Representations at Key Stage 3
The NCETM has recently launch a series of materials and videos to support the use of using different representations in Key Stage 3 using a variety of resources.
Commonly used in primary mathematics or have been historically considered the reserve of lower prior attainers, representations can be equally valuable for revealing mathematical structures at secondary level for all students.
Below we outline some of the resources used. For further guidance on using each research take a look at all the NCETM materials at www.ncetm.org.uk/resources/53609.
Algebra Tiles
Algebra
Tiles are a collection of square and rectangular tiles that can be used to
represent expressions containing both numerical and algebraic terms. There are
three different files: 1, x and x2 and are based on an area model.
Each tile is reversed so that it can denote the additive inverse of the original.
They can be used to represent expressions and show how expressions can be manipulated. One key idea is that of the zero pair where we can use a tiles and its additive inverse to make zero. This leads on to the idea of how we can represent working with directed numbers.
For more about Algebra Tiles visit www.ncetm.org.uk/files/110720772/ncetm_ks3_algebra_tiles.pdf
Arrays and Area Models
s are a way of organising set of countable objects. From the product of two integers to the product of two binomials, arrays can be used to provide a consistent model for representation that students will be familiar with.
For more about Arrays and Area Models
visit
www.ncetm.org.uk/files/110720774/ncetm_ks3_arrays_area_models.pdf
Cuisenaire Rods
Cuisenaire Rods are proportional number sticks ranging from 1cm to 10cm. Each stick is represented by a different colour and because they have no markings each stick can represent a ‘unit’.
Most commonly used in the primary setting Cuisenaire Rods can be used across the age range here, for example, with the binomial theorem at A Level.
For more about Cuisenaire Rods visit
www.ncetm.org.uk/files/110720776/ncetm_ks3_cuisenaire_rods.pdf
Other representations include:
- Bar models
- Dienes (and place-value counters)
- Double number lines (and ratio tables)
- The Gattegno chart
- Place-value charts
- Single number lines



