After everyone finished the test I decided to try and be a teacher and scaffold a bit. I put the point right in the middle of AB and asked them if the perimeters were the same here. They said “yes”. Then I asked them if the areas were the same – and this when they made me proud and told me “no” because if the perimeters of the square and circumference were the same, then the circle would have a larger area because the circle has the largest possible area for a fixed perimeter. Thus point P had to be closer to B then A. I think that realization made them more proud that it would have if I had already provided them the visual evidence.

http://hypersensitivecranky.wordpress.com/2014/03/09/the-panda-problem/

I translated it, but showed the students the abstraction too. Thought the problem was fun simply to solve, but it had wonderful links to systems, transformations, and geometric relationships.

Thanks for the idea.

This is no replacement for a really good first act. But if you don’t have one, a vague question may be a suitable substitute.

I’ve been going around a couple different places and claiming that “asking questions about the question” is an essential modeling act. But this requires a certain level of comfort from teachers with questions that aren’t fully specified. Not just questions where the information exists but has been withheld for a minute, but questions where the question requires follow-up questions. Like, “What do you mean by that? What form is the answer going to take? What does that word mean in this context?”

I find teachers evenly split in their perception of those ill-defined questions as advantages or disadvantages in class, as features or bugs.

But I also want to give a shout-out to Mr K (27 Feb, #54, above), and here is the link:
http://mathpl.us/posters/circlesquare.001.jpg

What I love is the precise level of vagueness of the question. If that makes sense. You cut the shoelace, make the circle and square, and the question is, “What are the two areas?” It may be a commonplace here (sorry if I’ve missed it), but I think this type of question is important, if only because a really great answer is, “it depends.”

Which motivates the payoff question: “Depends on what?” (Followed by, HOW does it depend…”). I made a bunch of physics labs where the vague question was an essential part of the intro — enough so that I eventually got field-test students to recognize the ploy and shout, “It depends” in unison whenever they sensed one of these.

This is no replacement for a really good first act. But if you don’t have one, a vague question may be a suitable substitute.

What does that second solution mean? Pose that extension question to your Pre-calculus and above students. I have some initial thoughts, but I haven’t checked up on them with any thorough reasoning.

Or can they?

Let’s say the rope is 48 inches long. The square will end up being 12 x 12, or 144 sq in, and the circle will end up having a circumference of 48, so a radius of about 7.64. The area of this circle is then approximately 183.4 square inches. That’s 27% bigger! But by shifting the cut just under an inch and a half (3% of the rope’s length), the areas become equal.

That’s amazing to me. How does 3% = 27%? Why are the areas of a square and a circle with congruent perimeter/circumference measurements so far apart? And why is it only 3% of the rope that makes the difference? And why do so many fencelines intersect at 90º instead of being rounded?

Maybe this problem in and of itself isn’t interesting, but I bet there are other approaches that could be taken to make it interesting – either to you or to your students.

Sometimes giving a problem to solve just feels like same-ol same-ol. Perhaps you could give them an answer and having them find out how to get it.

I just think there is *something* you can do with just about every problem to make it accessible and interesting.