5th Grade · 5.OA.A.2

Numerical expressions in 5th grade

Numerical expressions are math phrases made of numbers and operation signs, such as $3 \times (10 - 4)$ , with no equal sign. In 5th grade, your child learns to write numerical expressions from words and to read them without working them out.

What the standard says

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

5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

What it means, in plain words

An expression is like a phrase in a sentence. It names an amount but does not say anything about it. $3 \times (10 - 4)$ names 18, and $3 \times (10 - 4) = 18$ is a full math sentence, an equation. Numerical expressions in 5th grade have two jobs: writing them from words and reading them without calculating.

Writing numerical expressions means turning words into symbols. "Subtract 5 from 20, then divide by 3" becomes $(20 - 5) \div 3$ . The parentheses show that the subtraction happens first. Four words tell children which sign to use: sum is the answer to an addition, difference to a subtraction, product to a multiplication and quotient to a division. Writing expressions also works in reverse: children read $(15 + 3) \div 2$ aloud as "half the sum of 15 and 3".

Reading an expression without working it out means seeing its shape. $5 \times (1{,}248 + 376)$ is 5 times as large as $1{,}248 + 376$ , and there is no need to find the sum to know that. This habit helps children check answers and compare amounts quickly.

Your child should be able to:

Write a numerical expression from words such as "add 5 and 8, then double the result"
Use parentheses to show which step happens first
Read an expression aloud in words, using sum, difference, product and quotient
Say how two expressions compare without working them out

Example

Write an expression for "divide 36 by 4, then add 11".

"Divide 36 by 4" is $36 \div 4$ .
"Then add 11" puts $+ 11$ after it.
Division comes before addition anyway, so no parentheses are needed.
The expression is $36 \div 4 + 11$ , and its value is 20.

Example

Without calculating, compare $6 \times (219 + 58)$ with $219 + 58$ .

Both expressions contain the same sum, $219 + 58$ .
The first one multiplies that sum by 6.
So $6 \times (219 + 58)$ is 6 times as large as $219 + 58$ . (The values are 1,662 and 277.)

Where children usually get stuck

For "add 4 and 5, then multiply by 6", your child writes $4 + 5 \times 6$ .
The child writes the numbers in the order the words say them. Without parentheses, the multiplication happens first, so the expression means something else.
What helps. Have your child do the words step by step: 4 plus 5 is 9, and 9 times 6 is 54. Then work out $4 + 5 \times 6$ , which is 34. The two answers differ, so the parentheses are needed: $(4 + 5) \times 6$ .
For "subtract 3 from 10", your child writes $3 - 10$ .
The 3 comes first in the words, so it goes first on paper. "Subtract from" flips the order.
What helps. Say it with real things: "take 3 cookies from 10 cookies." You start with 10 and take 3 away, so it is $10 - 3$ .
Asked how $8 \times (350 + 75)$ compares with $350 + 75$ , your child says it is 8 more, or works out both sums in full.
The child reads the 8 as an amount to add, not as 8 copies of the whole sum in the parentheses.
What helps. Read the expression aloud as "8 groups of 350 plus 75". Then ask how 8 groups compare with 1 group.

Try four practice questions

Real 5th 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
Mia is trying to show what the parentheses tell you to do in the math shown below.
Original expression
$\displaystyle 3 \times (8 + 2)$
Mia's first rewrite
$\displaystyle = 24 + 2$
Select the statement that best describes Mia's work.
A
Mia multiplied 3 by both 8 and 2 instead of adding first.
B
Mia added 3 to 8 and left 2 unchanged.
C
Mia multiplied 3 by only 8 and left 2 unchanged.
D
Mia's rewrite is correct because the addition inside the parentheses was done first.
Show the answer and how to solve it
The correct statement says Mia multiplied 3 by only 8 and left 2 unchanged instead of adding $8 + 2$ first.
1. 1Compare the original expression with Mia's first rewrite.
2. 2The parentheses make $8 + 2$ one group, so that addition should happen first.
3. 3The shown rewrite changes 8 to 24 but leaves 2 as 2. That means Mia multiplied 3 by 8 and skipped the 2, so the outside multiplication was applied to only one number inside the parentheses.
4. 4So the correct statement is the one that says Mia multiplied 3 by only 8 and left 2 unchanged.
Common mistake. A common mistake is thinking $3 \times (8 + 2)$ can be rewritten as $24 + 2$ by applying the outside multiplication to the first number. The parentheses mean $8 + 2$ must be done first, so the correct first rewrite is $3 \times 10$ .
The idea underneath. Parentheses group the inside numbers into one quantity. Calculate that group first, or apply a valid rule to the whole group.
2On level
Maya and Jayden each described the numerical expression shown below.
$\displaystyle 5 + 4 \times 3$
Maya's description: "Add 5 and 4, then multiply by 3."
Jayden's description: "Add 5 to the product of 4 and 3."
Select the best description of their work.
A
Only Maya is correct.
B
Only Jayden is correct.
C
Both students are correct.
D
Neither student is correct.
Show the answer and how to solve it
Only Jayden is correct because Jayden keeps $4 \times 3$ together as one group before adding 5.
1. 1Start by reading the expression itself.
2. 2Multiplication comes before addition when nothing changes the order, so $4 \times 3$ stays together first.
3. 3Maya's phrase 'Add 5 and 4, then multiply by 3' groups the addition before the multiplication, changing the structure.
4. 4Jayden's phrase 'Add 5 to the product of 4 and 3' names $4 \times 3$ as one group and then adds 5, which matches the expression.
5. 5Only Jayden matches the expression, so that is the best description of their work.
Common mistake. A common mistake is reading the expression left to right and describing it as add 5 and 4, then multiply by 3. That changes $5 + (4 \times 3)$ into $(5 + 4) \times 3$ .
The idea underneath. Words that name a group, such as quotient or sum, tell which numbers stay together before the next operation.
3On level
Select the expression that matches the phrase "4 times as many as the difference of 15 and 9."
A
$\displaystyle 4 \times (15 - 9)$
B
$\displaystyle 4 + {}$
$\displaystyle (15 - 9)$
C
$\displaystyle 4 \times 15 - 9$
D
$\displaystyle 4 \times (15 + 9)$
Show the answer and how to solve it
The matching expression is $4 \times (15 - 9)$ because you find the difference first and then multiply that whole result by $4$ .
1. 1The words difference of $15$ and $9$ tell you to write $15 - 9$ first.
2. 2The words 4 times as many as mean multiply that whole difference by $4$ , so the difference needs parentheses.
3. 3So the matching expression is $4 \times (15 - 9)$ .
Common mistake. A common mistake is choosing $4 \times 15 - 9$ , which multiplies only $15$ by $4$ instead of multiplying the whole difference $15 - 9$ .
The idea underneath. In phrases like "times as many as the sum or difference," the named sum or difference is one grouped quantity, so parentheses keep that whole quantity together.

4Stretch

Each row in the table below shows two expressions. Decide whether the first expression is greater than, less than, or equal to the second expression without finding every value.

Row	First expression	Second expression
A	$\displaystyle 9 \times (400 + 60)$	$\displaystyle 9 \times (400 + 30)$
B	$\displaystyle 3 \times (300 + 70)$	$\displaystyle 6 \times (300 + 70)$
C	$\displaystyle 7 \times (500 + 40)$	$\displaystyle 7 \times 500 + 7 \times 40$
D	$\displaystyle 5 \times (200 + 80)$	$\displaystyle 5 \times 200 + 80$

In the comparison chart, mark one box in each row.

Show the answer and how to solve it

Mark Row A with >, Row B with <, Row C with =, and Row D with > by comparing the parts that differ in each expression pair.

1Compare one row at a time. Look for what stays the same and what changes in each pair of expressions rather than calculating the full values.
2In Row A, both expressions are multiplied by 9 and both have 400 inside the parentheses. Only the added amount differs: 60 versus 30. Since 60 > 30, the first expression is greater.
3In Row B, both expressions have the same sum inside the parentheses, $(300 + 70)$ . The only difference is the outside factor: 3 versus 6. Since 3 < 6, the first expression is less.
4In Row C, the second expression is the first expression written in distributed form. Multiplying 7 by the sum gives 7 groups of 500 and 7 groups of 40, which is exactly $7 \times 500 + 7 \times 40$ . The two expressions are equal.
5In Row D, distributing the left expression gives $5 \times 200 + 5 \times 80$ . The right expression has only 80 where $5 \times 80 = 400$ should be. Since 400 > 80, the first expression is greater.

Common mistake. A common mistake in Row D is thinking the two sides are equal because both use 5, 200, and 80. But the 80 is multiplied by 5 on the left and added only once on the right.

The idea underneath. When two expressions share most of the same structure, compare the part that changed instead of calculating the whole expressions.

Original expression	$\displaystyle 3 \times (8 + 2)$
Mia's first rewrite	$\displaystyle = 24 + 2$