6th Grade · 6.EE.A.3

The distributive property in 6th grade

The distributive property says that multiplying a sum is the same as multiplying each part and then adding: $5(x + 2) = 5x + 10$ . In 6th grade, your child uses the distributive property to rewrite expressions in a new form with the same value. Your child also learns combining like terms and factoring, which runs the property backward.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.EE.A.3 Apply the properties of operations to generate equivalent expressions.

What it means, in plain words

An expression is a math phrase made of numbers, letters and operation signs, with no equals sign. Two expressions are equivalent when they give the same value for every number the letter stands for. In 6th grade, your child makes equivalent expressions with a few rules that always hold, called the properties of operations.

The distributive property in 6th grade works in two directions. Expanding multiplies the number outside the parentheses by every term inside: $4(y + 6) = 4y + 24$ . A term is one part of an expression joined to the rest by a plus or minus sign. Factoring goes the other way. In $6y + 9$ , both terms can be divided by 3, so $6y + 9 = 3(2y + 3)$ . Children usually take out the greatest common factor, the largest number that divides every term.

Combining like terms means adding terms that have the same letter part. Like terms are terms with the same variable, such as $6a$ and $3a$ . Numbers without a letter are like terms too. $6a + 3a = 9a$ , just as 6 apples and 3 apples make 9 apples. $6a + 3$ stays as it is, because 3 is not a number of $a$ 's.

Your child should be able to:

Expand an expression such as $4(y + 6)$ by multiplying every term inside the parentheses
Factor an expression such as $6y + 9$ using the greatest common factor
Combine like terms, and explain why $6a + 3$ cannot become $9a$
Expand and then combine like terms to simplify a longer expression

Example

Simplify $2(3m + 5) + 4m$ .

Expand the parentheses: $2 \times 3m = 6m$ and $2 \times 5 = 10$ . The expression becomes $6m + 10 + 4m$ .
Combine the like terms $6m$ and $4m$ : $6m + 4m = 10m$ .
The 10 has no $m$ , so it stays as it is.
The simplified expression is $10m + 10$ .

Example

Factor $12x + 18$ using the greatest common factor.

The factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of 18 are 1, 2, 3, 6, 9 and 18.
The greatest common factor is 6.
Divide each term by 6: $12x \div 6 = 2x$ and $18 \div 6 = 3$ .
Check by expanding: $6 \times 2x + 6 \times 3 = 12x + 18$ . The factored form is $6(2x + 3)$ .

Where children usually get stuck

Your child expands $5(x + 3)$ as $5x + 3$ .
The 5 sits right next to the $x$ , so it looks as if it belongs only to the $x$ .
What helps. Put $x = 1$ into both. $5(1 + 3) = 20$ , but $5 \times 1 + 3 = 8$ , so the two forms do not match. Then draw an arrow from the 5 to each term inside the parentheses.
Your child simplifies $8b + 5$ to $13b$ .
Combining like terms gets learned as "add all the numbers". The 5 has no letter, so it does not count $b$ 's.
What helps. Read each term as a count of something: 8 bananas and 5 dollars. You cannot add them into 13 bananas. Ask what each term counts before joining any two.
Your child factors $14x + 21$ as $7(2x + 21)$ .
Only the first term was divided by the common factor. The second term was copied over unchanged.
What helps. Have your child expand the answer. $7(2x + 21) = 14x + 147$ , which is not where the problem started. Expanding is a quick check for any factoring.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Look at the expression below.
$\displaystyle 3x + 4x$
Type the simplified expression.
Loading...
Show the answer and how to solve it
Answer. 7x
The simplified expression is 7x because 3x and 4x make 7x.
1. 1Both terms are x-terms, so they are already grouped together and ready to combine.
2. 2Add the coefficients: $3 + 4 = 7$ , so the result is 7x.
Common mistake. A common mistake is writing 34x or 7 instead of 7x. Remember to add only the coefficients and keep the variable x.
The idea underneath. Like terms have the same variable part, so you add or subtract their coefficients while keeping the variable.
2On level
Look at Sofia's expression and rewrite below.
$\displaystyle 3(x - 4)$
$\displaystyle = 3x + 12$
Select the statement that best describes the mistake.
A
The outside factor was multiplied by only one term.
B
The terms inside the parentheses were multiplied together.
C
The outside number was added instead of distributed.
D
The sign on the second term was changed incorrectly.
Show the answer and how to solve it
The correct statement is that the sign on the second term was changed incorrectly. Distributing 3 over x - 4 gives 3x - 12, not 3x + 12.
1. 1Look at the two lines. The first line is the original expression and the second line is Sofia's rewrite.
2. 2Distribute 3 to both terms inside the parentheses to find the correct expansion.
3. 3The first product, 3x, is correct. The second product must stay negative because 3 times negative 4 is negative 12, not positive 12.
4. 4Sofia multiplied both terms but wrote +12 instead of -12. The mistake is that the sign on the second term was changed incorrectly.
Common mistake. A common mistake is multiplying 3 and 4 to get 12 but forgetting that the 4 is being subtracted. The subtraction stays with the second product, so the result should be -12.
The idea underneath. In a(b - c), the outside factor multiplies both terms, and the subtraction stays attached to the second product.
3On level
A student named Kai's factoring is shown below.
$\displaystyle 15x - 10$
$\displaystyle = 5(3x + 2)$
Select the statement that best evaluates Kai's work.
A
No mistake — this is correctly factored by the greatest common factor.
B
A term was not divided by the common factor.
C
The sign inside the parentheses is wrong.
D
The expression is equivalent but not fully factored by the greatest common factor.
Show the answer and how to solve it
The correct statement is that the sign inside the parentheses is wrong. Distributing 5 over 5(3x + 2) gives 15x + 10, so the inside sign should be minus.
1. 1Look at Kai's factoring below.
2. 2To check the factored form, distribute the outside factor back through the parentheses.
3. 3The original expression is 15x - 10, but distributing Kai's rewrite gives 15x + 10. The first term matches, but the second term changed from negative to positive.
4. 4So Kai's work is not correct. The sign inside the parentheses should be minus, not plus.
Common mistake. A common mistake is checking only the numbers and ignoring the sign. Since 10 divided by 5 is 2, some students accept 5(3x + 2), but the negative 10 means the inside term must be −2.
The idea underneath. A factored form is correct only if distributing the outside factor recreates the original expression exactly, including every sign.
4Stretch
Look at the expression below.
$\displaystyle 4(x - 2) + 3 + 2(x - 1)$
Select the expression that is equivalent and fully simplified.
A
$\displaystyle 6x - 7$
B
$\displaystyle 6x - 1$
C
$\displaystyle 6x + {}$
$\displaystyle 9$
D
$\displaystyle 6x - 3$
Show the answer and how to solve it
The answer is 6x - 7 because 4(x - 2) becomes 4x - 8 and 2(x - 1) becomes 2x - 2, and the like terms combine to 6x - 7.
1. 1Look at the original expression. There are two grouped parts, 4(x - 2) and 2(x - 1). Each one must be distributed before like terms can be combined.
2. 2Distribute each factor across its parentheses. For 4(x - 2): 4 times x gives 4x and 4 times -2 gives -8. For 2(x - 1): 2 times x gives 2x and 2 times -1 gives -2.
3. 3Combine the x terms and the constants separately.
4. 4Put those results together to get the simplified expression.
Common mistake. A common mistake is writing 4(x - 2) as 4x - 2. The 4 must multiply both terms inside the parentheses, so the correct expansion is 4x - 8.
The idea underneath. Like terms do not need to be next to each other; after distributing, you can regroup them and then combine them.