Using mathematical representations at KS3

Guidance for some of the most useful representations for Key Stage 3 mathematics.

Secondary KS3
Mastery PD Materials
Using mathematical representations at KS3

Although manipulatives and representations are commonly used in primary mathematics or have been historically considered the reserve of lower prior attainers, they can be equally valuable for revealing mathematical structures at secondary level for all students.

We have produced guidance for some of the most useful representations for Key Stage 3 mathematics. Each guidance document details:

  • what they are
  • why they are important
  • how they might be used.

Click the following links to download the guidance documents.

Please note some of the links to NCETM content in the documents below may not work. You can quickly find where the links point using the search tool on this website. If you cannot find what you’re looking for, contact

We have also produced explainer videos for algebra tiles and Cuisenaire® rods, which you can view via the links below.

Algebra tiles

  1. What are algebra tiles?
  2. Using algebra tiles: adding and subtracting directed numbers using zero pairs
  3. Using algebra tiles: simplifying expressions by collecting like terms
  4. Using algebra tiles: expanding brackets
  5. Using algebra tiles: factorising expressions
  6. Using algebra tiles: solving linear equations

Cuisenaire® rods

  1. What are Cuisenaire® rods?
  2. Using Cuisenaire® rods: modelling additive and multiplicative relationships
  3. Using Cuisenaire® rods: modelling decimals and fractions
  4. Using Cuisenaire® rods: exploring factors and primes


The NCETM and Maths Hubs would like to thank the following for their contribution to the materials:

  • Secondary mathematics specialists: Clare Dawson; Pete Griffin; Alison Hopper; Carol Knights; Pete Sides
  • Educational consultant: Dr Alf Coles, University of Bristol
  • Editorial and production: Natasha Chippendale; Charlotte Christensen; Tracey Cowell; Bronwyn Enright; Jackie Mace; Bill Mantovani; David Mantovani; Andrew Young
  • Illustrations: Steve Evans
  • Video development and production: Pam Jones, Eight Interactive Ltd