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Unit 4, Lesson 11

Solving Problems with Trigonometry

Geometry

11.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

The purpose of this Warm-up is to ensure all students are able to find the area and perimeter of a triangle. In subsequent activities students will be decomposing polygons into triangles to find the areas and perimeters of the shapes.

Launch

None

Activity

None

Find the area and perimeter of this triangle.

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

The goal of this discussion is for students to consider multiple strategies for finding the height of the equilateral triangle and how this value helps them find the area and perimeter.

Display the image, and invite students to share their strategies for calculating . There is no need to emphasize the exact value of because decimal approximations will be more useful for subsequent activities.

Remind students of the area formula for a triangle and the definition of perimeter as needed. These ideas will continue to be relevant in subsequent activities.

11.2

Activity

Instructional Routines

Card Sort

Materials

To Copy (from Blackline Masters)

Circles and Polygons Cards

Activity Narrative

Students sort different images during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP7).

Launch

Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2, and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:

Give students 1 minute to place all the cards face up and start thinking about possible ways to sort the cards into categories.
Pause the class and select 1–3 students to share the categories they identified.
Discuss as many different categories as time allows.

Attend to the language that students use to describe their categories and images, giving them opportunities to describe their images more precisely. Highlight the use of terms like “inscribed,” “radius,” “circumference,” “perimeter,” and “altitude.” If the word “inscribed” does not come up during the conversation, ask students if they remember the word and to record the definition near a relevant item on the list of categories. (A polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. A circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.)

After a brief discussion, invite students to open their books or devices and continue with the activity. Tell students to predict the order of the cards for now. They will calculate precise perimeters in subsequent activities.

11.3

Activity

Instructional Routines

MLR1: Stronger and Clearer Each Time

Materials

To Gather

Geometry toolkits (HS)Scientific calculators Tracing paper

Activity Narrative

During this activity students are asked to compute the perimeter of inscribed regular polygons given only the radius. They will need to figure out that drawing in the altitude will form a right triangle. Drawing this auxiliary segment shows that students recognize the important structure of right triangles to calculate unknown information (MP7). In this activity students are building towards an equation to approximate the value of , which they will complete in the next lesson.

The digital version of this activity includes an applet that may be helpful to display for the class during the Synthesis.

11.4

Activity

Optional

Instructional Routines

Aspects of Mathematical Modeling

Materials

To Gather

Scientific calculators

Activity Narrative

This activity requires more interpretation than a basic right triangle problem and requires students to use skills that will help them in mathematical modeling (MP4). First, students need to draw a diagram and figure out how to label it. Second, students need to recognize that the units do not match and figure out how to convert to a common unit. At this point the problem looks as if there are several basic right triangles, and students will be ready to use their usual techniques of trigonometry and the Pythagorean Theorem.

Launch

Display Mount Everest on a map. Note that there are two main routes up the mountain. This activity uses the one that starts in Nepal. Ask students if they have been hiking or know anything about altitude acclimation.

After a few minutes of quiet work time, ask students which is further for Base to Camp 1, hiking distance or elevation change? (hiking distance, because 2,087 meters is 2.087 kilometers) Display a right triangle for all to see. Invite students to discuss how to label the diagram with important information including the unknown angle of elevation.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 3 rows of the table to complete, including at least one row with an unknown angle and one row with an unknown distance.
Supports accessibility for: Attention, Social-emotional Functioning

Lesson Synthesis

Use the applet to show a circle with polygons both inside and outside the circle, along with the perimeter of each polygon. Demonstrate increasing the number of sides of the polygons.

Ask students, “What do you notice? What do you wonder?” (The circle always stays the same. The perimeter of the inside polygon is getting bigger. How many sides can the polygons have? What number are they going toward?)

Add students’ observations and questions to the list from a previous activity. Inform them they will be generalizing some of these ideas in a later lesson.

to access Cool-down.

Student Lesson Summary

We know how to calculate the unknown sides and angles of right triangles using trigonometric ratios and the Pythagorean Theorem. We can use the same strategies to solve some problems with other shapes—for example, given a regular hexagon with side length 10 units, find its area.

Decompose the hexagon into 6 isosceles triangles. The angle at the center is 60 degrees because . That means we created 6 equilateral triangles because the base angles of isosceles triangles are congruent.

<p>hexagon decomposed into 6 isosceles triangles. angle at center = 60 degrees. Each side of hexagon = 10.</p>

<p> </p>

<p>Equilateral triangle side lengths of 10 units. Altitude, labeled h, is drawn creating a right angle at the base and a 30 degree angle to the left of the altitude.</p>

To find the area of the hexagon, we can find the area of each triangle. Drawing in the altitude to find the height of the triangle creates a right triangle, so we can use trigonometry. In an isosceles (and an equilateral) triangle, the altitude is also the angle bisector, so the angle is 30 degrees. That means , so is about 8.7 units. The area of one triangle is about , or 43.5, square units. So the area of the hexagon is 6 times that, or about 261 square units.

Mount Everest is the tallest mountain on Earth. The peak is 8,849 meters above sea level. It is a challenging hike that is completed in sections.

section	hiking distance (km)	elevation change (m)	angle of elevation ()
Base to Camp 1	6	2,087
Camp 1 to 2	2.8	1,315
Camp 2 to 3		2,625	30–45
Camp 3 to 4		2,460	40
Camp 4 to Summit		2,944	60

Complete the table.
Write an equation relating distance, , elevation, , and angle, , that could represent any section of a hike.

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Lesson 11

Solving Problems with Trigonometry

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

G-GMD.1

G-SRT.8

G-SRT.8.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

Activity Narrative

Launch

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

Building Toward

G-GMD.1

Standards Alignment

Building On

Addressing

G-SRT.8.1

G-SRT.C

Building Toward

G-GMD.1

Standards Alignment

Building On

Addressing

G-SRT.8

N-Q.2

Building Toward