Applications of Mathematics 12 -
Problem Solving

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 describe and apply operations on matrices to solve problems, using technology as required.

It is expected that students will:

• solve problems that involve a specific content area such as geometry, algebra, trigonometry, statistics, or probability
• solve problems that involve more than one content area
• solve problems that involve mathematics within other disciplines
• analyse problems and identify the significant elements
• develop specific skills in selecting and using an appropriate problem-solving strategy or combination of strategies chosen from, but not restricted to, the following:
  • guess and check
  • look for a pattern
  • make a systematic list
  • make and use a drawing or model
  • eliminate possibilities
  • work backward
  • simplify the original problem
  • develop alternative original approaches
  • analyse keywords
• demonstrate the ability to work individually and co-operatively to solve problems
• determine that their solutions are correct and reasonable
• clearly communicate a solution to a problem and the process used to solve it
• use appropriate technology to assist in problem solving

 

SUGGESTED INSTRUCTIONAL STRATEGIES

Problem solving is a key aspect of any mathematics course. Working on problems involving a range of mathematical disciplines can give students a sense of the excitement involved in creative and logical thinking. It can also help students develop transferable real-life skills and attitudes. Multi-strand and interdisciplinary problems should be included throughout Applications of Mathematics 12.

• In a class discussion, define problem solving with students, pointing out that in mathematics problem solving involves many disciplines, including algebra, geometry, trigonometry, statistics, and probability.
• Introduce new types of problems directly to students (i.e., without demonstration), and facilitate as they attempt to solve them.
• Have students work in small co-operative groups of three to five when introducing new types of problems.
• Model and reinforce student use of a variety of approaches to problem solving (e.g., algebraic and geometric).
• Emphasize that many problems may not be solved in one try and that it is sometimes necessary to revisit the problem, start over, and try again.
• Have students or groups discuss their thought processes as they attempt to solve a problem. Point out the strategies inherent in their thinking (e.g., guessing and checking, looking for a pattern, making and using a drawing or model).
• Ask leading questions such as:
  • What are you being asked to find out?
  • What do you already know?
  • Do you need additional information?
  • Have you ever seen similar problems?
  • What else can you try?
• When students arrive at solutions to particular problems, encourage them to generalize from or extend the problem situations.

Note: See Appendix G for examples of multi-strand and interdisciplinary problems that most students should be able to solve. These problems are indicated with an asterisk (*)

SUGGESTED ASSESSMENT STRATEGIES

Students analyse problems and solve them using a variety of approaches. Assessment of problem-solving skills is made over time, based on observations of many situations.

Observe

• Have students present solutions to the class individually, in pairs, or in small groups. Note the extent to which they clarify their problems and how succinctly they describe the processes used.

Question

• To check the approaches students use when solving problems, ask questions that prompt them to:
  • paraphrase or describe the problem in their own words
  • explain the processes used to derive an answer
  • describe alternative methods to solve a problem
  • relate the strategies used to new situations
  • link mathematics to other subjects and to the world of work.

Collect

• For selected problems, have students annotate their work to describe the processes they used. Alternatively, have them provide brief descriptions of what worked and what did not work for particular problems.

Self-Assessment

• Ask students to keep journals to describe the processes they used in dealing with problems. Have them include descriptions of strategies that worked and those that did not.
• Develop with students a set of criteria to self-assess problem-solving skills. The reference set, Evaluating Problem Solving Across Curriculum, may be helpful in identifying such criteria.

RECOMMENDED LEARNING RESOURCES

Please see the introduction to Appendix B for a list of suggested utility software that supports this course.

The Western Canadian Protocol Learning Resource Evaluation Process has also identified numerous teacher resources and professional references. These are generally cross-grade planning resources that include ideas for a variety of activities and exercises.

These resources, while not part of the Grade Collections, have Provincially Recommended status.

Appendix B includes an annotated bibliography of these resources for ordering convenience.

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

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