7th Grade · 7.EE.A.1

Simplifying expressions in 7th grade

Simplifying expressions means rewriting an expression in a shorter form that has the same value for every number. In 7th grade, your child adds, subtracts, expands and factors expressions, now with negative numbers, fractions and decimals. The goal is to see that two expressions can look different and still be equal.

What the standard says

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

7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

What it means, in plain words

An expression is a math phrase made of numbers, letters and operations, with no equals sign, such as $4x + 3 - x$ . Each part joined by a plus or minus sign is a term. Like terms have the same letter, so $4x$ and $-x$ are like terms and combine to $3x$ . The number in front of a letter is its coefficient. In 7th grade, coefficients can be negative numbers, fractions or decimals, such as $-\frac{1}{2}x$ or $0.4x$ .

Expanding expressions means clearing parentheses with the distributive property: multiply the number outside by every term inside. So $-3(2x - 5)$ becomes $-6x + 15$ . Subtracting a whole expression works the same way, because a minus sign in front of parentheses acts like $-1$ . So $-(2x - 9)$ becomes $-2x + 9$ .

Factoring linear expressions is expanding in reverse. A linear expression has no squared letters. Your child finds a number that divides every term, called a common factor, and writes it outside the parentheses. Using the greatest common factor, the biggest number that divides every term, leaves nothing more to pull out. When asked, your child can also factor out a negative number: $-4x + 12 = -4(x - 3)$ . Simplifying expressions in 7th grade often uses all three moves: expand, combine like terms and factor.

Your child should be able to:

Combine like terms, including terms with negative, fraction or decimal coefficients
Expand an expression with the distributive property, including a negative number outside the parentheses
Subtract an expression in parentheses by changing the sign of every term inside
Factor a linear expression using the greatest common factor

Example

Simplify $5(x - 2) - (3x - 4)$ .

Expand both parentheses: $5x - 10 - 3x + 4$ .
Combine like terms: $5x - 3x = 2x$ and $-10 + 4 = -6$ .
Check with $x = 4$ : $5(2) - (12 - 4) = 10 - 8 = 2$ , and $2(4) - 6 = 2$ .
The simplified expression is $2x - 6$ .

Example

Factor $24x + 36$ using the greatest common factor.

The greatest number that divides both 24 and 36 is 12.
$24x \div 12 = 2x$ and $36 \div 12 = 3$ .
So $24x + 36 = 12(2x + 3)$ . Expanding gives back $24x + 36$ .
Pulling out only 6 gives $6(4x + 6)$ . That is equal, but it is not fully factored, because $4x$ and $6$ still share a factor of 2. The answer is $12(2x + 3)$ .

Where children usually get stuck

Your child rewrites $8 - (2x + 3)$ as $8 - 2x + 3$ .
The minus sign gets used on the first term inside the parentheses and forgotten on the second. It belongs to everything inside, so the 3 is subtracted too, and the answer is $5 - 2x$ .
What helps. Try $x = 1$ in both. $8 - (2 + 3) = 3$ , but $8 - 2 + 3 = 9$ . When the two values differ, a step went wrong.
Your child simplifies $3x + 4$ to $7x$ .
Your child sees a plus sign and adds every number in sight. Only like terms combine, and 3 groups of $x$ plus 4 ones cannot be counted together.
What helps. Put $x = 2$ into both. $3(2) + 4 = 10$ , but $7(2) = 14$ , so the two expressions are not equal.
Your child combines $\frac{1}{2}x + \frac{1}{3}x$ into $\frac{2}{5}x$ .
The numerators and the denominators get added as if they were separate numbers. Fraction coefficients follow the usual fraction rules, so they need a common denominator: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ .
What helps. Ask whether half a pizza plus a third of a pizza can be less than half a pizza. $\frac{2}{5}$ is less than $\frac{1}{2}$ , so it cannot be the answer.

Try four practice questions

Real 7th 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
Which choice shows the expression below with -2 factored out?
$\displaystyle -6x - 8$
A
$\displaystyle -2(3x - 4)$
B
$\displaystyle -2(-3x - 4)$
C
$\displaystyle -2(2x + 4)$
D
$\displaystyle -2(3x + 4)$
Show the answer and how to solve it
The correct choice is -2(3x + 4) because -6x divided by -2 is 3x and -8 divided by -2 is 4.
1. 1Look at the expression.
2. 2To factor out -2, divide each term by -2 to find what goes inside the parentheses.
3. 3Put the quotients inside the parentheses: 3x + 4.
Common mistake. A common mistake is flipping the sign of the inside constant. Since -8 divided by -2 is +4, the constant inside must be +4, not -4.
The idea underneath. Factoring out a negative is the reverse of distributing a negative, so each term inside the parentheses is the original term divided by that negative factor.
2On level
Look at the expression below. Which choice shows it fully factored using the greatest common factor?
$\displaystyle \frac{3}{2}x + 1$
A
$\displaystyle \frac{1}{4}(6x + 4)$
B
$\displaystyle \frac{1}{2}(3x + 1)$
C
$\displaystyle \frac{1}{2}(4x + 2)$
D
$\displaystyle \frac{1}{2}(3x + 2)$
Show the answer and how to solve it
The correct choice is $\frac{1}{2}(3x + 2)$ because $\frac{1}{2}$ is the greatest common factor of $\frac{3}{2}$ and $1$ .
1. 1Start by looking at the two coefficients in the expression: $\frac{3}{2}$ and $1$ .
2. 2The coefficients $\frac{3}{2}$ and $1$ both have a factor of $\frac{1}{2}$ , so the greatest common factor is $\frac{1}{2}$ .
3. 3Divide each term by $\frac{1}{2}$ to find what stays inside the parentheses.
4. 4The fully factored form is $\frac{1}{2}(3x + 2)$ . A smaller factor like $\frac{1}{4}$ also expands back correctly, but it is not fully factored because 6 and 4 still share a factor of 2.
Common mistake. A common mistake is choosing $\frac{1}{4}(6x + 4)$ . It does expand back to the original expression, but it is not fully factored because 6 and 4 still have a common factor of 2.
The idea underneath. A form is fully factored only when the number outside is the largest factor shared by both coefficients, so the numbers left inside the parentheses do not still have a common factor.
3On level
Look at Jordan's expansion below. Which statement best describes the work?
$\displaystyle \frac{3}{4}(8x - 4)$
Step 1
$\displaystyle = \frac{3}{4} \cdot 8x - \frac{3}{4} \cdot 4$
Step 2
$\displaystyle = 6x + 3$
A
The constant term was not multiplied.
B
The outside factor was added to both terms.
C
The x-term was not multiplied.
D
The sign changed incorrectly.
Show the answer and how to solve it
The answer is "The sign changed incorrectly." The signed constant product is $\frac{3}{4} \cdot (-4) = -3$ , so the correct expansion is $6x - 3$ .
1. 1Look at the original expression. The factor $\frac{3}{4}$ should be distributed to both terms inside the parentheses.
2. 2Check the x-term first.
3. 3Because $\frac{3}{4}$ is positive, the subtraction in 8x - 4 should stay subtraction after distributing.
4. 4The constant term is $-4$ . Multiply it by $\frac{3}{4}$ , including its minus sign.
5. 5So the correct last line should be 6x - 3, not 6x + 3. Both product sizes are right, but the minus sign became a plus sign.
Common mistake. The work changes subtraction to addition. Since $\frac{3}{4}$ is positive, $\frac{3}{4} \cdot (-4) = -3$ , so the constant in the expansion must be negative.
The idea underneath. Distributing a positive factor changes the coefficients, not the plus or minus sign between the two terms.
4Stretch
A classmate factored the expression below. Which statement best describes the work?
$\displaystyle 0.18x + 0.30$
$\displaystyle = 0.03(6x + 10)$
A
The work is equivalent, but it is not fully factored.
B
The work is correct and fully factored.
C
The x-term was divided incorrectly.
D
The constant term was divided incorrectly.
Show the answer and how to solve it
The correct statement is 'The work is equivalent, but it is not fully factored.' It expands to the original expression, but the 6 and 10 inside the parentheses still share a common factor of 2.
1. 1Look at the classmate's factorization.
2. 2Distribute the outside factor to check whether the factored form expands back to the original expression.
3. 3The factored form is equivalent. Now check whether the inside numbers share a common factor greater than 1. Since gcd(6, 10) = 2, factor out 2 from the inside to find the fully factored form.
4. 4The classmate's work expands correctly but is not fully factored, because 6 and 10 still share a factor of 2.
Common mistake. A common mistake is accepting 0.03(6x + 10) as fully factored because it expands correctly. However, 6 and 10 share a factor of 2, so the greater common factor 0.06 can still be pulled out.
The idea underneath. A factorization can be correct but incomplete; if the whole numbers inside the parentheses still share a common factor, then the outside factor was not the greatest common factor used for full factorization.

	$\displaystyle \frac{3}{4}(8x - 4)$
Step 1	$\displaystyle = \frac{3}{4} \cdot 8x - \frac{3}{4} \cdot 4$
Step 2	$\displaystyle = 6x + 3$