Principles of Mathematics 10 -
Shape and Space (Measurement)

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:

• demonstrate an understanding of scale factors, and their interrelationship with the dimensions of similar shapes and objects.
• solve problems involving triangles, including those found in 3-D and 2-D applications.

It is expected that students will:

• calculate the volume and surface area of a sphere, using formulas that are provided
• determine the relationships among linear scale factors, areas, the surface areas and the volumes of similar figures and objects
• solve problems involving two right triangles
• extend the concepts of sine and cosine for angles through to 180°
• apply the sine and cosine laws, excluding the ambiguous case, to solve problems

SUGGESTED INSTRUCTIONAL STRATEGIES

Many worthwhile and practical problems in mathematics involve shape and space and are most easily solved by labeling a drawing. Some drawings work best when reduced to scale while in other cases an annotated sketch is sufficient. Visualizing shape and space is a critical component of learning mathematics.

• Bring a collection of spherical balls related to various sports into the classroom. Have students find the circumference of each ball using a fabric tape, and, from the circumference, calculate radius-diameter-area of the greatest circle. Ask them to place the results in a table and find the relationship between each column and classify it as linear or non-linear.
• Discuss the limitations in manufacturing that large structures increase in weight faster than the surface area increases which results in the need for special bracing to keep them from collapsing.
• Discuss why smaller animals require faster metabolisms to maintain a constant body temperature, and require more food as volume decreases faster than the surface area decreases.
• Have students use equipment such as a parallax viewer (for range) and clinometer (for height) to analyse the motion of a ball thrown on a field, resolving the motion into horizontal and vertical components. Point out that the horizontal motion vs. time is linear while the vertical motion vs. time is described using a quadratic function (i.e., non-linear).
• Provide students with pictures of non-right triangles that show either two sides and one angle or two angles and one side. Have students calculate the missing side or angle. A practical example might involve using an orienteering map.
• Give instructions to students (in terms of angles and sides) to move to a certain point in two steps (two sides of a triangle), using Law of Cosines or Law of Sine to calculate how to return to the starting point.

SUGGESTED ASSESSMENT STRATEGIES

Assessment of students’ measurement skills reflects their abilities to calculate using formulas, and to see and use the relationships that exist among scale factors, areas, and volumes. Assessment should focus on students’ abilities to extract relevant triangle information from drawn or written problems, as well as their abilities to solve simple triangle problems.

Observe

• While students are solving problems involving scale, note their ability to transfer skills to new situations. Ask them, for example, to estimate the change in surface area that occurs as a result of increasing one of the dimensions of an object. Check their results and ask them to explain any differences.
• Note students’ ability to translate written problems into sketches, including those whose angles range from 0° to 180°. Observe students’ ability to use calculators and manipulate angle representations to solve problems.

Collect

• Check students’ solutions to problems for clarity and presentation. Students should be able to explain the strategy they used to solve selected problems.
• Post a number of sketches around the classroom. Give students a worksheet with word problems on it. Have them match sketches to appropriate problems. To make it interesting, give out different sheets with the problems scrambled. Provide one or two problems with no sketch and post more sketches than problems.

Peer/Self-Assessment

• Have students create two or three problems involving sine and cosine laws, including solutions. Students should exchange problems with a partner and have the partner complete, and then comment on, the construction of the problem and its solution.

Presentation

• Have students work in pairs on problems from the construction or other industry, and have them write up and present solutions to the rest of the class.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Pure Mathematics 10 (Distance Learning Package)

Multi-Media

• The Learning Equation: Mathematics 10 Lessons 12 - 16
• Mathematics 10, Western Canadian Edition
  Ch. 6 (Sections 6.1, 6.2)
  Ch. 8 (Sections 8.4 - 8.6)
• MATHPOWER 10, Western Edition
  Ch. 7 (Sections 7.2, 7.3, 7.5 - 7.8)

Software

• The Geometer’s Sketchpad
• Understanding Math Series

CD-ROM

• Geometry Blaster

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

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