My favorite lessons build an hour of complicated, engaging mathematics from a simple picture, question, or anecdote. This is one of those lessons.
- The Question
How many Styrofoam cups would you have to stack to reach the top of your math teacher's head?
- Mess With Your Students
Tell them you're 200 centimeters tall (if you're me). Measure a cup in front of them and tell 'em it's around 10 centimeters tall.
Act like you blew it and overestimated the question's difficulty. Ask them for a fast answer.
Someone will divide quickly and tell you "twenty cups," at which point you hold up a stack of 20 cups and let them wonder how they underestimated so grossly.
Let them figure out which math problem they actually solved:
- Offer Them Materials
Ask them what they need from you. Some will ask for hundreds of cups. Offer them ten.
They'll want a ruler. Offer that.
Some will chase you around the room, running after your feet with their stack of cups, asking you to hold still so they can eyeball the answer. Don't offer them that.
- Let It Go
The rest largely runs itself. Just walk around, ask good questions, and correct faulty assumptions.
- Good Questions
- How many parts of the cup are there? Two.
- Which part of the cup matters most over the long run? The lip. The base only counts once but you count the lip every time.
- If I asked you to tell me how tall a stack of sixty cups would be, what would you do? Add the height of sixty lips to the height of the base.
- If I asked you to go backwards and tell me how many cups are in a 200-centimeter-tall stack, what would you do? Subtract the height of the base and then divide by the height of the lip.
- Does it matter if you round to the nearest centimeter? It definitely does.
- Get A Graph And An Equation
Kids will solve this pretty well without either — two groups socked the answer right on the nose — but this is pretty meaningful context for graphs and equations. The lip-height is the slope and the base-height is the y-intercept.
- Actually Stack Them
After you've a) taken secret-ballot estimates from each group, and b) written them down on the board in descending order, have one member from each group i) count her cups, ii) stack them by your feet, and iii) call out the quantity for the rest of the class to tally up.
If, just for instance, you're twice as tall as some of your students, have one student stand on a chair to eyeball the answer. ("One more. Okay, one more. Nope, too much.")
The winning team receives fabulous cash and prizes.
- Extend It
This project has legs. My kids ran outta interest at different points after we announced the winners1 but these extensions are all gold.
- Ask them the same question with a different cup. A red Solo cup2, plastic, a thin lip, and tall base.
- Toss up this graphic.
Have them measure the lip and base of each.
Ask them, "Which will be taller after three cups?" (A: Cup B.)
Ask them, "Which will be taller after one hundred cups?" (A: Cup A.)
And then — respect, if you see what's coming — ask them, "How many cups does it take stack A to rise above stack B?" Wham. You're solving three-step equations.
These are my favorite projects3: easily scaffolded, easily differentiated, easily assessed, and arising completely from a simple question, a simple prop, and a single image.
- Note to self: postpone that announcement until after the extensions. *smacks forehead*
- Don't pretend like you don't know the ones I'm talking about.
- One, again, to which I can only claim certain flourishes. The rest comes out of ed-school at UC Davis.