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)
• 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  $$\, \dfrac{2ab}{3} \times \dfrac{6}{2b}, \,$$  $$\dfrac{3x^2}{8y^5} \div \dfrac{15x^3}{4y}, \,$$  $$\dfrac{a^2b^4}{6} \times \dfrac{9}{a^2b^2}, \,$$  $$\dfrac{3}{x} - \dfrac{1}{2x}$$
• 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)
• 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  $$\, \left( x+2 \right) \left(x-3 \right), \,\,$$ $$\left( 4a-1 \right) \left(3a+2 \right)$$
• 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)
• 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)