Applications of Mathematics 12 -
Patterns and Relations (Patterns)

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 analyze cyclic, recursive, and fractal patterns

It is expected that students will:

• describe periodic events, including those represented by sinusoidal curves, using the terms amplitude, period, maximum and minimum values, vertical and horizontal shift
• collect sinusoidal data; graph the graph using technology, and, represent the data with a best fit equation of the form:
  • 
• use best fit sinusoidal equations, and their associated graphs, to make predictions (interpolation, extraction)
• use technology to generate and graph sequences that model real-life phenomena
• use technology to construct a fractal pattern by repeatedly applying a procedure to a geometric figure
• use the concept of self-similarity to compare and/or predict the perimeters, areas and volumes of fractal patterns

SUGGESTED INSTRUCTIONAL STRATEGIES

The study of cyclic, recursive, or fractal patterns requires the use of spatial reasoning to model and analyse complex natural patterns such as biological structures, coastlines, or snowflakes. Periodic events and fractal geometry also offers students a chance to explore the aesthetic aspects of mathematics.

• Have students analyse patterns found in computer simulations that model transformations of geometric figures. Ask them to identify examples of divergent, convergent, oscillating, or static patterns.
• Encourage students to use their geometric and analytical skills to construct pencil-and-paper drawings (e.g., Koch snowflake, Sierpinski triangle), paper-and-scissors constructions (e.g., pop-up fractals), and computer software (e.g., fractal generators and drawing software). Ask students to record the changes observed in the perimeters, areas, and volumes of the fractals they construct. Ask them to use their observations to make generalizations such as the volume-to-area ratio found in naturally occurring fractal-like objects (e.g., sponges, lungs).
• Invite students to conduct a group or individual research project on fractal geometry examining:
  • its historical development
  • specific career applications (e.g., botany, animation, cartography, climatology, medicine, art)
• Have students use periodic functions to model and graph common periodic patterns (e.g., tidal flow, the beating of a heart, and seasonal changes), and to analyse and make predictions about them.
• Have students identify and collect data about a number of events that are cyclic over time. They then sketch graphs of these periodic events with time represented on the x-axis. Examples that may help stimulate ideas include:
  • the depth of water at a beach over a 48-hour period
  • the temperature in a house where the heat is controlled by a thermostat
  • the height of a pendulum in a grandfather clock as the pendulum swings
  • the firing sequence of cylinders in a four-cylinder car
  • the current in a household circuit

SUGGESTED ASSESSMENT STRATEGIES

Many complex phenomena can only be modeled using periodic or fractal patterns. Fractal geometry provides a way to solve problems involving irregularity and similarity under changes of scale. Assessment of students’ representations of cyclic data should focus on the accuracy of their graphs of the data, their abilities to interpret and make predictions based on their graphs, and the extent to which they accurately represent sinusoidal graphs with equations.

Observe

• While students are working with computer software that generates sequences or drawings, look for evidence of students’ abilities to:
  • use computers with relative ease and confidence
  • analyse patterns in the simulation runs
  • use the software application to calculate lengths, areas, and volumes of resulting patterns
• While students are constructing paper-and-pencil drawings, computer-generated images, or 3-D constructions, look for evidence that they use the concept of self-similarity.
• While students are graphing sinusoidal data, look for evidence that they:
  • use correct terminology
  • accurately construct the graphs from different sets of cyclic data
  • use the graphs to make predictions
  • represent the graphs with the correct equations
• Have small groups of students each research a problem that requires the collection of data about naturally occurring events. Have them graph the data and use the graphs to make predictions to solve the problems. As groups present their work, look for evidence of the extent to which they:
  • understand periodic functions
  • accurately graph the data
  • use the graphs to solve problems (e.g., make predictions)
  • present their solutions to the problems in a clear and logical manner

RECOMMENDED LEARNING RESOURCES

Print Materials

• Applied Mathematics 12 available June 2001
• Exploring Advanced Algebra with the TI-83
• An Introduction to the TI-82 Graphing Calculator

Software

• Secondary Math Lab Toolkit
• GrafEq (Macintosh & Windows Version 2.09)

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: November 22, 2000

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