If you have never rolled a cup across a flat surface and marveled at how precisely it returns to the same place you rolled it from, it's possible you're the wrong audience for this post.
There is math here, certainly, but I have made it a goal this year to stall the math for as long as possible, focusing on a student's intuition before her calculation, applying her internal framework for processing the world before applying the textbook's framework for processing mathematics.
Bad First Question
This one sucks the air right out of the room. We're into the math immediately, having bypassed several easy opportunities to pull in our students who hate math1.
Jason's First Question
Jason Dyer suggests handing out plastic cups, letting students roll them around, then asking "why do they do that?" I have no problem with this approach. I would like to start from a position of stronger student investment, though.
My First Question
Have them roll some plastic cups around. Then toss up this slide and ask them a question that has a correct answer, yes, but which attaches little stigma to the wrong answers. It's an educated guess and different students will make persuasive cases for all three of these. Ask them to write their guesses down, to put them on the record2.
A Lesson Sketch
The conversation can then proceed along some interesting lines where you ask the student to:
- justify her guess.
- draw the kind of cup that will roll the largest circle using a fixed amount of plastic. This is fun. Many will draw a really tall cup, which isn't the best use of limited material. A two-inch-tall cup can roll a circle that's a mile wide.
- make their ideal cup from a page of card stock. The fixed size of the card stock will normalize the results.
- draw a complete picture of their cup including the auxiliary lines. Can they find the invisible center of the circle it will roll? What's the method?
We do all of this before we start separating triangles, before we write up a proof, before we generalize a formula. We ask for all this risk-free student investment before we lower the mathematical framework down onto the problem.
A cool feature of this formula is how well it handles degenerate cases. For example these two:
- A cone's roll-radius is the same as its slant height so letting d = 0 should (and does) eliminate D from the formula.
- A cylinder will roll forever so letting D = d should (and does) return an undefined answer.
From there you can pull out of your cupboard (digital or otherwise) any random set of cups and the students should be able to predict the roll-radius within a small margin of error.
And the framework grows stronger.
A Parting Swipe At Textbooks
I didn't dig this out of a textbook3 but this (hypothetical) scan highlights the difference between my math pedagogy and my textbook's.