Math resources Number and quantity

Properties of equality

# Properties of equality

Here you will learn about the properties of equality, including what they are and how to use them to solve problems.

Students will first learn about the properties of equality as part of operations and algebraic thinking in 3rd grade and continue to build on this knowledge in operations and algebraic thinking in 5th grade and expressions and equations in 6th grade.

## What are properties of equality?

Properties of equality are general rules that can be applied to an equation to create equivalent equations, which can be useful when comparing or solving equations.

The associative property states that numbers can be grouped in different ways when they are added or multiplied and the answer will stay the same.

For example,

When adding 7 + 8 + 10, the addends can be grouped in different ways:

For example,

When multiplying 4 \times 6 \times 2, the numbers can be grouped in different ways:

The associative property works for both addition and multiplication, because it creates equivalent equations.

(4 \times 6) \times 2=4 \times(6 \times 2) \quad and \quad (7+8)+10=7+(8+10)

Step-by-step guide: Associative property

The commutative property states that the order of numbers can be changed when they are added or multiplied and the answer will stay the same.

For example,

When adding 11 + 15, you can change the order of the addends:

For example,

When multiplying 6 \times 5, you can change the order of the numbers:

Notice that even with a different order, the sum and the product are the same.

Step-by-step guide: Commutative property

The distributive property says that multiplying the sum of two or more numbers is equivalent to separately multiplying the addends.

For example,

When multiplying 2 \times 7, you can break 7 up into 2 + 5.

The distributive property allows us to multiply each part and then add the products together.

Any way you solve the equivalent expressions, the product is the same.

Initially, the distributive property is used to think flexibly about numbers. In later grades, students learn to simplify algebraic expressions using the distributive property.

Step-by-step guide: Distributive property

Order of operations is the sequence of steps followed to evaluate an expression.

All the mathematical operations must be completed in a specific order, following the acronym PEMDAS.

PEMDAS stands for:

Parentheses, Exponents, Multiplication and Division, Addition and Subtraction

The operations of multiplication and division are given equal priority, and addition and subtraction are also given equal priority.

Always solve these operations starting from the left side and working to the right side, unless there are parentheses.

For example,

Step-by-step guide: Order of operations

## Common Core State Standards

• Grade 3 – Operations and Algebraic Thinking (3.OA.B.5)
Apply properties of operations as strategies to multiply and divide.
Examples: If 6 \times 4 = 24 is known, then 4 \times 6 = 24 is also known. (Commutative property of multiplication.)
3 \times 5 \times 2 can be found by 3 \times 5 = 15, then 15 \times 2 = 30, or by 5 \times 2 = 10, then 3 \times 10 = 30. (Associative property of multiplication.)
Knowing that 8 \times 5 = 40 and 8 \times 2 = 16, one can find 8 \times 7 as 8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56. (Properties of equality.)

• Grade 5 – Operations and Algebraic Thinking (5.OA.1)
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

• Grade 5 – Operations and Algebraic Thinking (5.OA.2)
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

• Grade 6 – Expressions and Equations (6.EE.1)
Write and evaluate numerical expressions involving whole-number exponents.

## How to use properties of equality

There are a lot of ways to use properties of equalities. For more specific step-by-step guides, check out the individual pages linked in the “What are properties of equalities?” section above or read through the examples below.

## Properties of equality examples

### Example 1: simple associative property with addition

Use the associative property to solve 17 + 6 + 34.

1. Check to see that the operation is addition or multiplication.

The associative property can be used since all the numbers are being added.

2Change the grouping of the numbers and solve.

\begin{aligned} & 17+6+34 \\\\ & =17+(6+34) \quad \text{ *Group and add these numbers first} \\\\ & =17+40 \\\\ & =57 \end{aligned}

### Example 2: associative property – addition with friendly numbers

Use the associative property with friendly numbers to solve 77 + 16.

Check to see that the operation is addition or multiplication.

Change the grouping of the numbers and solve.

### Example 3: commutative property – multiplication with friendly numbers

Use the commutative property to create a friendly number and solve 3 \times 9 \times 2.

Check to see that the operation is addition or multiplication.

Change the order of the numbers and solve.

### Example 4: distributive property with basic facts

Show how to solve 8 \times 11 using the distributive property.

Identify an equation multiplying two numbers.

Show one of the numbers being multiplied as a sum of numbers.

Multiply each number in the sum.

Add the partial products together to find the final product.

### Example 5: PEMDAS with division and subtraction

Evaluate 40-8 \div 4.

Solve any calculations within parentheses.

Solve for any exponents.

Solve any division and multiplication calculations.

Solve any addition and subtraction calculations.

### Example 6: PEMDAS with brackets, an exponent, and division

Evaluate 12+(16 \, \div \, 4 \times 5)^2.

Solve any calculations within parentheses.

Solve for any exponents.

Solve any division and multiplication calculations.

Solve any addition and subtraction calculations.

### Teaching tips for properties of equality

• The properties are not useful in all solving situations. Be intentional about choosing practice problems that lend themselves to being solved with the associative, commutative, and distributive properties.

• Incorporate models (array or area) into instruction when working with the distributive property, especially in the beginning. This will help students build a conceptual understanding and make it easier for them to adapt their thinking to variables when the time comes.

• Instead of just giving students the definitions of each property, draw attention to examples of them as they come up in daily math activities. You may even create and update an anchor chart each time a student notices one of the properties naturally. Over time, students will start recognizing them on their own. Once this happens, you can introduce the property name and mathematical definition by using the student examples.

• Include plenty of student discourse around the properties so that students understand that using the properties creates equivalent equations, even though they are written differently. This could include students sharing their thinking and analyzing the thinking of their classmates.

• Find ways for students to practice the order of operations besides just using worksheets. For example, you can physically separate the answer choices and the word problems in the classroom. Students will be required to walk around to solve the problem and then find the answer that matches.

### Easy mistakes to make

• Thinking there is only one way to use each property to solve
Often there is more than one way to use the associative, commutative, or distributive property when solving.
For example,

\begin{aligned} & 3 \times 5 \hspace{4cm} 3 \times 5\\ & =(1+2) \times 5 \hspace{2.8cm} =3 \times(4+1) \\ & =(1 \times 5)+(2 \times 5) \hspace{0.7cm} OR \hspace{0.8cm} =(3 \times 4)+(3 \times 1)\\ & =5+10 \hspace{3.4cm} =12+3 \\ & =15 \hspace{3.95cm} =15 \end{aligned}

• Using the associative property or commutative property for subtraction or division
The associative and commutative properties only work when changing the grouping or the order of the numbers doesn’t affect the answer. This does not work for subtraction or division.
For example,
\begin{aligned} & 10-5-2 \ & =5-2 \ & =3 \end{aligned}

or

\begin{aligned} & 10-(5-2) \ & =10-3 \ & =7 \end{aligned}

or

\begin{aligned} & 5-10-2 \ & =-5-2 \ & =-7 \end{aligned}

• Not decomposing by place value
When using the distributive property, break apart a number being multiplied so that the overall value stays the same. Otherwise, the products will not be equivalent. This includes correctly using the place value of the number.
For example,

• Not following the order of operations correctly
Equations are always solved by moving from left to right and follow the sequence of PEMDAS. This also means that multiplication and division are done in order from left to right and addition and subtraction are done in order from left to right. Not following this order can lead to the wrong answer.
For example,

### Practice properties of equality questions

1. Which of the following equations shows how to solve 4 \times 4 \times 10 using the associative property?

4 \times 4 \times 10=16 \times 10

(4 \times 4) \times 10=(4 \times 4) \times 10

4 \times(4 \times 10)=4 \times 40

4 \times 4 \times 10=10 \times 4 \times 4

All the numbers are being multiplied, so the associative property can be used.

Change the grouping of the numbers and solve.

\begin{aligned} & 4 \times 4 \times 10 \\ & =4 \times(4 \times 10) \quad \text{ *Group and multiply these numbers first}\\ & =4 \times 40 \\ & =160 \end{aligned}

2. Which of the following equations shows how to solve 53 + 12 + 17 using the commutative property?

\begin{aligned} & 53+12+17 \\ & =53+(12+17) \\ & =53+29 \end{aligned}

\begin{aligned} & 53+12+17 \\ & =53+17+12 \\ & =70+12 \end{aligned}

\begin{aligned} & 53+12+17 \\ & =65+17 \end{aligned}

\begin{aligned} & 53+12+17 \\ & =(53+12)+17 \\ & =65+17 \end{aligned}

The commutative property says that changing the order in the equation does not change the sum.

\begin{aligned} & 53+12+17 \\ & =53+17+12 \;\; \text{ Change the order of 12 and 17} \\ & =70+12 \quad \quad \text{ *Adding 53 and 17 first gives us 70 – a friendly number} \end{aligned}

3. Which of the following equations shows how to solve 6 \times 3 \times 5 using the commutative property to create a friendly number?

\begin{aligned} & 6 \times 3 \times 5 \\ &= 6 \times(3 \times 5) \\ &= 6 \times 15 \end{aligned}

\begin{aligned} & 6 \times 3 \times 5 \\ & =18 \times 5 \end{aligned}

\begin{aligned} & 6 \times 3 \times 5 \\ & = (6 \times 3) \times 5 \\ & = 18 \times 5 \end{aligned}

\begin{aligned} & 6 \times 3 \times 5 \\ & = 6 \times 5 \times 3 \\ & = 30 \times 3 \end{aligned}

The commutative property says that changing the order in the equation does not change the product.

Friendly numbers are numbers that are easy to multiply mentally – like multiples of 10.

\begin{aligned} & 6 \times 3 \times 5\\ & =6 \times 5 \times 3 \quad \text{ *Change the order of 3 and 5} \\ & =30 \times 3 \quad \quad \text{ *Multiplying 6 and 5 first gives us 30 – a friendly number} \end{aligned}

4. Which of the following equations shows 6 \times 7 using the distributive property?

(5+1) \times 7=(5 \times 7)+(5 \times 1)

6 \times(2+2+3)=(6 \times 2)+(6 \times 2)

6 \times(1+3+3)=(6 \times 1)+(6 \times 3)+(6 \times 3)

(6 \times 1) \times 7=6 \times(1 \times 7)

The numbers are being multiplied, so the distributive property can be used.

5. Calculate 7+3 \times 12.

120

43

48

252

To solve, follow the order of operations by using PEMDAS.

6. Calculate 458+(5 \div 2 \times 4)^3.

1,458

458.24

488

2,778

To solve, follow the order of operations by using PEMDAS.

## Properties of equality FAQs

Do the properties of equality work with all real numbers?

Yes, these properties can be applied to any equation, expression or inequality that includes real numbers (which includes fractions, decimals, negative and irrational numbers).

What are inverse operations?

As students are learning the operations, they learn that subtraction is the inverse of addition and division is the inverse of multiplication.

Are there other properties of equality?

As students work with the operations, they will begin to notice other patterns that are later formalized. These include the addition property of equality, subtraction property of equality, multiplication property of equality, and the division property of equality, where students learn that the same operation can be done to both sides of an equation without jeopardizing equality.

This is because the equal sign can be thought of as a balance, so as long as the same operation is done to both sides of the equation, equivalence is maintained.

How are properties of equality useful in other math topics?

As students begin to work with equations with unknown variables, they will learn that these are also algebraic properties that can be used when working with algebraic equations.

For example, students can use the distributive property to simplify linear equations and then use the addition property of equality and the division property of equality to solve for a specific value.

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