Lesson 7

Applying Ratios in Right Triangles

7.1: Tilted Triangle (5 minutes)

Warm-up

Students are asked to calculate side lengths of a right triangle. They can apply their new understanding from the previous lesson and use trigonometry.

Student Facing

Calculate the lengths of sides \(AC\) and \(BC\).

Right triangle A B C. A B is 6 units, angle A C B is 90 degrees, angle A B C is 20 degrees.

Student Response

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

Activity Synthesis

Remind students that \(\sin(20)\) is shorthand for “the length of the side opposite the 20 degree angle divided by the length of the hypotenuse for any right triangle with an acute angle of 20 degrees.” So when they ask the calculator to display \(\sin(20)\), they are asking the calculator for the ratio that is constant for all the triangles similar to this one by the Angle-Angle Triangle Similarity Theorem.

7.2: Info Gap: Trigonometry (20 minutes)

Activity

This info gap activity gives students an opportunity to determine and request the information needed to calculate side lengths of right triangles using trigonometry.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Infogap cards.

Launch

Tell students they will continue to use trigonometry to solve for side lengths of right triangles. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems calculating side lengths of right triangles using trigonometry. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)?”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation 
Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity. 
Supports accessibility for: Memory; Organization 

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

  1. Silently read the information on your card.
  2. Ask your partner, “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
  3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
  4. Read the problem card, and solve the problem independently.
  5. Share the data card, and discuss your reasoning.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to answer the question.
  2. Ask your partner for the specific information that you need.
  3. Explain to your partner how you are using the information to solve the problem.
  4. When you have enough information, share the problem card with your partner, and solve the problem independently.
  5. Read the data card, and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Student Response

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

Activity Synthesis

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

  • “What information did you need to ask for?” (What letters and where they go. An acute angle and a side length.)
  • “What could you figure out once you knew angle \(C\) was the right angle?” (The legs of the right triangle both have the letter \(C\) and the hypotenuse uses the other two letters)

Highlight for students that drawing a diagram and labeling it with the information provided is an important strategy. It’s too easy to forget something or mix up information without a clear diagram to work from.

7.3: Tallest Tower (10 minutes)

Activity

Students continue to use trigonometry to calculate side lengths of right triangles. In this case they apply that skill to a real world context and engage in some error analysis during the synthesis.

Launch

Consider showing where Dubai and Philadelphia are on a map and defining masonry.

Writing, 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 their response to the first question. Students should first check to see if they agree with each other about the height of the building. Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “What did you do first?” or “How did you know to use tangent?” Next, provide students with 3–4 minutes to revise their initial draft based on feedback from their peers. This will help students explain how to use trigonometry to calculate side lengths of right triangles.
Design Principle(s): Optimize output (for explanation)
Action and  Expression: Internalize Executive Functions. Provide students with a four-column table to organize. Use these column headings: angle, adjacent side, opposite side, and hypotenuse. The table will provide visual support for students to identify ratios.
Supports accessibility for: Language; Organization

Student Facing

  1. The tallest building in the world is the Burj Khalifa in Dubai (as of April 2019).

    If you’re standing on the bridge 250 meters from the bottom of the building, you have to look up at a 73 degree angle to see the top. How tall is the building? Explain or show your reasoning.

    Picture of Burj Khalifa with right triangle A B C drawn on top. Right angle C B A. Height of building, A B. Bottom leg B C labeled 250. Angle B C A labeled 73 degrees.
  2. The tallest masonry building in the world is City Hall in Philadelphia (as of April 2019). If you’re standing on the street 1,300 feet from the bottom of the building, you have to look up at a 23 degree angle to see the top. How tall is the building? Explain or show your reasoning.

Student Response

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

Student Facing

Are you ready for more?

You’re sitting on a ledge 300 feet from a building. You have to look up 60 degrees to see the top of the building and down 15 degrees to see the bottom of the building. How tall is the building?

Student Response

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

Anticipated Misconceptions

If students struggle with the city hall question prompt then to draw a diagram with the information they know. 

Activity Synthesis

Display this image of the Philadelphia City Hall:

A photo of Philadelphia's City Hall building

“The exact heights are 829.8 meters for the Burj Khalifa and 548 feet for City Hall. Why don’t those numbers match your calculations?” (Rounding or measurement error. The difference between \(\tan(23)\) and \(\tan(23.2)\) makes a big difference when you work with large numbers.) "Is it reasonable to assume you are accurate to the nearest tenth in this case?" (No, the provided measurements were rounded to the nearest ten meters or hundred feet so our usual rounding scheme doesn't apply.)

Lesson Synthesis

Lesson Synthesis

Tell students, “In addition to measuring the heights of objects that are too tall to reach, professionals also use trigonometry to calculate the heights of objects they are designing. For example, here is some information a billboard designer knows about a new site.”

Display this information:

  • The local law says the maximum height from the ground to the top of any billboard is 50 feet.
  • To see over the trees from the highway people need to look up at least 40 degrees.
  • The highway is 47 feet from the billboard.

Invite students to discuss what they can do with this information. (How tall—from bottom to top of the image—can the billboard be?) Then invite students to answer the questions they generated. (To be above the trees the bottom of the billboard must be 39.4 feet off the ground, so the billboard can be up to 10.6 feet tall.)

7.4: Cool-down - Tallest Tree (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Using trigonometry and properties of right triangles, we can calculate and estimate measures in different right triangles. We can use these skills to estimate unknown heights of objects that are too tall to measure directly. For example, we can't reach the top of this tree with a measuring tape.

Triangle A B C where B C is the height of a tree. Angle A is 10 degrees, angle C is a right angle, and segment A C is 100 feet.

To calculate the height of the tree, we could stand where the angle between the top and bottom of the tree is 10 degrees. Since we know the distance to the tree (the adjacent leg) and would like to know the height (the opposite leg), we need to use tangent. So \(\tan(10)=\frac{h}{100}\). In the calculator we can look up that \(\tan(10)\) is 0.176. Then we can calculate that \(h\) is about 17.6. That means the tree is 17.6 feet tall.