Download Algebra 1 Unit 3: Systems of Equations

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.