I heard of a lovely activity for pseudo-discovering the Intermediate Value Theorem at a recent compulsory workshop for the calculus course I teach. It has everything i like in a thing. I do not have a record of the name of the teacher who presented it (and even if I did, I don't know if he wants to be famous on the Internet) so if you are he, please email me if you want credit.
(begin basic text of student handout/activity:)
The intermediate value theorem states: If a function y = f(x) is continuous on a closed interval [a,b], then f(x) takes on every value between f(a) and f(b).
Think about what you remember of conditional statements (from your geometry course:)
1) State the hypothesis of the IVT.
2) State the conclusion of the IVT.
3) In the following, be sure to use the endpoints (a, f(a)) and (b, f(b)).
A. Sketch a diagram where both the hypothesis and the conclusion hold true.
B. Sketch a diagram where the hypothesis is false, but the conclusion is true.
C. Sketch a diagram where the hypothesis and the conclusion are false.
D. Sketch a diagram where the hypothesis is true, but the conclusion is false.
(on to the back of the page)
4) Which one is impossible to do? Explain why.
5) Compare your diagrams with a partner. How are they similar? Different? If they are different, are they both valid?
6) Is any real number exactly 1 less than its cube?
A. Create a function whose roots satisfy the equation.
B. Find f(1) and f(2). How do you know there is a point (c, 0)? What do you know about c?