High Impact Tutoring Built By Math Experts
Personalized standards-aligned one-on-one math tutoring for schools and districts
In order to access this I need to be confident with:
Add and subtract within 100
Integers Order of operations
Here you will learn about equivalent expressions, including what they are and how to recognize them.
Students will first learn about equivalent expressions as part of expressions and equations in 6th grade.
Equivalent expressions are expressions in algebra that are equal in value, even though they look different. Equivalent expressions will have the same value when we use the same value(s) for the variable(s).
An algebraic expression is two or more numbers and letters that are combined with mathematical operations such as +, -, or \div .
To do this, use properties of operations to create and identify equivalent algebraic expressions.
When stating that two expressions are equivalent we use the equivalence symbol \equiv to show that they are identical.
Addition is commutative which means changing the order of the terms does not change the sum.
For example, 3x+7y \equiv 7y+3x.
3x+7y is the same as 7y+3x because the coefficients of x are the same, both positive 3x and the coefficients of y are the same, both positive 7x.
This means they are equivalent expressions.
For example, 3a-8b = 3a+(-8b) \equiv -8b+3a.
The negative sign belongs to the term after it. Subtracting a term is the same as adding a negative term.
3a-8b is the same as 3a+(-8b), because subtracting 8b is the same as adding -8b. And by the commutative property, 3a+(-8b) is the same as -8b+3a.
This means they are equivalent expressions.
Multiplication has a distributive property which allows us to break up a number into simpler parts and multiply each part separately. This can be used with variables as well.
For example, 3(x+7) \equiv 3x+21.
3(x+7) is the same as 3x+21 because 3 \times x=3x , and when you multiply the constants 3 and 7 , you get 21. The right side is the same as the left side.
This means they are equivalent expressions.
How does this relate to 6th and 7th grade math?
Use this worksheet to check your 6th grade studentsβ understanding of equivalent expressions. 15 questions with answers to identify areas of strength and support!
DOWNLOAD FREEUse this worksheet to check your 6th grade studentsβ understanding of equivalent expressions. 15 questions with answers to identify areas of strength and support!
DOWNLOAD FREE
In order to identify equivalent expressions:
Are the expressions 4a+7b and 7b-4a equivalent?
Take the first expression and change the order of terms being added: 4a+7b becomes 7b+4a.
Now compare 7b+4a and 7b-4a.
2Compare the two expressions.
The two expressions both have β4aβ and β7bβ, but in the second expression the β4aβ is negative. It is β-4aβ.
3Make your decision.
The two expressions are not the same. They are NOT equivalent expressions.
Are the expressions 4(x-5) and 4x-20 equivalent?
Use the properties of operations to create equivalent expressions.
Take the first expression and multiply the terms within the parentheses by the term on the outside:
\begin{aligned} & 4(x-5) \\\\ & =4 \times x-4 \times 5 \\\\ & =4 x-20 \end{aligned}
Compare the two expressions.
The two expressions are the same, 4(x-5) \equiv 4x-20.
Make your decision.
The two expressions are the same. They are equivalent expressions.
Are the expressions 2g + 3t+8g +6 and 10g + 9t equivalent?
Use the properties of operations to create equivalent expressions.
Take the first expression and combine the like terms:
\begin{aligned} & 2 g+3 t+8 g+6 \\\\ & =2 g+8 g+3 t+6 \\\\ & =10 g+3 t+6 \end{aligned}
The terms β10gβ and β+3tβ and β+6β are not like terms. This expression cannot be simplified further.
Now compare 10g + 3t + 6 and 10g + 9t.
Compare the two expressions.
The two expressions both have β10gβ, but β3t + 6β is not the same as 9t.
Make your decision.
The two expressions are not the same. They are NOT equivalent expressions.
Are the expressions 3(4+s) + 2 and 3s + 14 equivalent?
Use the properties of operations to create equivalent expressions.
Take the first expression and multiply the terms within the parentheses by the term on the outside:
3(4+s)+2=12+3 s+2
Then combine the like terms:
\begin{aligned} & 12+3 s+2 \\\\ & =12+2+3 s \\\\ & =14+3 s \end{aligned}
Compare the two expressions.
Now compare 14 + 3s and 3s + 14. The two expressions have β14β and β3sβ and are addition.
Make your decision.
The two expressions are the same. They are equivalent expressions.
Are the expressions \frac{1}{2} p+5 p-\frac{3}{4} and 4 \frac{3}{4} p equivalent?
Use the properties of operations to create equivalent expressions.
Take the first expression and combine the like terms:
\begin{aligned} & \frac{1}{2} p+5 p-\frac{3}{4} \\ & =5 \frac{1}{2} p-\frac{3}{4} \end{aligned}The terms '5 \frac{1}{2} p' and '-\frac{3}{4}' are not like terms. This expression cannot be simplified further.
Compare the two expressions.
Now compare 5 \frac{1}{2} p-\frac{3}{4} and 4 \frac{3}{4} p .
Both expressions have the variable p , but not the same amount. The first expression also has -\frac{3}{4} .
Make your decision.
The two expressions are not the same. They are NOT equivalent expressions.
Are the expressions y(y-4) and -4 y+y^2 equivalent?
Β Use the properties of operations to create equivalent expressions.
Take the first expression and multiply the terms within the parentheses by the term on the outside:
\begin{aligned} & y(y-4) \\ & =y \times y-4 \times y \\ & =y^2-4y \end{aligned}Β Compare the two expressions.
Now compare y^2-4 y and -4 y+y^2 . The two expressions have a positive βy^2β and β-4 yβ .
Β Make your decision.
The two expressions are the same. They are equivalent expressions.
See also: Shape patterns
1. Which expression is equivalent to 3a-4b?
The commutative property says that the order in which the two terms are added can be reversed and the expressions are still equivalent.
3a-4b=3a+(-4b) \equiv -4b+3a
2. Which expression is equivalent to 3(a+4)?
The commutative property says that the order in which the two terms are added can be reversed and the expressions are still equivalent.
\begin{aligned} & 3(a+4) \\ & =3 \times a+3 \times 4 \\ & =3 a+12 \end{aligned}
3. Which expression is equivalent to 5x+6y?
The commutative property says that the order in which the two terms are added can be reversed and the expressions are still equivalent.
5x+6y \equiv 6y+5x
4. Which expression is equivalent to -4k + 11h+9k +4h?
Combine the like terms:
\begin{aligned} & -4 k+11 h+9 k+4 h \\\\ & =-4 k+9 k+11 h+4 h \\\\ & =5 k+15 h \end{aligned}
5. Which is the equivalent expression to 8(z-4) + 4z?
To create an equivalent expression, multiply the terms within the parentheses by the term on the outside:
\begin{aligned} & 8(z-4)+4 z \\\\ & =8 \times z-8 \times 4+4 z \\\\ & =8 z-32+4 z \end{aligned}
Combine the like terms:
\begin{aligned} & 8 z-32+4 z \\\\ & =8 z+4 z-32 \\\\ & =12 z-32 \end{aligned}
As students continue to learn about equivalent expressions in middle school, they will be introduced to algebraic expressions that have fractions, decimals, and exponents, including quadratic equations. However, it is important that students do not work on more complicated polynomials until they thoroughly understand how to work with simpler expressions.
Yes, the same strategies for creating equivalent expressions and comparing them can be applied to inequalities.
Both involve two more equivalent entities, but expressions are mathematical statements that do no include the equal sign. An equation on the other hand, includes an equal sign.
An equivalent expression is an expression in algebra that has the same value as another expression but does not look the same. For example, 3x+7y \equiv 7y+3x.
Expressions are equivalent if the same values are substituted in for the variable and both arrive at the same answer.
At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.
Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.
Find out how we can help your students achieve success with our math tutoring programs.
Prepare for math tests in your state with these 3rd Grade to 8th Grade practice assessments for Common Core and state equivalents.
Get your 6 multiple choice practice tests with detailed answers to support test prep, created by US math teachers for US math teachers!