Use the tiles to reveal something specific.
Algebra tiles make mathematical relationships easier to inspect. Congruent tiles show like terms. The reverse face of a tile represents its additive inverse. A complete rectangular array can connect multiplication with area.
Begin by helping pupils distinguish the tile types and the relationships between them. Once these are secure, connect the arrangements to notation. This allows the visual representation to establish meaning before symbols are used to describe it. Some uses can become awkward or misleading, so the limits of the representation must also be made explicit. [1]
Make the conventions explicit.
An algebra tile is a representation of a mathematical object. It is not the object itself. Pupils need to know what each shape stands for and which visual features carry mathematical meaning.
| Tile or action | Meaning to establish |
|---|---|
| Unit tile | A 1 × 1 square has area 1. |
| 𝑥-tile | A 1 × 𝑥 rectangle has area 𝑥. Its rectangular shape supports an area interpretation but its displayed length does not fix a numerical value of 𝑥. |
| 𝑥2-tile | An 𝑥 × 𝑥 square has area 𝑥2. It is not simply a larger version of an 𝑥-tile. |
| Congruent tiles | Congruent tiles represent like terms. They have the same shape and size even when their signs are different. |
| Rotation | Rotating any tile changes its orientation on the board. Its dimensions stay the same and the term it represents does not change. |
| Inverse | Turning a tile to its reverse face changes the term to its additive inverse. Moving or rotating a tile does not change its sign. |
An 𝑥-tile is drawn at a convenient size. Its displayed length does not fix the numerical value of 𝑥. This needs saying plainly so that pupils do not treat the representation as a scale drawing. [1]
Connect the tiles to notation when the relationships are secure.
Pupils can learn to move tiles successfully without understanding the matching algebra. A tidy arrangement is not evidence that they can name an expression, justify a cancellation or reproduce the reasoning symbolically.
Introduce notation once pupils are clear about the different tiles and the relationships between them. TILES displays the expression represented on the board. Ask pupils to state the expression before a change, state the expression afterwards and explain what the change has done. If the value has been preserved, they should explain why.
Once notation has been introduced, ask pupils to write the algebra represented by an arrangement or change. If they cannot do this, they may know which tiles to move without understanding the algebra.
The tiles should help pupils move towards fluent symbolic algebra, not become a substitute for it. Pupils should connect each change on the board with the corresponding notation. [1] [2]
Zero pairs represent additive inverses.
A positive tile and the matching negative tile have a sum of zero. Removing that pair preserves the value of the expression because the two terms are additive inverses. For example, 𝑥 + (−𝑥) = 0 and 1 + (−1) = 0. The same reasoning applies to every matching positive and negative tile. Remove the pair because its terms have a sum of zero. The tiles disappearing from the board should show that reasoning, not replace it.
A negative tile should not be described as a piece with negative physical area. Area is being used to organise the algebra while the colour and negative sign shown on the tile indicate whether the term is positive or negative. That convention is useful but it contains a genuine tension. Pupils should not be left to resolve it from the representation alone. [1]
Arrays can be used for multiplication and grouping can be used for division. With negative terms, teachers should make clear what the colour, negative sign and any change of face represent. The explanation should come from the operation and the algebra, not from a rule about how the representation happens to look.
An array only works when its dimensions mean something.
For expansion or factorisation, the tiles must form a complete rectangular array and the side lengths must match the factors. A collection that merely looks rectangular is not enough. Gaps, overlaps or mismatched edge lengths break the area argument.
The dimensions need to be named. An 𝑥-tile contributes 𝑥 × 1 or 1 × 𝑥. An 𝑥2-tile contributes 𝑥 × 𝑥. Unit tiles contribute 1 × 1. Without these lengths, pupils may read a factor from the appearance of an arrangement rather than its mathematical structure. Label the dimensions whenever arrays are used for expanding or factorising. [1]
Negative terms require care because geometric area is not itself negative. The array can still represent the algebra, but the sign convention must be explained separately. A product of three binomials would require a three-dimensional model, while further factors cannot be represented by the same area model. At these points the tiles may still support part of the reasoning, but the representation no longer provides 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 size so we can see it. Its length does not tell us the value of 𝑥.” |
| A negative tile has negative area | “The tile still has an ordinary area. Its colour and negative sign tell us that it represents a negative term.” |
| Like terms have the same colour | “Like terms are shown by congruent tiles. Their tiles have the same shape and the same size.” |
| Rotating a tile changes the term | “Rotating the tile changes how it sits on the board. It does not change the term.” |
| Touching opposites makes them cancel | “Touching is only a visual clue. The pair can be removed because the two terms have a sum of zero.” |
| Any rectangle proves a factorisation | “The tiles must fill a complete rectangular array. The lengths around the outside tell us the factors.” |
| A completed arrangement proves understanding | “You still need to say what the tiles represent and write the same thing in 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 relationship clearly without them.” |
When to set aside the tiles
The aim is not to remove the representation quickly. It is to help pupils become independent of it. Pupils can return to the tiles whenever a new structure needs to be inspected, but fluent symbolic manipulation should not depend on rebuilding every step on the board. The tiles can be set aside when they are no longer the most appropriate representation. [1]