Organizing Archimedes' Method

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Objective

SWBAT identify key measurements of polygons inscribed in circles and identify patterns in the data as the number of sides increases.

Big Idea

Give students a scaffold to help organize their calculations so that they can start to make generalizations and find shortcuts in this complicated process.

Warm-Up

30 minutes

The Warm-Up straddles the two big ideas of this unit, so as students are getting started, I encourage them to think about which problems they need the most work with. Students who have already mastered some of the more basic problems are encouraged to skip them. Other students should spend more time on the first three problems to solidify their understandings.

Problem (2) offers students the opportunity to confirm that the lattice points they find satisfy the equation for the circle. Though this may seem like a pretty obvious connection, I find these questions are helpful for students to think about the same idea in multiple ways.

Problem (4) is one of the essential skills for students to be successful in today’s lesson, so I tell them this explicitly several times during warm-up work time. I like to empower them to decide for themselves when to work on a problem. If students skip Problem 4 or don’t know how to tackle it, I may group them together and teach them all at once. I might also partner them with students who do know how to solve the problem. This is more of a last resort, before this, I ask:

  • What kinds of triangles do we know the most about? How can we use those triangles to understand this situation?
  • Which triangle tools apply to this situation?

As long as all students have tackled Problem (4), then everyone should be ready to transition. Problems (5) and (6) are provided for students to start thinking about the approximating investigation.

Investigation and New Learning

30 minutes

Closing

10 minutes

The closing of this lesson is more open-ended because students have accomplished different things during the investigation. As I distribute the Approximate Pi Exit Ticket , I remind students of two big ideas:

(1) The goal of this investigation is to find a method to approximate the value of pi.

(2) The purpose of today's class is to find as many shortcuts and generalizations to do this.

Reminding students of these two foci can help them make sense of the work they were doing during the investigation. It is also a good time for students to share their ideas so that they can learn from each other.