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 represent and analyse situations that involve expressions, equations and inequalities.

It is expected that students will:

• graph linear inequalities, in two variables
• solve systems of linear equations, in two variables:
  • algebraically (elimination and substitution)
  • graphically
• solve systems of linear equations, in three variables:
  • algebraically
  • with technology
• solve non-linear equations, using a graphing tool
• solve non-linear equations:
  • by factoring
  • graphically
• use the Remainder Theorem to evaluate polynomial expressions, the Rational Zeros Theorem, and the Factor Theorem to determine factors of polynomials
• determine the solution to a system of nonlinear equations, using technology as appropriate

SUGGESTED INSTRUCTIONAL STRATEGIES

Polynomial functions can be used to model many complex, real-world situations. Students can use these functions and the appropriate technology to solve problems that would otherwise be too difficult.

• Have students graph two equations on the same coordinate plane, identify points of intersection, and check whether these satisfy each equation. Discuss the limitations of graphing by referring to a system for which the solution values cannot be accurately determined by graphing.
• Discuss solving a system of two equations in two variables by reducing to one equation in one variable (start with simple examples).
  

  Substitution Method	Elimination Method
  3x+y=6	2x+y=7
  y=x+2	2x-y=3

  
  
• As enrichment, discuss possible ways of solving systems of four equations in four variables, up to n equations in variables. Information Technology students could be encouraged to create programs that do this.
• Have students graph equation families such as:
  
  y=2x, y=2x+3, and y=2x+4
  x+2y+4, 2x+4y=8, and 3x+6y=12
  
  Have students identify the patterns that group these functions into families. Ask them what relationships between coefficients and constants would result in systems having:
  • no solutions (parallel lines)
  • infinite solutions (concurrent lines)
  • a unique solution (intersecting lines)
• Have students use graphing calculators or computer software to introduce concepts related to polynomial functions, such as:
  • the effect on the graph when the sign of the leading term is changed
  • the effect on the graph when multiple factors are introduced
  • the relationship between factors of a polynomial and zeros of the associated polynomial function
  • the relationship between the degree of a polynomial and its general shape
  • the relationship of the constant term to the y-intercept of the graph
• Have students create functions that contain specified characteristics (e.g., tangent, multiple zeros). Use these examples to illustrate the connections between the function and its graph.

SUGGESTED ASSESSMENT STRATEGIES

In assessing student performance involving relations and functions, it is important to observe students’ abilities to recognize patterns and functional relationships. In working with polynomial functions and equations, students demonstrate their reasoning through the application of algebraic skills and graphical skills and by making interpretations, generalizations, and representations.

Observe

• As students solve systems of linear equations, watch for and provide feedback on the extent to which they are able to:
  • use a variety of methods to solve problems
  • determine the domain and range for given relations
  • solve applied problems
• Note students’ efforts to create functions or graphs that adhere to prescribed conditions, using paper and pencil or graphing technology. For example, have students write a polynomial function with a double zero at (-2,0), single zero at (30,0), and passing through the point (1,5)). Observe to what extent students give logical reasons for the choices they made.

Self-Assessment

• Pose questions such as the following to help students review what they learned:
  • What are the most important ideas you learned about solving linear equations?
  • How are these ideas connected to what you already knew and to the world around you?
• To assess students’ command of nonlinear functions and equations, have them list common misconceptions or difficulties they encounter in solving the equations, or while graphing the functions.

Collect

• Discuss with students the relative merits of the various methods of analyzing and displaying solutions for nonlinear equations. Have them summarize in writing their understanding of the merits of each. Work with them to develop a set of criteria to assess the summaries.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Explore Quadratic Functions with the TI-83 or TI-82
• Exploring Functions with the TI-82 Graphics Calculator
• An Introduction to the TI-82 Graphing Calculator
• Mathematics 11, Western Canadian Edition
  Ch. 4 (Sections 4.1 - 4.8)
  Ch. 5 (Sections 5.1 - 5.8)
• MATHPOWER 11, Western Edition
  Ch. 1 (Sections 1.2 - 1.6)
  Ch. 2 (Sections 2.1, 2.3)
  Ch. 4 (Sections 4.1, 4.8 - 4.10)
• Modeling Motion: High School Math Activities with the CBR

Software

• Green Globs & Graphing Equations
• Secondary Math Lab Toolkit

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 1, 2000

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