Principles of Mathematics 10 -
Patterns and Relations (Variables and Equations)

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 generalize operations on polynomials to include rational expressions.

It is expected that students will:

• factor polynomial expressions of the form and
• find the product of polynomials
• divide a polynomial by a binomial and express the result in the forms:
  • 
  • 
  • 
• determine equivalent forms of simple rational expressions with polynomial numerators, and denominators that are monomials, binomials or trinomials that can be factored
• determine the nonpermissible values for the variable in rational expressions
• perform the operations of addition, subtraction, multiplication, and division on rational expressions
• find and verify the solutions of rational equations that reduce to linear form

SUGGESTED INSTRUCTIONAL STRATEGIES

Extensive exposure to factoring will help students increase their skills in manipulating algebraic expressions and consolidate their understanding of algebraic ideas and procedures. Appropriate technology can enhance student learning and extend skill development.

• Model the use of algebra tiles to represent quadratic equations. Have students work in pairs to manipulate the tiles to do factoring.
• Demonstrate types and methods of factoring. Provide or have students create examples of factoring and ask them to create the types and therefore the processes to be used. Have students justify and discuss their solutions in groups.
• Discuss as a class the ideas that not all data in real-world problems fit a linear model and that there may be more than one solution to a problem. Point out that solutions for such problems are usually more difficult to calculate than those for problems that fit the linear model
  Present a quadratic that does not factor and discuss possible solutions with the class. Are there any solutions?
• Have students use available technology to compare graphs of the quadratic functions with the equation, and to relate the zeros to the factors.
• Ask students to suggest reasons for simplifying expressions (e.g., to solve complex problems involving polynomial functions and equations).
• Have students factor polynomial expressions (trinomials) using trial and error. Give examples of polynomial equations that students would have difficulty recognizing without the skills of factoring or simplifying.
• Encourage students to develop their skills in factoring perfect square trinomials, by practising squaring binomials by sight.
• Ask students to graph rational functions on their graphing calculator to link the excluded value with the asymptote .
• Shuffle a card deck consisting of factored and non-factored pairs of polynomials, give each student a card, and ask them to find their partners.

SUGGESTED ASSESSMENT STRATEGIES

In order to solve equations and problems based on quadratic and rational expressions, students must be able to factor polynomials and understand when a root is extraneous. Assessment strategies should focus on observing student proficiency of these two concepts.

Observe

• Develop a concentration card game using cards that consist of polynomials in factored and non-factored form. Observe the speed at which students can match cards and win the game.
• Have students model factored polynomials using algebra tiles.

Question

• Encourage students to verbalize the patterns involved in factoring various types of polynomials.
• Have students draw connections between restrictions for a rational expression and the asymptotes on its graph.

Collect

• Give students a worksheet of non-simplified rational expressions. Post simplified answers and restrictions around the room on posters and have students record the locations of correct answers.
• Have each student develop a one-page summary of skills involved in factoring. Alternatively, have them write the summaries in flow chart form.

Peer/Self Assessment

• Have students work in pairs to develop a factoring puzzle or rational expression. Photocopy the work of each pair and give it to another pair to solve and assess by criteria supplied by the teacher or developed as a class activity.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Exploring Advanced Algebra with the TI-83
• Exploring Functions with the TI-82 Graphics Calculator
• Graphing Calculator Activities for Enriching Middle School Mathematics
• An Introduction to the TI-82 Graphing Calculator
• Modeling Motion: High School Math Activities with the CBR
• Pure Mathematics 10 (Distance Learning Package)
• A Visual Approach to Algebra

Multi-Media

• The Learning Equation: Mathematics 10
  Lessons 25 - 33
• Mathematics 10, Western Canadian Edition
  Ch. 6 (Sections 6.4 - 6.6, 6.10)
  Ch. 7 (Sections 7.1 - 7.7)
• MATHPOWER 10, Western Edition
  Ch. 3 (Sections 3.3, 3.4, 3.6, 3.8 - 3.10)
  Ch. 4 (Sections 4.2 - 4.9)

Software

• Green Globs & Graphing Equations
• Secondary Math Lab Toolkit
• Understanding Math Series

Games/Manipulatives

• Radical Math: Math Games Using Cards and Dice (Volume VII)

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: November 30, 2000

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