## dpp24gon.jpg - Section 2: Whole Class Review: Isosceles Triangles

# The Defining Pi Project, Day 4

Lesson 8 of 17

## Objective: SWBAT complete Part 2 of the Defining Pi Project, and Part 3 is introduced.

After using parts of the last three classes to set up our first construction and stage the rest of the Defining Pi Project, today students are given time to work on - and hopefully complete - their next two constructions (see Part 2 Constructions).

To add a little spice, I say they have **12π minutes** to get their work done, which serves a few purposes: **it's fun, and it's a quick check in to see if kids get it.** They naturally want to figure it out, and they don't even think of it as working on a problem. It's building their number sense, and their concept of writing a number in terms of π. Kids also find it so nerdy that it's actually cool, which I really dig. For most of my students, this time frame is pretty tight, and it’s a challenge to get the other two constructions done this quickly. In order to build a sense of urgency in a non-threatening way, I offer silly badges for anyone who can get through the third construction in a short amount of time. A few students achieve this goal, so I hand them the badge and I challenge them to make a 9-gon, 15-gon or 45-gon for their (optional) 4^{th}construction.

The work time itself is a great chance for me to circulate and get a picture of how much kids really understand about the project. Some of them are on a roll, others aren't really sure where to start, and my differentiation strategy is to circulate, answer questions and to facilitate the growth of community expertise by referring students to each other as much as possible.

As students work, I have the opportunity to help them build abstract and quantitative reasoning skills (**MP2**) by frequently asking them to estimate the value that a calculation with yield. “Don’t press enter yet!” I’ll say. “What percentage of the circle is covered by that hexagon?” or “How long do you think that outer side might be?”

After the hexagon construction, a common misconception is that the "length of outer side" is always equal to the radius. I ask students to return to their first construction, and notice that because our central angle was 60 degrees, there was something special about that. With an octagon (or whatever polygon they might follow with), this is no longer the case. I also ask them to notice that in the first construction, this outer side appears to be the same length as the radius. In subsequent constructions of 8 or more sided polygons, simple observation should make it clear that the outer side is not quite as long as the radius.

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After giving students time to work on their own, it's back to a lesson we've seen before:** finding the missing measurements in an isosceles triangle**. It's different this time, because now the lesson is embedded in an important context. I mention this by asking students to recall their problem set from a week earlier (link), and asking them to recognize why we spent time building that background knowledge.

On the board, I write:

**Suppose I split a circle into 24 slices. How many little isosceles triangles will I have?**

Some kids can conceptualize this. I look around the room to see if students can visualize what I’m talking about. If I see a lot of confusion, I take a few moments **to sketch a circle** with 24 slices on the board, and then I highlight one triangle in a different color.

Two notes about running through this example:

- I chose 24 slices just so we could have an example with an odd central angle of 15 degrees. I want students to be comfortable with the idea of 7.5 degrees and 82.5 degrees. There are always a few students in each class who had still been working with the assumption that angles must have whole number measurements.
- I must steadily reiterate that everyone has a different radius and therefore side length for their triangles. I structure the work on the board in such way to emphasize this and to use it to continue to push toward the algebraic generalization for this problem. I then leave this up for the remainder of the class, so students can use it as an example in their work.

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I hand out Part 3 of the Defining Pi Project project so students can start to think about it, and optionally work on before our next class. This part of the project is to be completed with a partner, so I make sure that everyone finds someone who chose a different radius than they did. I go more into depth on this part of the project in my narrative for Day 5**.**

Trig Problem Set 10 is the homework for the week. This is a review problem set that covers many of the basics from the semester so far. I tell students that their Unit 1 and 2 exam will look very similar to this.

We have a little bit of time for students to read through and to ask clarifying questions about the Problem Set #10 or Part 3 of the project.

**Finally, here are today’s Record Sheet Prompts**:

- Write today’s date.
- Write a sentence or two about what you did today.
- Write at least one question that you wondered about in class today.

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- UNIT 1: Statistics: Data in One Variable
- UNIT 2: Statistics: Bivariate Data
- UNIT 3: Statistics: Modeling With Exponential Functions
- UNIT 4: Statistics: Using Probability to Make Decisions
- UNIT 5: Trigonometry: Triangles
- UNIT 6: Trigonometry: Circles
- UNIT 7: Trigonometry: The Unit Circle
- UNIT 8: Trigonometry: Periodic Functions

- LESSON 1: Circles on the Coordinate Plane
- LESSON 2: Manipulating Circles on Geogebra
- LESSON 3: Old and New Knowledge of Circles
- LESSON 4: Problem Solving with Isosceles Triangles and Circles
- LESSON 5: The Defining Pi Project, Day 1
- LESSON 6: The Defining Pi Project, Day 2
- LESSON 7: The Defining Pi Project, Day 3
- LESSON 8: The Defining Pi Project, Day 4
- LESSON 9: The Defining Pi Project, Day 5
- LESSON 10: The Defining Pi Project, Day 6
- LESSON 11: The Defining Pi Project, Day 7
- LESSON 12: The Defining Pi Project, Day 8 (Wrapping Up)
- LESSON 13: Arc Length and Sector Area
- LESSON 14: Radian Measure
- LESSON 15: Review of Circles Learning Targets
- LESSON 16: Defining Pi Project Revisions and Circles Review
- LESSON 17: Circles Review: Open Spaces