Parts of an Expression and Writing Expressions from Words | Grade 7 Math | LevelKite

A climbing gym charges 25 dollars to join and 12 dollars for each visit. You do not know yet how many times you will go, so let the letter $v$ stand for the number of visits, a whole number such as 0, 1 or 2. A letter that stands for a number is called a variable.

Each visit adds 12 dollars, so $v$ visits cost $12 \cdot v$ dollars, and the joining fee adds 25 dollars once. The total cost in dollars is $12v + 25$ . Here $12v$ is a short way to write $12 \cdot v$ : a number written right next to a variable multiplies it.

The cost $12v + 25$ is an expression: numbers, variables and operation signs written together, with no equals sign. In this lesson you will name the parts of an expression, and then write expressions from word phrases.

Start with what is being added. The parts of an expression that are added together are its terms. In $12v + 25$ the terms are $12v$ and $25$ .

Subtraction hides the terms a little. Take $6x - 4y - 9$ , where $x$ and $y$ each stand for a number. Subtracting a quantity always gives the same result as adding its opposite: $10 - 4$ and $10 + (-4)$ both equal $6$ . So you can rewrite every subtraction as an addition, and the expression keeps the same value for every $x$ and $y$ :

6x - 4y - 9 = 6x + (-4y) + (-9)

Now you can see three parts added together, so the terms are $6x$ , $-4y$ and $-9$ . The sign in front of a term belongs to that term. Rewriting like this shows the terms of every expression whose parts you name in this lesson, because each one is a chain of additions and subtractions with no parentheses.

Each kind of term has a name.

In a term with a variable, the number that multiplies the variable is the coefficient. The coefficient of $6x$ is $6$ . The coefficient of $-4y$ is $-4$ , sign included.
A term with no variable is a constant. Its value stays the same whatever the variables stand for. Here the constant is $-9$ .

At the gym, the coefficient $12$ in $12v + 25$ is the cost in dollars of each visit, and the constant $25$ is the joining fee, which stays the same however many visits you make.

Worked example

Naming the parts of an expression

Name the terms, the coefficients and the constant in $2x - y + \frac{3}{4}z - 11$ , where $x$ , $y$ and $z$ each stand for a number.

Rewrite each subtraction as adding the opposite. The value of the expression does not change, because subtracting a quantity always gives the same result as adding its opposite.
$2x - y + \frac{3}{4}z - 11 = 2x + (-y) + \frac{3}{4}z + (-11)$
List the parts that are now added together. By definition these are the terms, and each one keeps the sign in front of it.
$2x, \quad -y, \quad \frac{3}{4}z, \quad -11$
In each term that has a variable, find the number that multiplies the variable. The second term shows no number, but the opposite of y equals negative 1 times y, so its coefficient is negative 1.
$2x = 2 \cdot x, \qquad -y = -1 \cdot y, \qquad \frac{3}{4}z = \frac{3}{4} \cdot z$
Find the term with no variable. Its value does not depend on x, y or z, so it is the constant.
$-11$

Answer

The terms are $2x$ , $-y$ , $\frac{3}{4}z$ and $-11$ . The coefficients are $2$ , $-1$ and $\frac{3}{4}$ , and the constant is $-11$ .

Now go the other way, from words to symbols. A word phrase such as “6 more than a number” describes an operation on a number you do not know yet. To write it as an expression, first choose a variable and say what it stands for, such as “let $n$ stand for the number”. Then turn each operation word into its symbol. The table below shows common phrases.

Order needs care. Adding and multiplying give the same result in either order: $n + 6 = 6 + n$ and $7 \cdot n = n \cdot 7$ for every number $n$ . With subtraction and division, the order usually changes the result: $8 - 3 = 5$ but $3 - 8 = -5$ , and $12 \div 4 = 3$ but $4 \div 12 = \frac{1}{3}$ . So for those two operations, work out which quantity comes first before you write anything.

Common word phrases. In each row, the letter n stands for the number.

Word phrase	What it tells you to do	Expression
the sum of a number and 8	add the number and 8	$n + 8$
6 more than a number	add 6 to the number	$n + 6$
a number decreased by 3	subtract 3 from the number	$n - 3$
5 less than a number	subtract 5 from the number	$n - 5$
the difference of 10 and a number	subtract the number from 10	$10 - n$
the product of 7 and a number	multiply 7 by the number	$7n$
twice a number	multiply the number by 2	$2n$
the quotient of a number and 4	divide the number by 4	$\frac{n}{4}$
the quotient of 12 and a number	divide 12 by the number	$\frac{12}{n}$ , where $n$ is not $0$ , because you cannot divide by $0$

Worked example

Writing a phrase with two operations

Write a variable expression for “7 less than the product of 3 and a number”. Then name its terms, its coefficient and its constant.

Choose a variable and say what it means, so that every symbol you write has a clear meaning. Let n stand for the number.
Write the product first, because the phrase takes 7 away from it. “The product of 3 and a number” means 3 multiplied by n, written with the number in front of the variable.
$3 \cdot n = 3n$
“7 less than” the product means start with the product and take 7 away. So the product comes first and 7 is subtracted from it.
$3n - 7$
Name the parts. Rewrite the subtraction as adding the opposite, which keeps the same value. Two parts are added: one has the variable, with 3 multiplying it, and the other has no variable.
$3n - 7 = 3n + (-7)$

Answer

You write the phrase as $3n - 7$ . Its terms are $3n$ and $-7$ , its coefficient is $3$ , and its constant is $-7$ .

Some phrases do an operation to a whole sum or difference. Try it with numbers first. “4 times the sum of 5 and 6” tells you to add first and then multiply all of the sum by 4: $4(5 + 6) = 4 \cdot 11 = 44$ . A number written directly in front of parentheses multiplies the whole quantity inside, so the parentheses keep the sum together. Without them, you multiply before you add, and only the 5 is multiplied by 4: $4 \cdot 5 + 6 = 20 + 6 = 26$ , a different value.

A variable works the same way. When a phrase multiplies a whole sum or difference by a number, write that sum or difference inside parentheses and put the number in front. So “4 times the sum of a number and 6” becomes $4(n + 6)$ , where $n$ stands for the number.

Worked example

Keeping a sum together

Write a variable expression for “5 less than twice the sum of a number and 3”.

The phrase is about an unknown number, so choose a variable for it. Let n stand for the number.
Start with the innermost quantity, the one that the other words act on. “The sum of a number and 3” means the number and 3 added together.
$n + 3$
“Twice the sum” multiplies the whole sum by 2, so write the sum in parentheses with 2 in front.
$2(n + 3)$
“5 less than” that result means start with the result and take 5 away, so the result comes first and 5 is subtracted from it.
$2(n + 3) - 5$

Answer

You write the phrase as $2(n + 3) - 5$ , where $n$ stands for the number.

Key concept

Every part of an expression has a name and a job

Terms are the parts of an expression that are added together. Each term keeps the sign in front of it, because subtracting is the same as adding the opposite. In a term with a variable, the coefficient is the number that multiplies the variable. A term with no variable is a constant.

Writing an expression from words builds these same parts. Choose a variable and say what it stands for. Then let the phrase tell you which operation to use, which quantity comes first, and which quantities stay together in parentheses.

For example, “9 less than 5 times a number” is $5n - 9$ , where $n$ stands for the number. Since $5n - 9 = 5n + (-9)$ , its terms are $5n$ and $-9$ , its coefficient is $5$ and its constant is $-9$ .

An expression is one or more terms added together, and words and symbols are two ways to say the same thing. You can now go in both directions. Given an expression, you can name its terms, coefficients and constants, keeping each sign with its term. Given a word phrase, you can build the expression by watching the operation, the order and the grouping. Next, you will find the value of an expression when you know the number its variable stands for, and you will write expressions for word problems.