You’ve got your solutions. They’re in a sack. Don’t hold on to them too tightly. Don’t be afraid to cut a few loose.

Dan, can I ask you to more fully explain what you mean by “Keep a loose grip on your own sack of solutions?” I’m not quite sure how to interpret that suggestion.

Thanks for the endless intrigue.

Going to have to think about how this works with 3D shapes: surface area vs volume.

If you write the equation of a circle in the form x^2 + y^2 – 2ax – 2by + c = 0, there are all sorts of interesting things that happen when you study the point (a,b,c) graphed in 3 dimensions.

“Off the top, I’d like students to see that if a polygon has a fixed perimeter and a fixed number of sides, you can get the most area by making every side congruent.”

(1) Great. Let’s simplify to a rectangle. Why does making the sides closer in length (with the same perimeter) increase the area?

(Here’s my explanation.)

“And if you have a fixed perimeter but you let the number of sides do as they please, you enclose more and more area by adding sides to the polygon, which constitutes a limit that ends in a circle.”

(2) Lots here — talking informally about limits will be great.

“Students should understand what points on the x-axis mean. Students should understand why the impossible points are impossible points and why the values for perimeter can increase endlessly even for a fixed small value for area.”

(3) Now we can have fun with fractals, etc!

There’s even good stuff for college-level topology students: Define a topology on figures in the plane. Is this map from there to the plane continuous?

For Dan, DavidC and Joshua:
I love the generation of goals! There are mathematics content goals, representation goals, problem solving goals, and self-reflection goals. All of these are worthy goals for a mathematics lesson and several serve to create a productive classroom climate. However, I don’t agree with Joshua that the “math content is the less important point”. Maybe I’m misinterpreting that statement from him, but I think the math content goal (what students should understand, not just what they should be able to do) should be the most important. Otherwise, how will I know whether students understood the math I wanted them to learn?

Dan’s list of what students should understand is a great start. The first two sentences in his list would make for a super summary at the end of the work on this problem. Making that connection explicit through the goal of what students should understand helped me to think about what to look for in student work, and what to listen for or introduce into a discussion of the problem.

I wholeheartedly agree with emphasizing “the lesson that you can understand what’s going by representing things somewhere you can visualize” — and multiple representations in general. But collecting data and visualizing it is certainly an important habit to get into, and one that will be useful outside of the math classroom, too (and I suspect increasingly useful in the future as well!)

To me the most important thing is that we explicitly point out these strategies to the students, so that they can start learning to use them on their own, rather than continuing to use them only when they’re following directions or being led that way with a carefully designed prompt.