Math Core

Math has its own short way of writing ideas. "Add 8 and 7, then double the result" becomes $2 \times (8 + 7)$ . Learning to switch between words and expressions helps you solve word problems and compare amounts without doing a lot of arithmetic.

From words to expressions

Definition

Numerical expression

A numerical expression is a group of numbers and operation signs, such as $4 \times 9 - 3$ or $(12 + 5) \div 2$ . It has no equal sign.

Some words point to each operation:

Operation	Words you might see
$+$	add, sum, plus, more than, total
$-$	subtract, difference, minus, less than, take away
$\times$	multiply, product, times, double (times 2), triple (times 3)
$\div$	divide, quotient, split equally, half of (divided by 2)

Pay attention to the word then. It tells you the order. When the first step is adding or subtracting and the next step is multiplying or dividing, you need parentheses to show that the first step happens first.

"Multiply 6 by 4, then add 9" is $6 \times 4 + 9$ . Multiplying already comes first, so no parentheses are needed.
"Add 5 and 3, then multiply by 7" is $(5 + 3) \times 7$ . Without parentheses, $5 + 3 \times 7$ would multiply first.

Common mistake

Watch out for "subtract 5 from 20." The number after "from" comes first: $20 - 5$ , not $5 - 20$ . The same goes for "4 less than 10," which is $10 - 4$ .

Worked example: Write it, then evaluate it

Write an expression for "subtract 6 from 20, then divide by 2." Then find its value.

Subtracting comes first, so put it in parentheses: $(20 - 6) \div 2$ .

Value: $20 - 6 = 14$ , and $14 \div 2 = 7$ .

Reading an expression without solving it

You can often tell a lot about an expression just by looking at its parts.

Look at $3 \times (18{,}932 + 921)$ . You don't need to add those big numbers. The expression takes the sum $18{,}932 + 921$ and multiplies it by 3. So it is 3 times as large as $18{,}932 + 921$ .

Think about the parts

$\text{(number)} \times (\text{something})$ is that many times as large as the something.
$(\text{something}) \div 2$ is half as large as the something.

Worked example: Compare without calculating

How does $(4{,}650 - 1{,}275) \div 5$ compare to $4{,}650 - 1{,}275$ ?

The first expression takes the difference $4{,}650 - 1{,}275$ and divides it into 5 equal parts. So it is one fifth as large as $4{,}650 - 1{,}275$ . You didn't have to subtract at all.

Worked example: A word problem

Ruby has 3 bags with 15 marbles in each bag. She gives 8 marbles to a friend. Write an expression for how many marbles Ruby has left, then evaluate it.

She starts with 3 groups of 15, which is $3 \times 15$ . Then she gives away 8: $3 \times 15 - 8$ .

Value: $3 \times 15 = 45$ , and $45 - 8 = 37$ marbles.

Tip

After you write an expression, read it back in words, following the order of operations. If it doesn't match the story, move or add parentheses.

Practice

Practice 1

Which expression means "multiply 6 by 4, then add 9"?

Practice 2

Which expression means "add 5 and 3, then multiply by 7"?

Practice 3

Which expression means "subtract 4 from 15"?

Practice 4

Write an expression for "add 9 and 12, then divide by 3." What is its value?

Type a number.

Practice 5

The expression $5 \times (389 + 127)$ is how many times as large as $389 + 127$ ?

Type a number.

Practice 6

How does $(7{,}250 - 1{,}318) \div 2$ compare to $7{,}250 - 1{,}318$ ?

Practice 7

Maya buys 3 packs of markers with 12 markers in each pack. She gives 5 markers to her brother. Write an expression for the number of markers she has left and find its value.

Type a number.

Practice 8

Without calculating, which expression is 4 times as large as $48 + 25$ ?