This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will solve exponential, logarithmic and trigonometric equations and identities.

It is expected that students will:

• solve exponential equations having bases that are powers of one another
• solve and verify exponential and logarithmic equations and identities
• distinguish between degree and radian measure, and solve problems using both
• determine the exact and the approximate values of trigonometric ratios for any multiples of 0°, 30°, 45°, 60° and 90° and 0,
  
  
• solve first and second degree trigonometric equations over a domain of length 2¹:
  • algebraically
  • graphically
• determine the general solutions to trigonometric equations where the domain is the set of real numbers
• analyse trigonometric identities:
  • graphically
  • algebraically for general cases
• use sum, difference,and double angle identities for sine and cosine to verify and simplify trigonometric expressions

SUGGESTED INSTRUCTIONAL STRATEGIES

Many real-world problems (e.g., in navigation, engineering, surveying, chemistry) require the application of exponential, logarithmic, or trigonometric equations. Trigonometry also links geometry and algebra (e.g., through its applications in matrix representations of rotations and direction angles of vectors).

• Provide students with a list of logarithmic equations. Have students categorize the equations in terms of the equation-solving technique required (e.g., ) would be solved by the power law and equating arguments).
• Have students generate multiple-choice questions with emphasis on creating good distracters. Use these questions created by the class as a review.
• Present examples illustrating the need for radian measure when modeling real-world situations (e.g., amount of air in lungs as a function of time, tide problems).
• As students simplify trigonometric expressions, encourage them to develop and state useful strategies such as:
  • express in terms of sine and cosine only
  • simplify fractions using conjugates, factoring, and common denominators
  • make substitutions using equivalent identities
• Relate solving trigonometric equations to solving algebraic equations with similar forms:
  Algebraic Equations
  
  
  
  Trigonometric Equations
  Solve for O to the nearest degree
  
  
  
• Have students analyse and discuss incorrect solutions to trigonometric equations.

SUGGESTED ASSESSMENT STRATEGIES

The ability to solve problems involving exponential, logarithmic, and trigonometric functions is important in fields such as engineering. Knowing how well students can manipulate equations and identities of these types is important. Assessment strategies should focus on the demonstration of algebraic skills and, to a lesser degree, the translation of word problems into algebraic equations.

Observe

• While students are working on trigonometric and logarithmic functions, observe that:
  • when solving bases other than 10, students convert them to base 10 or are comfortable with the base they are in
  • where necessary, students can convert all bases to a single appropriate base
  • students memorize tables of special angles or they can derive them quickly as needed
  • when proving trigonometric identities, students should apply appropriate techniques such as converting everything to sine and cosine
• Provide samples of student work on proofs that are not complete. Observe the extent to which students are able to:
  • determine the errors in the formal proof
  • correct the errors in the formal proof
  • assign a mark to the proof based on a criterion-based rubric

Question

• Can students differentiate between specific solutions and general solutions to trigonometric equations?

Self-/Peer Assessment

• Design a puzzle sheet which requires the solution to many trigonometric or logarithmic equations to find the answer. Encourage students to have a peer solve their puzzle and check their work for accuracy.
• Ask students to select three proofs from their homework assignment and have partners mark each others’ work using a criterion-based rubric.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Exploring Advanced Algebra with the TI-83
• Exploring Functions with the TI-82 Graphics Calculator
• An Introduction to the TI-82 Graphing Calculator
• Modeling Motion: High School Math Activities with the CBR

Multi-Media

• Mathematics 12, Western Canadian Edition
  Ch. 2 (Sections 2.1, 2.11, 2.12)
  Ch. 3 (Sections 3.2, 3.5)
  Ch. 5 (Section 5.1 - 5.6)
• MATHPOWER 12, Western Edition
  Ch. 2 (Sections 2.4, 2.7)
  Ch. 4 (Sections 4.1, 4.2)
  Ch. 5 (Sections 5.1 - 5.5)

Software

• Secondary Math Lab Toolkit

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 4, 2000

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