One of the big mathematical practices of the AP Calculus course is justification. Students must be able to explain, give reasons for, and justify conclusions. Often students will have to apply a definition or theorem that gives a rationale for a particular result.

Throughout the course, students learn about three existence theorems: the Intermediate Value Theorem (__Topic 1.16__), the the Mean Value Theorem (__Topic 5.1__), and the Extreme Value Theorem (__Topic 5.2____).__ These theorems explain that if the conditions for the theorem are met, certain values with given properties must exist. While the IVT and MVT must be identified by name when justifying a conclusion in a free response question, the EVT shows up more covertly, whenever a question asks for an absolute maximum or minimum. The EVT guarantees that the absolute extremum must exist, and its location can be determined using the Candidate's Test.

One of the biggest struggles students face is knowing when to use each theorem when they start doing cumulative review at the end of the year. It's easy to know that the question on the Unit 5 test probably requires the MVT, not the IVT, but at the end of the year, students can't use these same clues. This is why students must have deep understandings of each of these theorems: what they say, when they are used, how to write a proper conclusion using the theorem, and what contextual clues to look for in a problem that would indicate which theorem is appropriate.

We have a review activity that does just that! In this Desmos card sort we've created, students must match statements to the appropriate theorem. The first sort helps students review the conditions and conclusions of each theorem and the second sort has students classify a variety of scenarios/question stems based on what theorem should be used, similar to what students would have to do on an FRQ.

Simply follow the link below, and click "Assign to your classes" or "Single session code" to share the activity with your students. You can then use the teacher dashboard to see all of your students' responses.

### Facilitating a Desmos Card Sort

We recommend having students complete this activity in pairs sharing one device. This encourages rich conversation and helps students practice explaining and defending their thinking to each other! In the teacher dashboard, you will be able to see which cards are being sorted correctly and receive information about the most commonly missed card. You can also take snapshots of student work and pause the activity to highlight certain students' work. Student responses can be shown anonymously if you choose.

Note that students will NOT see which cards are correct and we have done this __on purpose__. Immediate feedback in the form of right/wrong can quickly halt discussion and sometimes encourage guessing and checking rather than sense making. Instead, consider pausing the activity after 5-10 minutes and revealing to students the most commonly missed card, then asking them to think about their sort again. Later you can pair two groups together and have them discuss any disagreements they have. You can also project a card that different groups put in different categories and have each group explain their reasoning, eventually coming to class consensus.

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