Using Triangles to Understand Circles

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Objective

SWBAT explain how to use polygons and the triangles that make up these polygons to approximate a circle and the value of pi.

Big Idea

We all know that pi relates to circles--but where does this number come from? Students use and understand Archimedes' method for approximating pi, and in doing so learn an important application of limits.

Warm-Up

30 minutes

Today’s Warm-Up attempts to straddle the two big ideas of this unit. Students have the opportunity to review the key skills and ideas related to circle equations and they are exposed to the first few problems about isosceles triangles.

To ensure that students engage effectively in the review (problems 1 through 4), I tell them many times, "Focus on the problems that you struggle with the most." Though this seems obvious, students are not used to being able to choose the problems they want to do, so they need a lot of urging to skip problems. The idea is to treat the first 4 problems as a menu of options. If students seem to be wasting time on problems that are not challenging for them, tell them to try more challenging problems as you circulate. I like to ask students:

  • Do you feel like you need more practice with that problem? Or do you think it would be a better use of your time to try a more challenging problem?

Everyone should do Problem (5), however, so if students lose focus or end up in very different places after 30 minutes, I make sure to tell everyone to transition to looking at Problem (5) at the same time.

The purpose of Problem (5) is to prime students’ prior knowledge on isosceles triangles. The idea is that this problem is open-ended: “Find out as much as you can,” which will hopefully encourage students to start thinking about anything they know. It also gives me the chance to do a quick, informal pre-assessment as I circulate. I will be looking to see where students stand with respect to the different Levels of Prior Knowledge of Isosceles Triangles that I anticipate they may bring to this problem.

If most of your students do not have all the necessary tools to proceed, I may decide to spend this entire lesson reviewing triangle tools, without looking at the larger task of approximating pi.

 

Prior Skill Review

20 minutes

This lesson has an extra section, which can either be used as an entirely separate lesson or as a short section of this lesson, depending upon students’ prior knowledge in a particular class. I like to frame this review by telling students that we will be using triangles to understand circles, and the eventual goal is to make meaning of the number.

Ideally, my students are able to:

  • Find missing angles in triangles using the Triangle Angle Sum Theorem
  • Use the Pythagorean Theorem to find missing sides of right triangles
  • Use trigonometric ratios to find missing sides of right triangles
  • Use inverse trigonometric ratios to find missing angles of triangle triangles
  • Use equivalent ratios to find missing sides of similar triangles

My goal is to review these skills as a coherent whole, rather than in isolation of each other. For this reason, the Triangle Tools Review worksheet has assorted problems that involve different tools. The idea is for students to be able to determine which tool to apply and when to apply it. I distribute this worksheet to students and ask them to figure out as much as possible about each triangle. The guiding question is:

  • What tools do you have to tackle this? Which tools apply to this situation?

This is a time when it may be necessary to do a few mini-lessons with small groups of students. Likely students have forgotten about the inverse trigonometric ratios, even if they learned these previously. So you can circulate and fill in some gaps. Basically, the essential idea students need to understand is when we can apply the Pythagorean Theorem and when we must use a trigonometric ratio. They then need to know how to choose which ratio to use and how to actually use it. As long as students know how to do these things, they will be ready for the main investigation.

 

 

Investigation and New Learning

20 minutes

Today’s investigation is the first in a series of 3-4 days of work on this problem: How can we approximate the value of ? What is  and where does it come from?

First of all, I make it clear to students that they will have many days to tackle this challenge. I repeat this over and over again at the start of the day’s investigation because I have found that students are more willing to engage in these challenging investigations when they trust that they have enough time to think and ask questions.

I start the investigation by asking students:

  • How can we use triangles to help approximate the value of ?

Though this is a very open-ended question, it gets students thinking (hopefully). Then I show them the images of the Shapes Inscribed in Circles and ask them:

  • What is going on here? How does this help us?

My priority is to leave the questions as open-ended as possible today, and to give students time to really make sense of the question (MP1). The big question is: How do these polygons help us approximate ?

Even if students don’t make a lot of progress today, as long as they are thinking about the problem, asking questions and making notes, I think it is important that I give them this less-structured time, so that they really think about the problem. I remind my students that they will have several days to tackle this, so the more deeply they understand it today, the more successful they will be in the upcoming lessons.

The basic approach is to start with the fact that pi is defined as being the ratio of the circle’s circumference to its diameter. So then—how can find the circumference of a circle whose diameter we know? This is where the inscribed polygons come in. If we set the radius of the circle to be 1 unit, then the we break the polygons into isosceles triangles and use triangle tools to find the perimeter of the inscribed polygon. As the number of sides of the polygon increases, the perimeter of the inscribed polygon gets closer and closer to the circumference of the circle.

I have created a resource with blanks for students to fill-in. I use this summary of Archimedes' Method with students who need some scaffolding to help understand the big ideas—they can think about it for a while and then discuss it in small groups. You could also show this on a projector towards the end of class and use it to facilitate a brief whole class discussion. This paragraph captures the big ideas that we want students to come away with.

 

Closing

5 minutes