Mathematics 9 -
Shape and Space (3-D Objects and 2-D Shapes)

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 spatial problem solving in building, describing, and analysing geometric shapes and specify conditions under which triangles may be similar or congruent and use these conditions to solve problems, it is expected that students will:

• draw the plan and elevation of a 3-D object from sketches and models
• sketch or build a 3-D object, given its plan and elevation views
• recognize when, and explain why, two triangles are similar, and use the properties of similar triangles to solve problems
• recognize when, and explain why, two triangles are congruent, and use the properties of congruent triangles to solve problems
  

SUGGESTED EXTENSIONS

To extend students' understanding of 3-D objects and 2-D shapes, they could:

• recognize and draw the locus of points in solving practical problems
• relate congruence to similarity in the context of triangles
• draw the image of a 2-D shape as a result of:
  • a single transformation
  • a dilatation
  • a combination of translation and/or reflections
• identify the single transformation that connects a shape with its image
• demonstrate that a triangle and its dilatation image are similar
• demonstrate the congruence of a triangle
  with its:
  • translation image
  • rotation image
  • reflection image

SUGGESTED INSTRUCTIONAL STRATEGIES

Developing an understanding of triangle congruency can strengthen reasoning and problem-solving skills. Visualization skills and spatial relations abilities are developed when students move back and forth between a 3-D object and its 2-D representation.

• Display a triangle on the overhead. Move the overhead closer to and farther from the screen and discuss what changes occur.
• Have students use grid paper or cardboard models to solve problems involving similar and congruent triangles related to stair-stringer, truss, and rafter designs.
• Invite students to investigate the role of the triangle in engineering, architecture, and construction. Discuss:
  • Are some shapes used more often than others?
  • Are particular shapes used to accomplish particular purposes?
  • What part does congruence play in the use of triangles for construction?
  • Why are rectangles and squares not so common?
• Use dynamic geometry software to construct similar or congruent triangles and then use the software to explore the conditions necessary for similarity and congruence.
• Arrange to have the class co-ordinate with technology education students and teachers to explore software for creating design plans (e.g., drawing programs, CAD).
• Have students work in pairs to consider locus problems such as the following:
  • design of a safety fence around a camel's cage, given the locus and range of the camel's spit
  • efficiency of a kitchen in terms of accessibility to sink, stove, and refrigerator
  • where to place shade-loving plants around a house, given the locus of the sun's shadow
  • area around a supermarket cashier, in terms of accessibility for both employee and customer

  Ask students to suggest additional problems and applications of locus (e.g., design of car consoles or airplane cockpits in a range of cars or aircraft). Where possible, conduct field studies to observe the application of their suggestions.

SUGGESTED ASSESSMENT STRATEGIES

To solve many geometric and physical problems, an understanding of the properties of triangles and their application is essential. It is often helpful to be able to move between a 3-D shape and its 2-D representation to correctly analyse and solve problems as well.

Collect

• Ask students to imagine they are going to give classmates directions over the telephone for drawing simple 3-D objects. Because the other students cannot see the objects, students must record or write out the instructions to be sure they are specific and clear. Check students' recorded instructions for accuracy, clarity, precision, and proper use of terminology. Alternatively, have two students, sitting back-to-back, take turns reading their written instructions to each other (one student reading while the other attempts to sketch or build the described 3-D object) to see if the directions are accurate. Have students modify their instructions before submitting them to the teacher.
• Have students draw plans and elevations for 3-D objects. Students could exchange plans with other students, who could sketch or build the objects and provide feedback to allow for corrections. When they are satisfied with their plans, have students submit them to the teacher for comments.
• Give students problems (see examples in Appendix F) that require them to recognize and draw the locus of points in solving practical problems. Ask students to present their solutions to the class and explain their reasoning. Assess the accuracy of students' answers, the clarity and logic of their presentations, and the reasonableness of their conclusions.

Peer Assessment

• Have students work individually to solve problems requiring the application of the properties of similar and congruent triangles. Then have students exchange their work with other students, check each other's answers, and resolve any differences. Ask students to work with different classmates until everyone has agreed on the same answers. Give the class the correct answers and have them compare results.
  

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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