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 quadratic, polynomal and rational functions, using technology as appropriate
• examine the nature of relations with an emphasis on functions

It is expected that students will:

• determine the following characteristics of the graph of a quadratic function:
  • vertex
  • domain and range
  • axis of symmetry
  • intercepts
• perform operations on functions and compositions of functions
• determine the inverse of a function
• connect algebraic and graphical transformations of quadratic functions, using completing the square as required
• model real-world situations, using quadratic functions
• solve quadratic equations, and relate the solutions to the zeros of a corresponding quadratic function, using:
  • factoring
  • the quadratic formula
  • graphing
• determine the character of the real and non-real roots of a quadratic equation, using:
  • the discriminant in the quadratic formula
  • graphing
• describe, graph, and analyse polynomial and rational functions, using technology
• formulate and apply strategies to solve absolute value equations, radical equations, rational equations, and inequalities

SUGGESTED INSTRUCTIONAL STRATEGIES

Describing functional relationships is essential to interpreting and predicting the behaviour of the world around us. Students will explore how to translate between algebraic and graphical representations and use these skills to make inferences and solve problems.

• Demonstrate the concept of a function as a machine (e.g., g(x) = x2 + 3, the "g" function always takes what you give it, squares it, and adds three to it).
• To demonstrate substitution of functions, always show the same function in the same color, especially when substituting into another.
• To develop the concept of the inverse of a function, provide five or six functions and have students:
  • make tables of values for them and graph them all in blue ink
  • switch x and y in the original equations and call the new equations the "inverse" equations
  • graph the new "inverse" equations on the same grid as the originals in red ink, pointing out that it may be easier to first solve the equations for y and then to make a table of values to graph the inverses
  • analyse the relationship between the original blue function and the new red function on each grid, describe the transformation,and draw the "mirror" line as a black dashed line
  • describe the "mirror" line as an algebraic equation
• Have students:
  • use programmable calculators or calculators with built-in programs to solve quadratic equations
  • use graphing calculators or computer graphing programs to illustrate the solutions of quadratic equations
• Suggest that students create a question comparing ticket price to revenue. Display the graph of the function on the graphing calculator and analyse key values (e.g., maximum value, intercepts, curve shape). As algebraic techniques are introduced, compare the algebraic results with the graphing calculator results.
• Obtain non-linear data through experiments or calculator based laboratory (e.g., heights of falling objects) and solve them for the roots, vertices, and time at a given height.

SUGGESTED ASSESSMENT STRATEGIES

Students should demonstrate the ability to describe functional relationships. Assessment focusses on students’ abilities to move easily between graphical and algebraic representations.

Observe

• As students work with non-linear equations, circulate, asking questions and observing how effectively they are able to:
  • solve non-linear equations using a variety of methods
  • understand the concept of extraneous root
  • test for extraneous roots
  • determine if solutions are reasonable within the context of the problem

Question

• Probe students’ problem-solving strategies as they work on quadratic equations by asking questions such as:
  • How did you decide which method to use to solve the equation?
  • How did you identify the proper form of the equation for each method?

Collect

• Ask students to find examples of real-world situations that involve quadratic relations. Have them present their situations, using some type of electronic technology. Observe the extent to which students:
  • make use of quadratic relations
  • use real-world situations
  • identify the correct quadratic to fit each situation
  • use research to verify their results

Self- and Peer Assessment

• Have students summarize and assess their results from a set of review exercises, noting what they do well and where they need to improve. Ask them to record how they plan to improve in the identified areas.
  Ask students to reflect on the various methods they use to solve quadratic equations:
  • Which methods do they clearly understand?
  • Which forms of the equations do they have trouble dealing with?
  • To what extent can they generalize from what they have learned by restating results in more widely applicable terms?

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. 2 (Sections 2.1 - 2.7)
  Ch. 3 (Sections 3.1 - 3.9)
• MATHPOWER 11, Western Edition
  Ch. 3 (Sections 3.1, 3.3, 3.5)
  Ch. 4 (Sections 4.1, 4.2, 4.4, 4.6, 4.7)
  Ch. 5 (Sections 5.1 - 5.7)
• Modeling Motion: High School Math Activities with the CBR
• Using the TI-81 Graphics Calculator to Explore Functions
• What If ...?: The Straight Line: Investigations with the TI-81 Graphics Calculator

Software

• GrafEq (Macintosh & Windows Version 2.09)
• Green Globs & Graphing Equations
• Secondary Math Lab Toolkit
• ZAP-A-GRAPH

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

Ministry of Education Home Page