Defining Key Angle Relationships

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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.

Geogebra Modules

40 minutes

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

 

Defining Angles

20 minutes

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.