Geometry Mathematics Curriculum Guide Unit 6 Trig & Spec. Right Triangles


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1 Unit 6: Trigonometry and Special Right Time Frame: 14 Days Primary Focus This topic extends the idea of triangle similarity to indirect measurements. Students develop properties of special right triangles, and use properties of similar triangles to develop their understanding of trigonometric ratios. These ideas are then applied to find unknown lengths and angle measurements. Common Core State Standards for Mathematical Practice Standards for Mathematical Practice MP1 Make sense of problems and persevere in solving them. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP8 Look for and express regularity in repeated reasoning. How It Applies to this Topic Analyze given information to develop possible strategies for solving the problem. Use observations and prior knowledge (stated assumptions, definitions, and previous established results) to make conjectures and construct arguments. Use a variety of methods to model, represent, and solve real world problems. Generalize the process to create a shortcut which may lead to developing rules or creating a formula. Unit 6 Clover Park School District Page 1
2 Stage 1 Desired Results Transfer Goals Students will be able to independently use their learning to Use the properties of special right triangles to solve real world geometric situations. Solve geometric problems involving the basic trigonometric ratios of sine, cosine, and tangent. UNDERSTANDINGS Students will understand that The angles in right triangles are related to the ratios of the side lengths. The sine and cosine of complementary angles are related. Right triangles properties can be applied to solve problems. Meaning Goals ESSENTIAL QUESTIONS How do the ratios of the side lengths of right triangles relate to the angles in the triangle? What is the relationship of the cosine and the sine of two complementary angles? What does it mean to "solve" a triangle? Acquisition Goals Students will know and will be skilled at Naming the sides of right triangles as related to an acute angle Recognizing that if two right triangles have a pair of acute, congruent angles that the triangles are similar Comparing common ratios for similar right triangles and develop a relationship between the ratio and the acute angle leading to the trigonometry ratios Using the relationship between the sine and cosine of complementary angles Identifying the sine and cosine of acute angles in right triangles Identifying the tangent of acute angles on right triangles Explaining how the sine and cosine of complementary angles are related to each other Recognizing which methods could be used to solve right triangles in applied problems Solving for an unknown angle or side of a right triangle using sine, cosine, and tangent Applying right triangle trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems Unit 6 Clover Park School District Page 2
3 Stage 1 Established Goals: Common Core State Standards for Mathematics Cluster: Standard(s) Prove theorems involving similarity G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Define trigonometric ratios and solve problems involving right triangles G SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. Specific modeling standard (versus an example of the modeling Standard for Mathematical Practice). Explanations, Examples, and Comments Generalize this theorem to prove that the figure formed by joining consecutive midpoints of sides of an arbitrary quadrilateral is a parallelogram. (This result is known as the Midpoint Quadrilateral Theorem or Varignon s Theorem.) Use cardboard cutouts to illustrate that the altitude to the hypotenuse divides a right triangle into two triangles that are similar to the original triangle. Then use AA to prove this theorem. Then, use this result to establish the Pythagorean relationship among the sides of a right triangle and thus obtain an algebraic proof of the Pythagorean Theorem. Prove that the altitude to the hypotenuse of a right triangle is the geometric mean of the two segments into which its foot divides the hypotenuse. Prove the converse of the Pythagorean Theorem, using the theorem itself as one step in the proof. Some students might engage in an exploration of Pythagorean Triples (e.g., 3 4 5, , etc.), which provides an algebraic extension and an opportunity to explore patterns. What students should know prior to this unit and may need to be reviewed Fluency with ratios and proportional reasoning Fluency with dilations Stage 3 MATERIALS BY STANDARD(S): Teacher should use assessment data to determine which of the materials below best meet student instructional needs. All materials listed may not be needed. Holt Geometry Lesson 5 7 The Pythagorean Theorem Holt Geometry Lesson 5 8 Applying Special Right Holt Geometry Lesson 8 1 Similarity in Right Holt Geometry Lesson 8 2 Trigonometric Ratios Holt Geometry Lesson 8 3 Solving Right Holt Geometry Lesson 8 4 Angles of Elevation and Depression Or use EngageNY Lessons Listed Below EngageNY Geometry Module 2: Lesson 21,24 30 Teacher Student Supplemental Resources Pythagorean Theorem Unit 6 Clover Park School District Page 3
4 Recognize a situation s connection to a mathematical model Basic ability to mathematically support a prediction or hypothesis Explanations, Examples, and Comments Students may use applets to explore the range of values of the trigonometric ratios as θ ranges from 0 to 90 degrees. opposite sine of θ = sin θ = hypotenuse hypotenuse adjacent cosine of θ = cos θ = hypotenuse opposite tangent of θ = tan θ = adjacent θ Adjacent to θ opposite of θ hypotenuse cosecant of θ = csc θ = opposite hypotenuse secant of θ = sec θ = adjacent adjacent cotangent of θ = cot θ = opposite Discovering Geometry 9.1The Theorem of Pythagoras Discovering Geometry 9.2 The Converse of the Pythagorean Theorem Special Right Discovering Geometry 9.3 Two Special Right Discovering Geometry 9.4 Story Problems Trigonometry Discovering Geometry 12.1Trigonometric Ratios Discovering Geometry 12.2 Problem Solving in Right Performance Tasks Georgia CCGPS Analytic Geometry Unit 2: Right Triangle Trigonometry Geometric simulation software, applets, and graphing calculators can be used to explore the relationship between sine and cosine. Students may use graphing calculators or programs, tables, spreadsheets, or computer algebra systems to solve right triangle problems. Example: Find the height of a tree to the nearest tenth if the angle of elevation of the sun is 28 and the shadow of the tree is 50 ft. Unit 6 Clover Park School District Page 4
5 Evaluative Criteria/Assessment Level Descriptors (ALDs): Claim 1 Clusters: Define trigonometric ratios and solve problems involving right triangles Claim 2 Clusters: Define trigonometric ratios and solve problems involving right triangles Claim 3 Clusters: Prove theorems involving similarity Go here for Sample SBAC items Sample Assessment Evidence Stage 2 Evidence Concepts and Procedures Level 3 students should be able to use the Pythagorean Theorem, trigonometric ratios, and the sine and cosine of complementary angles to solve unfamiliar problems with minimal scaffolding involving right triangles, finding the missing side or missing angle of a right triangle. Level 4 students should be able to solve unfamiliar, complex, or multistep problems without scaffolding involving right triangles Problem Solving Level 3 students should be able to map, display, and identify relationships, use appropriate tools strategically, and apply mathematics accurately in everyday life, society, and the workplace. They should be able to interpret information and results in the context of an unfamiliar situation. Level 4 students should be able to analyze and interpret the context of an unfamiliar situation for problems of increasing complexity and solve problems with optimal solutions. Communicating Reasoning Level 3 students should be able to use stated assumptions, definitions, and previously established results and examples to test and support their reasoning or to identify, explain, and repair the flaw in an argument. Students should be able to break an argument into cases to determine when the argument does or does not hold. Level 4 students should be able to use stated assumptions, definitions, and previously established results to support their reasoning or repair and explain the flaw in an argument. They should be able to construct a chain of logic to justify or refute a proposition or conjecture and to determine the conditions under which an argument does or does not apply. Go here for more information about the Achievement Level Descriptors for Mathematics Common Assessment Unit 6 Clover Park School District Page 5
6 Stage 3 Learning Plan: Sample Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning LEARNING ACTIVITIES: Suggested Sequence of the Unit (See NOTES section for more detail for Extended Geometry.) Recommended to use EngageNY Geometry Module 2: Lessons 21,24 30 Include Performance Task: Example Trig Performance Tasks Daily Lesson Components Learning Target Warm up Activities Whole Group: Small Group/Guided/Collaborative/Independent: Whole Group: Checking for Understanding (before, during and after): Assessments NOTES: Connection to Prior Grades: 8.G.6 8: These standards develop understanding of Pythagorean Theorem and it converse. Also, the Pythagorean Theorem is applied to develop distance formula. Extended Geometry classes should select two or three of the Discovering Geometry lessons shown above to support student success in the core curriculum. Additionally, Extended Geometry teachers should consider the algebra review lessons below if additional algebra support is needed. Algebra Review: Solving for a variable in a formula A CED.4 Holt Algebra 1: Chapter 2, Lesson 5 Solving for a variable. Simplifying Radical Expressions: N RN.1,2 Holt Algebra 1: Chapter 11, Lesson 6 Radical Expressions. Unit 6 Clover Park School District Page 6
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