NSW Syllabuses

# Mathematics K–10 - Stage 4 - Number and Algebra Indices

## Outcomes

#### A student:

• MA4-1WM

communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-3WM

recognises and explains mathematical relationships using reasoning

• MA4-9NA

operates with positive-integer and zero indices of numerical bases

## Content

• describe numbers written in 'index form' using terms such as 'base', 'power', 'index', 'exponent'
• use index notation to express powers of numbers (positive indices only), eg $$8 = 2^3$$
• evaluate numbers expressed as powers of integers, eg $$2^3 = 8, \,\,\, \left( -2 \right) ^3 = -8$$
• investigate and generalise the effect of raising a negative number to an odd or even power on the sign of the result (Communicating)
• apply the order of operations to evaluate expressions involving indices, with and without using a calculator, eg $$3^2+4^2$$, $$4^3+2 \times 5^2$$
• determine and apply tests of divisibility for 2, 3, 4, 5, 6 and 10
• verify the various tests of divisibility using a calculator (Problem Solving)
• apply tests of divisibility mentally as an aid to calculation (Problem Solving)
• express a number as a product of its prime factors, using index notation where appropriate
• recognise that if a given number is divisible by a composite number, then it is also divisible by the factors of that number, eg since 660 is divisible by 6, then 660 is also divisible by factors of 6, which are 2 and 3 (Reasoning)
• find the highest common factor of large numbers by first expressing the numbers as products of prime factors (Communicating, Problem Solving)
• Investigate and use square roots of perfect square numbers (ACMNA150)
• use the notations for square root $$\left( \sqrt{\,\,\,} \right)$$ and cube root $$\left( \sqrt[3]{\,\,\,} \right)$$
• recognise the link between squares and square roots and between cubes and cube roots,
eg $$2^3 = 8\,$$ and $$\, \sqrt[3]{8} = 2$$
• determine through numerical examples that:
$$\left( ab \right)^2 = a^2b^2, \,\,\,$$ eg $$\, \left( 2 \times 3 \right) ^{2} = 2^{2} \times 3^{2}$$
$$\sqrt{ab} = \sqrt{a} \times \sqrt{b}, \,\,\,$$ eg $$\sqrt{9\times4} = \sqrt{9} \times \sqrt{4}$$
• express a number as a product of its prime factors to determine whether its square root and/or cube root is an integer
• find square roots and cube roots of any non-square whole number using a calculator, after first estimating
• determine the two integers between which the square root of a non-square whole number lies (Reasoning)
• apply the order of operations to evaluate expressions involving square roots and cube roots, with and without using a calculator, eg $$\sqrt{16+9}, \,\, \sqrt{16} + 9, \,\, \frac{\sqrt{100-64}}{9}, \,\, \sqrt{\frac{100-64}{9}}$$
• explain the difference between pairs of numerical expressions that appear similar, eg 'Is $$\sqrt{36} + \sqrt{64} \,$$ equivalent to $$\sqrt{36+64}\,$$ ?' (Communicating, Reasoning)
• Use index notation with numbers to establish the index laws with positive-integer indices and the zero index (ACMNA182)
• develop index laws with positive-integer indices and numerical bases by expressing each term in expanded form, eg
$$3^2 \times 3^4 = \left( 3\times3 \right) \times \left( 3\times3\times3\times3 \right) = 3^{2+4} = 3^6$$
$$3^5 \div 3^2 = \frac{3\times3\times3\times3\times3}{3\times3} = 3^{5-2} = 3^3$$
$$\left( 3^2 \right)^4 = \left(3\times3\right) \times \left(3\times3\right) \times \left(3\times3\right) \times \left(3\times3\right) = 3^{2\times4} = 3^8$$
• verify the index laws using a calculator, eg use a calculator to compare the values of $$\left(3^{4}\right)^{\!2}$$ and $$3^{8}$$ (Reasoning)
• explain the incorrect use of index laws, eg explain why $$3^{2} \times 3^{4} \ne 9^{6}$$ (Communicating, Reasoning)
• establish the meaning of the zero index, eg by patterns
• verify the zero index law using a calculator (Reasoning)
• use index laws to simplify expressions with numerical bases, eg $$5^2 \times 5^4 \times 5 = 5^7$$

### Background Information

Students have not used indices prior to Stage 4 and so the meaning and use of index notation will need to be made explicit. However, students should have some experience from Stage 3 in multiplying more than two numbers together at the same time.

In Stage 3, students used the notion of factorising a number as a mental strategy for multiplication. Teachers may like to make an explicit link to this in the introduction of the prime factorisation of a number in Stage 4, eg in Stage 3, 18 × 5 would have been calculated as 9 × 2 × 5; in Stage 4, 18 × 5 can be factorised as a product of primes, as 3 × 3 × 2 × 5.

The square root sign signifies a positive number (or zero). So, $$\sqrt{9} = 3\,\,$$ (only). However, the two numbers whose square is 9 are $$\sqrt{9}\,\,$$ and $$-\sqrt{9}\,\,$$, ie 3 and –3.

#### Purpose/Relevance of Substrand

Indices are important in mathematics and in everyday situations. Among their most significant uses is that they allow us to write large and small numbers more simply, and to perform calculations with large and small numbers more easily. For example, without the use of indices, 21000 would be written as 2 × 2 × 2 × 2 …, until '2' appeared exactly 1000 times.

### Language

Students need to be able to express the concept of divisibility in different ways, such as '12 is divisible by 2', '2 divides (evenly) into 12', '2 goes into 12 (evenly)'.

A 'product of prime factors' can also be referred to as a 'product of primes'.

Students are introduced to indices in Stage 4. The different expressions used when referring to indices should be modelled by teachers. Teachers should use fuller expressions before shortening them, eg 24 should be expressed as '2 raised to the power of 4', before '2 to the power of 4' and finally '2 to the 4'. Students are expected to use the words 'squared' and 'cubed' when saying expressions containing indices of 2 and 3, respectively, eg 42 is 'four squared', 43 is 'four cubed'.

Words such as 'product', 'prime', 'power', 'base' and 'index' have different meanings outside of mathematics. Words such as 'base', 'square' and 'cube' also have different meanings within mathematics, eg 'the base of the triangle' versus 'the base of 32 is 3', 'the square of length 3 cm' versus 'the square of 3'. Discussing the relationship between the use of the words 'square' and 'cube' when working with indices and the use of the same words in geometry may assist some students with their understanding.