Generating Equivalent Expressions

Examples, videos, and solutions to help Grade 7 students learn how to generate equivalent expressions by using additive inverse relationships and multiplicative inverse relationships.

New York State Common Core Math Grade 7, Module 3, Lesson 2

Worksheets for Grade 7

Lesson 2 Student Outcomes

• Students generate equivalent expressions using the fact that addition and multiplication can be done in any order (commutative property) and any grouping (associative property).
• Students recognize how any order, any grouping can be applied in a subtraction problem by using additive inverse relationships (adding the opposite) to form a sum and likewise with division problems by using the multiplicative inverse relationships (multiplying by the reciprocal) to form a product.
• Students recognize that “any order” does not apply to expressions mixing addition and multiplication, leading to the need to follow the order of operations.

Lesson 2 Summary

• Rewrite subtraction as adding the opposite before using any order, any grouping.
• Rewrite division as multiplying by the reciprocal before using any order, any grouping.
• The opposite of a sum is the sum of its opposites.
• Division is equivalent to multiplying by the reciprocal.

Lesson 2 Vocabulary

An Expression in Expanded Form: An expression that is written as sums (and/or differences) of products whose factors are numbers, variables, or variables raised to whole number powers is said to be in expanded form. A single number, variable, or a single product of numbers and/or variables is also considered to be in expanded form.

Term: Each summand of an expression in expanded form is called a term.
Coefficient of the Term: The number found by multiplying just the numbers in a term together.
An Expression in Standard Form: An expression in expanded form with all its like terms collected is said to be in standard form.

NYS Math Module 2 Grade 3 Lesson 1 Opening Exercise

Additive inverses have a sum of zero. Multiplicative inverses have a product of . Fill in the center column of the table with the opposite of the given number or expression, then show the proof that they are opposites. The first row is completed for you.

Example 1: Subtracting Expressions

Example 2: Combining Expressions Vertically
a. Find the sum by aligning the expressions vertically.
b. Find the difference by aligning the expressions vertically.

Example 3: Using Expressions to Solve Problems
A stick is x meters long. A string is 4 times as long as the stick.
a. Express the length of the string in terms of x.
b. If the total length of the string and the stick is 15 meters long, how long is the string?

Example 4: Expressions from Word Problems
It costs Margo a processing fee of $3 to rent a storage unit, plus $17 per month to keep her belongings in the unit. Her friend Carissa wants to store a box of her belongings in Margo’s storage unit and tells her that she will pay her $1 toward the processing fee and $3 for every month that she keeps the box in storage. Write an expression in standard form that represents how much Margo will have to pay for the storage unit if Carissa contributes. Then, determine how much Margo will pay if she uses the storage unit for 6 months.

Example 5: Extending Use of the Inverse to Division
Find the multiplicative inverses of the terms in the first column. Show that the given number and its multiplicative inverse have a product of . Then, use the inverse to write each corresponding expression in standard form. The first row is completed for you.

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NYS Math Module 2 Grade 3 Lesson 1 Examples

Examples
Example 1: Any Order, Any Grouping Property with Addition
a. Rewrite 5x + 3x and 5x - 3x by combining like terms.
Write the original expressions and expand each term using addition. What are the new expressions equivalent to?
b. Find the sum of 2x + 1 and 5x.
c. Find the sum of -3a + 2 and 5a - 3.

Example 2: Any Order, Any Grouping with Multiplication
Find the product of 2x and 3

Example 3: Any Order, Any Grouping in Expressions with Addition and Multiplication
Use any order, any grouping to find equivalent expressions.
a. 3(2x)
b. 4y(5)
c. 4 • 2 • z
d. 3(2x) + 4y(5)
e. 3(2x) + 4y(5) + 4 • 2 • z

NYS Math Module 2 Grade 3 Lesson 2 Examples

Examples
Example 1
a. Subtract: (40 + 9) - (30 + 2)
b. Subtract: (3x + 5y - 4) - (4x + 11)

Example 4: Expressions from Word Problems
It costs Margo a processing fee of $3 to rent a storage unit, plus $17 per month to keep her belongings in the unit. Her friend Carissa wants to store a box of her belongings in Margo’s storage unit and tells her that she will pay her $1 toward the processing fee and $3 for every month that she keeps the box in storage. Write an expression in standard form that represents how much Margo will have to pay for the storage unit if Carissa contributes. Then, determine how much Margo will pay if she uses the storage unit for months.

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