Distinguish variables from constants and define the key vocabulary of algebra: term, coefficient, factor, and expression. Translate between word phrases and algebraic expressions with precision.
A lesson of Algebra 1, in Foundations of Algebra, about 30 minutes.
By the end of this lesson, you will be able to name every part of an algebraic expression, translate between words and algebra in both directions, and write a precise expression for a real situation by choosing and defining a variable.
One Letter Stands In for Every Number at Once
For thousands of years, mathematicians could only answer one question at a time.
$$3 + 5 = 8$$
That sentence is true, and it is true about exactly one pair of numbers. It says
nothing about $4 + 5$, and nothing at all about a number nobody has measured
yet.
Then someone wrote a letter where a number used to go.
$$n + 5$$
Now the same short piece of writing covers every number there is. Put $3$ in and
it means $8$. Put $40$ in and it means $45$. One line of algebra does the work
of an infinite list of arithmetic.
That letter is a variable, and it is the idea the whole subject is built on.
Everything this year sits on top of it, so this lesson is about reading and
writing the language variables live in.
That one sentence is the difference between algebra and guessing.
Constant
A constant is a fixed number in the expression. In $3x + 7$ the constant is
$7$. Whatever $x$ turns out to be, $7$ is still $7$.
Term
A term is a number, a variable, or numbers and variables multiplied
together. Terms are separated by $+$ and $-$ signs.
In $5x^2 - 3x + 8$ there are three terms:
$$5x^2 \qquad -3x \qquad 8$$
Notice the middle one. The minus sign belongs to the term that follows it, so
the term is $-3x$ and not $3x$.
Coefficient
The coefficient is the number multiplying the variable in a term. In $5x^2$
the coefficient is $5$. In $-3x$ it is $-3$, sign included.
A variable with no number written in front of it still has one, and it is $1$:
$$x \text{ means } 1x \qquad -y \text{ means } -1y$$
Factor
Factors are the things being multiplied. In the term $5x$, the factors are
$5$ and $x$.
All of it at once
Expression
Terms
Variables
Constants
Coefficients
$3x + 7$
$3x$, $7$
$x$
$7$
$3$
$2a - b + 4$
$2a$, $-b$, $4$
$a$, $b$
$4$
$2$, $-1$
$x^2 + 5x - 1$
$x^2$, $5x$, $-1$
$x$
$-1$
$1$, $5$
Turning English into Algebra
The skill this lesson is really about is translation: reading a situation in
words and writing it in symbols.
Words
Operation
Example
sum, more than, increased by, plus
$+$
"5 more than $x$" gives $x + 5$
difference, less than, decreased by, minus
$-$
"7 less than $n$" gives $n - 7$
product, times, of, multiplied by
$\times$, written $ab$
"three times $y$" gives $3y$
quotient, divided by, per, ratio
fraction
"the ratio of $x$ to 4" gives $\dfrac{x}{4}$
squared, cubed
exponent
"$n$ squared" gives $n^2$
"Less than" runs backwards. "7 less than $n$" is $n - 7$, never $7 - n$.
The phrase tells you what to take away, then what to take it away from, in that
order. Subtraction does not commute, so the order is the whole answer.
Four moves, every time:
Name the unknown, and say what it means
Find the operation words
Write the expression in the right order
Test it with a number and check it does what the sentence said
Example 1: Building an Expression One Phrase at a Time
Here is the whole method on one sentence, worked all the way through.
Write an expression for "eight less than three times a number".
Start by naming the unknown. Let $x$ be the number. Then read the sentence from
the inside out: the phrase "three times a number" describes a quantity, and
"eight less than" acts on that whole quantity.
Write "three times a number" in symbols 3x
Now take eight away from that whole quantity 3x - 8
Worked Example 1: Your Turn to Translate
Write an expression for "the quotient of a number and five, increased by two".
Let $n$ be the number. Again there are two phrases, and the second one acts on
the result of the first.
Example 2: When the Words Wrap Around a Whole Group
Some phrases act on a group rather than on a single term, and that is what
brackets are for.
Write an expression for "twice the sum of a number and four".
Let $x$ be the number. The sum is a single quantity, and the word "twice"
doubles all of it.
Write "the sum of a number and four", keeping it together as one quantity (x + 4)
Double the whole group 2(x + 4)
Worked Example 2: Two Phrases and a Group
Write an expression for "three times the difference of a number and seven, then
decreased by two".
Let $n$ be the number. Three phrases, applied from the inside out.
Example 3: An Expression for a Real Situation
A car park charges $$5$ to enter, plus $$3$ for every hour you stay.
The cost changes with the time, so the time is the variable. Let $h$ be the
number of hours parked.
Write the part of the cost that depends on the hours 3h
Add the entry fee, which is the same no matter how long you stay 3h + 5
Worked Example 3: A Gym Membership
A gym charges a $$25$ joining fee, then $$12$ per month.
Example 4: Reading Algebra Back Into English
Translation runs both ways, and being able to say an expression out loud is how
you check you have understood it.
$$5n - 3$$
"Five times a number, decreased by three." Or, said the other way round, "three
less than five times a number."
$$\frac{x + 4}{2}$$
"The sum of a number and four, all divided by two." The word "all" is doing the
job the fraction bar does: everything on top is divided.
$$n^2 + 2n$$
"The square of a number, plus twice that number." Both terms are about the same
number, which is why the same letter appears twice.
Common Mistakes
Mistake 1: Reversing a "less than" phrase
Wrong: "8 less than $x$" written as $8 - x$
Right: "8 less than $x$" is $x - 8$
"Less than" subtracts from the thing named after it. Start with $x$, take away $8$.
Mistake 2: Dropping the brackets when a phrase covers a group
Wrong: "twice the sum of $x$ and $3$" written as $2x + 3$
Right: "twice the sum of $x$ and $3$" is $2(x + 3)$
"Twice the sum" doubles the whole sum, not just the first thing in it.
Mistake 3: Writing a coefficient of 1
Wrong: $1x$
Right: $x$
The $1$ is always there and never written. It matters when you are counting
coefficients, and it disappears when you are writing them.
Mistake 4: Losing the sign of a coefficient
Wrong: in $5x^2 - 3x + 8$, the coefficient of the second term is $3$
Right: it is $-3$
The minus sign in front of a term is part of that term. Read the sign as
belonging to what follows it and this stops happening.
Mistake 5: One letter doing two jobs
If a problem has two unknowns, hours on Monday and hours on Tuesday, they need
two letters. Calling both of them $x$ says the two days were the same length,
which is a claim the problem never made.
Where This Goes Next
This lesson is about expressions, which have a value but make no claim. Two
close relatives are coming.
An equation sets two expressions equal:
$$3x + 5 = 20$$
That is a claim, and a claim can be true or false. It is true here only when
$x = 5$, and finding that value is what solving means.
A function turns an expression into a rule with an input and an output:
$$f(x) = 3x + 5$$
Same expression on the right. What is new is the name on the left, which lets
you talk about the output for a particular input, like $f(4)$.
Both are built out of expressions, which is why this lesson comes first.
Check Your Understanding
Four quick ones. No instructions this time.
Summary
Word
What it means
In $5x^2 - 3x + 8$
Variable
A letter standing for an unknown or changing quantity
$x$
Constant
A fixed number
$8$
Term
A number, a variable, or a product of them
$5x^2$, $-3x$, $8$
Coefficient
The number multiplying the variable, sign included