Exploring Circumference, Area and Cavalieri's principle
Lesson 1 of 4
Objective: Students will be able to discover properties of circumference, area and Cavalieri's principle in stations.
Students will complete two Do Now questions that review using the formula of a circle to find the radius, center and circumference. This will be the first attempt to get an understanding of students’ comfort level with the main CCSS for this lesson: CCSS.Math.Content.HSG-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
During the Do Now, you may notice that some students are confused between the circumference and area formula, which indicates that students need more work on developing informal arguments for deriving these formulas (ultimately, the goal for this lesson!) The goal for the Do Now is to help bridge our last standard, deriving the formula for a circle, with our new goal, deriving the formulas for area and circumference of a circle.
One way to extend this Do Now is to ask students to also find the area of the circle. This would be another great way to determine if students have prior knowledge of GMD.A.1.
After completing the Do Now, you can ask a student to read the objective and agenda for the day prior to jumping into an explanation of stations.
Introduction to Stations
You can provide students with a brief overview of each of the three stations for today’s lesson and split students into groups of 2-3. I would suggest taking some time to put students together who will work well and help engage each other in deeper conversation. For me, this sometimes means sacrificing a little bit time with side conversations by pairing two friends together but knowing that these students will work well together and push each other. Further, I always first group students by thinking of my students with the lowest skill level. I make sure that these students have supportive partners who hopefully have a stronger knowledge base of the material that we are starting.
Below you will find a brief description of each station. Each group should have 15 minutes for each station, and teachers may need to extend this time depending on student pace. After completing their task for each station, students will move to another station and complete all 3 by the end of the class.
Overall, our goal in this lesson is to address how viable arguments (Math Practice #3) can be made to derive and explain geometric formulas. The first station will focus on deriving the value of pi from circumference. Then using the formula and concepts established for circumference, students will use dissection techniques to derive the formula for area of a circle as shown below.
Lastly, we will ask students to use the area of circles to explore Cavalieri’s Principle (GMD.A.1). More specifically, students will create cylinder shapes and compare how cross sections for area of a circle relate to finding the volume of a cylinder.
Homework and Exit Ticket
In the student notes, you can find an exit ticket/homework where students are asked to write an email to a classmate who is sick explaining at least one of the stations in today’s class. Typically at the end of this lesson, we are struggling to find time to complete an exit ticket and get cleaned up so I typically will assign this letter as homework. This is a great assignment to collect as it allows for teachers to get a real idea of what stood out in students’ minds during the activity. Further, by writing the letter, students have a chance to reflect on the activity and solidify the key parts of the lesson.