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 represent and analyse exponential and logarithmic functions, using technology as appropriate.

It is expected that students will:

• model, graph, and apply exponential functions to solve problems
• change functions from exponential form to logarithmic form and vice versa
• model, graph, and apply logarithmic functions to solve problems
• explain the relationship between the laws of logarithms and the laws of exponents
• describe the three primary trigonometric functions as circular functions with reference to the unit circle and an angle in standard position
• draw (using technology), sketch, and analyse the graphs of sine, cosine, and tangent functions, for:
  • amplitude, if defined
  • period
  • domain and range
  • asymptotes, if any
  • behavior under transformations
• use trigonometric functions to model and solve problems

SUGGESTED INSTRUCTIONAL STRATEGIES

The inverse relationship between exponential and logarithmic functions makes possible the solution of a number of equations not readily solved algebraically (e.g., investment analysis, population growth, earthquake magnitudes, radioactive decay).

• Give students an exponential function and its inverse (logarithmic). Have them work in groups, using graphing calculators or computer software to identify the relationship between the two.
• Demonstrate for students the impossibility of graphing an exponential function with a negative base, using anything but integer values of x. Refer to the resulting difficulties to emphasize the need for domain and range restrictions when dealing with exponential and logarithmic functions. As a follow up, discuss base e and natural logarithms.
• Use the graphing calculator to discuss the effects of changing A, B, C, and D on the graph of
• Provide students with real sinusoidal data. Ask them to graph the data, estimate the amplitude and suggest possible equations to represent the data. The data can then be entered into the graphing calculator and the sinusoidial regression function can be applied. Students can compare the calculator results to their own predicted equation.
• Have students display the parallels of the log laws and exponent laws on an 81/2"x11" poster
• Develop a model of the wrapping function (e.g., sine, cosine) using a circle of cardboard and an acetate overlay to show how x, y and x/y change as the angle changes. Alternatively use geometry software to model the relationships of the inscribed right angle and the change in the central angle.
• Use a graphing calculator or graphing software to match a given graph to a trigonometric equation.
• Given a solution, have students write a problem that matches it.
• • Play trigonometric graph bingo Have students choose a number of equations from a list to make up their card, then graph equations on overhead until someone gets 5 in a row.

SUGGESTED ASSESSMENT STRATEGIES

Trigonometric, exponential, and logarithmic functions enable students to solve more complex problems in areas such as science, engineering, and finance. Students should be able to demonstrate their knowledge of the relationship between logarithms and exponential functions, both in theory and in applications, in a variety of problem situations. Assessment of students’ understanding of trigonometric concepts includes the observation of the processes they use when they relate the physical world to mathematical representations.

Observe

• Ask students to outline how they would teach a classmate:
  • the inverse relationship between exponential and logarithmic functions
  • the restrictions associated with exponential and logarithmic functions
    Note the extent to which the outlines:
  • include general steps to follow
  • use mathematical terms correctly
  • provide clear examples
  • describe common errors and how they can be avoided
• When reviewing students’ work, note the extent to which they can:
  • generate several different equations to describe the graph of a given cosine or sine function
  • graph a particular sine or cosine function using pencil and paper
  • check their solutions using graphing technology

Collect

• Assign a series of problems that require students to apply their knowledge of the relationship between logarithms and exponential functions. Check their work for evidence that they:
  • clearly understood the requirements of the problem
  • used efficient strategies and procedures to solve the problem
  • recognized when a strategy or procedure was not appropriate
  • verified that their solutions were accurate and reasonable

RECOMMENDED LEARNING RESOURCES

Print Materials

• 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
• Using the TI-81 Graphics Calculator to Explore Functions

Multi-Media

• Mathematics 12, Western Canadian Edition
  Ch. 2 (Sections 2.1 - 2.6, 2.10)
  Ch. 3 (Sections 3.1, 3.3 - 3.9)
  Ch. 4 (Sections 4.1, 4.4 - 4.6)
  Ch. 5 (Section 5.6)
• MATHPOWER 12, Western Edition
  Ch. 2 (Sections 2.2, 2.5, 2.6, 2.8)
  Ch. 4 (Sections 4.2 - 4.6)

Software

• GrafEq (Macintosh & Windows Version 2.09)
• Secondary Math Lab Toolkit
• ZAP-A-GRAPH

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

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