HCF and LCM – KS4 maths lesson plan

Key Stage

KS4

Age

Years 10-11

Subjects

This lesson plan takes the ideas of HCF and LCM (highest common factor and lowest common multiple), and deals with them together. In maths, we often teach these concepts separately, but looking at them together helps learners to appreciate the connections.

Sometimes learners encounter the ideas of highest common factor (HCF) and lowest common multiple (LCM) as procedures to follow to obtain answers, without much sense of what these quantities represent or why the procedures work. This maths lesson builds up to having learners find numbers to fit a specified HCF and LCM, giving them an opportunity to explore what possibilities are allowed by these constraints. The trial and error involved also gives plenty of practice at the techniques.

HCF and LCM learning objectives

In this lesson, students will learn about factors and multiples that numbers have in common with each other.

Starter activity

Put these Questions to Think About on the board as (or before, if possible) learners enter the room. Some learners may start talking about them as they get seated, but it doesn’t matter if they don’t.

What is a factor? (For example, 2 is a factor of 6.)
Does every number have factors?
Can you think of a number that has…
– exactly 2 factors?
– exactly 3 factors?
– exactly 4 factors?

When everyone is settled, you could ask:
Does anyone have any comments on, questions about or answers to any of these? You can start with whichever one you want.

A factor is a number (a positive integer) that goes into another number. The ‘goes into’ aspect has sometimes led people to call them ‘guzintas’, which can be fun, but learners need to know that the proper mathematical name is ‘factor’.

Every number has at least 1 factor, because 1 is a factor of every number. The only number with exactly 1 factor is 1, and all prime numbers have exactly 2 factors, which learners might realise. Numbers with an odd number of factors are always square numbers; in particular, numbers with exactly 3 factors are the squares of prime numbers. For example, 25 has 3 factors, because it is the square of 5, which is prime.

There are two possibilities for numbers with 4 factors: either they are semiprimes (the product of two different prime numbers), such as 6, which is 2 x 3, or the cube of a prime number, such as 8, which is 23.

KS4 maths curriculum

In the UK Key Stage 4 (KS4) mathematics curriculum, the highest common factor (HCF) and lowest common multiple (LCM) are part of the broader topic of number theory. They specifically fall under the content area of ‘Number’ where students explore various concepts related to factors, multiples, and primes.

Students need to know the following:

Number

Number operations and integers
- Understand and use the concepts and vocabulary of factors (divisors), multiples, common factors, common multiples, highest common factor, and lowest common multiple.
- Apply systematic listing strategies, including using lists, tables, and diagrams, to identify common factors and common multiples.

Specific objectives

Highest Common Factor (HCF):
- Be able to find the HCF of two or more numbers.
- Use the HCF to solve problems involving factors and multiples, such as simplifying fractions or solving problems with constraints based on divisibility.
Lowest Common Multiple (LCM):
- Be able to find the LCM of two or more numbers.
- Use the LCM in problem-solving contexts, such as finding common time periods for repeating events or combining different cycles.

Colin Foster (@colinfoster77) is a reader in mathematics education in the Department of Mathematics Education at Loughborough University. He has written numerous books and articles for mathematics teachers.

For more KS4 maths lesson plans from Colin Foster, click here.