Develop an understanding of bounding values and bounding functions
Confirm the hypotheses of the Squeeze Theorem (Sandwich Theorem, Pinching Theorem, etc.) and use the theorem to justify a limit result
I can order functions from least value to greatest value on a given interval.
I can confirm that the hypotheses of the Squeeze Theorem have been met.
I can use appropriate bounding functions in the Squeeze Theorem to evaluate limits.
Quick Lesson Plan
This lesson rewards individuals who can determine reasonable bounds on an unknown value. Students are shown a large jar filled with coffee beans and are asked to make two estimates for the number of beans: an estimate that they know is too low and an estimate that they know is too high. Groups then refine their estimates and finally guess the exact number of beans in the jar. This concept is then transferred to function values and the hypotheses of the Squeeze Theorem
We mention that the group with the smallest interval containing the true number of coffee beans will be rewarded, to focus their thoughts on “squeezing” upper and lower bounds to get closer and closer to the true value. Have a table on the front board to record the “too low” and “too high” guesses for each group. We then reveal the correct number of beans at the end of the period.
During the activity, direct student conversation to the fact that the ideal “too low” and “too high” estimates would be the actual value of beans in the jar, thus guaranteeing that the estimate squeezed between those two estimates is also the actual number of beans in the jar.
The condition of continuity will need to be addressed informally since students have not learned the definition of continuity yet. We suggest using intuitive language like “not having to pick up your pencil to trace the graph” as a working definition. Additionally, we are intentionally not including in the box for “Important Ideas” the fact that these functions do not have to be continuous at x = a. This is a detail that may, at this point, overwhelm students, although casual conversation in class might explain this concept.
Confirming that the conditions of this theorem are met is a requirement of MP4: Communication and Notation, which is tested in the FRQ section of the exam. Practicing this skill with the Squeeze Theorem will prepare students well for dealing with the IVT, MVT, L’Hopital’s Rule, and other theorems coming up later in the year.
Taking shortcuts or forgetting all together to confirm the conditions of the Squeeze Theorem must be addressed early and often. Referring to the “Coffee Bean” activity reminds students of the importance of bounding functions.