Algebra 1 Unit 3: Systems of Equations s7

Geometry Unit 6: Trigonometry

Enduring understanding (Big Idea): Students will be able to argue and justify the Pythagorean Theorem using right triangle similarity (from the previous unit), as well as identify, use, and apply trigonometric functions as they relate to right triangles. Essential Questions: 1) How do you prove the Pythagorean Theorem using similar triangles? 2) What are special right triangles? How are their ratios used as short cuts for finding side lengths? 3) What are trigonometric ratios? How are they used to solve right triangle problems? 4) What are an angles of elevation and angles of depression? How are they related to and used in right triangle trigonometry? Please note that students should already have a solid foundation of the Pythagorean Theorem from 8th grade.  8.G.6 – Explain a proof of the Pythagorean Theorem and its converse  8.G.7 – Apply Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions  8.G.8 – Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane BY THE END OF THIS UNIT: Students will know… Students will be able to…  Pythagorean Theorem  Prove the Pythagorean Theorem using triangle similarity  Special Right Triangles  Apply the properties of special right triangles to find side lengths  Trigonometric ratios  Set up trigonometric ratios in right triangles and use these ratios  Angle of Elevation & Depression to solve problems (find side lengths and angle measures). Vocabulary:  Use the Pythagorean Theorem and trigonometric ratios to solve  Pythagorean Theorem, Special Right triangles, Sine, Cosine, Tangent, adjacent right triangles in applied math problems and word problems. leg, opposite leg, hypotenuse, Angle of Elevation, Angle of Depression

Unit Resources Mathematical Practices in Focus: Performance Task: Chapter 8 Performance task from 1. Make sense of problems and persevere in solving them www.pearsonsuccessnet.com (Omit Task #4) 2. Reason abstractly & quantitatively Project: 3. Construct viable arguments and critique the reasoning of 1. #2 Trigonometry Project 2. Unit Project from www.pearsonsuccessnet.com others Unit Review Game: 4. Model with mathematics 1.Culture and Math: Math and the Greeks at discoveryeducation.com 8. Look for and express regularity on repeated reasoning http://player.discoveryeducation.com/index.cfm?guidAssetId=9AC20560- 711E-4BE3-84ED-9D4ADE77D1A1 2.8.1-8.3 Jeopardy Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Geometry Unit 6: Trigonometry

CORE CONTENT Cluster Title: Prove the Pythagorean Theorem using Similarity Standard: G.SRT.4 – Prove theorems involving similarity. Prove the Pythagorean Theorem using triangle similarity. Concepts and Skills to Master  Use right triangle similarity and geometric mean to prove the Pythagorean Theorem  Learn 30˚-60˚-90˚ and 45˚-45˚-90˚ triangle ratios  Use these ratios to find side lengths in special right triangles SUPPORTS FOR TEACHERS Critical Background Knowledge  Ability to argue and write proofs  Triangle similarity theorems  Knowledge and background of geometric mean and the following theorem: The altitude to the hypotenuse of a right triangle divides the triangle into 2 triangles that are similar to the original triangle and to each other  Ability to simplify radicals Academic Vocabulary Pythagorean Theorem, Special Right Triangles, Leg, Hypotenuse Suggested Instructional Strategies Resources  Make sure that students are relatively a.Textbook Correlation: 8.1 Pythagorean Theorem and its converse; 8.2 Special Right comfortable with and have had Triangles sufficient practice with writing proofs. b.This proof can be found at the following websites:  http://en.wikipedia.org/wiki/Pythagorean_theorem  http://www.khanacademy.org/math/geometry/triangles/v/pythagorean-theorem-proof-using- similarity  http://www.youtube.com/watch?v=Ng2EpkKooo4 c.The following site/document is another way to help students understand and prove the Pythagorean Theorem (it does not use similar triangles) - (“Proofs of the Pythagorean Theorem” Activity)  http://map.mathshell.org/materials/download.php?fileid=804 d.Section C: Special Right Triangles (Located at www.discoveryeducation.com)  http://player.discoveryeducation.com/index.cfm?guidAssetId=64D09380-A852-470D-

Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Geometry Unit 6: Trigonometry

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Sample Formative Assessment Tasks Skill-based task Problem Task 1. Write a proof (two-column, paragraph, etc.) of the 1. Michelle is preparing to build a fence around her backyard and Pythagorean theorem using triangle similarity finds that the length of the diagonal is 36 ft. 2. “Special Right Triangles Half Sheet Review” a. If her yard is the shape of a square and she wants to put 3. Solve for x and y in each problem below fencing around three sides (not including the side that a. b. connects with her home), how much fencing will she need to buy? b. She decides to section off part of her yard to create a vegetable garden (as in the diagram below). How long is each side of the vegetable garden? c. d.

CORE CONTENT Cluster Title: Right Triangle Trigonometry and Pythagorean Theorem Standard: 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. Concepts and Skills to Master  Define trigonometric ratios for acute angles  Set up trigonometric ratios for various triangles using the given angle SUPPORTS FOR TEACHERS Critical Background Knowledge  Order of operations  Basic knowledge of right triangles Academic Vocabulary Sine, Cosine, Tangent, Opposite side, Adjacent side, Hypotenuse Suggested Instructional Strategies Resources  This standard should be taught in conjunction with others in a) Textbook Correlation: this cluster (G-SRT.7 and G-SRT.8) o 8.3 Trigonometry  Use a mnemonic device to help students remember trig o 8.4 Angles of Elevation & Depression functions (ex. SOH-CAH-TOA) b) “Trigonometric Functions” Activity – Omit #3 unless students have already taken Algebra 2. o http://map.mathshell.org/materials/download.php? fileid=846 Sample Formative Assessment Tasks

Skill-based task 1. Find the following ratios. a. Sin S b. Cos S c. Tan S d. Sin R e. Cos R f. Tan R CORE CONTENT Cluster Title: Right Triangle Trigonometry and Pythagorean Theorem Standard: G-SRT.7 – Explain and use the relationship between the sine and cosine of complementary angles. Concepts and Skills to Master  Set up trigonometric ratios for right triangles  Use these trigonometric ratios to find angle measures and side lengths SUPPORTS FOR TEACHERS Critical Background Knowledge  Order of operations  Basic knowledge of right triangles Academic Vocabulary Sine, Cosine, Tangent, Opposite side, Adjacent side, Hypotenuse Suggested Instructional Strategies Resources

 This standard should be taught in conjunction with a) Textbook Correlation: others in this cluster (G-SRT.6 and G-SRT.8) o 8.3 Trigonometry o 8.4 Angle of Elevation & Depression

Sample Formative Assessment Tasks Skill-based task Problem Task 1. A square has a diagonal length of 15 cm. 1. “Hopewell Geometry” Activity a. Find the perimeter of the square o http://map.mathshell.org/materials/download.php?fileid=499 b. Find the area of the square 2. David made a ramp for a toy car. The ramp is 3.2 ft long and rises a 2. Solve for x in the problems below. vertical distance of 1.5 ft. a. What is the measure of the angle formed between the ramp and a. the ground?

Teacher Created Argumentation Tasks (W1-MP3&6) 1. In complete sentences, answer the following questions (include special right triangles, if applicable) a. What is the difference between the Pythagorean Theorem and Trigonometry? b. How would you explain these concepts to a future geometry student? c. How do you determine when to use each concept? 2. Your classmate John makes the following claim: “The sine and cosine values of Angle A should be the same for this Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabeticaltriangle.” /numerical You order notice not suggested that teachingthe right order triangle. PLC’s must he order is theexamining standards to form is scalene. a reasonable unit for instructional purposes. a. Do you agree or disagree with John’s claim? If yes, explain why he is correct. If no, what would you say to John to help him understand why his claim is incorrect? Geometry Unit 6: Trigonometry

CORE CONTENT Cluster Title: Right Triangle Trigonometry and Pythagorean Theorem Standard: G-SRT.8 – Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* Concepts and Skills to Master  Read word problems to construct an accurate/applicable picture  Use the word problem and the constructed picture to solve the problem. SUPPORTS FOR TEACHERS Critical Background Knowledge  Order of operations  Basic knowledge of right triangles Academic Vocabulary Sine, Cosine, Tangent, Opposite side, Adjacent side, Hypotenuse, Angle of Elevation, Angle of Depression Suggested Instructional Strategies Resources  This standard should be taught in conjunction with others in a) Textbook Correlation: this cluster (G-SRT.6 and G-SRT.7) o 8.3 Trigonometry  If time permits, teach 8.4 Law of Sines and Law of Cosines to o 8.4 Angle of Elevation & Depression honors geometry students. It is a topic that will be covered in b) “8.4 Review” – Two versions included (one with Law of the fourth math, but exposure in geometry would be beneficial. Sines and Law of Cosines) Sample Formative Assessment Tasks Skill-based task Problem Task 1. Joanne, who is 5 ft tall, is watching her friend Marjan parasail. 1. 8.4 Enrichment task from www.pearsonsuccessnet.com a. If Marjan is 400 ft high, and Joanne is 250 ft away from the 2. The task below is a very high level thinking task that also requires point directly below Marjan, what is the angle of elevation? knowledge of circles. Therefore, it should be completed after 2. In ΔABC, the angle of elevation from C to A is (5m – 37)˚ and the covering Unit 8. angle of depression from A to B is (3m – 1)˚. http://illustrativemathematics.org/illustrations/607 a. Solve for m 3. Crystal and Curtis are standing on opposite sides of an 18ft tree, b. Find the measure of Angle A observing a bird’s nest at the top. If Crystal is 5.5 ft tall and uses an c. Find the measure of Angle B angle of elevation of 55˚ and Curtis, who is 6 ft tall, uses an angle of d. Find the measure of Angle C elevation of 43˚, how far apart are Crystal and Curtis standing?

Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.