Outcomes
A student:

 MA5.21WM
selects appropriate notations and conventions to communicate mathematical ideas and solutions

 MA5.23WM
constructs arguments to prove and justify results

 MA5.26NA
simplifies algebraic fractions, and expands and factorises quadratic expressions
Content
 Students:
 Apply the four operations to simple algebraic fractions with numerical denominators (ACMNA232)

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)
 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)

simplify expressions that involve algebraic fractions, including algebraic fractions that involve pronumerals in the denominator and/or indices,
eg \( \, \dfrac{2ab}{3} \times \dfrac{6}{2b}, \, \) \( \dfrac{3x^2}{8y^5} \div \dfrac{15x^3}{4y}, \, \) \(\dfrac{a^2b^4}{6} \times \dfrac{9}{a^2b^2}, \, \) \( \dfrac{3}{x}  \dfrac{1}{2x} \)
 Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (ACMNA213)
 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 \( \, 2y \left( y5 \right) + 4 \left(y5 \right), \,\, \) \( 4x\left(3x+2\right)  \left( x1 \right) \)
 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)
 Expand binomial products and factorise monic quadratic expressions using a variety of strategies (ACMNA233)

expand binomial products by finding the areas of rectangles, eg
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 \( \, \left( x+2 \right) \left(x3 \right), \,\, \) \( \left( 4a1 \right) \left(3a+2 \right) \)
 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)
 explain why a particular algebraic expansion or factorisation is incorrect, eg 'Why is the factorisation \( \, x^26x8 = \left(x4\right)\left(x2\right) \, \) incorrect?' (Communicating, Reasoning)