These are the kinds of things I want them to understand and/or be able to do:

- what physical property you are actually calculating when you calculate a volume or a surface area
- why the formulas are what they are
- how changing a 2D or 3D figure's dimensions affects its area or volume. for example, I think they understand that if you order a pizza that has twice the diameter, you get way more than double the amount of pizza. But I don't think that intuition has any ties to math class.
- isolate a variable in a formula. for example, solve S = lw + wh + lh for w.

1. Design a new label for a given tennis ball canister, oatmeal canister, or soda can. (a) Create a prototype label so that it covers the entire lateral surface of the canister with little to no overlapping paper. (b) Congratulations! The company chose your design and wants to produce 100,000 labels. Calculate how much material (paper, aluminum, whatever) you will need to order.

2. This game at NLVM is quite nice for challenging your intuition about how volumes are related to dimensions.

3. This video features people with charming accents complaining about how the volume of their chocolate bar decreased even though it appears that the surface area stayed the same or possibly increased. I've shown this in the past and found that students are unable to articulate what these people are upset about using the word "volume" (much less intelligently discuss surface area.) The word "volume" from math class is not connected in their brains to "how much stuff inside."

4. Starting with a piece of copier paper, roll it into a cylinder both the long way and the short way. Will it contain the same amount either way? If not, which way holds more? Mathematically justify your response.

5. Starting with a sheet of copier paper, cut four congruent squares out of the corners and fold up the sides to make a box. Who can make the box that holds the most? Kristen Fouss did something like this but in pre-calculus. Geometry probably doesn't need to get into deriving and optimizing a polynomial equation.

6. Starting with a sheet of copier paper, design, cut out, and assemble a right pyramid with a square base. First pass: any pyramid will do. Second pass: make the area of the square exactly ___. Third pass: make the overall height of the pyramid a specified length. Present your best-looking pyramid, including the area of its base, its overall height, its lateral surface area, its total surface area, and its volume.

7. Investigate what happens to area when dimensions change. What happens to volume when dimensions change. (Somehow.)

8. The car talk fuel-tank problem.

9. Some version of the PCMI volume/surface area problems. (If you know the perimeter and area of a rectangle, can you determine its dimensions? Are there any rectangles whose perimeter = area? If you know the surface area and volume of a rectangular prism, can you determine its dimensions? Are there any rectangular prisms whose volume = surface area?)

10. Derive the formula for the volume of a sphere without calculus. From Exeter Book 3. Would pose quite a challenge for my students. They would not be able to do it on their own. In fact, as it is written, it would completely mystify them.

What I am struggling with and probably will be for the next week or so is, how do I take any of these things and fit them into a logical, coherent unit of study of surface area and volume? NY/my district/my school does not provide us with a curriculum. We have : a list of standards, a collection of previous exams, a pacing calendar, and a kind-of crappy textbook, which are all useful in their own limited ways, but none of them tells you what to do in class. I have lessons already written that get it done, so there is no incentive to bother, other than it bothers me when I feel I could be doing a better job. Part of the dilemma is, I feel that any of this would have to be added to what I already do, not replace it. I still need them to be able to, for example, identify that the bases of a prism are the parallel sides, even if they are not on the top and the bottom. And I'm already about two weeks behind in this course.

How do you take a compelling resource and turn it into an effective lesson?