Generating Equivalent Expressions

Examples, videos, and solutions to help Grade 7 students learn how to generate equivalent expressions by using commutative property and associative property.

New York State Common Core Math Grade 7, Module 3, Lesson 1

Download worksheets for Grade 7, Module 3, Lesson 1

Lesson 1 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 1 Summary

• Terms that contain exactly the same variable symbol can be combined by addition or subtraction because the variable represents the same number. Any order, any grouping can be used where terms are added (or subtracted) in order to group together like terms. Changing the orders of the terms in a sum does not affect the value of the expression for given values of the variable(s).

Lesson 1 Vocabulary

Variable (description): A variable is a symbol (such as a letter) that represents a number, i.e., it is a placeholder for a number.

Numerical Expression (description): A numerical expression is a number, or it is any combination of sums, differences, products, or divisions of numbers that evaluates to a number.

Value of a Numerical Expression: The value of a numerical expression is the number found by evaluating the expression. Expression (description): An expression is a numerical expression, or it is the result of replacing some (or all) of the numbers in a numerical expression with variables.

Equivalent Expressions: Two expressions are equivalent if both expressions evaluate to the same number for every substitution of numbers into all the letters in both expressions.

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. Examples of expressions in expanded form include: 324, 3x, 5x + 3 - 40, x + 2x + 3x, etc.

Term (description): Each summand of an expression in expanded form is called a term. For example, the expression 2x + 3x + 5 consists of 3 terms: 2x, 3x, and 5.

Coefficient of the Term (description): The number found by multiplying just the numbers in a term together. For example, given the product 2 • x • 4, its equivalent term is 8x. The number 8 is called the coefficient of the term 8x.

An Expression in Standard Form: An expression in expanded form with all its like terms collected is said to be in standard form. For example, 2x + 3x + 5 is an expression written in expanded form; however, to be written in standard form, the like terms 2x and 3x must be combined. The equivalent expression 5x + 5 is written in standard form.

NYS Math Module 2 Grade 3 Lesson 1 Opening Exercise

Each envelope contains a number of triangles and a number of quadrilaterals. For this exercise, let t represent the number of triangles, and let q represent the number of quadrilaterals.
a. Write an expression, using t and q, that represents the total number of sides in your envelope. Explain what the terms in your expression represent.
b. You and your partner have the same number of triangles and quadrilaterals in your envelopes. Write an expression that represents the total number of sides that you and your partner have. If possible, write more than one expression to represent this total.
c. Each envelope in the class contains the same number of triangles and quadrilaterals. Write an expression that represents the total number of sides in the room.
d. Use the given values of t and q, and your expression from part (a), to determine the number of sides that should be found in your envelope.
e. Use the same values for t and q, and your expression from part (b), to determine the number of sides that should be contained in your envelope and your partner’s envelope combined.
f. Use the same values for t and q, and your expression from part (c), to determine the number of sides that should be contained in all of the envelopes combined.
g. What do you notice about the various expressions in parts (e) and (f)?

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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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