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NSW Syllabuses

Mathematics K–10 - Stage 5.1 - Number and Algebra Indices

Outcomes

A student:

  • MA5.1-1WM

    uses appropriate terminology, diagrams and symbols in mathematical contexts

  • MA5.1-3WM

    provides reasoning to support conclusions that are appropriate to the context

  • MA5.1-5NA

    operates with algebraic expressions involving positive-integer and zero indices, and establishes the meaning of negative indices for numerical bases

Content

  • use the index laws previously established for numerical bases with positive-integer indices to develop the index laws in algebraic form, eg 
    \( \begin{align} 2^2 \times 2^3 &= 2^{2+3} = 2^5 &\text{ leads to } & \,\,\, a^m \times a^n = a^{m+n} \\ 2^5 \div 2^2 &= 2^{5-2} = 2^3 &\text{ leads to } & \,\,\, a^m \div a^n = a^{m-n} \\ (2^2)^3 &= 2^{2 \times 3} = 2^6 &\text{ leads to } & \,\,\, (a^m)^n = a^{mn} \end{align} \)
  • explain why a particular algebraic sentence is incorrect, eg explain why \( a^3 \times a^2 = a^6 \) is incorrect (Communicating, Reasoning) CCT
  • establish that \(x^0 = 1\) using the index laws, eg
    \( \begin{align} a^3 \div a^3 &= a^{3-3} = a^0\\ \textrm{ and }~ a^3 \div a^3 &= 1 \\ \therefore ~ ~ a^0 &= 1 \end{align} \)
  • explain why \(x^0 = 1\) (Reasoning) CCT
  • simplify expressions that involve the zero index, eg \(5x^0 + 3 = 8\)
  • simplify expressions that involve the product and quotient of simple algebraic terms containing positive-integer indices, eg 
    \( \begin{align} (3x^2)^3 &= 27x^6 \\ 2x^2 \times 3x^3 &= 6x^5 \\ 15a^6 \div 3a^2 &= 5a^4 \\ \frac{3a^2}{15a^6} &= \frac{1}{5a^4} \end{align} \)
  • compare expressions such as 3a2 × 5a and 3a2 + 5a by substituting values for a (Communicating, Reasoning) CCT
  • Apply index laws to numerical expressions with integer indices (ACMNA209)
  • establish the meaning of negative indices for numerical bases, eg by patterns
    A table showing the number 3 to the powers of 3 to -2 and the resulting number or fraction. CCT
  • evaluate numerical expressions involving a negative index by first rewriting with a positive index, 
    eg \( 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \)
  • write given numbers in index form (integer indices only) and vice versa