We’re back from break, and I wanted to start with some review of sine and cosine functions to refresh students’ memories. I was inspired by this recent post by Nat Banting to try a “trigonometry menu.” It looked like this:
It was fun! Here’s what I like about it:
- It requires students to think in the opposite direction they usually do. Nat Banting calls this “upstream thinking.” Rather than being given a model and reasoning about it or solving something with it, students are building models to certain specifications.
- It has a low floor and a high ceiling; students can access questions they are more comfortable with first, but solving the task with only three or four functions requires some pretty sophisticated thinking.
- It elicits thinking about relationships. Graphing functions or building functions to match a graph or data can feel formulaic, and encourage students to follow a set procedure without much thinking: find the midline, amplitude, phase shift, and period, set up the graph or equation, rinse, repeat. With the trig menu, I heard students talking about the relationship between the midline and the amplitude, visualizing what different constraints might look like together, and reasoning about which constraints are mutually exclusive. I think this type of flexible thinking is a really valuable opportunity for students to apply their knowledge in a new way.
At the same time, it was tough to get all students to move past writing individual functions for each constraint and think about which constraints can be combined and which cannot. I want to try this type of task again in the future, but I also think it needs to fit in a particular place in the curriculum. Review after a break, when some students lacked the confidence and fluency to work flexibly, probably wasn’t the best place for it. I think this task can be an important stepping stone between typical practice and more sophisticated reasoning, but I think it functions most equitably when students have a solid foundation of fluency with the basic components of trig functions, rather than pausing to review how to find the period from an equation halfway through. This task provides some useful opportunities for thinking, but I want to use it for more sophisticated reasons than just because it feels fun and different.
Five Twelve Thirteen 
Love this activity!! We are working on quadratics, so now I’m trying to think of the same type of activity… your parabola must have only one x-intercept, must open down, must not go through quadrant 1, must have an ‘a’ value of 2, etc.
Check out Nat Banting’s blog post, he has just the thing!
http://natbanting.com/thinking-upstream-with-a-quadratics-menu/
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I just tried this with my 8th graders, I had them design cones and cylinders based on certain specifications (I told them to imagine that they were engineers designing shipping containers for a company that wanted to purchase as few different containers as possible.) The level of engagement was incredible! I heard shouts around the room like, “Yes! Mine works!” and a pair of boys that were working together suddenly ran back to their own seats — they decided to compete against each other to see who could design a better cylinder! One struggling student look up with such relief when I explain that she could design as many cylinders as she wanted (rather than try to combine all the criteria into one), as long as she could give me the volumes of the cylinders she created. So instead of being lost and confused, she was happily designing shapes and practicing finding volumes of solids all period. Thank you for such a great idea!
Glad to hear! And credit goes to Nat Banting (linked above) for sharing it originally — he’s collecting more examples of menus like these on his site!