Mathematics 9 -
Patterns and Relations (Variables and Equations)

This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Sugested Extensions
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

In order to prepare students to evaluate, solve, and verify linear equations in one variable and to
generalize arithmetic operations from the set of rational numbers to the set of polynomials. It is expected that students will:

• illustrate the solutions process for a first-degree, single-variable equation, using concrete materials or diagrams
• solve and verify first-degree, single-variable equations of forms such as: ax = b + cx; a(x+b)=c; ax + b = cx + d where a, b, c, and d are integers, and use equations of this type to model and solve problems
• identify constant terms, coefficients, and variables in polynomial expressions
• evaluate polynomial expressions, given the value(s) of the variable(s)
  

SUGGESTED EXTENSIONS

To extend students' understanding of variables and equations, they could:

• solve and verify first-degree, single-variable equations of forms such as: a(bx + c) = d(ex + f);
  = b where a, b, c, d, e, and f are rational numbers
• solve, algebraically, first-degree inequalities in one variable, display the solutions on a number line, and test the solutions
• perform the operations of addition and subtraction on polynomial expressions
• represent multiplication, division, and factoring of monomials, binomials, and trinomials of the form x² + bx + c using concrete materials and diagrams
• find the product of two monomials, a monomial and a polynomial, and two binomials
• determine equivalent forms of algebraic expressions by identifying common factors and factoring of the form x² + bx + c
• find the quotient when a polynomial is divided by a monomial
  

SUGGESTED INSTRUCTIONAL STRATEGIES

Skills in manipulating algebraic expressions can be developed by connecting new ideas and operations to arithmetic skills and concrete representations. Extending algebra beyond the linear relationships enhances students' understanding of the 2-D and 3-D world. Drill and practice techniques help students to internalize the abstractions and improve their skills in the language of algebra.

• Have students substitute numbers in expressions involving different powers and the same variable base, to verify that the terms are not "like terms." Students can also verify their factoring by substitution of numbers.
• Give students abstract forms of algebraic expressions or equations and ask them to create numerical examples. Then reverse the activity, challenging them to write the abstract forms of numerical examples.
• Ask students to use a mathematics glossary or encyclopedia (either print or electronic) to define the terms constant, coefficient, and variable.
• Invite students to explore ways of solving equations with their programmable or graphing calculators. Alternatively, have them use computer spreadsheet applications to practise "programming" standard equations, using them to perform multiple calculations for a range of data.
• Ask students to work individually to create a first-degree, single-variable equation and determine its solution. Then have them work in pairs to solve each other's equations.
• Display an inequality on the board or overhead (e.g., 3x + 2 > 8). Ask students to determine a value for x that satisfies this inequality. Plot and compare students' answers. Discuss as a class why there are many solutions. Can students display all solutions efficiently?
• Use algebra tiles, algebra lab gear, or diagrams to demonstrate expansion of polynomials. For example, may not be readily apparent without tiles.
  

SUGGESTED ASSESSMENT STRATEGIES

Students should have an understanding of various forms of linear equations and how to use them to model and solve problems.

Observe

• As students use algebra tiles or diagrams to model equations concretely, check their work and provide feedback. Determine the extent to which they:
  • use tiles or diagrams to represent various operations
  • are willing to modify their efforts based on their experiences

Question

• As students solve and verify linear equations in one variable, have them explain the processes they are using.
• As students work individually and in small groups to perform operations on polynomials and factor binomials and trinomials, discuss their work with them to determine the extent to which they:
  • use correct terminology to identify constants, coefficients, and variables in polynomial expressions
  • persist in their efforts to solve the difficult problems
  • use a variety of resources such as textbooks, other students, and technology to solve equations requiring them to perform operations

Collect

• Give students study sheets containing examples of monomials, binomials, and polynomials that have been factored, some correctly and some incorrectly. Ask students to identify the incorrect examples, identify the mistakes, and correct them.

Peer Assessment

• Have students develop algebraic expressions and challenge each other to represent them using algebra tiles or diagrams. Have students identify and correct errors in each other's work.
  

RECOMMENDED LEARNING RESOURCES

Print Materials

• Interactions (Level 9)
• MATHPOWER 9, Western Edition

Multimedia

• The Learning Equation Mathematics 9 (TLE)
• Math Tools
• Mathematics 9 (Distance Education Package)
• Minds on Math 9, Revised Edition
• Understanding Math Series

© Copyright 2001. All Rights Reserved. BC MOE Standards Department.
Maintained by: Mathematics Coordinator
Revised: September 1, 2001

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