Unit 1, Lesson 19
Evidence, Angles, and Proof
Geometry
19.1
Warm-up
Materials
NoneActivity Narrative
This Math Talk focuses on angle measures of intersecting lines. It encourages students to think about angle relationships and to rely on what they know about straight angles to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students explain why vertical angles are congruent.
In this activity, students have an opportunity to notice and make use of structure (MP7) when they identify supplementary angles.
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. (Start with the 50-degree angle, then 65 degree, 30 degree, and 135 degree). For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneMentally evaluate all of the missing angle measures in each figure.
Activity Synthesis
To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone use the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because .” or “I noticed , so I.” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Advances: Speaking, Representing
19.2
Activity
Materials
NoneActivity Narrative
The purpose of this activity is for students to generate an informal conjecture and study how to describe it more precisely by labeling a figure. Students begin by describing three examples of angle bisectors and forming a conjecture. By engaging with this explicit prompt to take a step back and become familiar with a context and the mathematics that might be involved, students are making sense of problems (MP1).
In this partner activity, students take turns describing the reasoning of the characters in the scenario. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
Launch
Arrange students in groups of 2. Tell students to close their books or devices (or to keep them closed). Display three examples of angle bisectors of linear pairs for all to see:
Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Things students may notice:
- There are solid and dashed lines.
- There is a horizontal line in all three.
- The two solid angles make a linear pair.
Things students may wonder:
- Are the dashed lines angle bisectors?
- Do the dashed lines make a right angle?
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. If the conjecture that the angle between the angle bisectors is always a right angle does not come up during the conversation, ask students to discuss this idea.
Tell students that they will read a conversation between Elena and Diego and then take turns explaining what they read. The first partner explains the first paragraph while the other listens and works to understand. When both partners agree, they switch roles and move on to the next paragraph.
MLR8 Discussion Supports. Display question frames to support small-group discussion: “Can you say more about ?” and “What does this part of mean?”
Advances: Conversing, Representing
Advances: Conversing, Representing
Action and Expression: Internalize Executive Functions. To support organization, provide students with a table to record what they notice and wonder.
Supports accessibility for: Language, Organization
Supports accessibility for: Language, Organization
Lesson Synthesis
Tell students, “In everyday life, it is often complicated to understand all of the reasons why statements are true or false. For example, think about the economic system, international relations, or the history of a country. In geometry, the objects of study are not as complex: angles, lines, points, triangles, and so on. In this way, geometry is a great training ground for understanding the reasons why ideas are true and communicating those reasons to others.”
Inform students that they will be making conjectures and proving them throughout this course. Sometimes they will need to convince their partner. Other times they will need to write a more formal argument that can convince a skeptic. A proof starts with some given information and then carefully details each reason additional facts must be true. At the end there is enough justification to state the conjecture with confidence, at which point we call the statement a theorem.
Add the following theorem to the class reference chart, and ask students to add it to their reference charts:
Vertical angles are congruent.
(Theorem)
Provide the following tip: Look for vertical angles whenever two lines intersect.
Student Lesson Summary
In many situations, it is important to understand the reasons why an idea is true. Here are some questions to ask when trying to convince ourselves or others that a statement is true:
- How do we know this is true?
- Would these reasons convince someone who didn’t think it was true?
- Is this true always, or only in certain cases?
- Can we find any situations where this is false?
In this lesson, we reasoned that pairs of vertical angles are always congruent to each other:
We saw this by labeling the diagram and making precise arguments having to do with transformations or angle relationships. For example, label the diagram with points:
Rotate the figure 180 degrees around point . Then ray goes to ray , and ray goes to ray . That means the rotation takes angle onto angle , so angle is congruent to angle .
Many true statements have multiple explanations. Another line of reasoning uses angle relationships. Notice that angles and together form line . That means that . Similarly, . That means that both and are equal to , so they are equal to each other. Since angle and angle have the same degree measure, they must be congruent.