YEAR 8 MATHS FOCUS
NUMBER AND ALGEBRA
INDICES
OUTCOME
A student:
MA4-9NA:
operates with positive-integer and zero indices of numerical bases
| TEACHING POINTS | 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, e.g. in Stage 3, 18×5 The square root sign signifies a positive number (or zero). So, √9=3 (only). However, the two numbers whose square is 9 are √9 and −√9, i.e. 3 and –3. |
| 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, e.g. 2^4 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, e.g. 4^2 is ‘four squared’, 4^3 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, e.g. ‘the base of the triangle’ versus ‘the base of 3^2 |
| PURPOSE RELEVANCE | 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. |
Expectations of Attainment
Investigate index notation and represent whole numbers as products of powers of prime numbers (ACMNA149) | describe numbers written in ‘index form’ using terms such as ‘base’, ‘power’, ‘index’, ‘exponent’ {Literacy} |
| use index notation to express powers of numbers (positive indices only), e.g. 8=23 {Literacy} | |
| evaluate numbers expressed as powers of integers, e.g. 2^3=8, (−2)^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, Critical and creative thinking} | |
| apply the order of operations to evaluate expressions involving indices, with and without using a calculator, e.g. 3^2+4^2, 4^3+2×5^2 {Information and communication technology capability} | |
| determine and apply tests of divisibility for 2, 3, 4, 5, 6 and 10 {Critical and creative thinking} | |
| – verify the various tests of divisibility using a calculator {Problem Solving, Information and communication technology capability} | |
| – apply tests of divisibility mentally as an aid to calculation {Problem Solving, Critical and creative thinking} | |
| 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, e.g. since 660 is divisible by 6, then 660 is also divisible by factors of 6, which are 2 and 3 {Reasoning, Critical and creative thinking} | |
| – find the highest common factor of large numbers by first expressing the numbers as products of prime factors {Communicating, Problem Solving, Critical and creative thinking} |
Investigate and use square roots of perfect square numbers (ACMNA150) | use the notations for square root  and cube root  {Literacy} |
| recognise the link between squares and square roots and between cubes and cube roots, e.g. 2^3=8 and ^3√8=2 {Critical and creative thinking} | |
| determine through numerical examples that: | |
| 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 {Information and communication technology capability} | |
| – determine the two integers between which the square root of a non-square whole number lies {Reasoning, Critical and creative thinking} | |
apply the order of operations to evaluate expressions involving square roots and cube roots, with and without using a calculator, e.g. Figure 2 {Information and communication technology capability} | |
explain the difference between pairs of numerical expressions that appear similar, e.g. ‘Is √36+√64 equivalent to √36+64 ?’ |
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, e.g. Figure 3 {Literacy} |
| – verify the index laws using a calculator, e.g. use a calculator to compare the values of (3^4)^2 and 3^8 {Reasoning} | |
| – explain the incorrect use of index laws, e.g. explain why 3^2×3^4≠9^6 {Communicating, Reasoning, Literacy, Critical and creative thinking} | |
establish the meaning of the zero index, e.g. by patterns {Critical and creative thinking} | |
| – verify the zero index law using a calculator {Reasoning, Information and communication technology capability} | |
| use index laws to simplify expressions with numerical bases, e.g. 5^2×5^4×5=5^7 |
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