## Purposeful Practice & Dandy Candies

August 5th, 2014 by Dan Meyer

Practice may not always be fun, but it *can* be purposeful. Some of my favorite tasks lately *contain* purposeful practice.

For instance, Dandy Candies tells students they’re going to package up 24 cubical candy boxes. It asks them, “Which of four packages uses the least amount of packaging? Which uses the least amount of ribbon?”

This is the usual house style. Concrete imagery. No abstraction. Contrasting cases. Predictions. Students make their guesses. Then they get the dimensions from the video. They calculate surface area and ribbon length. (Ribbon length is a little bit more interesting than perimeter but not by a lot.) They validate their predictions with their calculations.

But then we ask them to find out if *another* package dimension will use even *less* material.

So now the students have to think systematically, tabling out their work so they don’t waste effort finding the surface area of a lot of different prisms.

Contrast that against a worksheet like this, which is practice also, though rather less purposeful:

Where else have you seen purposeful practice?

I’d look to:

- Robert Kaplinsky and Nanette Johnson’s Open Middle project.
- Malcolm Swan’s tasks: impossible points & break 25.
- Bryan Meyer’s scientific notation task.

**BTW**. Given any number of cubical candies, what is the best way to minimize packaging? Can you prove it? I can handle real number side lengths but when you restrict the sides to integers, my mind explodes a little.

A workshop participant gave this algorithm. I have no reason to believe it works. I also have no reason to believe it doesn’t work.

- Take the cube root of the volume.
- Floor that to the nearest integer factor.
- Square root the remainder factor.
- Floor that to the nearest integer factor.
- With the remainder factor, you have three factors now.
- The smallest of all three factors is your height.
- The other two are your length and width. Doesn’t matter which.

Note to self: test this against a bunch of cases. Find a counterexample where it falls apart.