Lesson 10

Solving Problems with Trigonometry

10.1: Notice and Wonder: Practicing Perimeter (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that the more sides an inscribed polygon has the closer it comes to approximating a circle, which will be useful when students calculate perimeter and eventually \(\pi\) in later activities. While students may notice and wonder many things about these images, approximating a circle is the important discussion point.

This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

Launch

Display the images 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 the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

A square inscribed in a circle, radius 1 unit.
A hexagon inscribed in a unit circle.
A decagon inscribed in a unit circle.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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. After all responses have been recorded without commentary or editing, ask students “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the word inscribed does not come up during the conversation, ask students if they remember the word and record the definition near a relevant item on the list of things students noticed. (We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.) Leave the responses visible throughout the lesson, as students will refer to them during the next activity and the lesson synthesis.

Demonstrate for students how the inscribed polygons change as the number of sides increases using the applet.

10.2: Growing Regular Polygons (20 minutes)

Activity

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 students recognize the important structure of right triangles to calculate missing information (MP7). In this activity students are building towards an equation to approximate the value of \(\pi\) which they will complete in the next lesson.

Launch

Provide the definitions of a regular polygon (a polygon where all of the sides are congruent and all the angles are congruent), pentagon (5-sided polygon), and decagon (10-sided polygon) as needed.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. During the launch, take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of: regular polygons, pentagons, and decagons. Include definitions as well as diagrams for students to refer to throughout the lesson.
Supports accessibility for: Conceptual processing; Language

Student Facing

  1. Here is a square inscribed in a circle with radius 1 meter. What is the perimeter of the square? Explain or show your reasoning.

    Square inside of circle with radius =1. Distance from center of circle and square to each vertex of square =1.
  2. What is the perimeter of a regular pentagon inscribed in a circle with radius 1 meter? Explain or show your reasoning.

  3. What is the perimeter of a regular decagon inscribed in a circle with radius 1 meter? Explain or show your reasoning.

  4. What is happening to the perimeter as the number of sides increases?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Here is a diagram of a square inscribed in a circle and another circle inscribed in the same square.

diagram of square inscribed in circle and another circle inscribed in the same square. Radius of large circle = 1 unit 
  1. How much shorter is the perimeter of the small circle than the perimeter of the large circle?
  2. If the square was replaced with a regular polygon with more sides, would your previous answer be larger, smaller, or the same? Explain or show your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are spending too long attempting to draw the pentagon or decagon, provide them with a circle that has 10 equally spaced points.

If students are stuck after sketching the diagram ask them what helpful auxiliary lines they could draw. (Connect the center to the other vertices to make triangles. The altitude would create right triangles so I can use trigonometry.)

A circle with 10 points equidistance along the edge.

Activity Synthesis

Invite students to share their strategies. Focus on students who repeated the same strategy for multiple questions.

Use the applet to demonstrate what happens with the perimeter of the inscribed polygon as the number of sides increases. 

Ask students what they noticed. Connect the generalizations they make with the noticing and wondering they did in the warm-up. If not mentioned by students, ask students what the circumference of the circle is. (\(2\pi\)) Build anticipation for calculating the value of \(\pi\) in the next lesson.

If students are not using workbooks tell them to be extra careful to put this work in a safe place because they will use it in the next lesson.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their writing, by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 2–3 partners to share and get feedback on their work. Display prompts for feedback that students can use to help their partner strengthen and clarify their ideas. For example, "Your explanation tells me . . .", "Can you say more about why you . . . ?", and "A detail (or word) you could add is _____, because . . . ." Give students with 3–4 minutes to revise their initial draft based on feedback from their peers. 
Design Principle(s): Optimize output (for explanation); Cultivate conversation

10.3: Gentle Descent (10 minutes)

Activity

This activity requires more interpretation than a basic right triangle problem. First, students need to draw the 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 like a basic right triangle (with large numbers) and students will be ready to use their usual techniques of trigonometry and the Pythagorean Theorem.

Monitor for groups that convert to feet and groups that convert to miles.

Launch

If angle of descent isn’t familiar vocabulary, after 1 minute of quiet work time ask students which side will be longer, the horizontal or the vertical? (Horizontal because it is in miles.) Draw a diagram. Invite students to discuss how to label the diagram with important information including the angle of descent.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because. . .”, “Then/Next, I. . .”, “I noticed _____ so I. . .”, “Another strategy would be _____ because. . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

An airplane travels 150 miles horizontally during a decrease of 35,000 feet vertically.

  1. What is the angle of descent?
  2. How long is the plane's path?
A photo of an airplane landing

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Prompt students to attend to precision with units.

Activity Synthesis

Invite the previously selected groups to share their process of converting to one of the common units. Acknowledge that both methods are valid. Ask which unit is clearer for answering the question of how far the plane traveled (miles).

Speaking: MLR8 Discussion Supports. Use this routine to support small-group discussion. At the appropriate time, give groups 2–3 minutes to plan what they will say when they present their strategy and ensure that everyone in the group can explain each step or part of the solution. Encourage students to consider what details are important to share and to think about how they will explain their reasoning using mathematical language. Then make sure to vary who is selected to represent the work of the group so that students get accustomed to preparing each other to fill that role.
Design Principle(s): Support sense-making; Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Display the images and ask students “What do you notice? What do you wonder?” (All the images have circles. Some circles are inside the polygon and other circles are outside the polygon. How big are the polygons? Which is closest to the circumference of the circle?)

Square inscribed in a circle, radius shown.
A hexagon inscribed in a circle, radius shown.
A decagon inscribed in a circle, radius shown.
A circle inscribed in a square, radius shown.
A circle inscribed in a hexagon, radius shown.
A circle inscribed in a decagon, radius shown.

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

Add students’ observations and questions to the list from the warm-up. Inform them they will be generalizing some of these ideas in the next lesson.

10.4: Cool-down - Again with Area (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We know how to calculate the missing 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.

Hexagon with side length = 10

Decompose the hexagon into 6 isosceles triangles. The angle at the center is \(360^\circ \div 6=60^\circ\). That means we created 6 equilateral triangles because the base angles of isosceles triangles are congruent.

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

 
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.

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 \(\cos(30)=\frac{h}{10}\) so \(h\) is about 8.7 units. The area of one triangle is \(\frac12(10)(8.3)\), or 43.3 square units. So the area of the hexagon is 6 times that, or about 259.8 square units.