Use the tiles when there is something specific you want pupils to notice.
Algebra tiles can make mathematical relationships easier for pupils to see and discuss. Congruent tiles represent like terms. Flipping a tile over shows its additive inverse. Matching positive and negative tiles can also be paired to show a sum of zero, helping pupils to see why zero pairs can be removed without changing the value of an expression.
A complete rectangular array can help connect multiplication with area, which becomes particularly useful when exploring expanding and factorising.
Start by giving pupils time to become familiar with the different tiles and the relationships between them. Once these relationships are secure, begin connecting what pupils can see to algebraic notation.
This gives pupils a chance to make sense of the structure visually before using symbols, which can otherwise feel abstract and disconnected from anything tangible. Some uses of the tiles can become awkward or potentially misleading, so it is also important to make the limitations of the representation explicit. [1]
Make the conventions explicit.
It is important that we, as teachers, make it clear from the beginning that each tile is a representation. An 𝑥-tile is not 𝑥 itself; it is only a way of representing 𝑥. Its physical length is also part of that representation and does not tell us the numerical value of 𝑥.
Pupils need to understand what each tile represents and which parts of its appearance carry mathematical meaning.
| Tile or action | Meaning to establish |
|---|---|
| Unit tile | A 1 × 1 square represents an area of 1. |
| 𝑥-tile | A 1 × 𝑥 rectangle represents an area of 𝑥. Its rectangular shape shows an area interpretation, but its physical length does not tell us the numerical value of 𝑥. |
| 𝑥2-tile | An 𝑥 × 𝑥 square represents an area of 𝑥2. It is not simply a larger version of an 𝑥-tile. |
| Congruent tiles | Congruent tiles, those with the same shape and size, represent like terms, even when their signs are different. |
| Rotation | Rotating a tile changes its orientation on the board, but its dimensions and the term it represents stay the same. |
| Inverse | Flipping a tile over changes the term to its additive inverse. For example, 𝑥 becomes −𝑥, while −𝑥 becomes 𝑥. Moving or rotating a tile does not change its sign. |
An 𝑥-tile has to be drawn at a convenient size for the teacher or pupil using it, whether that is in a book, on a whiteboard or on a digital board. This means its size might vary depending on who is drawing it. It does not have a fixed length in all representations. Its physical length is simply part of the representation; it does not tell us the numerical value of 𝑥.
It is worth saying this plainly so that pupils do not begin to treat the tiles as a scale drawing. For example, a pupil may notice that an 𝑥-tile might appear to be about 4.5 times the length of a unit tile and conclude that 𝑥 = 4.5. The proportions used to draw the tiles should never be interpreted in this way. [1]
Connect the tiles to notation when the relationships are secure.
It is possible for pupils to become quite good at moving the tiles without making the connection to the algebra underneath.
A pupil arranging the tiles correctly does not necessarily mean that they can accurately describe the expression, explain why terms cancel or reproduce the same reasoning symbolically.
Once pupils are clear about the different tiles and the relationships between them, we should begin connecting their actions to notation. TILES displays the expression represented on the board, which can help make this connection explicit.
Before changing an arrangement, ask pupils what expression is represented. Ask again afterwards, then ask them to explain what changed and why. If the value of the expression has been preserved, pupils should be able to explain why.
Once notation has been introduced, regularly ask pupils to write the algebra represented by an arrangement, whether the tiles are on the board, on a table in front of them or being manipulated digitally.
If they cannot do this, they may know which tiles to move without yet understanding the algebra those movements represent.
The aim is for the tiles to support pupils towards fluent symbolic algebra, rather than becoming something they need for every step. Look for every opportunity to connect what pupils do with the tiles to the corresponding notation. [1] [2]
Zero pairs represent additive inverses.
A positive tile and its matching negative tile have a sum of zero. This means that the pair can be removed without changing the value of the expression.
For example,
𝑥 + (−𝑥) = 0
and
1 + (−1) = 0.
The same reasoning applies to every matching positive and negative pair.
When a positive tile and its matching negative tile are removed, make clear that this is because their values sum to zero. Avoid relying on the language of terms simply “cancelling”. Pupils should understand that a zero pair can be removed because together the two terms have a value of zero. The physical or digital removal of the tiles should support the explanation rather than becoming the explanation itself.
A negative tile should not be described as having a negative physical area. The tile still has an ordinary positive area. Its red colour and negative sign are being used as a representation of a negative algebraic term only. This is a limitation of the representation, although in practice it is unlikely to cause difficulty for most pupils. It is still worth addressing explicitly if the issue arises, rather than expecting the tiles themselves to resolve it. [1]
Arrays can be used to support multiplication and grouping can be used to support division. When negative terms are involved, make clear what the colour, negative sign and any change of face represent. The explanation should come from the operation and the algebra, rather than from a rule based only on what the arrangement happens to look like.
An array only works when its dimensions mean something.
When algebra tiles are used for expanding or factorising, they need to form a complete rectangular array where the side lengths match the factors.
A collection of tiles that simply looks rectangular is not enough. Gaps, overlaps when using physical tiles, or mismatched edge lengths break the area argument.
Without the lengths of the tiles being made explicit, pupils can begin to read factors from what an arrangement looks like rather than from the mathematical structure that gives the rectangle its dimensions.
Take every opportunity to label or discuss the dimensions of the tiles whenever arrays are used for expanding or factorising. This is a key part of helping pupils understand where the factors come from. [1]
A product of three binomials would require a three-dimensional model, while further factors cannot be represented using the same area model at all. At these points the tiles may still support part of the reasoning, but the representation no longer gives a complete explanation of the algebra.
Misconceptions.
| Possible reading | What the teacher needs to make clear |
|---|---|
| The length of 𝑥 shows its value. | “The tile needs a physical size so that we can see and use it. Its length does not tell us the value of 𝑥.” |
| The value of 𝑥 can be found by estimating how many unit tiles can be put together to make the same length as the 𝑥-tile. | “The physical length of the 𝑥-tile is chosen for convenience. We do not use unit-tile lengths to determine the numerical value of 𝑥.” |
| A negative tile has negative area. | “The tile still has an ordinary positive area. Its red colour and negative sign tell us that it represents a negative term.” |
| Like terms have the same colour. | “Like terms are represented by congruent tiles. Their tiles have the same shape and the same size.” |
| Rotating a tile changes the term. | “Rotating the tile only changes how it sits on the board. It does not change the term it represents.” |
| Touching opposites makes them cancel. | “A positive tile and its matching negative tile form a zero pair. They can be removed because together their value is zero.” |
| Any rectangle proves a factorisation. | “The tiles need to fill a complete rectangular array. The lengths around the outside tell us the factors.” |
| A completed arrangement proves understanding. | “You still need to explain what the tiles represent and connect the arrangement to the algebra once notation has been introduced.” |
| Tiles must be used for every step. | “The tiles can be set aside when you can explain the same mathematical relationship clearly without them.” |
When to set aside the tiles
There is no need to rush pupils away from the tiles.
The aim is to use the representation for as long as it is helping pupils see and understand the mathematics, while gradually building their confidence to work without it.
Pupils can return to the tiles whenever a new structure or relationship would benefit from being explored visually. However, fluent symbolic manipulation should not depend on rebuilding every step using the tiles.
The tiles can be set aside when they are no longer the most useful representation for the mathematics being explored. [1]