standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
problem_activity: Activity
This activity is a culmination of all the work students have done with angles in this unit. With less support than in previous activities, students come up with equations that represent the relationships between angles in a figure. Then, students solve their equation to find each unknown angle measure.
Launch
Encourage students to write an equation for each problem. Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge by asking students to start by looking for any vertical, complementary and supplementary angles. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Find the unknown angle measures. Show your thinking. Organize it so it can be followed by others.
[IMG]
[IMG]
Lines
\(\ell\)
and
\(m\)
are perpendicular.
[IMG]
[IMG]
Student Response
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Student Facing
Are you ready for more?
The diagram contains three squares. Three additional segments have been drawn that connect corners of the squares. We want to find the exact value of
\(a+b+c\)
.
[IMG]
Use a protractor to measure the three angles. Use your measurements to conjecture about the value of
\(a+b+c\)
.
Find the exact value of
\(a+b+c\)
by reasoning about the diagram.
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
If students struggle to see the angle relationships in the figures, prompt them to look for any angles that are vertical, complementary, or supplementary to get them started.
Activity Synthesis
The goal of this discussion is for students to see different equations that can be used to represent and solve for the same unknown angle measures.
Select students to share their answers to each problem. Consider asking some of the following questions:
“Did anyone use a different equation for this same problem? If so, did you get the same answer?”
“Were any of the questions harder than others? Why?”
“Were there any questions you used a strategy that was new to you?”
Speaking: MLR8 Discussion Supports.
When selected students share how they calculated the angle measurements, invite other students to challenge an idea, elaborate on an idea, or clarify the idea using improved mathematical language. Encourage students to demonstrate central concepts of the angle relationships (e.g., complementary, supplementary, vertical) multi-modally by explaining their reasoning using images of the angles as well as gestures. This will support student understanding about how to write equations that represent the relationships between angles in a figure.
Design Principle(s): Optimize output (for explanation); Support sense-making


aligns: no