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New NSW Syllabuses

Mathematics K–10 - Stage 5.2 - Number and Algebra Algebraic Techniques

Outcomes

A student:

  • MA5.2-1WM

    selects appropriate notations and conventions to communicate mathematical ideas and solutions

  • MA5.2-3WM

    constructs arguments to prove and justify results

  • MA5.2-6NA

    simplifies algebraic fractions, and expands and factorises quadratic expressions

Content

  • simplify expressions that involve algebraic fractions with numerical denominators,
    eg \(\, \dfrac{a}{2} + \dfrac{a}{3}, \, \)  \(\dfrac{2x}{5} - \dfrac{x}{3}, \, \)  \(\dfrac{3x}{4} \times \dfrac{2x}{9}, \, \)  \(\dfrac{3x}{4} \div \dfrac{9x}{2} \)
  • connect the processes for simplifying expressions involving algebraic fractions with the corresponding processes involving numerical fractions (Communicating, Reasoning) CCT
  • Apply the four operations to algebraic fractions with pronumerals in the denominator
  • simplify algebraic fractions, including those involving indices, eg \(\, \dfrac{10a^4}{5a^2}, \, \)  \(\dfrac{9a^2b}{3ab}, \, \)  \(\dfrac{3ab}{9a^2b} \)
  • explain the difference between expressions such as \(\dfrac{3a}{9} \) and \(\dfrac{9}{3a} \) (Communicating) LCCT
  • simplify expressions that involve algebraic fractions, including algebraic fractions that involve pronumerals in the denominator and/or indices,
    eg  \( \, \dfrac{2ab}{3} \times \dfrac{6}{2b}, \, \)  \( \dfrac{3x^2}{8y^5} \div \dfrac{15x^3}{4y}, \, \)  \(\dfrac{a^2b^4}{6} \times \dfrac{9}{a^2b^2}, \, \)  \( \dfrac{3}{x} - \dfrac{1}{2x} \)
  • expand algebraic expressions, including those involving terms with indices and/or negative coefficients, eg \( \, -3x^2 \left( 5x^2 + 2x^4y \right) \)
  • expand algebraic expressions by removing grouping symbols and collecting like terms where applicable, eg expand and simplify \( \, 2y \left( y-5 \right) + 4 \left(y-5 \right), \,\, \)  \( 4x\left(3x+2\right) - \left( x-1 \right) \)
  • Factorise algebraic expressions by taking out a common algebraic factor (ACMNA230)
  • factorise algebraic expressions, including those involving indices, by determining common factors, eg factorise \( \, 3x^2 - 6x, \,\, \)  \( 14ab + 12a^2, \,\, \)  \(21xy - 3x + 9x^2, \,\, \) \( 15p^2q^3 - 12pq^4 \)
  • recognise that expressions such as \( \, 24x^2y + 16xy^2 = 4xy \left( 6x+4y \right) \, \) may represent 'partial factorisation' and that further factorisation is necessary to 'factorise fully' (Reasoning) CCT
  • expand binomial products by finding the areas of rectangles, eg 
    Image shows strategy to find area of rectangle by expanding binomial equation.
    hence,
    \( \begin{align} \left( x+8 \right) \left(x+3 \right) &= x^2 + 3x + 8x + 24 \\ &= x^2 + 11x + 24 \end{align} \)
  • use algebraic methods to expand binomial products, eg  \( \, \left( x+2 \right) \left(x-3 \right), \,\, \) \( \left( 4a-1 \right) \left(3a+2 \right) \)
  • factorise monic quadratic trinomial expressions, eg  \( \, x^2 + 5x + 6, \,\, \) \( x^2 + 2x - 8 \)
  • connect binomial products with the commutative property of arithmetic, such that \( \, \left(a+b\right)\left(c+d\right) = \left(c+d\right)\left(a+b\right) \) (Communicating, Reasoning) CCT
  • explain why a particular algebraic expansion or factorisation is incorrect, eg 'Why is the factorisation \( \, x^2-6x-8 = \left(x-4\right)\left(x-2\right) \, \) incorrect?' (Communicating, Reasoning) LCCT