Algebra Tiles

Algebra tiles model algebra with area: a yellow 1-by-1 square is 1, a green 1-by-x strip is x, and a blue x-by-x square is x². Red tiles are negative. To show 2x² − 3x + 4, lay out two blue squares, three red strips and four yellow squares. A red and a yellow unit tile together make a zero pair worth 0.

How to use these algebra tiles

1. Tile mat: type an expression such as 2x^2 – 3x + 4 and press Build tiles, or add tiles one at a time. Drag tiles with a finger or mouse. Select a tile to rotate it, flip its sign, duplicate it or delete it. Zero pairs get a dashed outline. Press Simplify to combine like terms.
2. Equation mat: type an equation such as 3x + 2 = x + 4. Use the balance steps (add or remove the same tile on both sides, remove zero pairs, flip all tiles, divide into equal groups) or press Next step. The step log writes each move as algebra and checks the answer at the end.
3. Multiply & factor: type two binomials such as (x + 2)(x − 3) and fill the frame tile by tile. Or switch to Factor a trinomial, change the sides of the rectangle until the tiles match, and press Show the solution if you get stuck.

Everything also works by keyboard (Tab to a tile, then arrow keys, R, F, D or Delete), with no login or download.

What algebra tiles are and what each tile means

Algebra tiles are flat pieces whose area stands for a term. The small square has sides 1 and 1, so its area is 1. The long strip has sides 1 and x, so its area is x. The big square has sides x and x, so its area is x². The x side is deliberately not a whole number of unit tiles, because x can be any number.

Every tile here also carries a text label (such as −x), so color is never the only cue. A rotated x tile is still worth x.

The zero pair principle

A positive tile and a negative tile of the same kind add to zero: 1 + (−1) = 0 and x + (−x) = 0. That pair is called a zero pair. You can remove a zero pair from a mat, or add one, and the value does not change. This one idea drives combining like terms, subtraction, solving equations and factoring.

For example, 3x − x + 2 − 5 is three green x tiles, one red x tile, two yellow tiles and five red unit tiles. One x zero pair and two unit zero pairs cancel, leaving 2x − 3.

Adding and subtracting polynomials with algebra tiles

To add, put both polynomials on the mat and remove zero pairs. To subtract, add the opposite: flip every tile of the polynomial you are subtracting, then combine.

Worked example: (2x² + 3x − 1) − (x² − x + 4). Flip the second group to −x² + x − 4. The mat now has 2 blue and 1 red x² tile, 4 green x tiles, and 1 plus 4 red unit tiles. One x² zero pair cancels, so the result is x² + 4x − 5. Try (2x + 3) − (x − 4) on the tile mat: the answer is x + 7.

Solving equations with algebra tiles

The equation mat is a balance. Whatever you do to one side, you do to the other. The goal: one x tile alone on one side, unit tiles on the other.

Worked example: 3x + 2 = x + 4.

1. Take one x tile off each side: 2x + 2 = 4.
2. Take two unit tiles off each side: 2x = 2.
3. Split each side into 2 equal groups. Each x matches 1 unit tile, so x = 1.
4. Check: 3(1) + 2 = 5 and 1 + 4 = 5.

A negative answer: −2x + 1 = 5. Take one unit tile off each side to get −2x = 4. Flip every tile (multiply both sides by −1) to get 2x = −4, then divide into 2 groups: x = −2. Check: −2(−2) + 1 = 5.

When you cannot take a tile away because the other side does not have one, add its opposite to both sides instead and remove the zero pairs that appear. For 5 = 2x − 3, add 3 unit tiles to both sides: 8 = 2x, so x = 4. If the x tiles cancel completely you are left with two numbers. For x + 3 = x + 5 you get 3 = 5, which is false, so there is no solution. For 2(x + 1) = 2x + 2 you get 2 = 2, which is always true, so every x is a solution.

Multiplying binomials with the area model

A product is the area of a rectangle. Put one factor along the top and the other down the side, then fill the inside. Each tile inside is (its column tile) × (its row tile): x × x = x², x × 1 = x and 1 × 1 = 1. Like signs give a positive tile; unlike signs give a red negative tile.

Worked example: (x + 2)(x − 3). The frame fills with one x², three −x tiles, two x tiles and six −1 tiles. Two x zero pairs cancel, so (x + 2)(x − 3) = x² − x − 6. With a leading coefficient, (2x + 1)(x + 3) = 2x² + 7x + 3, because the rectangle holds 2 x² tiles, 6 + 1 = 7 x tiles and 3 unit tiles.

Factoring trinomials with algebra tiles

Factoring runs the area model backwards. Start with the tiles of the trinomial and arrange them into one complete rectangle. The side lengths are the factors.

• x² + 5x + 6: the six unit tiles fill a 2-by-3 corner, so the sides are x + 2 and x + 3: (x + 2)(x + 3).
• x² − x − 6: no rectangle fits until you add two x zero pairs. Then the rectangle is (x − 3)(x + 2), using 2 x tiles and 3 −x tiles (net −x).
• 2x² + 5x + 3: two x² tiles go side by side, and the rectangle is (2x + 3)(x + 1).
• x² + x + 1: no rectangle can be made, even with zero pairs, so it does not factor over the integers. The tool says so instead of guessing.

Limitations of algebra tiles

• Fractions and decimals: a unit tile cannot be cut, so 2x = 3 ends with “x = 3/2” written as a fraction rather than shown as tiles.
• Large numbers: 47 unit tiles are slow to count, so switch to paper once the idea is clear.
• Higher powers: x³ would need a cube, so tiles stop at degree 2 (x², xy, y²).
• Negative area: a red tile is a bookkeeping signal, not a real negative area. Say this out loud to students.

Teacher tips for virtual algebra tiles

• Project the equation mat and ask students to call out the next balance step before you press it. The step log records the algebra for the board.
• Use the snap-back on the equation mat: when a student drags a tile across the = sign it bounces back. Ask why: it opens a talk about “moving terms” versus adding opposites.
• Nothing to print or cut out and no accounts: it runs on school phones and Chromebooks.

Step-by-step lessons with algebra tiles

• How to model and solve equations using algebra tiles
• Using algebra tiles to model and solve equations
• Use algebra tiles to identify equivalent expressions
• Adding and subtracting polynomials using algebra tiles
• How to multiply two binomials using algebra tiles
• Factoring quadratics using algebra tiles

Algebra tiles practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What do the colors of algebra tiles mean?

The color shows the kind of tile and its sign. Here, positive x² tiles are blue, x tiles green and unit tiles yellow, and every negative tile is red. Physical sets vary by brand; the shape tells you the value.

What is a zero pair in algebra tiles?

A zero pair is one positive and one negative tile of the same kind, such as x and −x. Together they are worth 0, so you can add or remove zero pairs without changing an expression.

How do you solve an equation with algebra tiles?

Build each side on the equation mat, add or remove the same tiles on both sides until one x tile is alone, and split the units into equal groups if there are several x tiles.

Can algebra tiles show fractions?

Not directly. Tiles show whole numbers, so for 2x = 3 the answer x = 3/2 is written as a fraction.

Related calculators

• Algebra calculator: solve and simplify with full steps
• Linear equation calculator: check equations you solved with tiles
• FOIL calculator: multiply binomials the symbolic way
• Factoring calculator: factor any polynomial with named techniques
• Completing the square calculator: the next step after area models