Experience First

The idea for this lesson came from my amazing math department colleagues Barb Montgomery and Kathy VanderBee. Students look at a variety of scenarios to determine if a particular output is guaranteed or not. They then categorize the scenarios based on if the value IS guaranteed (Yes) or if it is NOT guaranteed (No), then find similarities in the scenarios. Students were able to describe functions in the “Yes” category as being “smooth functions where you can’t skip values” and the “No” category as representing unpredictable functions that don’t follow a pattern. Students realized that while temperature and the altitude of the plane were continuous, the given outputs still were not guaranteed because they fell outside the range. Without doing any formal teaching, students discovered the IVT!

Before you teach this lesson, print the scenario cards, cut them up and put each set of cards in a bag or envelope so each group can have one set.

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Formalize Later

To debrief this lesson, have groups taking turns sharing which scenario they put into each category and why. Be sure to discuss at least 2 in the Yes category and 3 in the No category (at least one where the function is discrete, and one where the desired value is outside the given range). Be ready for some heated debates!

The idea behind the Intermediate Value Theorem is straightforward and students had on problems explaining it to a peer. The formal justification can be a bit trickier for students. Although we will not insist on the level of rigor required on the AP Calculus exam, it is still very important that students always check the condition first, identify the endpoints, and then reference the IVT in their conclusion. Students often don’t understand what is meant by “the closed interval [a,b]” part of the theorem. These values represent the interval of x-values where the particular output is going to be found on. The expressions f(a) and f(b) represent the corresponding outputs at those endpoints. It can be helpful to use the notation x=a and x=b to reinforce the fact that a and b are x-values.

The big idea of the IVT is that all function values (meaning y-values) between f(a) and f(b) are guaranteed AND that those outputs will occur somewhere between x=a and x=b.

Question 3 of the Check Your Understanding previews some ideas of instantaneous rate of change. D(t) represents the rate at which donuts are being produced, and some students assume this has to be a whole number or that there is only one rate that applies for the whole hour-long interval. Although Unit 9 will delve into these ideas in greater depth, it suffices to explain that the rate is constantly changing and is not a piecewise graph.