Summary
Students will investigate and discover the trigonometric ratios (sine, cosine, and tangent) through observations of right triangles. Students are expected to know the Pythagorean Theorem and its converse before beginning this lesson. This lesson is centered around the question: "How can we find missing measures in a right triangle if we cannot use the Pythagorean Theorem?"
Essential Question(s)
How are the angles and sides of a right triangle related?
Snapshot
Engage
Students complete a card sort recalling triangle vocabulary.
Explore
After seeing the need for a way to solve right triangles—other than the Pythagorean Theorem—students investigate the ratios of side lengths in similar right triangles.
Explain
Students define sine, cosine, and tangent based on the relationships discovered in the exploration.
Extend
Students apply their knowledge to solve problems involving trigonometric ratios.
Evaluate
Students demonstrate their knowledge of trigonometric ratios by solving for a missing side length.
Materials
Lesson Slides (attached)
Triangle Cards (attached; one per group; printed front only)
Right Triangle Exploration handout (attached; one per student; printed front only)
Right Triangle Exploration (Sample Responses) document (attached; for teacher use)
Making Connections handout (attached; one per student; printed front only)
Making Connections (Sample Responses) document (attached; for teacher use)
Using Trig Ratios handout (attached; one per student; printed front only)
Using Trig Ratios (Sample Responses) document (attached; for teacher use)
Pencils
Paper
Ruler (one per student)
Scientific calculator (one per student)
Engage
15 Minute(s)
Introduce the lesson using the attached Lesson Slides. Display slide 3 to show the lesson’s essential question. Slide 4 identifies the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.
Display slide 5 and have students get a scientific calculator; follow regular classroom procedures for this. Form students into groups of 2-3 and provide them with a set of the attached Triangle Cards.
Introduce the Card Sort instructional strategy to the class. Allow students 5-10 minutes to sort the cards in their groups. Encourage them to sort the cards in any manner that they find reasonable. Advise students to use scratch paper to assist in this task. You should not help students sort them, but instead, question students about the sorting scheme they used and have them justify it. Choose a few groups to explain/share their sorting with the class.
If necessary, discuss other sorting schemes that become obvious after students have shared.
Explore
30 Minute(s)
Display slide 6, which has a right triangle that is easily solved using the Pythagorean Theorem and knowledge of the sum of the interior angles of a triangle being 180°. Ask guiding questions and have volunteers help work through this problem as a class. Be sure to have students use academic vocabulary and explain not just how to do the next step, but also why. Use the attached Right Triangle Exploration (Sample Responses) document as needed.
After this discussion, display slide 7, which has a right triangle that cannot be solved using the Pythagorean Theorem. Pose the question to the class: "How can we solve this one?" The students will be puzzled, but they may have some idea or agree that they need more information. Because it cannot be solved with their current knowledge, let’s refer to this as the “unsolvable” triangle that we will return to later in the lesson.
Show slide 8 and provide each student with a copy of the attached Right Triangle Exploration handout and a ruler. Explain that by investigating right triangles, the class can figure out a way to solve the mysterious “unsolvable” triangle. Allow students to work in their groups on this investigation. Provide help if needed but try not to guide students explicitly. Instead, question the students about their process and help them come to their own conclusions. Consider asking some of the following questions:
Why do you think that?
What does your group think?
Can you tell me what you've tried already?
What do you think you should do/try next?
How did you find the result?
Do you think it holds if ___?
What did you notice?
What did you wonder?
As students finish gathering measurements, move to slide 9 and direct their attention to the second page of their handout: Comparing Data. Direct students to use their table from the first page and a calculator in order to write each ratio of segment lengths as a decimal.
After a few minutes, transition to slide 10 and have students discuss their observations from the table. Then have them complete the handout by making a prediction about the relationship between the ratios and the sides of the right triangles. As time allows, have students share their predictions with the class.
Have students keep their handout in a safe place as they need it for the Explain portion of the lesson.
Explain
40 Minute(s)
Give each student a copy of the attached Making Connections handout. Transition through slides 11-12 to preview the activity with the class. They need to use the first table from the Right Triangle Exploration handout from the Explore portion of the lesson to complete the first column of their table on their Making Connections handout. They are then to use their calculator to complete the remainder of the table with the goal of writing their own definitions for sine, cosine, and tangent.
Display slide 13 and have students find the buttons on their calculator needed to find the sine, cosine, and tangent ratios of given angles. Direct students’ attention to the first row of the table. Tell the class that for some triangle XYZ, they are given that angle Y is 33°. Have students use their calculator to find the sine, cosine, and tangent of 33°. Tell them to check their work with that first row. In other words, if they find that sin (33°) = 0.54, then they are using their calculator correctly, which is the purpose of this first row of the table. Once students understand how to use their calculator, have them complete the table. As students are working, circulate the room and answer any questions about how to use the calculator, but wait to answer other questions, as students will be noticing patterns in this activity as well.
As students are completing the table, transition to slide 14. Have groups compare their tables with the ones from the second page of their Right Triangle Exploration handout. Direct groups to discuss the similarities between the tables and jot down a few notes on their handout.
Move to slide 15 and introduce the vocabulary of opposite, adjacent, and hypotenuse, using the graphics on the slide. Ask for volunteers to share why they think the base of the two triangles are labeled differently. Facilitate a brief discussion but be sure that students understand that the angle is what determines which side is the opposite and which side is the adjacent side before continuing the lesson. Encourage students to take notes on the back of the Making Connections handout.
Show slide 16 and have groups write their own definitions for sine, cosine, and tangent using their new vocabulary: opposite, adjacent, and hypotenuse on the back of their handouts.
Move to slide 17 and have groups trade their handouts/definitions with another group. Then direct students to use the definition they were given—and only that definition—to find the sine, cosine, and tangent of angle A.
Show slide 18 and have students trade handouts/definitions back and then compare their results. Ask the class if they got the same results or not, and if they did not, have groups share with each other how they might clarify their definition.
After students revise their definitions, use slides 19-20 and instruct students to repeat this task with a different group.
Once students have their own handouts and have had time to again revise their definitions, transition through slides 21-23 to give students the definitions for the ratios of sine, cosine, and tangent. Give students time to edit their definitions as needed.
Use the attached Making Connections (Sample Reponses) document as needed.
Extend
35 Minute(s)
Place students into groups of four. By using the Numbered Heads Together strategy, assign each student a number 1-4 within each group. Display slide 24 and give each student a copy of the Using Trig Ratios handout. Explain to the class that after each question, everyone in each group should be ready to share their group’s answer to the problem and reasoning.
Explain to students that they have approximately three minutes to complete the first question. Remind the class that everyone needs to be ready to share. Then have everyone who was assigned number 2 take turns sharing how their group answered the first question. Repeat this with the remaining three questions on the handout, each time selecting a different assigned number.
Evaluate
5 Minute(s)
Ask the class: Could your new understanding of right triangles help you solve for a missing side? Display slide 25 and return to the "unsolvable" triangle. Use the Exit Ticket strategy to individually assess what students have learned from the lesson. Have students independently find the missing side length, now that they have trigonometric ratios in their toolkit. Allow students a couple of minutes to respond using an index card, sticky note, piece of paper, etc.
Use the hidden slide 26 to check students’ work. Consider unhiding the slide and using it the following day as bellwork. Use students’ responses as a formative assessment to see students’ understanding of this concept and if they are ready for the next topic or need additional practice.
Resources
K20 Center. (n.d.). Bell ringers and exit tickets. Strategies. https://learn.k20center.ou.edu/strategy/125
K20 Center. (n.d.). Card sort. Strategies. https://learn.k20center.ou.edu/strategy/147
K20 Center. (n.d.). Numbered heads together. Strategies. https://learn.k20center.ou.edu/strategy/2476
Wallace, K., Meyers, C. (2015). Discovering trigonometric ratios. [Lesson]. Florida State University. http://www.cpalms.org/Public/PreviewResource/Preview/46546