Multiples, Factors, Primes and Squares

Home → Curriculum Hub → Maths Lessons → Schemes of Work → Key Stage 3 → Multiples, Factors, Primes and Squares

This Key Stage 3 Multiples, Factors, Primes and Squares scheme of work supports Year 7 teaching in Term 2 by developing fluency with multiplicative structure: multiples and factors, prime numbers, prime factorisation, highest common factors and square numbers. The unit is designed to strengthen reasoning with integers and simple algebraic terms before students meet more demanding manipulation and index work later in Key Stage 3.

Use the sections below to align lessons with your department map, agree vocabulary (factor, multiple, prime, composite, HCF, prime factor, square and square root), and plan for common misconceptions—especially around primes, factorisation and index notation. The lesson list at the end matches the published sequence for this scheme.

Published: October 2022 | Last updated: April 2026

Jump to Lessons

Curriculum Hub

For more schemes, topic bundles and departmental planning support, visit the Curriculum Hub.

How to use this scheme

Use the prerequisite list to decide whether students need to consolidate times tables and factor thinking from Key Stage 2 before you increase demand. Teach in the order shown under Lessons where possible so multiples and factors are secure before prime factorisation and HCF, and introduce square numbers once students are confident multiplying integers. Use the key concepts and subject content sections to set objectives, plan mini-whiteboard checks, and agree consistent language across your department.

Mr Mathematics membership

Members get full access to editable PowerPoints, worksheets, solutions and additional revision resources that match Mr Mathematics lessons. If you want everything in one place for planning and departmental use, see Mr Mathematics membership.

Prerequisite knowledge

Students should be able to do the following before or during early lessons in this unit. Use the list to diagnose gaps and plan short consolidation if needed.

Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers.
Know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers.
Establish whether a number up to 100 is prime and recall prime numbers up to 19.
Identify common factors, common multiples and prime numbers.

Key concepts

These ideas are central to teaching Key Stage 3 Multiples, Factors, Primes and Squares effectively and link directly to later work on indices, fractions and algebra.

A multiple of a number or term is found by multiplying it by an integer (or by another term when working algebraically).
A factor of a number is an integer that divides exactly into that number (and similarly for algebraic terms when appropriate).
A prime number has exactly two factors. A common misconception is “a prime divides only by 1 and itself”; stated carelessly, that wording can wrongly suggest 1 is prime—be explicit that 1 has only one factor.
Any integer greater than 1 can be written as a product of prime factors; more confident students should use index notation where appropriate.
The highest common factor (HCF) of a set of numbers or terms is the largest factor common to each of them.
A square number can be represented as the area of a square; the square root relates the area to the side length when the context is suitable.

Working mathematically

The scheme supports students to work in line with Key Stage 3 expectations for fluency, reasoning and making connections across number and algebra.

Develop fluency
- Consolidate numerical and mathematical capability from Key Stage 2 and extend understanding of the number system and place value to include decimals, fractions, powers and roots.
- Select and use appropriate calculation strategies to solve increasingly complex problems.
- Use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships.
Reason mathematically
- Extend understanding of the number system; make connections between number relationships and their algebraic representations.

Subject content

Below is the Key Stage 3 content this scheme connects to, summarised from your original materials.

Number
- 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.
- Use conventional notation for the priority of operations, including brackets, powers and roots.
Algebra
- Simplify and manipulate algebraic expressions to maintain equivalence by:
  - collecting like terms
  - multiplying a single term over a bracket
  - taking out common factors
  - expanding products of two or more binomials

Frequently asked questions

What is included?
This page summarises the curriculum focus for the unit. Full teaching packs (lessons, questions and answers) are available through the site and membership where applicable.

Which skills are developed?
Students develop fluency with multiples, factors, primes, prime factorisation, highest common factors and square numbers, and learn to use precise vocabulary when reasoning and solving problems.

Can lessons be used independently?
Yes—individual lessons can be taught standalone if you adjust the prior-knowledge check, though teaching in sequence usually gives the smoothest progression.

Does this support GCSE?
Yes. Factors, multiples, primes and indices underpin fraction work, standard form, surds and algebraic factorisation at GCSE; this Key Stage 3 foundation supports confident calculation and reasoning later on.

Lessons in this Multiples, Factors, Primes and Squares scheme of work

These lesson titles match the scheme sequence on Mr Mathematics. Link or embed your departmental resources (PowerPoints, question generators and downloads) as needed.