The most amazing thing happened in 5th period today.
Setting the scene: We’re in Enhanced Math 3 Honors, which (to make a long story short) is an accelerated honors class that includes all of Honors Precalculus and most of Math 3. We started working on inverse trig functions before break, and this is the first day back from winter break, so we’re bound to be rusty.
The task: Inverse or not? from Underground Mathematics
We started as a Stand and Talk. Per Sara VanDerWerf (my fave!), students walked 17 steps away from their desks to find a partner from another table, no writing, just talking for a full 2 minutes. I handed them the prompt: One of the graphs shows 𝑦 = arctan(tan 𝑥). Another shows 𝑦 = tan(arctan 𝑥). Which graphs are these, and why?
Side note – or TANGENT (haha, see what I did there?!?) – this is a good example of a task that I like for a refresh of prior knowledge. I hate reviewing the same material from last year or last semester all over again in the same way. The kids are bored, I’m bored… it’s like we’re assuming they never learned it in the first place. And yet, it seems wrong just jumping into new material that builds on what they’ve learned without acknowledging that they may have forgotten something. What did we learn before break? We learned about inverse trig functions, specifically their domains and ranges. I could just re-teach the same thing in the same way (ugh) or just keep teaching like it was yesterday and not two weeks ago… or I can exercise the same concepts in a different way, like with this task.
Anyway… back to 5th period. It was awesome.
After students talked in pairs, I had them share out some of the ideas they had. Teams were unsure whether 𝑦 = arctan(tan 𝑥) matched graph A or graph B. One student said that it must be graph B because the domain of 𝑦 = arctan(𝑥) is (-π/2, π/2). Another student said that it was actually the range of 𝑦 = arctan(𝑥) that is (-π/2, π/2), which is why they had excluded graph C. One student said that the domain of 𝑦 = tan(𝑥) is (-π/2, π/2), which is why it must match graph B, while another student argued that 𝑦 = tan(𝑥) has a domain of all real numbers except odd multiples of π/2 and that (-π/2, π/2) is only the restriction to make the tangent function one-to-one. One student very eloquently stated, “the domain for this function 𝑦 has to be the same as the domain for tan(𝑥).” That convinced the class that graph A was the only possible logical conclusion.
Whew… already airing misconceptions, using rich vocabulary, strategic thinking… and we haven’t even gotten to the best part!
We moved on to the function 𝑦 = tan(arctan 𝑥). They had been working with the same partner for a while, so I sent them back to their teams of 4 to check in. Amazingly enough, exactly half of the teams thought this equation matched with graph B and half thought it matched with graph C. I gave them a minute to bolster their arguments with the “believers” aka their teammates who agreed with them. Then they had to find a “non-believer” (someone who thought differently) and convince them to join their side. A little trick I picked up from Andrew Stadel – students had to face their body toward opposite walls (one labeled B, one labeled C) so they (and I) could visibly see who agreed with which graph.
Amazing conversations. MP 3 all over the place. Lots of moving bodies, students visibly changing sides. I paused them after a few minutes and asked, “If you were convinced to change sides, what was the argument that convinced you?” One student and his partner explained that the domain of 𝑦 = arctan(𝑥) is all real numbers, then outputs a value between -π/2 and π/2, which 𝑦 = tan(𝑥) then takes in and outputs all real numbers. The only graph with both domain and range unrestricted is graph C.
I saw a lot of students nodding and moving their bodies… except one. D just looked at everyone and said, “but I just don’t understand how the range of arctan(𝑥) doesn’t matter!”
There was a pause… then every single student in the class started explaining and arguing and yelling to try to convince D. There’s absolutely no way he could have understood anything they were saying in the cacophony of voices. To any random passerby, I’m sure it was chaotic and loud, but to me it was the most incredible symphony. Students found their voices and felt confident to use them without any kind of prompting from me.
OK… so maybe it doesn’t sound as amazing as it felt during 5th period. TBH it wasn’t even as good in 6th period when I
did tried to do the exact same thing. But I overheard one student say as she walked out the door, “I’m going to convince you that I’m right during English next period” and another stayed after to ask me questions (which I did not answer) and walk through his thinking (to which I shrugged) until he felt confident that 𝑦 = tan(arctan 𝑥) “matches with graph C, darn it! my brain hurts…”
And isn’t that what we want for our students? To be so engrossed in a problem that they can’t let it go? To feel like their brain has been exercised to the point of exhaustion? To feel confident that they have reached the correct conclusion without verifying the answer with an “expert?” I don’t feel like I’m doing this every day, or even once a week, but it feels amazing for that one small moment when everything clicks.
I love teaching.