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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… Pythagorean Theorem Special Right Triangles Trigonometric ratios Angle of Elevation & Depression Vocabulary: Pythagorean Theorem, Special Right triangles, Sine, Cosine, Tangent, adjacent leg, opposite leg, hypotenuse, Angle of Elevation, Angle of Depression Unit Resources Performance Task: Chapter 8 Performance task from www.pearsonsuccessnet.com (Omit Task #4) Project: 1. #2 Trigonometry Project 2. Unit Project from www.pearsonsuccessnet.com Unit Review Game: 1.Culture and Math: Math and the Greeks at discoveryeducation.com http://player.discoveryeducation.com/index.cfm?guidAssetId=9AC20560711E-4BE3-84ED-9D4ADE77D1A1 2.8.1-8.3 Jeopardy Students will be able to… Prove the Pythagorean Theorem using triangle similarity Apply the properties of special right triangles to find side lengths Set up trigonometric ratios in right triangles and use these ratios to solve problems (find side lengths and angle measures). Use the Pythagorean Theorem and trigonometric ratios to solve right triangles in applied math problems and word problems. Mathematical Practices in Focus: 1. Make sense of problems and persevere in solving them 2. Reason abstractly & quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 8. Look for and express regularity on repeated reasoning 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 Make sure that students are relatively comfortable with and have had sufficient practice with writing proofs. Resources a.Textbook Correlation: 8.1 Pythagorean Theorem and its converse; 8.2 Special Right Triangles 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-usingsimilarity 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-A852-470DAB0D-01C012E4E8DD 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 Sample Formative Assessment Tasks Skill-based task 1. Write a proof (two-column, paragraph, etc.) of the Pythagorean theorem using triangle similarity 2. “Special Right Triangles Half Sheet Review” 3. Solve for x and y in each problem below a. b. c. Problem Task 1. Michelle is preparing to build a fence around her backyard and finds that the length of the diagonal is 36 ft. a. If her yard is the shape of a square and she wants to put fencing around three sides (not including the side that 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? d. 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: 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 a) Textbook Correlation: This standard should be taught in conjunction with others in o 8.3 Trigonometry this cluster (G-SRT.7 and G-SRT.8) o 8.4 Angles of Elevation & Depression Use a mnemonic device to help students remember trig b) “Trigonometric Functions” Activity – Omit #3 unless functions (ex. SOH-CAH-TOA) students have already taken Algebra 2. o http://map.mathshell.org/materials/download.php?fileid=8 46 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 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: 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 others in this cluster (G-SRT.6 and G-SRT.8) a) Textbook Correlation: o 8.3 Trigonometry o 8.4 Angle of Elevation & Depression 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 Sample Formative Assessment Tasks Skill-based task 1. A square has a diagonal length of 15 cm. a. Find the perimeter of the square b. Find the area of the square 2. Solve for x in the problems below. a. Problem Task 1. “Hopewell Geometry” Activity o http://map.mathshell.org/materials/download.php?fileid=499 2. David made a ramp for a toy car. The ramp is 3.2 ft long and rises a vertical distance of 1.5 ft. a. What is the measure of the angle formed between the ramp and the ground? b. 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 triangle.” You notice that the right triangle he is examining is scalene. 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? 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: 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 This standard should be taught in conjunction with others in this cluster (G-SRT.6 and G-SRT.7) If time permits, teach 8.4 Law of Sines and Law of Cosines to honors geometry students. It is a topic that will be covered in the fourth math, but exposure in geometry would be beneficial. Resources a) Textbook Correlation: o 8.3 Trigonometry o 8.4 Angle of Elevation & Depression b) “8.4 Review” – Two versions included (one with Law of Sines and Law of Cosines) Sample Formative Assessment Tasks Skill-based task 1. Joanne, who is 5 ft tall, is watching her friend Marjan parasail. a. If Marjan is 400 ft high, and Joanne is 250 ft away from the point directly below Marjan, what is the angle of elevation? 2. In ΔABC, the angle of elevation from C to A is (5m – 37)˚ and the angle of depression from A to B is (3m – 1)˚. a. Solve for m b. Find the measure of Angle A c. Find the measure of Angle B d. Find the measure of Angle C Problem Task 1. 8.4 Enrichment task from www.pearsonsuccessnet.com 2. The task below is a very high level thinking task that also requires knowledge of circles. Therefore, it should be completed after covering Unit 8. http://illustrativemathematics.org/illustrations/607 3. Crystal and Curtis are standing on opposite sides of an 18ft tree, observing a bird’s nest at the top. If Crystal is 5.5 ft tall and uses an angle of elevation of 55˚ and Curtis, who is 6 ft tall, uses an angle of 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.