Algebra · Variables and Equations · Lesson 3

Modelling and Solving Equations with Algebra Tiles

Algebra tiles turn letters and numbers into shapes you can see and move. A long strip stands for $x$ and a small square stands for 1. In this lesson you will build equations with tiles on an equation mat, read off the expressions they show, and solve them by removing tiles, using zero pairs and sharing equally.

NaCCA B7.2.3.1.3 ⏱ about 45 min

By the end you will be able to

✔Name the algebra tiles and state the value of each one.
✔Write the expression or equation shown by a set of algebra tiles.
✔Solve linear equations with tiles by removing equal tiles from both sides, using zero pairs and sharing.
✔Describe each step of a tile solution in words and in symbols.

Key words

x-tile: A long green rectangle standing for the unknown $x$ . A red one stands for $- x$ .
Unit tile: A small square standing for 1. Yellow means +1 and red means −1.
Equation mat: A sheet divided by a line into a left side and a right side; the line acts as the equals sign.
Zero pair: A positive tile and a matching negative tile (1 and −1, or $x$ and $- x$ ). Together they are worth 0.
Coefficient: The number multiplying the variable. In $3 x$ the coefficient is 3, shown by 3 x-tiles.

Curriculum reference

Content standard: B7.2.3.1 Demonstrate an understanding of linear equations of the form x + a = b (where a and b are integers) by modelling problems as a linear equation and solving the problems concretely, pictorially, and symbolically.

Indicator: Model linear equations, then write mathematical expressions and describe the process of solving the equation using algebraic tiles.

Starter

Mr Agbeko's cardboard tiles

In a JHS classroom in Ho, Mr Agbeko cuts old cereal boxes into long green strips and small yellow squares. He places two strips and three squares on the left of a line drawn on the table and seven squares on the right. "Each strip hides the same number of squares," he says. "Who can tell me what one strip is worth — without guessing?" Selorm moves a few tiles and has the answer in under a minute. Let us learn her method.

1. Meet the algebra tiles

Algebra tiles are a picture of algebra. Their sizes are chosen so you never mix them up: the x-tile is a long strip because its value is unknown and could be large, while the unit tile is a small square worth exactly 1. Colour shows the sign: yellow or green for positive, red for negative.

The four algebra tiles used in Basic 7
Tile	Looks like	Value
x-tile	long green rectangle	$x$
negative x-tile	long red rectangle	$- x$
unit tile	small yellow square	+1
negative unit tile	small red square	−1

We work on an equation mat: a sheet with a line down the middle. Everything on the left of the line is the left side of the equation; everything on the right is the right side. The line itself is the equals sign, so the two sides must always have the same value — exactly like a balance.

2. Reading and writing expressions from tiles

To write the expression on one side of the mat, count each kind of tile. The number of x-tiles is the coefficient of $x$ ; the unit tiles give the number term. Red tiles give minus signs.

Tiles on one side	Expression
3 green x-tiles and 2 yellow units	$3 x + 2$
1 green x-tile and 4 red units	$x - 4$
2 green x-tiles and 3 red units	$2 x - 3$
5 yellow units	5
1 red x-tile and 6 yellow units	$6 - x$

Put the two sides together with an equals sign and you have the equation. Three x-tiles and 2 units on the left with 11 units on the right is the equation $3 x + 2 = 11$ . Going the other way, to model $2 x - 3 = 5$ you place 2 green x-tiles and 3 red units on the left and 5 yellow units on the right.

3. Solving: keep both sides equal

To solve, we want one x-tile alone on one side. We get there with three kinds of move, each done to both sides so the mat stays balanced: remove matching tiles, add zero pairs, and share equally. Step through Mr Agbeko's puzzle, $2 x + 3 = 7$ .

▶ Watch 3 units leave both sides together. Then each x-tile is matched with an equal share of the units that remain.

1Model the equation with tiles on the mat.
2Remove the same tiles from both sides (or add zero pairs first if you need to) until only x-tiles are on one side and only unit tiles are on the other.
3Share the unit tiles equally among the x-tiles. One x-tile's share is the value of $x$ .
4Check by replacing every x-tile in the first model with your value.

Describe it like this: "I took 3 units from each side, so 2x-tiles equal 4 units. Sharing 4 units between 2 x-tiles gives 2 units each, so x = 2." In symbols: $2 x + 3 = 7 \Rightarrow 2 x = 4 \Rightarrow x = 2$ .

4. When negative tiles appear

In $x - 2 = 5$ the left side has an x-tile and 2 red units. We cannot simply "remove 2 units" — there are no yellow units on the left to remove. Instead we add 2 yellow units to both sides. On the left, each yellow unit makes a zero pair with a red unit, and the zero pairs disappear.

▶ Adding the same yellow tiles to both sides keeps the mat balanced. On the left they cancel the red tiles as zero pairs.

The same idea works when the answer is negative. For $x + 4 = - 2$ , add 4 red units to both sides. On the left they cancel the 4 yellow units; on the right there are now 2 + 4 = 6 red units, so $x = - 6$ .

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Writing and solving from a tile picture

On a mat, the left side has 4 x-tiles and 1 unit tile; the right side has 9 unit tiles. Write the equation and solve it.

Show the solution

Step 1Count the tiles: left = $4 x + 1$ , right = 9. Equation: $4 x + 1 = 9$ .
Step 2Remove 1 unit from each side: $4 x = 8$ .
Step 3Share 8 units among 4 x-tiles: 2 units each, so $x = 2$ .
Step 4Check: 4 × 2 + 1 = 9 ✓

Answer: $4 x + 1 = 9$ ; $x = 2$

Example 2

Three x-tiles

Use algebra tiles to solve

3 x + 2 = 14

, describing each move.

▶ The tile moves are the same as balance moves: remove from both sides, then share equally.

Show the solution

Step 1Model: 3 x-tiles and 2 units on the left; 14 units on the right.
Step 2"I remove 2 units from each side": $3 x = 12$ .
Step 3"I share 12 units among 3 x-tiles, 4 each": $x = 4$ .
Step 4Check: 3 × 4 + 2 = 14 ✓

Answer: $x = 4$

Example 3

Using zero pairs

Solve

2 x - 3 = 5

with algebra tiles.

Show the solution

Step 1Model: 2 x-tiles and 3 red units on the left; 5 yellow units on the right.
Step 2Add 3 yellow units to each side. On the left they form 3 zero pairs, which we remove: $2 x = 8$ .
Step 3Share 8 units between 2 x-tiles: $x = 4$ .
Step 4Check: 2 × 4 − 3 = 8 − 3 = 5 ✓

Answer: $x = 4$

Example 4

A negative answer

Solve

x + 4 = - 2

with algebra tiles.

Show the solution

Step 1Model: 1 x-tile and 4 yellow units on the left; 2 red units on the right.
Step 2Add 4 red units to each side. On the left, 4 zero pairs vanish, leaving $x$ .
Step 3The right side now has 2 + 4 = 6 red units, worth −6.
Step 4So $x = - 6$ . Check: −6 + 4 = −2 ✓

Answer: $x = - 6$

Example 5

x-tiles on both sides

Solve

3 x + 1 = x + 7

with algebra tiles.

Show the solution

Step 1Model: 3 x-tiles and 1 unit on the left; 1 x-tile and 7 units on the right.
Step 2Remove 1 x-tile from each side: $2 x + 1 = 7$ .
Step 3Remove 1 unit from each side: $2 x = 6$ .
Step 4Share: $x = 3$ . Check: 3 × 3 + 1 = 10 and 3 + 7 = 10 ✓

Answer: $x = 3$

Summary

A long green tile is $x$ , a small yellow tile is +1; red tiles are the negatives $- x$ and −1.
Count the tiles on each side to write the expression; the mat line is the equals sign.
Solve by doing the same to both sides: remove matching tiles, add zero pairs, then share equally.
A zero pair (+1 and −1, or $x$ and $- x$ ) is worth 0 and can be added or removed freely.
Describe each move in words and symbols, and check your answer in the original equation.

Mr Agbeko's cardboard tiles

1. Meet the algebra tiles

2. Reading and writing expressions from tiles

3. Solving: keep both sides equal

4. When negative tiles appear

Worked examples

Writing and solving from a tile picture

Three x-tiles

Using zero pairs

A negative answer

x-tiles on both sides

Summary

Practice: 15 questions