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# The Similar Triangles Project, Part 2: Empirical and Theoretical Data

Lesson 3 of 9

## Objective: SWBAT collect data about their measurements of rectangles, then use this data to make predictions about what's "right".

*70 minutes*

Today's class opens with two problems that help students review the idea that similar shapes are proportional in size. We begin with a pair of squares: students are given the size of one and scale factor, and they have to figure out the length of a side for the other. When they do this, the second part of the problem is then to compare the areas of these two squares. It's satisfying to give kids the chance to understand that 1-dimensional and 2-dimensional scale factors are different. This is not a big deal in today's lesson, but just a fun place to start.

The second problem gives a pair of similar triangles, and students must figure out the scale factor and the lengths of two missing sides. Problem Set #2 consists of problems like this one.

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**Post Data to Posters.**

At the end of the previous class, students used perpendicular lines to make similar triangles appear on rectangular pieces of card stock. Many of them finished this activity, and a few need a little extra time.

What we end up with is a data for everyone who measured this same rectangle.

**Receive Part 2 of the Project, Complete front side.**

As each student records their ratios values on the collection posters, I give them a copy of Part 2 of the Similar Triangles Project. The front side of this document is a learning target review. Throughout this course, I provide ways for students to make their own connections about how their work is related to the learning targets. That is the purpose of the front side of the handout.

**Find group, complete the back of Part 2. // Empirical vs. Theoretical Data**

When the front side of the handout is complete, I instruct students to find the data collection poster that matches their colored rectangle. Students should find their poster and any classmates who had the same rectangle. Once groups are formed, it is the task of each group to take the data on each poster and determine what the "right" value should be for each ratio. I explain to students that these right, or predicted values are theoretical values, and I contrast them with the empirical data we collected.

In order to find the theoretical value for each ratio, students determine the measures of central tendency for the sets of ratio values on their data collection posters. I do not offer guidance about how to use these values to make predictions, however - that conversation is left to students. There is a distinction between calculation and interpretation, and here students must do both. By participating in this task, **students have the chance to reason & argue with each other (MP3)**.

A lot of this is a review from the Fall semester, and this is important. I think it's important to show students that just because the name of a course may change, the math doesn't disappear. The tools that we practiced using in the first semester remain in play!

**Sources of Error**

I don't force the issue, but I look for opportunities to discuss sources of error with students. When students wonder why numbers vary, I answer by asking them to reflect on where these numbers came from. What tools did we use to measure these rectangles? How precise were these measurments? Remember that the similar triangles were made by drawing perpendicular lines on each rectangle. What would happen if someone was a little bit off in measuring their 90 degree angle?

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As will usually be the case this semester, there are currently two homework assignments that students should be working on.

There is a Delta Math assignment called *Trig Exercise 1: Background Knowledge*.

Also, Problem Set #2 is due in a few days.

I remind students of these deadlines and give them a chance to discuss their status with their classmates and to make a plan for getting everything done.

#### Resources

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To finish class, I distribute index cards and display the following Index Card Essay prompt:

**What is the difference between empirical data and theoretical data, and which one is more real? Use evidence from your work to support your answer.**

I look forward to reading what students tell me!

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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: Intro to Trigonometry / What is Similarity?
- LESSON 2: Course Structures and the Similar Triangles Project, Part 1
- LESSON 3: The Similar Triangles Project, Part 2: Empirical and Theoretical Data
- LESSON 4: Definitions of the Trig Ratios and a Fascinating Chart
- LESSON 5: Finishing Up the Similar Triangles Project
- LESSON 6: Solving Right Triangle Trig Problems
- LESSON 7: Constructing an Equilateral Triangle
- LESSON 8: Constructing Special Triangles on Geogebra
- LESSON 9: Using Special Right Triangles and Spotting Some Patterns