Chapter 8 · Expressions

Variables and One-Operation Expressions

Learn how a letter can stand for a number that is unknown or can change. Then write expressions with one operation from word phrases and real situations.

12 min read

You already write numerical expressions such as $6 \times 3$ to record a calculation with numbers. Today you will record a calculation when one of its numbers is not known yet, or can change.

Think about packs of markers. Each pack holds 6 markers. For 2 packs, you calculate $6 \times 2$ . For 3 packs, you calculate $6 \times 3$ . For 10 packs, you calculate $6 \times 10$ .

Look at what stays the same and what changes. The 6 is in every calculation. Only the number of packs changes.

Markers in packs of 6

Number of packs	Calculation	Number of markers
$2$	$6 \times 2$	$12$
$3$	$6 \times 3$	$18$
$10$	$6 \times 10$	$60$
any number of packs	6 times the number of packs	depends on the number of packs

The last row is the pattern behind every other row. Writing the number of packs each time is slow, so you choose a letter to stand for it.

A variable is a letter that stands for a number. Here, let $p$ be the number of packs.

Be clear about what $p$ is. It is a number, such as 2, 3 or 10. It is not the word packs, and it is not one pack. Before you use a variable, say what number it stands for. Give its unit, such as centimeters, when it has one.

A variable can do two jobs:

It can stand for one number you do not know yet. You bought some packs, and $p$ is how many.
It can stand for any number from a set of possible values. Here $p$ can be any whole number of packs: 0, 1, 2, 3 and so on.

Once you say what a letter stands for, it keeps that meaning for the whole problem.

6 \times 2 \qquad 6 \times 3 \qquad 6 \times 10 \qquad \longrightarrow \qquad 6 \times p

With $p$ in place of the number of packs, the one expression $6 \times p$ covers every row of the table. An expression that has at least one variable in it is a variable expression, also called an algebraic expression. Like a numerical expression, it records a calculation and has no equals sign.

Multiplication is written in a new way once letters appear. The sign $\times$ looks too much like the letter $x$ . So you write $6 \cdot p$ or, shorter, $6p$ . A number written right next to a variable means multiply. People write the number first: $6p$ , not $p6$ .

So $6p$ means 6 times $p$ . If $p$ is 3, then $6p$ is $6 \cdot 3 = 18$ . It is not the two-digit number 63.

Key concept

A letter can stand for a number

A variable is a letter that stands for a number. That number may be one value you do not know yet, or any value from a set of possible values. A variable expression records a calculation with that number. Because the letter can take any of its possible values, one expression describes the calculation for all of them.

Now you can turn words into a variable expression with one operation. Use three moves:

Name the variable. Say what number it stands for, with its unit if it has one. If the words already give a letter, use that letter.
Find the operation. Ask what is done with that number: adding, subtracting, multiplying or dividing.
Write and test. Write the expression. Then put an easy number in place of the variable. The expression should give the same result as the words do. If it does not, the expression is wrong.

The table shows all four operations. In it, $n$ stands for any number. Division is usually written as a fraction. For $n$ divided by 8, you write $\dfrac{n}{8}$ . It means the same as $n \div 8$ .

Word phrases with one operation

Word phrase	What happens	Expression
the sum of a number $n$ and 8	8 is added to $n$	$n + 8$
8 more than a number $n$	8 is added to $n$	$n + 8$
the difference of a number $n$ and 8	8 is taken away from $n$	$n - 8$
8 less than a number $n$	8 is taken away from $n$	$n - 8$
a number $n$ subtracted from 8	$n$ is taken away from 8	$8 - n$
the product of 8 and a number $n$	$n$ is multiplied by 8	$8n$
8 times a number $n$	$n$ is multiplied by 8	$8n$
the quotient of a number $n$ and 8	$n$ is divided by 8	$\frac{n}{8}$
a number $n$ divided by 8	$n$ is divided by 8	$\frac{n}{8}$

Order matters when you subtract. Take the phrase 8 less than a number $n$ . It is tempting to copy the words from left to right and write $8 - n$ . Test it with $n = 12$ . Eight less than 12 means you start at 12 and take away 8, which gives $12 - 8 = 4$ . But $8 - n$ becomes $8 - 12$ . That starts at 8 and takes away 12, which does not give 4. Copying the word order made 8 the starting number. The phrase really starts with $n$ and takes 8 away from it, so the correct expression is $n - 8$ .

Division needs the same care: $\dfrac{n}{8}$ and $\dfrac{8}{n}$ are different expressions. Addition and multiplication work in either order, because swapping the two numbers never changes a sum or a product: $n + 8 = 8 + n$ and $8 \cdot n = n \cdot 8$ for every number $n$ .

Worked example

Adding to an unknown number

Maya has some stickers in her album. She gets 7 more stickers. Write an expression for the number of stickers she has now.

Name the variable. You do not know how many stickers Maya starts with, so choose a letter for that number. Let s be the number of stickers she has at the start.
Find the operation. Getting 7 more stickers makes the starting number bigger by 7, so 7 is added to s.
Write the expression. Begin with s, the number she had, and add the 7 new stickers.
$s + 7$
Test with an easy number. Suppose Maya starts with 10 stickers. Getting 7 more gives her 17 stickers. Now put 10 in place of s. The expression gives 17 too, so it matches the story.
$10 + 7 = 17$

Answer

Maya now has $s + 7$ stickers, where $s$ is the number of stickers she had at the start.

Worked example

Dividing a length into equal pieces

A ribbon is cut into 4 pieces of equal length. Write an expression for the length of one piece, in centimeters.

Name the variable and its unit. The length of the whole ribbon is not given, so let r be the length of the whole ribbon in centimeters.
Find the operation. Cutting a length into 4 equal pieces shares it equally 4 ways, and equal sharing is division. So r is divided by 4.
Write the expression as a fraction. The number being divided, r, goes on top. The number of equal pieces, 4, goes on the bottom.
$\dfrac{r}{4}$
Test with an easy number. Suppose the ribbon is 20 centimeters long. Each piece is then 5 centimeters, because 4 pieces of 5 centimeters make 20 centimeters. Put 20 in place of r. The expression gives 5 too, so it matches.
$\dfrac{20}{4} = 5$

Answer

Each piece is $\dfrac{r}{4}$ centimeters long, where $r$ is the length of the whole ribbon in centimeters.

Worked example

Subtracting an unknown amount

A full water jug holds 64 ounces. You pour out some water. Write an expression for the number of ounces left in the jug.

Name the variable and its unit. The amount poured out is unknown, so let w be the number of ounces you pour out. Here w can be any amount from 0 to 64 ounces, because you cannot pour out more than the jug holds.
Find the operation and its order. Pouring out takes water away, so you subtract. The jug starts with 64 ounces, and w is taken away from 64. So 64 comes first.
$64 - w$
Test with an easy number. Suppose you pour out 10 ounces. The jug then has 54 ounces left. Put 10 in place of w. The expression gives 54 too, so it matches.
$64 - 10 = 54$

Answer

The jug has $64 - w$ ounces left, where $w$ is the number of ounces poured out.

Takeaway: A variable is a letter that stands for a number. It may be one number you do not know yet, or any number from a set of possible values. To write a variable expression with one operation, name the variable, find the operation, and write it in an order that matches what happens. Then test it with an easy number.

Next, you will write variable expressions with two operations and match expressions to word phrases.

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Two-Operation Phrases and Grouping

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