the legs as I traced the path of the orthocenter.
I made a video and narrated it, showing the steps as well.
Well, it turns out that when you do interesting things and share them on Twitter, you get a wonderful variety of interesting other things happening.
Mark Kaercher (@shskaercher) had all sorts of brilliant ideas to explore:
— Mark Kaercher (@shskaercher) December 5, 2017
Steve Phelps (@giohio) suggested tracing the incenters and excenters together, and later put a whole bunch together:
Do the paths really all intersect at these two points? pic.twitter.com/6J9j8Z3Yck
— Steve Phelps (@giohio) December 5, 2017
Henri Piccioto (@hpicciotto) modeled my original question in Cabri:
I did the orthocenter experiment you suggested in Cabri, because GeoGebra does not give me the equation of a locus, or at least I don’t know how to get it. I used a unit circle centered at the origin, and I put B on the x-axis. See below for the equation. pic.twitter.com/woXswfCPUP
— Henri Picciotto (@hpicciotto) December 5, 2017
And then Henri explored further:
I tried it, with that same experiment. (Fig 1.) In red, the cubic computed by Cabri, which matches the locus (Fig 2.) In blue, the quartic computed by GeoGebra (Fig 3). It too matches the locus, but notice the extra line. Equations are in Fig 4. I think Cabri wins this round? pic.twitter.com/zBllE84Vzp
— Henri Picciotto (@hpicciotto) December 6, 2017
I was thrilled to watch this thread, although I barely was able to take part. You see, I started this post on Monday morning before school, and partway through my second class on Monday, a family emergency came up and I’ve been otherwise occupied for most of the time since then. For anyone I haven’t told who reads this and is curious, please feel free to reach out and ask. Everything isn’t great or okay, but I’m being cautiously optimistic at the moment, and really yearning to get back to the classroom and back to the world of math and teaching and a sense of normalcy.