Intermediate Value Theorem (Lesson 8.4 Day 2)

Unit 8 - Day 11

​Learning Objectives​
  • Apply the Intermediate Value Theorem to prove the existence of roots

Quick Lesson Plan
Activity: Can We Guarantee Roots?



Lesson Handout

Experience First

We absolutely loved Desmos’ Intermediate Value Theorem activity found in their Calculus bundle. We thought this would be a great supplement to yesterday’s IVT lesson. The link to the activity can be found here.


Give students about 20 minutes to complete the Desmos slides and type in their responses. Hand out the student lesson page for students to fill out after they complete the Desmos activity. Using the teacher dashboard, look for responses that highlight main ideas about when a root is guaranteed and when it is not. You can take snapshots of these responses and refer to them later in the whole class debrief. Response will be anonymized so students will not know who gave a particular response.


After students have filled out their chart, give groups 5 minutes to share their ideas with their group members. You can have them do this round-robin style where every group member shares one item until all the lists are complete, or simply have groups facilitate the sharing on their own. Encourage students to add to their own chart and to hash out any disagreements.


As a whole class, allow groups to share their conclusions about one of the categories. Ask if other groups agree, disagree, have questions, or want to add on. Then move on to the next group for the next category.


Finish the discussion by asking students to discuss, first in their groups and then to the whole class, what this has to do with the IVT: “How is knowing whether there will be a root or not related to the IVT?”

We gave students the rest of the class period to work on homework related to the IVT and continuity.

Formalize Later

One important take-away from this lesson is that the IVT can be used to prove that a root exists, but not to prove that a root doesn’t exist. If a scenario or graph does not meet the conditions of the IVT, then the IVT can not be used to justify the result; it does not mean that the function with certainty DOESN’T attain the desired value. It is possible in some situations to know for sure there will not be a root (like when the circle did not touch the x-axis) but this result is not based on the IVT.