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Unit 8 - Day 17

​Learning Objectives​
  • Review Topics 8.4-8.12

Quick Lesson Plan
Activity: Volume Mash-up



Lesson Handout


Answer Key


As a final review before the Unit 8 test, students are asked to demonstrate their understanding of all the learning targets around area and volume, from solids of revolution to cross sections . Students will work in groups to craft solutions to given problems and then critique the work of others in a fun and revealing Gallery Walk. 


This activity would work well on VNPS (vertical non-permanent surfaces). Depending on your school setting, you may be able to distribute chalk to each group, assign a section of sidewalk, and simply let them begin working. A large window can also become the canvas for a group’s work. Other options include whiteboards or chart paper --- make the best choice for your environment!

Teaching Tips

Have students work in trios or groups of four on this activity. We recommend giving each group member a different writing utensil so you have a quick visual of who has done what. Monitor students and ask questions about their work, asking different students to explain their group’s thought process for how they chose to set up the integral. Avoid answering “is this right?” questions. Instead, you could have each group send a “spy” to check in on some other groups. 


As you are monitoring, keep track of groups that got different answers for their integral set-up. For example for Problem A part d, perhaps some students used the diameter, rather than the radius for the area of the semicircle. You could put a small star next to the students’ work you want to discuss in the debrief.


When debriefing the task, do not go over every part of every question. Instead, focus on those areas where there was disagreement. Have students decide which answer is better and why. Avoid being the final arbiter of which integral set-up is correct. Insist on viable arguments!


As we near the AP Exam, point out some test-taking strategies, like labeling the coordinates of the intersection point in Problem B so they don’t have to write down all the digits for the limits of integration.

Exam Insights

Finding areas of regions and the volumes associated with those regions (either by revolving the region around an axis or using the region as a base for given cross-sectional areas) is standard fare for the AP test each year.  As the last major topic in the AB calculus curriculum, the calculation of areas and volumes utilizes numerous calculus concepts. Student work on these questions will clearly communicate to the AP Readers the scope of their understanding and extent of their mathematical skills.

Student Misconceptions

Peer-to-peer conversations during this activity can alleviate many persistent misconceptions, and careful questioning by the teacher may resolve other specific areas of confusion. Proper use of grouping symbols is critical when setting up the integrals. Properly identifying the radius when the axis of revolution is not the x or y-axis continues to be challenging for students. We always encourage students to draw a sample radius from the axis of revolution to the edge of the region.

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