Algebra Tiles: Free Virtual Algebra Tiles Online

What algebra tiles teach

Algebra tiles make symbols into shapes. Before a student is asked to "combine like terms," they can see that two long rectangles and three small squares are just that: 2x + 3. An x tile and an x² tile can't be combined because they aren't the same shape, which is exactly why 2x + 3x² doesn't simplify. The tiles carry the rules of algebra in their geometry.

There are three tiles, each with a positive side and a red negative side:

The x length is deliberately not a whole number of units. If it were, a student could line up three unit tiles beside an x tile and decide x = 3. Keeping the lengths unrelated keeps x unknown.

Zero pairs and simplifying expressions

The key idea is the zero pair: a positive tile and a negative tile of the same kind add up to 0. With zero pairs, integer rules stop being rules to memorize. 3 + (−5) is three yellow units and five red units; three pairs cancel, and two red units are left: −2.

The same move simplifies expressions. Task card 3 builds 3x + 2 − x − 5. Pair one x with the −x, and two units with two of the negative units. What's left is 2x − 3. Remove zero pairs on this board does it in one press, and the readout confirms the value didn't change.

The Common Core standards ask 6th graders to generate equivalent expressions using properties of operations (6.EE.A.3) and 7th graders to add, subtract, factor and expand linear expressions with rational coefficients (7.EE.A.1). Tiles give both a physical meaning: equivalent expressions are different arrangements of the same tiles, after zero pairs are removed.

Solving equations with algebra tiles

The equation mat has a left side and a right side, separated by an equals sign. A balance is the right picture: whatever you do to one side, you do to the other, and the two sides stay equal.

1. Build the equation. For 2x + 1 = 7, put two x tiles and one unit on the left, seven units on the right.
2. Remove the same thing from both sides. Take one unit off each side: 2x = 6.
3. Split into equal groups. Two x tiles on the left; split the six units on the right into two equal groups. Each x matches 3.
4. Keep one group. One x tile is alone on the left and three units on the right: x = 3.

Negatives use zero pairs. For x − 2 = 3, add two positive units to both sides. On the left, they cancel the two negative units; on the right, the 3 becomes 5. For 3x − 1 = x + 5, take an x off both sides first, then finish as before. Seventh graders solve equations of the form px + q = r (7.EE.B.4), and 8th graders solve linear equations with variables on both sides (8.EE.C.7). The board checks the finish: one x tile alone on one side, only unit tiles on the other, and no zero pairs left over.

Multiplying binomials on the product mat

The product mat turns multiplication into area. Put one factor along the top and the other down the side. Every tile inside the rectangle has the width of the tile above it and the height of the tile beside it:

• x on top and x on the side meet in an x² tile.
• x and 1 meet in an x tile.
• 1 and 1 meet in a unit tile.
• If the two signs differ, the tile inside is negative.

For (x + 1)(x + 2), the rectangle holds one x², three x tiles and two units: x² + 3x + 2. For (x + 2)(x − 1), the negative unit on the side makes a row of negative tiles; after the zero pair, the product is x² + x − 2. In high school, the standards ask students to add, subtract and multiply polynomials and to understand that polynomials are closed under these operations (A-APR.A.1).

Factoring runs the same picture backward. Task card 10 gives the tiles for x² + 4x + 3 and asks for a rectangle. The only rectangle uses one x² in the corner, four x tiles split three and one, and three units in the corner opposite. Its top edge reads x + 3 and its side edge x + 1, so x² + 4x + 3 = (x + 3)(x + 1). Seeing the structure of an expression this way is the heart of A-SSE.A.2.

Algebra tiles by grade

Activities to try

• Integer warm-up. Show 4 + (−7) with units, cancel the zero pairs and read the answer. Then try −3 − (−5): start with three negative units, add two zero pairs so there are five negatives to take away, and see what's left (2).
• Same value, different tiles. Build 2x + 1 three different ways, using zero pairs. Ask a partner to simplify each.
• Distribute. Build 3(x + 2) as three groups of x + 2. Count the tiles: 3x + 6.
• Equation maker. One student builds an equation and solves it secretly; the other student solves it on the mat. Compare steps.
• Rectangle hunt. Give the tiles for x² + 5x + 6 and ask for a rectangle. Then try x² + 5x + 4. Which trinomials make rectangles?
• Perfect squares. Make the biggest square you can from one x², some x tiles and some units. What has to be true about the x tiles? (That's the idea behind completing the square.)

Tips for teachers and parents

• Say the tile names out loud. "Two x's and three ones" helps students connect the picture to the expression.
• Record each step in symbols. After every move on the mat, write the new equation. Connect the model to the written calculation. The What Works Clearinghouse guide recommends using concrete and semi-concrete representations.
• Let the zero pair do the thinking. When a student isn't sure how to remove −2, ask them to add zero pairs until they can.
• Stop using tiles when they stop helping. Tiles are a bridge. Once students solve and factor fluently in symbols, they no longer need them, though they're useful again for new ideas like completing the square.

What makes these algebra tiles different

This board has all three mats in one place: a free mat that simplifies as you build, an equation mat that knows when x is alone, and a product mat that reads both factors and the product. Tiles snap edge to edge, flip sign, cancel in zero pairs in one press, and rotate. An expression box builds any quadratic with up to 6 x² tiles, 12 x tiles and 20 units. Undo, redo, share links, a printable snapshot, keyboard controls and ten self-checking task cards round it out.

When students are ready to graph, move to the interactive coordinate plane. For programs that teach algebra conceptually, compare the best middle school math curriculum and the best high school math curriculum guides.

Choosing an activity

Grade bands describe possible curriculum connections, not an assessment of an individual learner. Start with an example the learner can explain, then change one feature at a time. The activity suggestions here are editorial teaching ideas, not results from a classroom trial of this software. Use the model alongside written work and physical materials when useful. The linked standards describe learning objectives; they do not endorse Edulist or require this particular tool.