Angle Chase Justification
Lesson 4 of 6
Objective: Students will be able to determine angle measurements in a given diagram.
I give this Angle Chase Warm-Up (which comes from the Discovering Geometry textbook) to students because it allows them to apply their understanding of angle relationships on parallel lines. When debriefing the Warm-Up, I make sure to ask students how they came up with the measure of angle q. This is important to surface this in the class discussion since finding the measure for this angle often is the most challenging (MP1) since students must use the transitive property to draw the correct conclusions.
I like to have students work in pairs on the Angle Chase Justification because it creates a safe environment for them to not only apply their understanding of angle relationships on parallel lines, but to write a convincing argument by giving a reason for every statement they make (MP3). I like to have pairs of students working on it together, each writing in a different color, taking turns to find a new angle's measure and giving a reason for how they know.
Angle Relationships Stations
To accommodate different learning styles in the classroom, I like to have students work at stations, where each station features a different problem to solve (MP1). When doing station work, students can choose whether they would like to work individually, with a partner who will go from station to station with him/her, or with the people who are currently at the station. Stations give students the chance to collaborate with different people, as well as get the moving around the room, which is important on a long, block schedule.
I write up my expectations for students' process on the whiteboard:
- Draw the diagram
- Solve the problem and show enough work so anyone can reproduce your results
- Justify your work by writing the angle relationships you used alongside your work
I tell students that their primary goals are:
- To understand the types of angle relationships posed in a problem and to be able to justify their work using angle relationships correctly
- To increase students' accountability for communicating clearly in their work and in their explanations
I tell them one of their peers will evaluate their work and provide them with feedback--I have found that explicitly telling students how they will be held accountable helps keep them focused and on task.
Since this is the first time I am asking students to give feedback to each other, I know we need to discuss what useful feedback looks like. I ask students:
- to be specific in their comments
- to ask questions when they need clarity
- to make suggestions for improvement
For example, instead of students saying "Good job!" I want students to say, "Your work makes sense because you wrote an equation that correctly represents the angles in the problem." Additionally, as so much of the reasoning in these kinds of problems depends on the fact that the lines given must be parallel, I want that to be a specific feature of the explanation that students look for. In this case, "corresponding angles are congruent" would not be a sufficient way to justify their work unless the students first stated that the lines are parallel. By having students engage in giving and receiving feedback, I give them a heightened awareness of what it means to attend to precision (MP6).
After about five silent minutes of students writing out their feedback directly on their peers' papers, I let them talk to each other about their comments and questions. Then I give everyone time to revise their work to make it better.
To assess students' understanding of basic angle relationship ideas, I write the following Exit Ticket questions on the whiteboard and ask students to answer the questions and justify their answers on their own paper, which they hand in before leaving class. I require students to use a picture to help them explain.
- When are alternate interior angles, alternate exterior angles, and corresponding angles congruent?
- When are alternate interior angles, alternate exterior angles, and corresponding angles not necessarily congruent?
- When are vertical angles congruent?
For this homework assignment, I want students to practice applying their understanding of angle relationships and to justify reasoning by writing two proofs. I scaffolded the homework by incorporating a "tool box" of reasons students can pair with their statements when writing their proofs.