Elaboration on this Achievement Objective

This means students will know that prime numbers are numbers divisible by only themselves and one, and apply this to the fundamental law of arithmetic that every counting number has a unique prime factorisation, for example 36 = 2 x 2 x 3 x 3 = 2^{2} 3^{2}. They should apply prime factorisation to problems that involve factors and multiples, including finding the least common multiple or highest common factor. For example, “What sized cuboids can be made using 105 unit cubes?”, or “What is 105 out of 231 in simplest form?”

They should understand and use the additive law of exponents, that is a^{b} x a^{c} = a^{b+c} and a ^{b} ÷ a^{c} = a ^{b - c} and compare powers relationally (without calculation) where this is appropriate, for example 3^{6} >6^{3} because (3x3)x(3x3)x(3x3)>6x6x6. Students should understand the arithmetic and geometric origin of square roots (for example, a square of area 144cm^{2} has a side length of 12cm) and use common square roots to estimate the value of other square roots. For example, √36 = 6 and √49 = 7 so √42 ≈ 6.5. They should also understand the convention for negative exponents through pattern. For example 2^{1}= 2 so 2^{0}= 1 so 2 ^{-1}= 1/2 since the effect of decreasing the exponent by one is to divide the previous power by two.