## Trig Unit1 CM.pdf - Section 3: Introduction to Capacity Matrix

# Course Structures and the Similar Triangles Project, Part 1

Lesson 2 of 9

## Objective: SWBAT understand some of the important structures in this class, and they will finish part 1 of the Similar Triangles Project.

#### Opener: Tech Set Up

*20 min*

One objective of today's class is to establish a few classroom structures. Use of technology is one of these structures. Throughout the semester, I will have a laptop cart available for use during some classes, and we will use a variety of computer-based tools throughout the semester. Some tools are for the sharing of information with students, others are **mathematical tools (MP5)** that enhance our understanding of new ideas.

Today's class opens with approximately 20 minutes for students to set up and practice using some of these tools. I post the Agenda on the board, I show students the routines for receiving and maintaining laptops, and I tell them that they can access today's agenda going to trig.validusprep.org.

Once computers are distributed, there are three things I want students to see and do:

- Class web site - http://trig.validusprep.org - this redirects to a Wordpress site on which daily agendas and class assignments are posted. Students should go to this web site to see links and information
- Introductory course survey - students use this link to access a survey that consists of a combination of disposition questions and mathematical background knowledge. I use this to get to know my students a bit as the course begins. Feel free to "submit" so you can peruse the entire survey.
- Delta Math - students should set up accounts for this Trigonometry class. There is an initial assignment they can try just to get the feel for this.

With any extra time, students can get started on the first Delta Math assignment of the semester, which is called Trig Exercise 1: Background Knowledge, and which consists of practice modules for identifying perfect squares, reducing algebraic fractions, and simplifying radicals.

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After technology, another important structure in this class is called a "Capacity Matrix." This is a document on which students can track their progress on math content learning targets. I distribute the Unit 1 Capacity Matrices to the class, and I run through the notes on today's Prezi.

**Teacher's Note**: Capacity Matrices may make sense to you or they may require some explanation. I plan to develop a resource explaining how I use them. If you have specific questions please add them as comments to this lesson. I will try to respond, and, address them when I complete the resource.

#### Resources

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**Up To Speed**

After beginning today's class with some explicit explanations of some of our course structures, I use the remainder of the class to show students how the class is going to work: projects will be central to our work, and students will take responsibility for constructing their own understandings of what we're doing.

With this goal in mind, I want students to understand the value of taking notes, so I start by asking, "who has good notes from the previous class? Where did we leave off?"

I'm hoping that someone is still puzzled by the question from the end of our previous class: we're trying to figure out if all rectangles are similar. I can usually count on it. Once the topic is raised, I say that the work we continue today is going to help us do that.

Most students have their initial measurements and ratios for their colored card-stock rectangles. If they need a little extra time on this, I'll give it to them.

**Using Perpendicular Lines to Make Similar Triangles**

Please see U1 L2 narrative for my description of what happens here.

After running through this initial explaination on the board, I circulate and explain it to small groups of students as necessary. When I see that some students get it, I have them help each other out.

The protractor will be a useful tool in this class. This is a first chance to practice using it, although we're only using the 0 and 90 lines here. To get kids thinking about this tool, I like to point out that this protractor can measure any angle from 0 to 180 degrees, and that for some reason 90 degrees is the only angle that doesn't appear twice on this protractor. I ask if anyone can explain why.

**New Table**

At the bottom of Part 1 of the similar triangles project, there is a table with one row that students can use to collect measurements and ratios for their colored rectangle. In order to record the values for both of the dilations of this rectangle, I tell students to copy this table into their notes, and this time give it three rows, one for their original rectangle/triangle, then one for each dilation.

It's always kind of amazing to me, but it's the moment that they fill in the table that causes most of them to have that eureka insight that these ratio values are close - or maybe even equivalent! - in value. This moment builds excitement to figure out what's going on.

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With about 15 minutes left in the class, I remind students that Problem Set 1 is due at the end of the day. They have time now to discuss it with each other, ask questions, and make sure their work is a clear indication of what they understand about the problem.

I also hand out Problem Set #2, which will be due in a week. Problem Set #2 is a pretty basic review of the idea that similar shapes are proportional to each other. I remix this handout, from JMAP. I'm including here a direct link to the source instead of my remixed copy.

As I distribute Problem Set #2, I tell students that this is a review of an important concept from geometry: that if two triangles are similar, their sides are proportional. With a little bit of theater, I point out that *somehow*, this is going to have something to do with the project that we're currently working on.

**Teacher's Note**: Problem Set #2 was taken from the www.jmap.org website, a source for free resources for New York State teachers. The guidelines for using materials on the site prohibit distributing the resources on a third party website.

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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