Applications of Mathematics 11 -
Shape and Space (3-D Objects and 2-D Shapes)

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 develop and apply the geometric properties of circles and polygons to solve problems.

It is expected that students will:

• use technology with dynamic geometry software to confirm and apply the following properties:
  • the perpendicular from the centre of a circle to a chord bisects the chord
  • the measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc
  • the inscribed angles subtended by the same arc are congruent
  • the angle inscribed in a semicircle is a right angle
  • the opposite angles of a cyclic quadrilateral are supplementary
  • a tangent to a circle is perpendicular to the radius at the point of tangency
  • the tangent segments to a circle, from any external point, are congruent
  • the sum of the interior angles of an n-sided polygon is equal to right angles
• use properties of circles and polygons to solve design and layout problems

SUGGESTED INSTRUCTIONAL STRATEGIES

Applications of geometry, which connects the physical world with students’ mathematical knowledge, will help students understand the practical importance of geometry and reinforce the concept of rules and patterns in mathematics. Exploring geometric ideas encourages the development of critical thinking and analysis skills, and helps students recognize geometry’s wide applicability in solving problems.

• Have students apply the properties of a circle to:
  • create a graphic design using computer graphics software
  • determine the centre of curvature of an arc in order to replicate it
• Encourage divergent thinking by providing students with geometry applications that include problems with more than one possible solution or that require multiple methods for solving the problem.
• Have students research and report on an area of geometry of personal interest or build a model that illustrates a geometric concept. Examples could include:
  • fractal geometry
  • transformational geometry
  • models of Platonic solids
  • exploring optical illusions
  • researching careers that use geometry
  • topology
  • tessellations
    Have them explain the model to the class.
• Have students develop experiments in which they measure chord and tangent circle properties and use inductive reasoning to draw conclusions from results.
• Show students how to construct hexagons and octagons using a straightedge and compass. Ask students to find ways to construct other regular polygons.
• Have students form cooperative groups and provide each group with a length of string that represents either the radius or the length of one side of a regular polygon. Have each group find a way to lay out a foundation for a small structure (e.g., a regular polygon-shaped patio, a gazebo foundation).
• Arrange a class field trip to a local community college or engineering office to observe a demonstration of computer-aided design.
• Using examples of applications of geometry (e.g., construction projects, landscape design, computer graphics, quilting or tiling) have students complete a layout project.

SUGGESTED ASSESSMENT STRATEGIES

Students demonstrate their understanding of geometric principles when solving problems that involve making connections between the physical world and the geometric models that represent it.

Observe

• Provide students with examples of art from different cultures, such as Aboriginal beadwork, mandalas from India and Mexico, Celtic knot designs, Icelandic sweater designs, Japanese and Chinese lattice designs, and patterns in Islamic art. Have them analyse the art for the extent to which the artist used geometric principles (e.g., symmetry, circle properties, transformations).

Collect

• Have students keep a scrapbook or math journal of practical relationships or applications in which geometry and corresponding terms are used. Require them to provide a brief written explanation of how each entry is an example of some aspect of geometry they have studied. Assess their collections based on criteria developed with students. Criteria might include:
  • knowledge of properties
  • representation of geometric ideas with models or diagrams
  • recognition of mathematical interrelationships
  • application of geometry to solve problems

Self/Peer Assessment

• Have students work in pairs to create geometry problems involving properties of circles. Each pair exchanges problems with another pair of students, solves the problems, then compares the different solutions and processes used.
• Ask students to exchange the written communication of a design with a partner. Have the partner reproduce the design from this communication. Was the communication clear enough for reproduction of the design? What would improve the communication?

RECOMMENDED LEARNING RESOURCES

Print Materials

• Applied Mathematics 11, Western Canadian Edition

Software

• The Geometer's Sketchpad

CD-ROM

• Geometry Blaster

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: November 29, 2000

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