The purpose of this Warm-up is for students to use structure to reason about equivalent equations. In this unit, students will write equations to represent how angles are related to each other, and this Warm-up helps prepare for that work.
All of the given statements could be true so students may be quick to say each of them must be true. Ask these students if there is a case in which that particular statement would not be true for possible values for and .
Student Lesson in Spanish
Launch
Arrange students in groups of 2.
Ask students, “If we know for sure that , what are some possible values of and ?” Give students 30 seconds of quiet think time, and then ask several students to share their responses. Some examples are and , and , and and . Tell students that in this activity, we know for sure that , but we don’t know the exact values of and .
Give students 2 minutes of quiet work time followed by 1 minute to discuss their responses with a partner. Follow with a whole-class discussion.
Activity
None
If is true, which statements also must be true?
and
Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share their reasoning for each statement. If students disagree, allow students to discuss until they come to an agreement. Consider asking some of the following questions while students discuss:
“Do you have an example that might support this statement being true (or untrue)?”
“What evidence do you have to support that statement being true (or untrue)?”
“What other values of and might work?”
“What was done to the equation to make the statement true (or untrue)?”
If students claim that or and must be true, explain that this may be true, but does not have to be true. Invite them to discuss this idea and think of examples where but their statement is not true.
If students claim that cannot be true, ask them to consider what would happen if one of the values is 0.
3.2
Activity
Standards Alignment
Building On
Addressing
7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
In this activity, students see that angles do not need to be adjacent to each other in order to be considered complementary or supplementary. Students are given two different polygons and are asked to find complementary and supplementary angles, using any tools in their geometry toolkit (MP5). The most likely approaches are:
Measure each angle with a protractor and and look for any that add up to 180 or 90 degrees.
Trace the legs of an angle with tracing paper and align its vertex and one leg with another angle to see if the two angles, when adjacent, form a straight angle or a right angle.
As students work, monitor for students who use either approach listed or some other strategy. Also, encourage students to use precise vocabulary and language that they learned in previous activities and lessons (MP6).
Launch
Invite students to share their definitions of complementary and supplementary, and consider displaying the meanings for all to see through the remainder of the class. Highlight ideas that involve angle measurements or sums, and leave ideas implying that they must be touching or adjacent as an open question for now.
Arrange students in groups of 2. Provide access to geometry toolkits. Give students 3–4 minutes of quiet work time, followed by partner and whole-class discussions.
Action and Expression: Internalize Executive Functions. To support organization, provide students with a printed enlarged version of the figures in the Task Statement to be able to measure the angles without extending the lines. Supports accessibility for: Language, Organization
Activity
None
Identify any pairs of angles in these figures that are complementary or supplementary.
Student Response
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Building on Student Thinking
Some students may struggle to use a protractor to measure angles when the rays are not drawn long enough to reach the edge of the protractor. Prompt them to extend the sides of the angle, using a straightedge.
Activity Synthesis
Invite previously selected students to share their strategies for finding a pair of complementary and supplementary angles. Ensure that correct use of a protractor to find the measure of an angle is clearly and carefully demonstrated. This will help all students prepare for the next activity, in which everyone will be using a protractor.
MLR8 Discussion Supports. For each method for finding pairs of angles that is shared, invite students to turn to a partner and restate what they heard, using precise mathematical language. Consider asking the original speaker if their peer accurately restated their method. Advances: Listening, Speaking
3.3
Activity
Standards Alignment
Building On
Addressing
7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
The purpose of this activity is for students to learn about vertical angles. Each student draws two intersecting lines and measures the four resulting angles. Then, students examine multiple examples to come up with a conjecture for any relationships they noticed (MP8).
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Arrange students in groups of 2–4. Provide access to geometry toolkits.
Display this image for all to see. Tell students that this is a pair of intersecting lines, or lines that cross at a point.
Ask students to identify 2 angles that are supplementary to angle ( and ). Invite a student to demonstrate measuring these angles with a protractor (135 degrees). If possible, leave the diagram with angle measures marked displayed throughout the activity.
Tell students they will draw their own diagram with a different set of intersecting lines. Remind them to draw arcs to label the degree measures of their angles.
Give students 2–3 minutes to draw and measure the figure. Remind them to draw arcs to label the degree measures of their angles. Each person in the group should draw a different diagram. Follow with small-group and whole-class discussions.
Use Collect and Display to direct attention to words collected and displayed from an earlier activity. Collect the language that students use to describe the relationships they notice between angles. Display words and phrases, such as “adjacent,” “right,” “straight,” “complementary,” and “supplementary,” as well as descriptions such as “across from each other,” “mirrors of each other,” or “add up to a straight angle.”
Activity
None
Use a straightedge to draw two intersecting lines. Use a protractor to measure all four angles whose vertex is located at the intersection.
Compare your drawing and measurements to the drawings and measurements created by others in your group. What patterns do you notice about the relationships between angle measures at an intersection?
Student Response
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Building on Student Thinking
Some students may label the angle measures toward the end of the rays, where they read the number from the protractor. This is not precise enough, because two different angles share each ray. Remind students about drawing arcs to clarify which angle they measured.
Activity Synthesis
Direct students’ attention to the reference created using Collect and Display. Ask students to share their observations. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “The angles across from each other look the same” already on it, which can be updated with the more precise phrase “The opposite angles have the same angle measure.”)
If there are students who use supplementary angles to explain why vertical angles have equal measures, ask them to share their observations last.
Define vertical angles as a pair of angles, formed by two intersecting lines, that are opposite each other.
Display the image, and ask students to identify four pairs of vertical angles. In particular, students may have trouble seeing that angles and are vertical angles.
Although students don’t need to know a proof that vertical angles always have the same measure, it may be helpful to show one way to understand why they are. If time allows, share that in the image. . .
Angles and are supplementary, so the sum of their measures is 180 degrees.
Angles and are also supplementary, so the sum of their measures is also 180 degrees.
If we take angle away from the straight angles, we see that angles and must have the same measure.
3.4
Activity
Optional
Standards Alignment
Building On
Addressing
7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
This activity gives students an opportunity to practice recognizing complementary, supplementary, and vertical angles and using what they know about those types of angles to find unknown angle measures. Some students may feel comfortable writing equations to show their reasoning, but it is not important that all students use this strategy at this point, as it will be the focus of future lessons. Encourage students to continue using the new vocabulary.
Launch
Arrange students in groups of 2. Make sure students know how to play a row game. Give students 5–6 minutes of partner work time followed by a whole-class discussion.
Activity
None
Find the measurement of the angles in one column. Your partner will work on the other column. Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to find the error and correct it.
column A
column B
is on Line . Find the value of .
Find the value of .
Find the value of .
In right triangle , angles and are complementary. Find the measure of angle .
column A
column B
Angle and angle are supplementary. Find the measure of angle .
is on line . Find the value of .
Find the value of .
is on line . Find the measure of angle .
Two angles are complementary. One angle measures 37 degrees. Find the measure of the other angle.
Two angles are supplementary. One angle measures 127 degrees. Find the measure of the other angle.
Student Response
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Building on Student Thinking
If students struggle to see relationships of angles in figures, prompt students to look for complementary, supplementary, or vertical angles.
Activity Synthesis
Ask students, “Were there any rows on which you and your partner did not get the same answer?” Invite students to share how they came to an agreement on the final answer for the problems in those rows.
Consider asking some of the following questions:
“Did you and your partner use the same strategy for each row?”
“What was the same and different about both of your strategies?”
“Did you learn a new strategy from your partner?”
“Did you try a new strategy while working on these questions?”
Lesson Synthesis
Explain that a conjecture is a statement that we think is true but aren’t certain about. It is more than just a guess. A conjecture could be a guess that is based on some evidence. Ask students to come up with a conjecture about vertical angles. Give students 1–2 minutes to draft a conjecture, then 1–2 minutes to share their conjecture with a partner. Invite a few students to share. (Vertical angles have the same measure)
Here are some additional questions that may be helpful to clarify their descriptions:
What are vertical angles? (A pair of angles across from one another where two lines cross.)
What is true about the measures of vertical angles? (The measures are always the same.)
Do supplementary or complementary angles need to be next to one another? (No.) Think of examples where they are not.
Display diagrams and definitions of new vocabulary somewhere in the classroom so that students can refer back to them during subsequent lessons. “Vertical angles” is new vocabulary. You might consider also adding “intersecting lines” and “conjecture.” As the unit progresses, new terms can be added.
Student Lesson Summary
When two lines cross, they form two pairs of vertical angles. Vertical angles are across the intersection point from each other.
Vertical angles always have equal measure. We can see this because they are always supplementary with the same angle. For example:
This is always true!
so .
so .
That means .
Standards Alignment
Building On
Addressing
7.EE.A
Use properties of operations to generate equivalent expressions.
