An equation is a mathematical sentence formed by setting two expressions equal. We say that the two expressions are equal because they have the same value. For example, 6+4=10 because the expression 6+4 evaluates to 10.

We don't need to know the value of every expression in order to write an equation. In fact, we can use algebra to represent equations with unknown values.

Model equations with algebraic tiles

There are many ways to represent an equation. One way is to use Algebra tiles to represent the algebraic expressions on each side.

Table showing tiles for positive and negative variables and units. Ask your teacher for more information.

Using the key above, let's represent a few equations with algebraic tiles.

Examples

Example 1

Write the equation represented by the algebraic tiles. Do not solve the equation.

An equation represented by 1 positive x tile and 4 positive 1 tiles on the left side and 5 positive 1 tiles on right side.

Worked Solution

Create a strategy

Count the number of positive x tiles and positive 1 tiles

Apply the idea

There is 1 \, \, +x tile and 4 \,\, +1 tiles on the left side. There are 5 \,\, +1 tiles on the right side. So the equation is:x+4=5

Example 2

Write the equation represented by the algebraic tiles. Do not solve the equation.

An equation represented by tiles. The left side has 2 positive x tiles. The right side has 5 positive 1 tiles.

Worked Solution

Create a strategy

Apply the idea

Remember that the value of five positive 1 tiles is 5 and the value of 2 positice x tiles is 2 x

So the equation is:2x=5

Idea summary

We can represent equations with algebraic tiles.

Balance equations

We want to keep equations balanced so that the two sides of the equals sign remain equivalent. If we don't we could change what the equation means.

Think of a balanced set of scales. The scale remains level when the weights on both sides of the scales are even. The same thing happens with equations.

Exploration

This applet represents the equation x=3.

You can click and drag algebraic tiles from the bottom to the gray part at the bottom to be on the scale. Click the reset button in the top right corner to go back to x=3.

What sorts of things make the scale imbalanced?
What sorts of things can you do to keep the scale balanced?
What are three different equations which are equivalent to x=3?

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For every variable x that we add in the left side of the scale, we should add 3 units at the right to keep it balanced.

We can add the same amount to both sides and the equation will stay balanced.

A scale having 1 positive x tile and 1 positive 1 tile on the left and 4 positive 1 tiles on the right

If we add the same amount to each side of the equation, it will remain balanced.

We can also take away the same amount from both sides.

A scale with positve x and positive 1 tiles. Ask your teacher for more information.

If we take away the same amount from each side of the equation, it will remain balanced.

If we double, triple, or even quadruple the amounts on both sides of a scale, the scale will stay balanced. In fact, we can keep it balanced by multiplying or dividing the amounts by any nonzero number - so long as it's the same on both sides.

A scale with 2 positve x tiles on the left and 6 positive 1 tiles on the right.

If we double the amount on each side, it will remain balanced.

We can also change what amount appears on each side. The same is true for equations. So, x=3 is the same equation as 3=x.

A scale with 3 positve 1 tiles on the left and 1 positive x tile on the right.

Examples

Example 3

Scale 1 is a balanced scale.

Scale 1:

A scale with 1 positive x tile on the left and 3 positive 1 tiles on the right.

Scale 2:

A scale with 2 positive x tiles on the left and a question mark on the right.

Which of the following options could go in place of the question mark to balance scale 2?

Worked Solution

Create a strategy

The balance shows that one positive x tile is the same as three positive 1 tiles.

Apply the idea

\displaystyle x	\displaystyle =	\displaystyle 3	Write the equation
\displaystyle 2x	\displaystyle =	\displaystyle 6	Multiply both sides by 2

We need 6 positive 1 tiles in place of question mark to balance scale 2, option B.

Example 4

Scale 1 is a balanced scale.

Scale 1:

A scale with 1 positive x tile on the left and 2 positive 1 tiles on the right.

Scale 2:

A scale having 1 positive x tile and 1 positive 1 tile on the left and a question mark on the right.

Which of the following options could go in place of the question mark to balance scale 2?

Worked Solution

Create a strategy

From Scale 1, we can say that 1 positive x tile is equivalent to 2 positive 1 tiles or simply x=2.

Apply the idea

We must add one positive 1 tile to the existing ones on the right side of scale 1

\displaystyle x	\displaystyle =	\displaystyle 2	Write the equation
\displaystyle x + 1	\displaystyle =	\displaystyle 2 + 1	Add 1 to both sides
\displaystyle x + 1	\displaystyle =	\displaystyle 3	Evaluate

We need 3 positive 1 tiles in place of question mark to balance scale 2, option D.

Idea summary

Equations must remain balanced. What we do to one side we must do to the other.

Algebraic tiles can be used to help balance and solve equations.

Outcomes

6.EE.A.4

Identify when two expressions are equivalent (i.e., When the two expressions name the same number regardless of which value is substituted into them).

6.02 Model balanced equations

Introduction

Ideas

Model equations with algebraic tiles

Examples

Example 1

Example 2

Balance equations

Exploration

Examples

Example 3

Example 4

Outcomes

6.EE.A.4

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