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# Defining Key Angle Relationships

Lesson 9 of 16

## Objective: SWBAT identify and discuss the different types of angles studied around lines, right angles, and parallel lines.

## Big Idea: We should let students observe the relationships between different types of angles and develop their intuition.

*60 minutes*

#### Geogebra Modules

*40 min*

The goal here was to give students the time they need to observe the relationship between different types of angles. I use the first few slides of the Transversals PowerPoint to introduce students to the GeogebraTube modules that I created for them to use today.

After my inroduction, students start with Module 1: Complementary and Supplementary angles, then work with Module 2: Vertical Angles. As they proceed they gain experience with identifying and reasoning about angles. Finally, when students reach Module 5, they work with all of the standard angles around transversals and parallel lines.

Each module is interactive and set up to minimize unnecessary distractions. This means that each diagram is set up so that students can simply observe things like, "these purple angles are always equal," instead of saying "vertical angles are always equal." The definitions come later in the lesson, but not until they can observe it for themselves.

In my class, I have students work in partnerships and use the Angle Intuition worksheet to guide them through each module. The worksheet asks students reflective questions as they work. I find it helps students to make meaning of angle relationships if they write about their observations as they work. I also believe that it helps to first form observations and explanations in their own words, without the burden of definitions and calculations.

**List of GeogebraTube Modules**:

- Module 1: Complementary and Supplementary Angles
- Module 2: Vertical Angles
- Module 3: Introduction to Alternate Interior Angles
- Module 4: Alternate Interior and Alternate Exterior Angles I
- Module 5 Alternate Interior and Alternate Exterior Angles II

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#### Defining Angles

*20 min*

This section was developed to follow a hands-on investigation of angle relationships using dynamic geometry. In my class, I will interrupt the students towards the end of class and say something like, "Let's name all the things you have been working with in class today." We will complete the Transversals PowerPoint and specifically name and define the angles that were highlighted in each demo applet.

I like to run through the images and of angle relationships and ask students to talk about what they saw in each module. My goal is to connect their work in the module to the formal definitions we are about to give.

It is important to ask students to rephrase each definition in their own words. When possible, I quote students when they describe an angle relationship in their own words. For example, with Module 1, I ask them, "what did you notice?" This opens up the conversation and encourages them to share what they have noticed. Sometimes students don't acknowledge the relationships between angles and instead make statements about each angle. Although the students are looking at complementary and supplementary angles, they might not notice that they add up to 180 or 90. Instead they might say, "I noticed that the blue and purple angles (the two complementary angles) are not equal." In this case, I will then push students a bit and say something like, "are they ever equal? When?" This helps students to realize that the angles are equal when they are both 45 degrees. With this established, the conversation often evolves toward the sum of the two angles.

After these conceptual conversations, the students are ready to define the angles because our definition started with their ideas. We began with their observations, progressed to their inferences, and finished with formal mathematical language. This is a very different approach from starting with formal language and finishing with inferences.

#### Resources

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- UNIT 1: Starting Right
- UNIT 2: Scale of the Universe: Making Sense of Numbers
- UNIT 3: Scale of the Universe: Fluency and Applications
- UNIT 4: Chrome in the Classroom
- UNIT 5: Lines, Angles, and Algebraic Reasoning
- UNIT 6: Math Exploratorium
- UNIT 7: A Year in Review
- UNIT 8: Linear Regression
- UNIT 9: Sets, Subsets and the Universe
- UNIT 10: Probability
- UNIT 11: Law and Order: Special Exponents Unit
- UNIT 12: Gimme the Base: More with Exponents
- UNIT 13: Statistical Spirals
- UNIT 14: Algebra Spirals

- LESSON 1: Developing Right and Straight Angle Intuition
- LESSON 2: Create Problems with Right and Straight angles
- LESSON 3: Why Are Vertical Angles Equal?
- LESSON 4: Create Vertical Angle Problems
- LESSON 5: Developing Transversal Intuition
- LESSON 6: Create Transversal Problems
- LESSON 7: Why Do Triangles Have 180 Degrees?
- LESSON 8: Walking Around a Triangle
- LESSON 9: Defining Key Angle Relationships
- LESSON 10: Triangle Sum Theorem Proof
- LESSON 11: Angles and Algebra
- LESSON 12: Super Practice with Angle Values
- LESSON 13: Super Practice with Angle Values - Feedback session
- LESSON 14: Super Practice with Angles and Algebra
- LESSON 15: Super Practice with Angles and Algebra - Feedback Session
- LESSON 16: My Little Transversal: A multi-day project lesson