May 24, 2010

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Students will develop an understanding of trigonometric ratios and how they are used to measure indirectly. Students will also develop an understanding of how the sine, cosine, and tangent functions and their inverses can solve for missing angle measures and side lengths in right triangles. Finally, students will investigate how to measure indirectly using trigonometry.

- Mathematics > General
- Mathematics > Geometry
- Mathematics > Trigonometry

- Grade 9
- Grade 10

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This unit should take about 3 weeks. It will be introduced in the 4th quarter of a high school geometry class. Students should have a strong understanding of the pythagorean theorem and the properties of right triangles prior to this unit. Students should also be able to set up and solve ratios and proportions.

**Lasting Ideas & Results:**

Enduring Understandings: Students will understand that...

...the trigonometric functions can solve for missing side lengths and angle measures.

...there are many real0world prolbmes that can be solved using trigonometric fucntions.

**Essential Question(s):**

- What are the right-triangle trigonometric functions/ratios for sine, cosine, and tangent?

**Desired Learner Outcomes:**

Students will be able to… | Students will know… |
---|---|

e.g., "compare and contrast…" | e.g., "key facts about…" |

compare the angles of elevation and depression | recognize right triangles and special right triangles |

apply trigonometric ratios to measure indirectly | how to apply the pythagorean theorem to find the missing side length in righr triangles |

identify missing side lenghts and angle measures of right triangles using the trigonometric raios and their inverses |

**Standards:**

Mathematics

, Grades 9-10, GeometryStandard 3: Geometry - The student uses geometric concepts and procedures in a variety of situations.Benchmark 1: Geometric Figures and Their Properties - The student recognizes geometric figures and compares and justifies their properties of geometric figures in a variety of situations.

Knowledge Base Indicators

5. uses the Pythagorean Theorem to:

a. determine if a triangle is a right triangle,

a. determine if a triangle is a right triangle,

5. uses the Pythagorean Theorem to:

b. find a missing side of a right triangle.

b. find a missing side of a right triangle.

6. recognizes and describes --and applies (Geometry Only):

b. the ratios of the sides in special right triangles: 30°-60°-90° and 45°-45°-90°. (Continuing Algebra Only)

b. the ratios of the sides in special right triangles: 30°-60°-90° and 45°-45°-90°. (Continuing Algebra Only)

6. recognizes and describes --and applies (Geometry Only):

d. relationship of sine, cosine, and tangent ratios; (Geometry Only)

d. relationship of sine, cosine, and tangent ratios; (Geometry Only)

6. recognizes and describes --and applies (Geometry Only):

e. draw, label, and solve right triangle problems involving sine, cosine, and tangent. (Geometry Only)

e. draw, label, and solve right triangle problems involving sine, cosine, and tangent. (Geometry Only)

7. recognizes, describes, and compares the relationships of the angles formed when parallel lines are cut by a transversal. (Continuing Algebra Only)

Mathematics

, Grades 9-10, GeometryS3, B1 Application Indicators

b. applying the Pythagorean Theorem, 2. uses deductive reasoning to justify the relationships between the sides of 30°-60°-90° and 45°-45°-90° triangles using the ratios of sides of similar triangles. (Continuing Algebra Only)Mathematics, Grades 9-10, GeometryS3, B2

Application Indicators

1. solves real-world problems by:

d. using the Pythagorean theorem,

Trade & Industry, Grades 9-12, Drafting IXIV. INTEGRATION OF ACADEMICSd. using the Pythagorean theorem,

4. Demonstrate basic use of geometry to solve problems.

**Summative Assessment(s):**

Describe Performance Tasks | Explain &/or Reference Criteria |
---|---|

e.g., quiz | e.g., use Answer Key attached |

Assessment for learning: written | create a mnemonic device to remember the trigonometric ratios |

Assessment for learning: class activity | Use Senteo clickers to answer questions |

Assessment for learning: homework | Prentice Hall textbook problems |

Assessment of learning: personal project | see project guidelines |

Assessment of learning: written test | see test and key |

**Pre-Requisite Knowledge & Skills:**

- Pythagorean theorem and properties of right triangles.
- Be able to set up and solve ratios and proportions.

*Lesson 1*

**Generalization: Trigonometric ratios are used to measure indirectly.**

**Guiding Question: What is the tangent ratio? How is the tangent ratio, and its inverse used to determine information about a right triangle?**

-Begin with a review of right triangle properties. Students will “Pair and Share” about what they already know about right triangles. Teacher will walk around the room and listen to comments. *Cooperative Learning*-Teacher will introduce the Performance Assessment Project.

-Teacher will introduce the Tangent Ratio (See attachment: Section 8.3 Notes #26). Students will write notes in their personal notebooks. *Summarizing and note taking*

-Teacher will illustrate several examples. Students will follow along and write examples at the top of their homework paper.

-Students will complete the lesson quiz on their own, then report back to class. *Student problem-solving*

-Teacher will introduce the Trig Table as a preview for Lesson #2.

-Students will work individually on the book assignment (8.3 page 434 #2-22 even). (Some students will have a modified assignment). *Differentiation/Individual practice/homework*

-Teacher will walk around the room and answer questions.

*Lesson 2*

**Generalization: The sine, cosine, and tangent functions and their inverses can solve for missing angles measures and side lengths.**

**Guiding Questions: What are the sine and cosine ratios? How are the sine and cosine ratios, and their inverses used to determine information about a right triangle? How can trig ratios be solved? What methods can be used?**

-Investigate the Trig Table. Teacher will show students how to use the trig table in their book.

-Allow time for students to ask questions about the assigned homework.

-Discuss multiple ways to solve trigonometric ratios. Ask for student volunteers. *Be sure to illustrate the proportion method.

**Technology**: Students will have the opportunity to use the classroom set of graphing calculators.

-Teacher will introduce the Sine and Cosine Ratios (See attachment: Section 8.4 Notes #27). Students will write notes in their personal notebooks. *Summarizing and note taking*

-Teacher will illustrate several examples. Students will follow along and write examples at the top of their homework paper.

-Students will complete the lesson quiz on their own, then report back to class. *Student problem-solving*

-Students will work individually on the book assignment (8.4 page 441 #2-16 even, 17, 41-43). (Some students will have a modified assignment). *Differentiation/Individual practice/homework*

-Teacher will walk around the room and answer questions.

*Lesson 3*

**Generalization: Generalization: The sine, cosine, and tangent functions and their inverses can solve for missing angles measures and side lengths.**

**Guiding Questions: **

-Allow time for students to ask questions about the assigned homework.

-Teacher will read the SOHCAHTOA Story. (See attachment) *Global experiences: stories*

-Students will create a mnemonic device to remember the three trigonometric ratios and share with class. (AFL)

-Practice problems in teacher assigned small groups *Cooperative Learning/Differentiation*

-Students will work individually on the worksheet assignment (Trig WS). Teacher will walk around the room and answer questions. *Individual practice/homework*

*Lesson 4*

**Generalization: Some things can only be measured indirectly.**

**Guiding Questions: Where are angles of elevation and depression found in the real-world?**

-Allow time for students to ask questions about the assigned homework.

-Illustrate angles of elevation and depression. Students will “Pair and Share” where angles of elevation and depression are found in the real-world.

-Teacher will discuss Angles of Elevation and Depression (See attachment: Section 8.5 Notes #28). Students will write notes in their personal notebooks. *Summarizing and note taking*

-Teacher will illustrate several examples. Students will follow along and write examples at the top of their homework paper.

-Students will answer several questions from the “In class Senteo Review” (See attachment) Reinforcing *effort and providing recognition* (AFL)

**Technology:** The Senteo “clickers” will provide the teacher instant feedback according to student responses. *Assessment for Learning*

-Students will work individually on the book assignment (8.5 page 447 #1-10, 12-14, 16, 17, 35-37). (Some students will have a modified assignment). *Differentiation/Individual practice/homework*

-Teacher will walk around the room and answer questions.

*Lesson 5*

**Guiding Questions: How can we use trigonometry to find the areas of regular polygons?**

-Allow time for students to ask questions about the assigned homework.

-Students will “Pair and Share” about what they already know about areas or regular polygons. *Cooperative Learning*

-Teacher will discuss Trigonometry and Area (See attachment: Section 10.5 Notes #29). Students will write notes in their personal notebooks. *Summarizing and note taking*

-Students will complete the lesson quiz on their own, then report back to class. *Student problem-solving*

-Students will work individually on the book assignment (10.5 page 561 #1-4, 6-9, 11-1621-26). (Some students will have a modified assignment). *Differentiation/Individual practice/homework*

-Teacher will walk around the room and answer questions.

*Lesson 6*

**Generalization: Trigonometric ratios are used to measure indirectly. The sine, cosine, and tangent functions and their inverses can solve for missing angles measures and side lengths. Some things can only be measured indirectly.**

**Guiding Questions: What is trigonometry used for in the real-world? What type of real-world problem would use trigonometry to help model and solve it? **

-Allow time for students to ask questions about the assigned homework.

-Teacher will discuss the Performance Assessment Project, present expectations, and explain the rubric. (AOL)

-Students will begin their research for the project.

**Technology:** The mobile lab will be used for students to search for pictures. Students can use Geometers Sketchpad, Microsoft Word, or another program to begin their project.

*Lesson 7*

-Allow time for students to ask questions about the Performance Assessment.

-Teacher will present practice problems for students to complete individually in order to review for the trigonometry test. Students will respond using their personal dry erase board.

-Students will work individually on the book assignment (Chapter 8 Review page 462 #14-30/Chapter 10 Review #19-25). *Individual practice/homework*

-Teacher will walk around the room and answer questions.

*Lesson 8*

-Trigonometry Test (Some students will have a modified test.) *Differentiation* (AOL)

**Materials:**

- SMART board
- Senteo Clickers

*See Procedures sections

**Required Attachments:**

8Geometry-Chapter8-Trigonometry.notebook |

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