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 generate and analyse exponential patterns.

It is expected that students will:

• derive and apply expressions to represent general terms and sums for geometric growth and to solve problems
• connect geometric sequences to exponential functions over the natural numbers
• estimate values of expressions for infinite geometric processes

SUGGESTED INSTRUCTIONAL STRATEGIES

Sequences and series can be used to model the patterns observed in phenomena such as a launched rocket, a bouncing ball, and plant or animal growth. Limit theory, a fundamental building block of calculus, extends from the concepts of sequences and series.

• Give students numerical and non-numerical patterns and ask them to define the rule(s) for each pattern. Have them use the rules to make predictions and classify the patterns (e.g., arithmetic, geometric, neither).
• Have students find examples and describe the type of pattern of geometric sequences and series occurring in real-world tools and scales (e.g., the f-stops on a camera lens, rule of five).
• Have students research and report on pattern-related topics such as;
  • recursive definitions of a sequence (e.g., computer programming)
  • development and significance of mathematical symbolism (e.g., summation notation)
  • the relationship between infinite geometric series and limits (e.g., Zeno’s paradox, Koch’s snowflake curve, fractal curves, the notion of infinity)
• Encourage students to use a variety of media and to refer to a range of real-life examples to show patterns (e.g., in history, fine arts, science, economics).
• Have students use concepts of sequence and series to solve compounding interest problems (e.g., If $1000.00 is deposited each year into an account that earns 8% compounded annually, how much money will have accumulated after 25 years?). Ask students to:
  • generate a sequence or series modeling each situation
  • use the appropriate format to answer the questions
  • discuss the reasonableness of the answers

SUGGESTED ASSESSMENT STRATEGIES

Students analyse problems and solve them using a variety of approaches. Assessment of problem-solving skills is made over time, based on observations of many situations.

Observe

• While students are working on problems involving patterns, look for evidence that they can:
  • recognize, classify, and make predictions and generalizations from functional relationships and patterns
  • recognize patterns in a sequence and generate successive terms of that sequence
  • classify different types of sequences and series

Question

• To discover how well students can recognize and explain a fallacy, give them a problem such as:
  S = 1 + 2 + 4 + 8 +...
  S = 1 + 2 (1 + 2 + 4 +...)
  therfore S = 1 + 2S
  and S - 2S = 1
  so S = -1
  
  In their analyses, look for evidence that they understand infinite geometric series when discussing the fallacy. For example, do they:
  • recognize that something is wrong
  • identify what it is that is wrong and explain why
• Have students brainstorm real world examples of geometric sequences and then encourage them to see that they are examples of exponential functions.

Collect

• Ask student to research fractal-like objects that are naturally occurring, such as broccoli or lung tissue. Ask them to explain, using geometric series, how lung tissue might benefit from a maximum surface area.

Collect/Present

• Have students state five geometric and five non-geometric sequences and defend why each is placed in a particular group. Alternatively, give students a list of sequences, and have them sort them and defend their choices.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Exploring Advanced Algebra with the TI-83
• 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 2.7 - 2.9)
• MATHPOWER 12, Western Edition
  Ch. 6 (Sections 6.3, 6.5, 6.6)

Software

• Secondary Math Lab Toolkit

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

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