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 apply the principles of mathematical reasoning to solve problems and to justify solutions.

It is expected that students will:

• differentiate between inductive and deductive reasoning
• explain and apply connecting words, such as and, or, and not, to solve problems
• use examples and counterexamples to analyse conjectures
• distinguish between an if–then proposition, its converse and its contrapositive
• prove assertions in a variety of settings, using direct and indirect reasoning

SUGGESTED INSTRUCTIONAL STRATEGIES

Students need to understand the language used in logic and reasoning. This understanding assists students in drawing conclusions distinguishing between facts and propaganda. These concepts of mathematical reasoning need to be reinforced in all areas of mathematical study.

• Give students examples of deductive and inductive reasoning and have students work in small groups to decide whether examples belong in either of two columns in a table, where one column will contain deductive examples and one will contain inductive examples.
• Have students use Venn diagrams to illustrate AND, OR, NOT and combinations of these. Use examples from law where deductive proof is used.
• Have students analyse the logic used in newspaper articles or videos of television news programs that report statistical results and that draw oversimplified conclusions.
• Provide examples of debatable or controversial issues argued in a short paragraph and have students find flaws in the arguments.

SUGGESTED ASSESSMENT STRATEGIES

Students should be able to recognize valid arguments and reject invalid arguments. These skills should be assessed in all areas of mathematics.

Observe

• While students are working in pairs and reading a newspaper article or report, observe the extent to which they are able to:
  • determine the valid arguments or facts
  • determine the point or conclusion of the argument
  • distinguish between points that are logically reached and conclusions that are sweeping statements or simply do not follow logically

Question

• Ask students to:
  • list examples of if-then propositions, converses and contrapositives
  • design an if-then game where, for example, a person’s turn ends when a connecting statement cannot be made or when a converse statement is made
  • explain how they would prove that the sum of the angles in a polygon of n sides is (n-2) x 180°
  • prove that objects with the same perimeter may have different areas

Collect

• Ask students to collect newspaper articles that present good arguments, some that jump to conclusions, and some that contradict themselves.
• Ask students to collect cartoon strips that show causal relationships and give examples of if-then situations

Self-Assessment

• Provide students with arguments that are either flawed or not flawed. Have a student try to convince the class of the conclusion and see if the group is correct in accepting or rejecting the arguments. This may lead to a class discussion (e.g., on how people can be swayed into believing others such as sales pitches).
• Have students list necessary criteria for a good argument.

RECOMMENDED LEARNING RESOURCES

Print Materials

• An Introduction to the TI-82 Graphing Calculator
• Mathematics 11, Western Canadian Edition
  Ch. 6 (Sections 6.1 - 6.7)
  Ch. 7 (Section 7.5)
  Ch. 9 (Sections 9.4, 9.5)
• MATHPOWER 11, Western Edition
  Ch. 6 (Sections 6.1 - 6.6)
• Modeling Motion: High School Math Activities with the CBR

Software

• Secondary Math Lab Toolkit

CD-ROM

• Geometry Blaster

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

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