Playing I Challenge You!

(1) Set-up: Explain to the students that in this activity they are to break up either into pairs (each of the two students will challenge the other) or into groups of four (each pair of students will challenge the other pair). Alternatively, you may specify groups of two or groups of four rather than giving students a choice.

For an odd number of students in class, you may select a student to be an Arbiter who must be convinced that each Challenge (see below) has been met. Alternatively, if you are using groups of 4 or mixed groups (some groups of two, some groups of four), use one group of five with two players on one team and three in another.

(2) Rules: Regardless of the number of players per team (1, 2, or sometimes 3), each team uses one sheet of I Challenge You! per game, as described below. Unless pressed for time, you will want to give each team several sheets, so that a series of games can be played with increasingly complex Challenges made.

The students are to fill out a multiple choice form in the top half of the sheet, indicating the requirements of their Challenge -- the challenge is for the other person or pair to come up with the graph of a function satisfying the requirements given in the Challenge.

You may tell the students they can add things like "at x=3 and x=-1" next to their choice of "two relative maxima" for example. They are to try to craft the requirements to make the task not trivial.

(3) Additional Rules: The Challenger(s) are to first graph a solution (or several) themselves, in the bottom half of the sheet, before they present it to the other team.

Each team is to then fold the paper in half so only the top half is visible. The two teams then switch folded papers. Each person/pair then tries to solve the other's Challenge.

(4) Facing the Challenge: Each team then sketches at least one graph to meet its Challenge. Once solution graphs have been attempted, the two teams compare the Challengers' intended solution(s) with the ones constructed by those challenged. Are they the same? Very different? Are there more possibilities than the designer(s) of the Challenge realized? Was the challenge unintentionally hard, or easier than intended? Why? Encourage and help students to discuss these issues and (if there is time and you use presentations) to prepare to present at the board the Challenges, their solutions, and a discussion of these issues.

(5) Extension (for teams having extra time): The students can repeat the "I challenge you" several times if they finish early. An optional extra challenge if they want to go even further: give formulas for your solution functions. Or, try to classify all functions meeting a given Challenge's requirements.

Optional: Student Presentations

Have groups of students present their solutions to the Activity. Try to select student groups so that they present a variety of different kinds of graphs on the board. Have them explain what the requirements were which they were given. Consider having the person who designed the Challenge explain what was challenging about the requirements they set, and the person presenting explain their thought processes and how they solved the problem.

It would be good to bring up the issue of continuity, and ask the students its relevance. In particular, mention to the point that for a continuous function on a closed interval both absolute extrema must exist (ask the students: by what what Theorem?), so, if a given Challenge's requirements excluded one of both of the absolute extremes from existing, then the domain must either not be a closed interval, or else the function must be non-continuous. Note also that a local maximum or minimum cannot exist (by definition) at an endpoint.