Math Core

A movie ticket costs $9. Two tickets cost $9 \times 2$ dollars and five tickets cost $9 \times 5$ dollars. But what if you don't know yet how many tickets you need? Algebra lets you write one expression that works for any number of tickets. The secret is a letter called a variable.

Variables

Definition

Variable

A variable is a letter that stands for a number. The number might change, or you might not know it yet.

If $t$ stands for the number of tickets, the cost is $9 \times t$ dollars. In algebra, a number written right next to a variable means multiply, so you can write this as $9t$ . You can also use a dot, $9 \cdot t$ . Algebra avoids the $\times$ sign because it looks so much like the letter $x$ .

Division is usually written as a fraction. " $m$ divided by 4" is written $\dfrac{m}{4}$ or $m \div 4$ .

An expression is a combination of numbers, variables and operation signs, like $9t$ , $n + 5$ or $\dfrac{m}{4} - 2$ . An expression has no equals sign.

Words that mean each operation

operation	words and phrases	example	expression
add	sum, plus, more than, increased by, total	6 more than $n$	$n + 6$
subtract	difference, minus, less than, decreased by, fewer	$n$ decreased by 6	$n - 6$
multiply	product, times, of, twice, triple	the product of 6 and $n$	$6n$
divide	quotient, divided by, per, split equally	the quotient of $n$ and 6	$\dfrac{n}{6}$

Common mistake

Watch out for "less than." It flips the order. "6 less than $n$ " means you start with $n$ and take 6 away, so it is $n - 6$ , not $6 - n$ .

Check with a number: if $n = 10$ , then "6 less than 10" is 4. And $10 - 6 = 4$ , so $n - 6$ is right.

Worked example: Translating phrases

Write an expression for each phrase.

8 more than a number $k$
the quotient of a number $w$ and 3
5 less than twice a number $y$

Solutions.

"More than" means add: $k + 8$ .
"Quotient" means divide, with $w$ first: $\dfrac{w}{3}$ .
"Twice $y$ " is $2y$ . Taking 5 less than that gives $2y - 5$ .

When you need parentheses

Some phrases ask you to do one operation to a whole group. Words like "the sum of" or "the difference of" tell you to make that group first, so wrap it in parentheses.

Worked example: Parentheses make a difference

Write expressions for "3 times the sum of $n$ and 4" and "4 more than 3 times $n$ ."

In the first phrase, you find the sum $n + 4$ first and then triple it: $3(n + 4)$ .
In the second phrase, you triple $n$ first and then add 4: $3n + 4$ .

Try $n = 2$ : $3(2 + 4) = 18$ but $3(2) + 4 = 10$ . Different phrases, different values.

Naming the parts of an expression

Mathematicians use special words to talk about the pieces of an expression. Look at

4x + 7 + y.

The terms are the parts that are added or subtracted: $4x$ , $7$ and $y$ . This expression has 3 terms.
A coefficient is the number multiplying a variable. The coefficient of $x$ is 4. The coefficient of $y$ is 1, since $y$ means $1y$ .
A constant is a term that is just a number: 7.

The words sum, difference, product and quotient can describe a whole expression, too. For example, $3(n + 4)$ is a product of two factors, 3 and $(n + 4)$ . The second factor is itself a sum of two terms.

Expressions from real situations

Worked example: A phone plan

A phone plan costs $15 a month plus $2 for every gigabyte of data used. Write an expression for the monthly cost when $g$ gigabytes are used.

Each gigabyte costs $2, so $g$ gigabytes cost $2g$ dollars. Add the $15 base price: the cost is $2g + 15$ dollars.

Practice

Practice 1

Write an expression for "12 more than a number $n$ ."

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 2

Which expression means "7 less than a number $x$ "?

Practice 3

Write an expression for "the quotient of a number $p$ and 5."

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 4

How many terms are in the expression $5a + 2b + 9$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

What is the coefficient of $m$ in the expression $6k + m + 10$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Write an expression for "twice the sum of a number $y$ and 9."

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

Leo has some baseball cards. His sister has 4 fewer than 3 times as many cards as Leo. If Leo has $c$ cards, which expression shows how many cards his sister has?

Practice 8

Tickets to a school play cost $6 each, and there is an $8 charge for the whole order. Write an expression for the total cost, in dollars, of $s$ tickets.

Enter an expression, e.g. 3x^2 - 2x + 1