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3. Numeric & Algebraic Expressions
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Grade 7

3.05 Equivalent algebraic expressions

Lesson
Practice

Equivalent algebraic expressions

Exploration

The same expression is on both sides of the scale.

You can click and drag algebraic tiles from the bottom to be on the right side of scale. Click the reset button in the top right corner to go back to 2x+3+x+1.

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Add or remove tiles to change the expression on the right side of the scale.

  1. What kinds of things can you do to throw the scale off balance?

  2. What kinds of things can you do to keep the scale balanced?

  3. What are two different expressions that are equivalent to 2x+3+x+1?

Algebra tiles: five positive variable tiles and three positive unit tiles plus two positive variable tiles and one positive unit tile.

\, \\Let's start by looking at the expression {5x+3+2x+1}. It is represented by the algebra tiles shown.

Equivalent expressions are expressions that have the same value but do not look the same. We can rearrange the algebra tiles to create an equivalent expression that uses the same amount of x tiles and 1 tiles.

Four positive variable tiles and two positive unit tiles plus three positive variable tile and two positive unit tiles

These tiles represent 4x+2+3x+2.

Because the expressions 4x+2+3x+2 and 5x+3+2x+1 use the same number of tiles for both +x and +1, the two expressions are equivalent.

If we add a variable or constant and its opposite we've added 0 or nothing, we call these zero pairs. Adding or subtracting a zero pair from an expression makes an equivalent expression.

Two positive variable tiles and two positive unit tiles is equal to three positive variable tiles, one negative variable tile, three positive unit tiles and one negative unit tile. One pair of positive and negative variable tiles and positive and negative unit tiles are encirled and labeled 'Zero pairs'.

\, \\Algebra tiles on the left side represent the expression 2x+2. The tiles on the right side represent the expression 2x+x-x+2+1-1. The zero pairs x and -x and 1 and -1 have added 0 to the original expression so the expressions are equivalent.

Examples

Example 1

These tiles represent the expression 4x+4+4x+5.

Algebra tiles: Four positive variable tiles, four positive unit tiles, four positive variable tiles and five positive unit tiles in order.

Create a model of an equivalent expression.

Worked Solution
Create a strategy

The original expression has eight x tiles and nine 1 tiles. We will create a model that has the same amount of each type of tile.

Apply the idea

This model represents an equivalent expression.

Algebra tiles: Six positive variable tiles, two positive unit tiles, two positive variable tiles and seven positive unit tiles in order.

This model represents the expression 6x+2+2x+7.

Idea summary

Equivalent expressions are expressions that have the same value but do not look the same.

We can create equivalent expressions with algebra tiles by:

  • Rearranging the tiles to use the same number of x tiles and 1 tiles

  • Adding tiles and their opposites to create zero pairs

Simplified algebraic expressions

A simplified expression is a special type of equivalent expression. When we write simplified expressions we are rewriting the same algebraic expression with no like terms and in the most compact way we can.

Algebra tiles: Four negative variable tiles, three positive variable tiles, four negative unit tiles, three negative variable tiles and five positive unit tiles.

Let's look at the expression {-4x+3x-4-3x+5} modeled with algebra tiles.

Notice that we have a lot of the same types of tiles. To simplify the expression, we can combine tiles of the same type. We call these like terms.

Like terms

Terms (or parts) of an expression that have the same variables and exponents

Algebra tiles: Four negative variable tiles, three positive variable tiles, four negative unit tiles, three negative variable tiles and five positive unit tiles. Three pairs of positive and negative variable tiles are encircled.

\, \\ \, \\We can start by looking for zero pairs of x tiles.

Algebra tiles: One negative variable tile, four negative unit tiles, three negative variable tiles and five positive unit tiles.All the negative variable tiles are encircled.

Once we eliminate the zero pairs, we are left with the expression -x-4-3x+5.

All of the -x tiles have the same variable, so we can combine the like terms by counting up the amount of -x tiles. We can simplify the expression to be -4x-4+5.

There are still some like terms, so let's keep going.

Algebra tiles: four negative variable tiles, four negative unit tiles and five positive unit tiles. The four negative and positive unit tiles are encircled.

Next, we can look at the constant terms. These are the terms with no variables. We can start by eliminating zero pairs of constant terms.

Algebra tiles: Four negative variable tiles and one positive unit tile.

Once there are no like terms remaining, we can write the simplified expression by counting up the different types of tiles. The expression {-4x+3x-4-3x+5} can be simplified to -4x+1. These are equivalent expressions.

We can also simplify this expression algebraically.

Steps showing how to simplify -4x+3x-4-3x+5

An expression can tell a story and different forms of the expression can tell us different things about the story. An expression that is not simplified tells the story of what happened whereas the simplified expression tells the outcome of the story.

Consider the following expressions:

  • Unsimplified: 4x+8+2x-4

  • Simplified: 6x+4

Let's say x represents the number of cupcakes in a pack and the constants represents the number of party hats.

You are planning to provide cupcakes and party hats for your friend's birthday party. You have 4 packs of cupcakes, each with the same number of cupcakes inside, and 8 party hats. Some more friends RSVP to your party at the last minute so you quickly go to the store and buy 2 more packs of cupcakes. On the way to the party, you lose 4 party hats. This is represented by the unsimplified version of the expression.

When you arrive at the party you have 6 packs of cupcakes and 4 party hats. This is shown by the simplified version of the expression. Each form tells part of the story.

Examples

Example 2

Use a model to determine which of these expressions is equivalent to 7s+2-4s.

A
3s-2
B
11s+2
C
3s+2
D
11s-2
Worked Solution
Create a strategy

Count the number of like variable tiles for each term. This will give us the coefficients for the variables. Count the number of constant tiles. This gives us the values of the constant terms.

Then we can identify and combine any like terms.

Apply the idea

This model represents the expression 7s+2-4s.

Algebra tiles: Seven positive variable tiles, two positive unit tiles and 4 negative unit variable tiles. The variable tiles are labeled '+s'.

Once removing all of the zero paris, the expression can be represented with this model.

Algebra tiles: Three positive variable tiles (+s) and two positive unit tiles.

Now, we will create models to determine which of the options are equivalent to the original expression.

Option A:

Three positive variable tiles (+s) and two negative unit tiles.

Option B:

Algebra tiles: eleven positive variable (+s) tiles and two positive unit tiles.

Option C:

Algebra tiles: three positive (+s) variable tiles and two positive unit tiles.

Option D:

Algebra tiles: Eleven positive variable (+s) tiles and two negative unit tiles.

Option C is equivalent to the original expression.

Reflect and check

We could have also simplified the original expression algebraically and compared it to the options given.

\displaystyle 7s+2-4s\displaystyle =\displaystyle 7s-4s+2Rearrange like terms (commutative property)
\displaystyle =\displaystyle 3s+2Combine the variable terms 7s-4s

Option C is not only equivalent to the original expression, but it is also fully simplified.

Example 3

Write the algebraic expression represented by the algebra tiles, then simplify.

a
Algebra tiles: two positive variable tiles and two positive unit tiles plus three positive variable tiles and four positive unit tiles.
Worked Solution
Create a strategy

Count the number of like variable tiles for each term. This will give us the coefficients for the variables. Count the number of constant tiles. This gives us the values of the constant terms.

Then we can identify and combine any like terms.

Apply the idea

There are two positive x tiles and two positive 1 tiles for the first part of the expression, and three positive x tiles and four positive 1 tiles for the second part of the expression.

We write the algebraic expression as:2x+2+3x+4

We can rewrite it by grouping the like terms 2x and 3x and the like terms 2 and 4:2x+3x+2+4

Simplify the expression by combining the like terms:

5x+6

Reflect and check

We can also use the algebra tiles to find the simplified algebraic expression by combining similar tiles.

Algebra tiles: Five positive variable tiles and six positive unit tiles

The algebra tiles represent 5x+6.

b
Algebra tiles: three negative variable tiles and six negative unit tiles plus four negative variable tiles and two negative unit tiles.
Worked Solution
Create a strategy

Count the number of like variable tiles for each term. This will give us the coefficients for the variables. Count the number of constant tiles. This gives us the values of the constant terms.

Then we can identify and combine any like terms.

Apply the idea

There are three negative x tiles and six negative 1 tiles for the first binomial, and four negative x tiles and two negative 1 tiles for the second binomial.

We can write the algebraic expression as:-3x-6+\left(-4x\right)-2

We can rewrite the expression by grouping like terms:-3x+\left(-4x\right)-6-2

Simplify the expression by combining like terms.

-7x-8

c
Four positive variable tiles, two negative variable tiles, five negative unit tiles and two positive unit tiles plus two negative variable tiles, two positive unit tiles, three positive unit tiles and one negative unit tile.
Worked Solution
Create a strategy

Count the number of like variable tiles for each term. This will give us the coefficients for the variables. Count the number of constant tiles. This gives us the values of the constant terms.

Remember to count positive and negative tiles separately.

Then we can identify and combine any like terms.

Apply the idea

Counting the positive and negative x and 1 tiles.

We can write the algebraic expression as:4x-2x-5+2+\left(-2x\right)+2x+3-1

We can rewrite the algebraic expression by grouping like terms:4x-2x+\left(-2x\right)+2x-5+2+3-1

Notice the zero pair of 2x and -2x:4x-2x+\cancel{\left(-2x\right)}+\cancel{2x}-5+2+3-1

4x-2x-5+2+3-1

Simplify the expression by combining like terms:

2x-1

Example 4

Simplify the expression. Justify each step.

a

8 x + 5- 3x+ x - 7

Worked Solution
Create a strategy

Rewrite to group the like terms and then combine the coefficients and constants.

Apply the idea
\displaystyle 8 x + 5- 3x+ x - 7\displaystyle =\displaystyle 8 x - 3x+ x +5- 7Rearrange like terms (commutative property)
\displaystyle =\displaystyle 6x+5- 7Combine the x terms 8x-3x
\displaystyle =\displaystyle 6x-2 Combine the constant terms 5-7
b

12 y + \left(- 9 + 5+11y\right)-17

Worked Solution
Create a strategy

Rewrite to group the like terms and then combine the coefficients and constants, starting with the terms inside the parentheses.

Apply the idea
\displaystyle 12 y + \left(- 9 + 5+11y\right)-17\displaystyle =\displaystyle 12 y - 4 +11y-17Combine the constants inside the parentheses -9+5
\displaystyle =\displaystyle 12 y +11y- 4-17Rearrange like terms (commutative property)
\displaystyle =\displaystyle 23 y - 4-17Combine the y terms 12y +11y
\displaystyle =\displaystyle 23 y - 21 Combine the constant terms -4-17

Example 5

Given the following unsimplified and simplified expressions, write a real-world context for each expression and describe what each equivalent form represents.

  • Unsimplified: 6x+4-x-2

  • Simplified: 5x+2

Worked Solution
Create a strategy

We can use x represent one object and the constants to represent another object. Then create a real-world scenario involving these objects.

Apply the idea

Let's say x represents a box fruits, and the constants represent fruit juices.

You are planning to provide fruits and vegetables for an event at the community center. You have 6 boxes of fruits (each with the same number of fruits inside) and four fruit juices. You realize when you get to the community center, you left 1 box of fruits and 2 fruit juices at your parents' house. This is represented by the unsimplified version of the expression.

You are now left with 5 boxes of fruits and 2 fruit juices for the event. This is shown by the simplified version of the expression.

Idea summary

Two algebraic terms are called like terms if they have exactly the same combination of variables.

To combine like terms means to simplify an expression by combining all like terms together through addition and/or subtraction.

Outcomes

7.PFA.2b

Represent equivalent algebraic expressions in one variable using concrete manipulatives and pictorial representations (e.g., colored chips, algebra tiles).

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