# CGE 3b 2 What s My Ratio? The Investigate the three primary trigonometric ratios for right-angled MT2.01 triangles. Summarize investigations.

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1 Unit 2 Trigonometry Lesson Outline Grade 10 Applied BIG PICTURE Students will: investigate the relationships involved in right-angled triangles to the primary trigonometric ratios, connecting the ratios to constants of proportionality between similar triangles developed in Unit 1; solve problems involving right-angled triangles, using the primary trigonometric ratios and the Pythagorean theorem, including problems that require using imperial and metric measurements. Day Lesson Title Math Learning Goals Expectations 1 What s My Ratio? Determine sine, cosine, and tangent ratios for right-angled triangles using concrete materials. MT2.01 CGE 3b 2 What s My Ratio? The Investigate the three primary trigonometric ratios for right-angled MT2.01 Sequel triangles. Summarize investigations. CGE 5a, 5e 3 What s My Angle? Consolidate investigations for sine, cosine, and tangent ratios of right-angled triangles. MT2.01, MT2.04 Determine angles given trigonometric ratios, using scientific calculators. CGE 3e, 7b, 7f Research a career that requires the use of trigonometry. 4 Figure Out the Triangle 5 Solving Right-Angled Triangles (Part 1) Determine the measures of the sides and angles of right-angled triangles using the primary trigonometric ratios and the Pythagorean theorem. Determine the measures of the sides and angles of right-angled triangles, using the primary trigonometric ratios and the Pythagorean theorem. MT2.01, MT2.04 CGE 4f MT2.02 CGE 5a (lesson not included) 6 Solving Right-Angled Triangles (Part 2) (lesson not included) 7 Solving Right-Angled Triangles (Part 3) (lesson not included) 8 Trigonometric Applications (lesson not included) 9 Who Uses Trigonometry? (Project Presentations) (lesson not included) 10 Summative Assessment Determine the measures of the sides and angles in right-angled triangles, using the primary trigonometric ratios and the Pythagorean theorem. Students make and use clinometers to measure angles. Determine the measures of inaccessible objects around the school using the primary trigonometric ratios and clinometers. Solve problems involving the measures of sides and angles in surveying and navigation problems. Solve trigonometric problems by performing conversions between and within the imperial and metric systems. Make a presentation to the class on careers that involve trigonometry. MT2.02, MT2.03 CGE 3f MT2.03 CGE 3c MT2.03, MT3.02 CGE 5b, 7b MT2.04 CGE 2a, 2d TIPS4RM: Grade 10 Applied: Unit 2 Trigonometry 1

3 2.1.1: Similar Triangles Template (Teacher) TIPS4RM: Grade 10 Applied: Unit 2 Trigonometry 3

4 2.1.1: Similar Triangles Template (Teacher) (continued) TIPS4RM: Grade 10 Applied: Unit 2 Trigonometry 4

5 2.1.2: What s My Ratio? Group Activity Fill in each of the columns with information for your triangle. TIPS4RM: Grade 10 Applied: Unit 2 Trigonometry 5

6 2.1.3: What s My Ratio? Individual Reflection 1. If you have a fifth triangle that is similar to your four triangles, what would your hypothesis be about the following ratios? Explain. opposite hypotenuse = adjacent hypotenuse = opposite adjacent Explanation: Explanation: Explanation: = 2. Identify a relationship between the ratios in the chart for: opposite hypotenuse and adjacent hypotenuse 3. Identify a relationship if you divide the ratio for for one of the angles. opposite hypotenuse by adjacent hypotenuse TIPS4RM: Grade 10 Applied: Unit 2 Trigonometry 6

7 Unit 2: Day 2: What s My Ratio? The Sequel Math Learning Goals Investigate the three primary trigonometric ratios for right-angled triangles. Summarize investigations. Grade 10 Applied Materials large chart (grid) paper 3 colours of stickers 75 min Minds On Groups of 4 Activity Students transfer information from the final three columns of BLM onto the appropriate large grid paper, labelled Sine, Cosine, and Tangent. They use stickers of three different colours, one for each ratio. Note: Developing the graph is used to represent the data collected and provides an example of a non-linear and non-quadratic relation. Assessment Opportunities Set up one large grid for each of the three ratios to graph results of the whole class. Horizontal axis degrees from 0 to 90 Vertical axis ratios from 0 to 1 in increments of 0.1 Action! Whole Class Discussion Ask: What do you think the ratio for the sine of 42 degrees is? Students write individual responses using the graph. Demonstrate how to obtain the sine, cosine, and tangent ratio using the scientific calculator. Practise with various ratios and angles. Individual Activity Students check answers from the Day 1 Home Activity using a calculator. Using stickers, students add more data points to the three large grids. Math Process/Using Tools/Observation/Mental Note: Observe how students use their calculator to find sine, cosine, and tangent of angles. Consolidate Debrief Whole Class Discussion Use the three large classroom grids that were created to discuss: type of relationship linear vs. non-linear; type of variable discrete vs. continuous. Exploration Home Activity or Further Classroom Consolidation Use guess and check and your calculator to determine the angle pertaining to the given ratios. Don t use the inverse key. Give students a value that represents a sine ratio, cosine ratio, and a tangent ratio. TIPS4RM: Grade 10 Applied: Unit 2 Trigonometry 7

10 Unit 2: Day 4: Figure Out the Triangle Math Learning Goals Determine the measures of the sides and angles of right-angled triangles using the primary trigonometric ratios and the Pythagorean relationship. Grade 10 Applied Materials BLM cardboard signs for sine, cosine, tangent, and Pythagorean relationship 75 min Minds On Whole Class Review Discuss any issues regarding the research assignment. Review the conventions for labelling triangles (opposite, adjacent, hypotenuse). Review the ratios sine, cosine, and tangent, using the terms opposite, adjacent, and hypotenuse. Pairs Investigation Draw a right-angled triangle on the board or overhead and provide the degrees of one of the acute angles and the length of one side. Students investigate how they might use what they have learned previously to find one of the missing sides. Circulate and ask leading questions, and listen to their dialogue to identify any misconceptions. Ask: How did you know to use that particular ratio? Pairs share their strategy for solving the problem with the rest of the class. Provide further examples and demonstration, as required. Curriculum Expectation/Oral Question/Anecdotal Note: Observe how students label the triangle and identify the ratio to determine the missing sides. Assessment Opportunities Word Wall ratio sine cosine tangent hypotenuse Action! Whole Class Guided Instruction Using questions 1 4, guide students to determine whether they would use sine, cosine, tangent ratios, or the Pythagorean theorem to solve for the unknown side or indicated angle (BLM 2.4.1). Start at the reference angle on the diagram and draw arrows to the two other pieces of information stated in the problem. One of the pieces will be unknown. Label the sides as opposite, adjacent, or hypotenuse and decide which is the appropriate ratio needed to solve the problem. As students complete questions 1 4, summarize the correct solution(s). Students then complete questions 5 8 individually. Note: Some of the questions can use more than one method. Students could use a mnemonic device or make up a sentence to help them to remember the primary trigonometric ratios, e.g., SOHCAHTOA Consolidate Debrief Pairs Share Solutions Pairs share their solutions for questions 5 8; identify incongruent solutions; and make corrections, as required. Concept Practice Home Activity or Further Classroom Consolidation Complete the practice questions. Provide students with appropriate practice questions. TIPS4RM: Grade 10 Applied: Unit 2 Trigonometry 10

11 2.4.1: What s My Triangle? 1. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find x. Solve for x. 2. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find C. Solve for C. 3. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find b. Solve for b. 4. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find x. Solve for x. TIPS4RM: Grade 10 Applied: Unit 2 Trigonometry 11

12 2.4.1: What s My Triangle? (continued) 5. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find x. Solve for x. 6. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find B. Solve for B. 7. Decide whether to use sine, cosine, tangent, or the Pythagorean relationship to find a. Solve for a. 8. Decide whether to use sine, cosine, tangent, or Pythagorean relationship to find C. Solve for C. TIPS4RM: Grade 10 Applied: Unit 2 Trigonometry 12

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