Project Summary

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Student ID # 29060

Sarah I. Nicholson

Mentor: Mingming Jiang

TIP3

Project Summary for Instructing a Unit on Trigonometric Ratios and Solving Triangles

(An introduction to Trigonometry)

Sarah Nicholson

Western Governors University

Abstract

This paper summarizes an instructional product for high school geometry students to learn to solve triangles using trigonometric formulas. The product includes lesson plans, worksheets, assessments, and an interactive website. Instruction is teacher-led with hands-on student activities and use of technology for clarification and review. Materials needed include a computer with internet to display the website connected to a TV or projector and screen, student computers for individual or group use, and geometry supplies, such as paper, pencils, calculators, protractors, scissors, glue, string, rulers, and tape measures.

Goal of Instruction

High school geometry students will be able to calculate the solutions to geometry problems (such as finding angles and sides of triangles) by using formulas. As part of this process, students should identify which formula to use, identify where given information fits into a formula, and then use a paper and pencil or calculator to manipulate the values put into the formula.

Audience

Learners are primarily high school sophomores, ages fifteen to seventeen. Students that enroll in Geometry in high school generally have a harder time in math classes than students that take Geometry prior to entering high school because they must be proficient in one math class before moving on. They have a variety of learning styles and need diverse instruction to address these styles.

Students should have a background in elementary algebra in order to enroll in geometry, but some of them come from junior high programs where they only get through half of the algebra curriculum in a year, and there are no classes available to take the second half of the curriculum, so they are enrolled into geometry before they are really ready. Others mistakenly end up in Geometry after failing elementary algebra. As a result, this instruction includes objectives for prerequisite algebra skills and calculator skills that may be included in the instruction if needed. They also need to have been enrolled in Geometry from the beginning of the course to have necessary entry skills in Geometry before learning trigonometry.

Type of Learning

This unit of instruction is for instructor-led classroom use. The web-based instruction included in the unit initially began as a self-directed learning opportunity, and may eventually be developed further to work as such.

Learning Objectives

The objectives are written for what the learners are to learn, but they are worded for the instructors to understand what is to be taught. Objectives should be simplified further before presentation to learners.

Entry Behavior Objectives

Algebra Skills.

1. Learners will demonstrate ability and confidence in adding, subtracting, multiplying and dividing integers by completing on paper problems containing these operations.

2. Learners will demonstrate understanding of the order of operations by accurately completing problems containing a combination of parenthesis, exponents, addition, subtraction, multiplication, and division.

3. Learners will demonstrate ability to use a scientific calculator or graphing calculator by accurately checking their answers to the problems for objective #2 above.

4. Given an algebra problem, learners will accurately solve for the unknown value (variable, such as x) by using addition, subtraction, multiplication and division properties when appropriate.

5. Learners will recognize that the first step to solving a proportion problem (fractions on each side of the equal sign) is to use the cross-product property.

6. Learners will use and simplify problems containing square roots correctly.

Geometry Skills.

1. Learners will demonstrate understanding of the theorem that every triangle adds up to 180 degrees by accurately finding the measure of an angle of a triangle given the measures of two of the angles.

2. Learners will define a right angle as a ninety-degree angle, and will accurately draw or find right angles in their environment.

3. Learners will describe characteristics of a right triangle, accurately draw a right triangle, and select all right triangles from a selection of both right and oblique (non-right) triangles.

4. Learners will identify the right angle of a right triangle and label each side of a right triangle as a hypotenuse or a leg.

Skills and Subordinate Skills Performance Objectives

1. Given a right triangle, learners will identify each side as the hypotenuse, the leg opposite a specific angle, or the leg adjacent to a specific angle.

2. Learners will state the meaning of each trigonometric ratio (sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent) by using the mnemonic device soh cah toa if necessary.

3. Given a right triangle with specific side lengths, learner will write down the ratios for sine, cosine, and tangent for either of the non-right angles.

4. Given a right triangle with information about 2 of the sides and/or angles, learner will state which trig ratio to use to find any of the missing sides or angles.

5. Given a right triangle with information about 2 of the sides and/or angles, learners will demonstrate on paper how to set up an equation using a trigonometric ratio.

6. Given an equation containing a trigonometric ratio, learners will solve the equation for the variable on paper.

7. Learners will explain how to tell if a calculator is in degree mode, and if it is not, demonstrate how to change it to degree mode.

8. Learners will demonstrate how to use a scientific or graphing calculator to calculate the answer to an equation with a trig ratio.

9. Learners will explain and draw pictures of the phrases “angle of elevation” and “angle of depression”.

10. Given a word problem that can be solved using trigonometry, learners will draw a picture that accurately displays a right triangle and its given measurements.

11. Given a non-right triangle (oblique) with at least 3 pieces of information (including at least one side), learners will identify the information as SSS, SAS, AAS (or ASA), or SSA.

12. After identifying which sides and angles are given for a non-right (oblique) triangle, students will accurately choose to use Law of Sines or Law of Cosines to solve it (and not soh cah toa).

13. Given the formulas for the Law of Sines and the Law of Cosines and given a non-right triangle with at least 3 pieces of information (including at least one side), learners will solve for any missing information in the triangle by using the formula and the calculator correctly.

Terminal Objective

Given an application problem in which there is a triangle and at least 3 pieces of information about the sides and/or angles of the triangle (including at least one side), learners will calculate the value of any unknown sides or angles using trigonometric ratios, Law of Sines, or Law of Cosines.

Media Selection Rationale

Media consists primarily of an interactive website that integrates graphics, animation, video, and text to illustrate concepts and examples. The website may be used by the instructor as a presentation or by individual students either as individual instruction prior to completing worksheets or as a review following presented instruction. Future plans for the website beyond the scope of this unit include interactive programmed practice problems with immediate feedback and a database driven account for each student that the instructor can access to evaluate the progress of each student and intervene with additional one-on-one instruction if necessary.

The rationale for this choice is based on research that shows that technology integration improves student learning and motivation, and has been designed to include the following specific considerations:

1. Information presented in the web site can be integrated with direct instruction and then reviewed by individual students as needed from the classroom or at home, thus providing self-paced reinforcement of concepts.

2. The Internet would be available to the instructor in multiple locations without needing to plan ahead with CD Rom installation.

3. The web site can be copied to individual computers in the event that an internet connection is not available.

4. The multi-media aspect of technology helps instruction to address various learning styles, thus helping to reach and motivate more of the students.

5. Future updates for the web are simpler and more cost-effective than for other media. Such updates would increase interactivity, allowing learners to move at their own pace through as few or many practice problems as individually needed with immediate feedback. Results of practice problems completed by each student could be recorded in a database and accessed by the instructor.

Lesson Content Outline

I. Prior Knowledge – Review

A. Naming Triangles

1. Angles named with capital letters (often A, B, C) (If there is a right angle, it is often named C)

2. Sides named 2 possible ways

a. With the two letters on the angles that are the endpoints of the side with a line over the top of them (indicating it is a segment).

b. With a lower case letter of the letter naming the opposite angle (so if a right angle is C, the hypotenuse is c

3. Triangle named with the names of the angles put together, such as ?ABC.

B. Right Triangles

1. Triangle with one right (90 degree) angle

2. Sides of a right triangle

a. Hypotenuse

a. Side opposite the right angle

b. Longest side

b. Legs

a. Sides adjacent the right angle

b. Shorter than hypotenuse

3. Special Right Triangles

a. 30-60-90 Triangles

a. Hypotenuse is twice the length of the short leg (opposite the 30 degree angle)

b. Long leg (opposite the 60 degree angle) is the square root of 3 (approx 1.7) times the length of the short leg.

b. 45-45-90 Triangles

a. Legs are congruent

b. Hypotenuse is the length of a leg times the square root of 2 (approx 1.4)

4. Pythagorean Theorem

a. The sides of a right triangle are related by the equation a squared plus b squared equals c squared, where a and b represent the lengths of the legs and c is the length of the hypotenuse.

C. Proportions

1. One fraction equal to another

2. Solve using the cross-product property

D. Calculators

1. Most scientific calculators do “order of operations” where they multiply and divide before add and subtract.

II. Solving Problems involving Triangles

A. Trigonometric Ratios – For use with right triangles only

1. Sine ratio is the opposite leg divided by the hypotenuse

2. Cosine ratio is the adjacent leg divided by the hypotenuse

3. Tangent ratio is the opposite leg divided by the adjacent leg.

4. Sohcahtoa is a mnemonic device to help remember the ratios.

B. Solving Oblique (non-right) Triangles

1. Law of Sines (use for AAS, ASA, or SSA): Sin A/a = Sin B/b = Sin C/c.

2. Law of Cosines (SSS or SAS): a squared = b squared plus c squared plus b times c times Cos A.

3. AAA cannot be solved because side lengths may vary

C. Story Problems

1. Angle of Elevation – angle above the horizontal from one object to another

2. Angle of Depression – angle below the horizontal from one object to another

3. Conversion factors – ratio used to change from one unit of measurement to another, such as feet to miles,which is 5280 to 1.

III. Procedure: To find missing angle or side of a triangle, a student should go through the following procedure: First, observe given information and wanted information. Then go through the following steps:

A. If 2 angles are known and want to find other angle, add the two known angles together and subtract from 180.

B. If the triangle is a right triangle…

1. and 2 sides are known, want to find other side, use Pythagorean theorem. (a squared plus b squared equals c squared, where a and b are legs and c is the hypotenuse. Be sure not to trade the measurements for legs with the hypotenuse!)

2. Otherwise, use sine, cosine, or tangent.

a. When trying to find either the opposite leg to a given angle or the hypotenuse, and the other measurement is given, use sine.

b. If finding side involving adjacent leg and hypotenuse, use cosine

c. If finding side involving both legs, use tangent.

d. When finding angles, use the inverse sine, cosine, tangent in the above situations.

C. If triangle is oblique, use law of sines or law of cosines.

1. Use law of sines when an angle and the side opposite that angle are known, or when any two angles and a side are known (because opposite angle can be found by subtracting from 180)

2. Use law of cosines if no given information are opposites (i.e. all 3 sides or 2 sides and angle between.

3. Note: sides cannot be determined if only angles are given (don’t know what size the triangle is)
IV. Calculators

D. Since calculators follow the order of operations, when entering complex fractions into the calculator, use parenthesis around the numerator and the denominator, or calculate each separately before dividing.

E. Entering Trigonometric Functions

1. Make sure that the mode of the calculator is set to the correct unit for the angle measurements being used (probably degree mode rather than radian mode)

2. Enter trig functions in some (most graphing calculators) in the order written.

3. In some calculators, the angle is entered before the trig function

4. When calculating angles, generally use the 2nd or shift key with the trig functions to obtain the inverse trig functions.

Materials Outline

Instructor should have the following materials to use during instruction:

1. White board or chalk board on which to demonstrate examples.

2. Computer display system for displaying the web site to the entire class for demonstration and introduction of the web site (such as computer attached to a television for display).

Students will need access to the following materials and equipment to complete the instruction:

1. Paper

2. Pencil

3. Scientific or graphing calculator

4. Computer with internet access

5. Other materials, such as protractor, scissors, glue, string, ruler, and tape measure