Mathematics 9 -
Shape and Space (Measurement)

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

In order to prepare students to use trigonometric ratios to solve problems involving right triangles, it is expected that students will:

• explain the meaning of sine, cosine, and tangent ratios in right triangles
• demonstrate the use of trigonometric ratios (sine, cosine, and tangent) in solving right triangles
• calculate an unknown side or an unknown angle in a right triangle, using appropriate technology
• model and then solve given problem situations involving only one right triangle
  

SUGGESTED EXTENSIONS

To extend students' understanding of measurement, they could:

• relate expressions for volumes of pyramids to volumes of prisms, and volumes of cones to volumes of cylinders
• calculate and apply the ratio of area to perimeter to solve design problems in two dimensions
• calculate and apply the ratio of volume to surface area to solve design problems in three dimensions

SUGGESTED INSTRUCTIONAL STRATEGIES

The ability to describe the world using direct or indirect measurement is an important skill for various careers. Students develop trigonometric concepts through hands-on measurement activities.

• Have students use protractors and straight edges, or dynamic geometry software, to construct right triangles (with given angles) that just fit on letter-size pieces of paper. Then have them measure the lengths of the opposite and adjacent sides of one angle and record the tangent ratio in tables. Then introduce the calculator and the definition of tangent ratio and compare students' measurements with the values shown on the calculator display. Repeat with the sine and cosine ratios.
• Have students use trigonometry to measure the heights of several trees to determine how tall a totem pole they could make.
• Have students solve right triangle problems using appropriate software (e.g., spreadsheet program, dynamic geometry software) or a graphing calculator.
• Ask students to estimate the volumes of models of cones, cylinders, prisms, and pyramids. Then have them fill the models (e.g., with sand, water, macaroni) and compare volumes. Have students work with different sizes of models to compare their estimates, strategies, and conclusions.
• Provide groups of students with a variety of volume-to-surface-area and area-to-perimeter problems such as:
  • minimum-maximum problems (e.g., maximum volume for given surface area, minimum surface area for given volume, maximum area for given perimeter, packing smaller boxes into larger ones)
  • relationships among volume, surface area, radius, and height of cones and cylinders and/or pyramids and prisms
    Assign different problems to different groups. Ask students to design several solutions to the problems and identify the ramifications of each solution (e.g., cost-effectiveness, aesthetics). Where appropriate, students could use nets and models to represent the situations. Have students present their findings to the class for evaluation and discussion.
    

SUGGESTED ASSESSMENT STRATEGIES

Students' ability to correctly relate the information from a right triangle to the trigonometric ratios is essential for the solution of a triangle. Assessment focusses on solving problems using their knowledge of right triangle properties.

Observe

• As students use trigonometric ratios to solve right triangles, look for patterns of errors that indicate a need for additional instruction. Ask the students to:
  • identify the adjacent and opposite sides and the hypotenuse, given a specific angle
  • identify the trigonometric relationship, given any two sides

Collect

• Have students develop charts that compare similarities and differences between the volumes and surface areas of pyramids and prisms and between the volumes and surface areas of cones and cylinders. Ask students to give examples, using drawings to illustrate their work and identifying related formulae.

Peer Assessment

• Have students use the criteria to evaluate their own or each other's presentations. Compare student and teacher ratings and discuss differences.
• Work with students to develop criteria to evaluate the presentations of their solutions to volume and surface area problems. Criteria might include:
  • development or correct application of the appropriate formula
  • logic of conclusions reached
  • effective organization of data
  • clarity of visual display
  • accuracy of calculations and measurements
  • accuracy of conversions among metric units, if appropriate
  • accuracy of nets (if used in problem)
    

RECOMMENDED LEARNING RESOURCES

Print Materials

• Interactions (Level 9)
• MATHPOWER 9, Western Edition

Multimedia

• The Geometer's Sketchpad
• The Learning Equation Mathematics 9 (TLE)
• Math Tools
• Mathematics 9 (Distance Education Package)
• Minds on Math 9, Revised Edition
• Understanding Math Series
  

CD-ROM

• Geometry Blaster
• Mirror Symmetry
  

© Copyright 2001. All Rights Reserved. BC MOE Standards Department.
Maintained by: Mathematics Coordinator
Revised: September 1, 2001

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