The day after my great “letting students be the teachers” day, we started class by checking the answers to the entire worksheet. I hadn’t done the solutions in full, so I just projected the final answers. I was having students give thumbs up to each question they had solved correctly so I could get a sense of where they were with the material. We got to #16: x^(2/3)=100. No thumbs. Why does the answer key only have 1000 as a solution? Why not +/- 1000?
My students and I both adamantly defended our work to show that it was +/- 1000. The book must be wrong! I thought I’d check WolframAlpha quickly to see what it said, and was surprised (and worried) to see that it only listed 1000 as a solution. I was sure our mathematical work was correct, and it was obvious that -1000 works as a solution because it satisfies the equation when you plug it back in. So… there must be some other reason why WolframAlpha (and the textbook) overlooked -1000 as a solution to the equation. I had students check graphically with their TI-84 calculators, graphing y=x^(2/3) and y=100. The TI-84 showed the graph we expected – symmetric across the y-axis, and intersection points at both values where we expected solutions.
What was going on?
Without me even noticing, a student whipped out his computer and started a reddit thread for our discussion. (This made me smile. Resourcefulness! Also, one of his comments on the thread, “can you explain this like I’m five?” still gives me a fit of the giggles.) The responses started rolling in, and as the day wore on several students from the class (including the one who posted the thread) kept emailing me, stopping by my classroom, and discussing the things we learned together. By the end of the day I was able to give a decent explanation of the issue to a couple of interested students (that WolframAlpha only considers principal nth roots, which is why the y=x^(2/3) graph looks so funny on WA for negative x.) One student emailed me several times over the weekend after continuing to explore the topics we had encountered: “I understand what polar coordinates are, what radians are, what a unit circle is, and I get how Euler’s formula works.” (<—- !!!) And he still wanted to keep exploring.
My students never cease to amaze. Their zeal for learning reinvigorates me.