KeyStage

Key Stage 3 maths · Number

KS3 primes, factors, multiples and ordering numbers

In Years 7, 8 and 9 pupils deepen what they know about the number system. They use place value with very large numbers and with decimals, order negative numbers, decimals and fractions on a number line, and learn the vocabulary of primes, factors, multiples, highest common factor and lowest common multiple. They also begin to see that number sets go on for ever.

Place value, ordering and number properties, explained

Place value is extended to integers of any size and to decimals. Integers are whole numbers, including negatives and zero. Pupils read and write numbers into the millions and billions and use the same idea at the other end: the digits after the decimal point are tenths, hundredths, thousandths and so on. This is needed for measures. 0.45 m is 45 hundredths of a metre, which is 45 cm, and 2.5 kg is 2,500 g. Multiplying or dividing by 10, 100 or 1,000 moves every digit 1, 2 or 3 places, with zeros as placeholders where needed.

Pupils put positive and negative integers, decimals and fractions in order, using the number line as a model. Numbers further to the right are greater, so −8 is less than −2. To order −3/4, 0.7, −0.8 and 3/5, they convert everything to decimals: −0.75, 0.7, −0.8 and 0.6. From smallest to largest that gives −0.8, −3/4, 3/5, 0.7. They use the symbols =, ≠ (is not equal to), < (is less than), > (is greater than), ≤ (is less than or equal to) and ≥ (is greater than or equal to). So x ≥ 3 includes 3 itself, while x > 3 does not.

A factor of a number divides into it exactly, and a multiple is in its times table. A prime number has exactly two factors, 1 and itself. The first few are 2, 3, 5, 7, 11 and 13. The number 1 is not prime because it has only one factor, and 2 is the only even prime. Common factors are shared by two numbers, and the highest common factor (HCF) is the biggest of these. Common multiples are shared multiples, and the lowest common multiple (LCM) is the smallest. Every integer greater than 1 can be written as a product of primes in only one way, apart from the order. This is the unique factorisation property. Pupils write 60 as 2 × 2 × 3 × 5, or 2² × 3 × 5 using powers, and use this to find the HCF and LCM.

Pupils also learn to appreciate that some sets of numbers are infinite. There is no largest integer, because you can always add 1. There are infinitely many primes too. The rational numbers, which can be written as fractions, are infinite as well and so tightly packed that between any two of them there is another. The real numbers include all of these and numbers such as π that cannot be written exactly as fractions. This is a first look at the idea of infinity.

Words your child will hear

Integer
A whole number, positive, negative or zero, such as −7, 0 or 15.
Prime number
A number with exactly two factors, 1 and itself. 13 is prime. 15 is not, because 3 × 5 = 15.
Factor
A number that divides exactly into another. The factors of 12 are 1, 2, 3, 4, 6 and 12.
Multiple
The result of multiplying a number by a whole number. Multiples of 12 include 12, 24 and 36, and they go on for ever.
Highest common factor (HCF)
The largest number that is a factor of both numbers. The HCF of 12 and 18 is 6.
Lowest common multiple (LCM)
The smallest number that is a multiple of both numbers. The LCM of 12 and 18 is 36.
Prime factorisation
Writing a number as a product of prime numbers, such as 60 = 2² × 3 × 5.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

  • Understand and use place value for decimals, measures and integers of any size
  • Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥
  • Use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property
  • Appreciate the infinite nature of the sets of integers, real and rational numbers

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Write 180 and 84 as products of prime factors. Use them to find the HCF and LCM of 180 and 84.

  1. Break 180 into smaller factors until every one is prime. A factor tree does this: 180 = 2 × 90, 90 = 2 × 45, 45 = 3 × 15, 15 = 3 × 5.
    180 = 2 × 2 × 3 × 3 × 5
    = 2² × 3² × 5
  2. Do the same for 84: 84 = 2 × 42, 42 = 2 × 21, 21 = 3 × 7.
    84 = 2 × 2 × 3 × 7
   = 2² × 3 × 7
  3. For the HCF, take the primes that appear in both numbers, each to the lowest power it appears with. These are 2² and 3.
    HCF = 2² × 3 = 4 × 3 = 12
  4. For the LCM, take every prime that appears in either number, each to the highest power it appears with. These are 2², 3², 5 and 7.
    LCM = 2² × 3² × 5 × 7
    = 4 × 9 × 35 = 1260
  5. Check. 180 ÷ 12 = 15 and 84 ÷ 12 = 7, so 12 is a common factor, and 15 and 7 have no common factor other than 1, so no larger common factor exists. Also 1260 ÷ 180 = 7 and 1260 ÷ 84 = 15, so both numbers divide into 1260.

180 = 2² × 3² × 5 and 84 = 2² × 3 × 7. The HCF is 12 and the LCM is 1,260.

Common mistakes and how to help

Thinking 1 is prime, or that every odd number is prime

Primes often look like “numbers that can’t be divided”, so 1 seems to fit, and 3, 5 and 7 suggest odd numbers are prime. But 9, 15 and 21 are odd and not prime.

What helps: Go back to the definition: exactly two factors. 1 has one factor. List the factors of 9 (1, 3, 9) to show it fails the test. 2 is prime and the only even prime.

Thinking a longer decimal is a bigger number, so 0.45 is more than 0.5

Pupils treat the digits after the point like a whole number and compare 45 with 5.

What helps: Line up the place values and add a zero: 0.45 and 0.50. Compare tenths first (4 tenths against 5 tenths). A number line or money (45p against 50p) makes it clear.

Believing −7 is greater than −3 because 7 is greater than 3

The size of the digits is noticed first, and the minus sign is treated as a label.

What helps: Use a number line or a thermometer. −7 °C is colder than −3 °C, and the further left a number is, the smaller it is.

Mixing up HCF and LCM, or factors and multiples

The words look similar, and “highest” and “lowest” are easy to swap round.

What helps: Factors are small and few because they have to fit into the number, while multiples go on for ever. The HCF can’t be bigger than the smaller number, and the LCM can’t be smaller than the larger one.

Saying there is no number between 0.5 and 0.6

Counting in tenths, 0.6 follows 0.5 so it seems there is nothing in between.

What helps: Ask for a number between them: 0.55. Then between 0.5 and 0.55, and so on. You never run out, which is the idea that there are infinitely many numbers.

How to help at home

  • Factor pairs. Choose a number such as 36 or 48 and find every pair of numbers that multiply to make it. Then list the primes under 50, crossing out multiples on a hundred square.
  • Washing-line number line. Write numbers such as −2.5, −1/3, 3/4, 0.7 and 1 on cards and ask your child to put them in order along a strip of paper or a length of string, and explain their reasoning.
  • When will they meet again? One bus leaves a stop every 12 minutes and another every 8 minutes. If both leave at 9:00, when do they next leave together? The answer is the LCM of 12 and 8, which is 24 minutes later.
  • Big and small numbers in the news. Look at populations, distances or costs in the millions and billions. Ask your child to read them aloud and say what each digit is worth, then round them.
  • The in-between game. Name a number between 0.3 and 0.4, then one between that and 0.3, and carry on. It is a friendly way to feel why decimals and fractions never run out.

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