Reading and Writing Algebraic Expressions | Grade 8 Math | LevelKite

You already know how to add, subtract, multiply and divide positive and negative numbers, including fractions and decimals. Algebra lets you describe a calculation before you know every number in it.

A variable is a letter, such as $x$ or $n$ , that stands for a number you do not know yet or a number that can change. An algebraic expression is a mathematical phrase built from numbers, variables and operations, such as $5x - 2y + 8$ . A number written next to a variable multiplies it, so $5x$ means $5 \cdot x$ .

An expression has no equals sign. It names a quantity; it does not claim that two quantities are equal.

In this lesson you will name the parts of an expression, write an expression from words, and write an expression that models a real situation.

A term is one of the parts of an expression that are added together. To see the terms of $5x - 2y + 8$ , where $x$ and $y$ may be any numbers, write the expression as a sum. Subtracting any number gives the same result as adding its opposite, as in $7 - 3 = 7 + (-3)$ , so subtracting $2y$ becomes adding $-2y$ .

Two expressions are equivalent when they allow the same values of their variables and give the same result for every one of those values. The line below joins two equivalent expressions with an equals sign: both are defined for every value of $x$ and $y$ , and they always give the same result.

5x - 2y + 8 = 5x + (-2y) + 8

Now the three terms are easy to see: $5x$ , $-2y$ and $8$ . The sign in front of a term belongs to that term.

Two more names describe a term:

A coefficient is the number that multiplies the variable in a term.
A constant is a term that is only a number, with no variable. Its value stays the same whatever values the variables take.

The table sorts the three terms of this expression.

The three terms of the example expression

Term	What it is	How you know
$5x$	term with a variable; its coefficient is $5$	$5$ multiplies $x$
$-2y$	term with a variable; its coefficient is $-2$	$-2$ multiplies $y$ ; the minus sign stays with the term
$8$	constant	it has no variable, so its value never changes

Worked example

Naming the parts of an expression

Identify the terms, the coefficients and the constant in $7m - n + \frac{p}{4} - 12$ , where $m$ , $n$ and $p$ may be any numbers.

Rewrite each subtraction as adding the opposite. Subtracting any number gives the same result as adding its opposite, so this is true for every value of m, n and p. The new form is equivalent to the original, and each sign now stays with its term.
$7m - n + \frac{p}{4} - 12 = 7m + (-n) + \frac{p}{4} + (-12)$
List the parts that are added, because each added part is a term. There are four of them.
$7m, \quad -n, \quad \frac{p}{4}, \quad -12$
Write each term that has a variable as a number times that variable, because the coefficient is the number that multiplies the variable. Minus n means negative 1 times n. Dividing p by 4 gives the same result as multiplying p by one fourth, for every value of p.
$7m = 7 \cdot m, \quad -n = -1 \cdot n, \quad \frac{p}{4} = \frac{1}{4} \cdot p$
Find the constant. It is the term with no variable, so its value stays the same whatever values m, n and p take.
$-12$

Answer

The terms are $7m$ , $-n$ , $\frac{p}{4}$ and $-12$ . The coefficients are $7$ , $-1$ and $\frac{1}{4}$ , and the constant is $-12$ .

Now work in the other direction: start with words and write the expression. First say what the variable stands for. Then let the words tell you the operations and their order. By convention, write a multiplication with the number first, as in $3n$ , and write a division as a fraction, as in $\frac{n}{5}$ .

Order needs care. In an addition or a multiplication you may switch the two parts: $n + 9$ and $9 + n$ are equal for every value of $n$ , and so are $3 \cdot n$ and $n \cdot 3$ . Subtraction and division are different. The expressions $n - 4$ and $4 - n$ are equal only when $n = 4$ , so they are not equivalent. Division needs the same care: when $n = 5$ , $\frac{20}{n} = 4$ but $\frac{n}{20} = \frac{1}{4}$ . In a subtraction or a division, the words decide which quantity comes first.

One condition always applies: you cannot divide by $0$ . When you divide by a variable, state that the variable is not $0$ .

Phrases written as expressions, where n stands for any number unless a condition is shown

Words	Operation	Expression
the sum of a number and 9	addition	$n + 9$
6 more than a number	addition	$n + 6$
a number decreased by 4	subtraction	$n - 4$
4 less than a number	subtraction	$n - 4$
the product of 3 and a number	multiplication	$3n$
twice a number	multiplication	$2n$
the quotient of a number and 5	division	$\frac{n}{5}$
20 divided by a number	division	$\frac{20}{n}$ , where $n \neq 0$

Worked example

Writing an expression that needs parentheses

Write a variable expression for "9 less than twice the sum of a number and 4".

Define the variable so the letter has a clear meaning. Let n stand for the number. The phrase does not divide by the number, so n may be any number.
Start with the part that must be calculated first. The sum of a number and 4 is an addition.
$n + 4$
Twice the sum means 2 times the whole sum, so put the sum in parentheses. Without parentheses, 2n + 4 would double only n, because multiplication is done before addition.
$2(n + 4)$
The words 9 less than a quantity mean: start with that quantity and take 9 away. So 9 is subtracted last.
$2(n + 4) - 9$
Test the expression with one value, n = 1. From the words: the sum is 1 + 4 = 5, twice that is 10, and 9 less than 10 is 1. The expression gives the same value. A matching test cannot prove the expression is right for every number, but a mismatch would prove it wrong.
$2(1 + 4) - 9 = 2 \cdot 5 - 9 = 10 - 9 = 1$

Answer

With $n$ standing for any number, the phrase is written as $2(n + 4) - 9$ .

An expression can also model a real situation: it shows how to calculate a quantity in that situation. Build a model in three moves.

Define the variable. Say what it counts or measures, its unit, and which values make sense.
Sort the quantities. A fixed amount that is counted once gives a constant. An amount that is added in the same way for each unit of the variable gives a term with the variable in it, and that amount is the coefficient, because adding the same amount again and again is multiplication. If the amount is taken away each time instead, the term is subtracted, so its coefficient is negative.
Check the units. Terms can be added or subtracted only when they are measured in the same unit.

Key concept

The structure of an expression carries meaning

An algebraic expression is made of one or more terms, the parts that are added together. In a term with a variable, the coefficient is the number that multiplies the variable. A constant is a term with no variable, so its value never changes.

To write an expression, first say what each variable stands for and which values it may take. Then the words or the situation decide the operations and their order. In a model where one amount is counted once and another amount is added or taken away for each unit of the variable, the constant is the one-time amount and the coefficient is the change for each unit.

Worked example

Modeling savings that grow

Lena has 40 dollars saved. At the end of each week she adds 6 dollars and takes nothing out. Write an expression for the total amount she has saved, in dollars, after any whole number of weeks.

Define the variable. Let w be the number of full weeks that have passed. She adds money once at the end of each week, so w is a whole number: 0, 1, 2 and so on.
Find the amount that is counted once. The 40 dollars she starts with does not depend on w, so it is the constant.
$40$
Find the amount that repeats. She adds 6 dollars for each week, and adding 6 a total of w times is 6 times w. The units agree: dollars per week times weeks gives dollars.
$6 \cdot w = 6w$
Add the two terms, because the total is the starting amount together with everything added since. Both terms are measured in dollars, so they can be added.
$6w + 40$
Test with w = 3. From the story, 40 + 6 + 6 + 6 = 58 dollars. The expression gives the same amount. One matching value supports the model but does not prove it; a mismatch would show an error.
$6 \cdot 3 + 40 = 18 + 40 = 58$

Answer

After $w$ weeks, where $w$ is a whole number, Lena has saved $6w + 40$ dollars. The coefficient $6$ is the amount added each week, and the constant $40$ is her starting amount.

Worked example

Modeling an amount that goes down

A full tank holds 240 liters of water. A valve opens and the water drains at a constant rate of 8 liters each minute. Write an expression for the amount of water left in the tank, in liters, and say which values of the variable make sense.

Define the variable. Let t be the number of minutes since the valve opened. Time can include parts of a minute, so t need not be a whole number, but it cannot be negative.
$t \ge 0$
Sort the quantities. The 240 liters is counted once, so it is the constant. The rate is constant, so the amount drained in t minutes is 8 times t. Liters per minute times minutes gives liters.
$8t$
The drained water is taken away from the starting amount, so the starting amount comes first in the subtraction. Both terms are in liters, so they can be subtracted.
$240 - 8t$
Find where the model stops. The tank cannot hold less than 0 liters. Draining all 240 liters at 8 liters each minute takes 240 divided by 8, which is 30 minutes, because liters divided by liters per minute gives minutes. So t is at most 30.
$240 \div 8 = 30, \quad 0 \le t \le 30$
Test with t = 2. From the story, taking 8 liters away twice from 240 leaves 224 liters. The expression gives the same amount. One matching value supports the model but does not prove it.
$240 - 8 \cdot 2 = 240 - 16 = 224$
Read the structure. Written as a sum, the terms are 240 and negative 8t. The coefficient is negative 8, which shows that the amount falls by 8 liters each minute. Subtracting 8t gives the same result as adding its opposite for every allowed value of t.
$240 - 8t = 240 + (-8t)$

Answer

The amount of water left is $240 - 8t$ liters, where $t$ is the number of minutes since the valve opened and $0 \le t \le 30$ .

Takeaway. You can now read an expression and write one. Reading means naming its terms, the coefficient in each term that has a variable, and any constant. Writing means saying what each variable stands for and which values it may take, then choosing the operations and their order from the words or the situation.

Next, you will evaluate expressions: you will replace each variable with a given number and calculate the value, including expressions with exponents, roots and absolute values.