Algebra and functions strand

Students should be able to:

AF.1

Patterns and relationships

Investigate patterns and relationships (linear, quadratic, doubling and tripling) in number, spatial patterns, and real-world phenomena involving change.

Working with patterns

  • represent these patterns and relationships in tables and graphs;
  • generate a generalised expression for linear and quadratic patterns in words and algebraic expressions, and fluently convert between representations;
  • categorise patterns as linear, non-linear, quadratic, and exponential (doubling and tripling) using their defining characteristics in different representations.
AF.2

Expressions and equations

Expressions

  • generate and interpret expressions in which letters stand for numbers;
  • find the value of expressions given the value of the variables.

Equations and equality

  • use the concept of equality to generate and interpret equations.
AF.3

Equivalent expressions and factorisation

Apply the properties of arithmetic operations and factorisation to generate equivalent expressions and develop appropriate strategies.

Add, subtract, simplify

  • simplify linear expressions in one or more variables with coefficients in ℚ;
  • simplify quadratic expressions in one variable with coefficients in ℤ;
  • simplify expressions of the form a/(bx + c), where a, b, c ∈ ℤ.

Multiply expressions

  • a(bx + cy + d); a(bx² + cx + d); and ax(bx² + cx + d), where a, b, c, d ∈ ℤ;
  • (ax + b)(cx + d) and (ax + b)(cx² + dx + e), where a, b, c, d, e ∈ ℤ.

Divide expressions

  • divide quadratic and cubic expressions by linear expressions where all coefficients are integers and there is no remainder.

Convert between forms

  • convert flexibly between factorised and expanded forms of:
    • axy, where a ∈ ℤ;
    • axy + byz, where a, b ∈ ℤ;
    • sx − ty + tx − sy, where s, t ∈ ℤ;
    • dx² + bx; x² + bx + c; and ax² + bx + c, where b, c, d ∈ ℤ and a ∈ ℕ;
    • x² − a² and a²x² − b²y², where a, b ∈ ℤ.
AF.4

Solving equations and inequalities

Solving equations

  • select and use strategies (graphic, numeric, algebraic, trial/improvement, working backwards) to solve:
  • linear equations in one variable with coefficients in ℚ and solutions in ℤ or in ℚ;
  • quadratic equations in one variable with coefficients in ℤ and solutions in ℤ, coefficients in ℚ and solutions in ℝ;
  • simultaneous linear equations in two variables with coefficients and solutions in ℤ or in ℚ.

Inequalities

  • solve linear inequalities in one variable of the form g(x) < k and graph the solution set on the number line for x ∈ ℕ, ℤ, and ℝ.
AF.5

Quadratic equations with integer roots

  • generate quadratic equations given integer roots.
AF.6

Changing the subject of a formula

  • apply the relationship between operations and an understanding of the order of operations (including brackets and exponents) to change the subject of a formula.
AF.7

Functions

Understanding functions

  • demonstrate understanding of the concept of a function.

Representing functions

  • represent and interpret functions graphically (for x ∈ ℕ, ℤ, ℝ, continuous functions where appropriate), diagrammatically, in words, and algebraically;
  • use the language and notation of functions (domain, range, co-domain, f(x), f : x ↦ y, etc.);
  • draw the graph of a function given its algebraic expression (limited to linear and quadratic functions at OL).

Solving using graphs

  • use graphical methods to find approximate solutions of equations such as f(x) = g(x);
  • find approximate solution sets of inequalities such as f(x) < g(x).

Interpreting graphs

  • make connections between the shape of a graph and the story of a phenomenon, including identifying and interpreting maximum and minimum points.