# NA5-2: Use prime numbers, common factors and multiples, and powers (including square roots).

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.

solve problems involving exponents

explore other number bases

- Understand, use, and calculate simple surd numbers.
- Show how to find the fractional form of rational numbers expressed as decimals.
- Understand the difference between rational and irrational numbers.
- Realise that there are more irrationals than there are rationals.
- Produce proofs by300

- Understand, use, and calculate simple surd numbers.
- Understand, use, produce, and work with prime numbers and prime factorisation.
- Produce simple proof.
- Work with numbers in bases other than 10.
- Use Heron’s method to calculate square roots.
- Understand the construction and point of300

solve addition problems using complementary numbers

solve subtraction problems using complementary numbers

explore complementary numbers in base 2

explore how computers solve mathematical operations

investigate prime and composite numbers

- Continue a pattern.
- Use a table of values.
- Be able to find the general rule for simple patterns.

investigate the properties of square numbers

find powers of numbers

Solve problems that involve exponents and square roots.

find cubes of numbers

Know what happens when a number is multiplied or divided by a power of 10.

- Express a power as multiplication of the base, e.g. 54 = 5 x 5 x 5 x 5.
- Recognise patterns in the last digits of powers for the same base.
- Multiply and divide powers with the same base by adding or subtracting the exponents, e.g. 24 x 23 = 27 and 27 ÷ 24 = 23.

investigate common factors

investigate patterns involving powers

Solve problems by finding the prime factors of numbers.

investigate powers of 2

apply exponents to solve problems

use addition and multiplication to solve problems (Problem 1)

continue a sequential patterns (Problem 2)

solve puzzles involving square roots and factorials

investigate prime numbers

find prime numbers up to 100

apply exponents to solve problems