Terms, Coefficients, and Expression Structure | Grade 6 Math | LevelKite

You already know how to write an expression: numbers, variables (letters that stand for numbers) and operations put together to name one quantity. Now you will go the other way. You will look at an expression that is already written and name its parts.

Start with a shopping trip. Notebooks cost 4 dollars each. You also buy one pen for 3 dollars. Let $n$ be the number of notebooks you buy. Your total cost, in dollars, is:

4n + 3

This expression has two parts that are added together. The part $4n$ is the cost of the notebooks, in dollars. The part $3$ is the cost of the pen, in dollars.

Each part that is added is called a term. So $4n + 3$ has two terms: $4n$ and $3$ .

Now look inside the term $4n$ . It means $4 \cdot n$ : 4 dollars for each of the $n$ notebooks. The number that multiplies the variable in a term is called the coefficient. Here the coefficient of $n$ is $4$ .

The term $3$ has no variable. A term that is only a number is called a constant, because its value stays the same no matter how many notebooks you buy.

The table shows these parts in four expressions, where each letter stands for a number. In the third row, nothing is added to $7y$ , so the whole expression is one term. In the fourth row, the constant comes first, and it is still the constant.

Parts of four expressions. Each letter stands for a number.

Expression	Terms	Coefficients	Constant
$4n + 3$	$4n$ and $3$	$4$ is the coefficient of $n$	$3$
$2a + 5b$	$2a$ and $5b$	$2$ is the coefficient of $a$ ; $5$ is the coefficient of $b$	none
$7y$	$7y$ (one term)	$7$ is the coefficient of $y$	none
$9 + 6k$	$9$ and $6k$	$6$ is the coefficient of $k$	$9$

Worked example

Naming terms and coefficients

The variables $x$ and $y$ stand for numbers. Name the terms, the coefficients and the constant in $6x + y + 9$ .

Find the parts that are added. The two plus signs split the expression into three parts, and each added part is a term.
$\underbrace{6x}_{\text{term}} + \underbrace{y}_{\text{term}} + \underbrace{9}_{\text{term}}$
Look at the term 6x. It means 6 times x, so the number that multiplies x is 6. That makes 6 the coefficient of x.
$6x = 6 \cdot x$
Look at the term y. No number is written in front of it. But y is the same as 1 times y for every value of y, because multiplying a number by 1 leaves it unchanged. So the coefficient of y is 1, not 0.
$y = 1 \cdot y$
Look at the term 9. It is only a number, with no variable, so it is the constant.

Answer

The expression $6x + y + 9$ has three terms: $6x$ , $y$ and $9$ . The coefficient of $x$ is $6$ , the coefficient of $y$ is $1$ , and the constant is $9$ .

An expression names a quantity. It does not say that the quantity is equal to anything. On the shopping trip, $4n + 3$ is your total cost in dollars, and that is all it tells you.

Now suppose you know that you spent 23 dollars in all. You can say so with an equals sign:

4n + 3 = 23

This is an equation: a statement that two expressions have the same value. An equation always has an equals sign with an expression on each side. Here the left side is $4n + 3$ and the right side is $23$ . A number by itself counts as an expression.

You can see the difference when you choose a value for $n$ .

Replace $n$ with $2$ in the expression $4n + 3$ . You get a number: $4 \cdot 2 + 3 = 8 + 3 = 11$ .
Replace $n$ with $2$ in the equation $4n + 3 = 23$ . You get a claim, $11 = 23$ , and that claim is false.
Replace $n$ with $5$ in the equation. The left side is $4 \cdot 5 + 3 = 20 + 3 = 23$ , so the claim is $23 = 23$ , which is true.

Once $n$ has a value, the expression gives you a number. The equation gives you a statement that is true or false.

So to tell the two apart, look for an equals sign. An equation always has one. An expression never does.

Worked example

Sorting expressions and equations

The variable $k$ stands for a number. Sort these four into expressions and equations: $5k + 2$ , $5k + 2 = 17$ , $9 = 4 + 5$ and $6 + 10$ .

Look at 5k + 2. It has no equals sign, so it cannot be an equation. It names one quantity, so it is an expression.
$5k + 2 \;\rightarrow\; \text{expression}$
Look at 5k + 2 = 17. It has an equals sign with an expression on each side: 5k + 2 on the left and 17 on the right. It states that the two sides have the same value, so it is an equation.
$5k + 2 = 17 \;\rightarrow\; \text{equation}$
Look at 9 = 4 + 5. It has no variable, but it does have an equals sign with an expression on each side. It states that 9 and 4 + 5 have the same value, so it is an equation.
$9 = 4 + 5 \;\rightarrow\; \text{equation}$
Look at 6 + 10. It has no equals sign, so it cannot be an equation. It names one quantity, which has the value 16, but it does not state that two things are equal. It is an expression.
$6 + 10 \;\rightarrow\; \text{expression}$

Answer

The expressions are $5k + 2$ and $6 + 10$ . The equations are $5k + 2 = 17$ and $9 = 4 + 5$ . The equals sign decides, not the variable.

You have used the words sum, product and quotient to write expressions. The same words describe an expression that is already written.

A sum is the result of adding. The parts you add are its terms.
A product is the result of multiplying. The parts you multiply are its factors.
A quotient is the result of dividing. The part you divide by is the divisor. The divisor must not be zero, because dividing by zero does not give a number.

Many expressions use more than one operation. To name the whole expression, imagine a number in place of each variable. Then think about the order of operations, the agreed order for finding a value. For the expressions in this lesson, that order is: any grouped part first (inside parentheses, or above or below a fraction bar), then multiplying and dividing, then adding.

The operation you would do last names the whole expression: adding last makes a sum, multiplying last makes a product, and dividing last makes a quotient. In this lesson, the last operation is always one of these three.

The table gives five examples. Each letter in it stands for a number, and $d$ is not zero because it is a divisor.

Naming five expressions by the operation done last. Here

d

is not zero.

Expression	What you would do last	Description
$n + 6$	add $n$ and $6$	the sum of $n$ and $6$
$7y$	multiply $7$ by $y$	the product of $7$ and $y$
$\frac{20}{d}$	divide $20$ by $d$	the quotient of $20$ and $d$
$4n + 3$	add $3$ , after multiplying $4$ by $n$	the sum of $4n$ and $3$
$2(8 + 7)$	multiply by $2$ , after adding $8 + 7 = 15$	the product of $2$ and the sum $8 + 7$

Key concept

An expression is built from named parts

The structure of an expression tells you what to call it and what to call its parts.

The parts that are added are terms. In a term, the number that multiplies the variable is the coefficient. A term that is only a number is a constant.
The operation you would do last names the whole expression. Adding last makes a sum. Multiplying last makes a product. Dividing last, by a divisor that is not zero, makes a quotient.
An equals sign between two expressions makes an equation: a statement that the two sides have the same value.

Worked example

Same numbers, three different structures

The variable $x$ stands for a number. Describe each of these as a sum, a product or a quotient: $3x + 4$ , $3(x + 4)$ and $\frac{x + 4}{3}$ .

Start with 3x + 4. Nothing is grouped, and multiplying comes before adding. So you would multiply 3 by x first and add 4 last. Adding last makes a sum. Its terms are 3x and 4.
$3x + 4 \;\rightarrow\; \text{sum}$
Next, 3(x + 4). The parentheses group x + 4, so you would add first and multiply by 3 last. Multiplying last makes a product. Its factors are 3 and x + 4.
$3(x + 4) \;\rightarrow\; \text{product}$
Last, the fraction. The fraction bar groups the top, so you would add x and 4 first and divide by 3 last. The divisor 3 is not zero, so the division is allowed. Dividing last makes a quotient.
$\frac{x + 4}{3} \;\rightarrow\; \text{quotient}$
Check that the structure matters by replacing x with 2 in all three. The values are all different. One value of x where expressions disagree is enough to show they are not equivalent, so the place of the grouping really does change the expression.
$3 \cdot 2 + 4 = 6 + 4 = 10 \qquad 3(2 + 4) = 3 \cdot 6 = 18 \qquad \frac{2 + 4}{3} = \frac{6}{3} = 2$

Answer

The expression $3x + 4$ is a sum, $3(x + 4)$ is a product, and $\frac{x + 4}{3}$ is a quotient.

Worked example

Putting the three ideas together

The variable $w$ stands for a number. Is $2(w + 4) = 3w + 5$ an expression or an equation? Describe each side as a sum, a product or a quotient. Then name the terms, the coefficient and the constant on the right side.

Look for an equals sign. There is one, with an expression on each side, so this is an equation.
$2(w + 4) = 3w + 5$
Describe the left side. The parentheses group w + 4, so multiplying by 2 comes last. The left side is a product with factors 2 and w + 4.
$2(w + 4) \;\rightarrow\; \text{product}$
Describe the right side. Multiplying 3 by w comes first, so adding 5 comes last. The right side is a sum.
$3w + 5 \;\rightarrow\; \text{sum}$
Name the parts of that sum. Its terms are 3w and 5. The number 3 multiplies w, so 3 is the coefficient of w. The term 5 has no variable, so it is the constant.
$\underbrace{3w}_{\text{coefficient } 3} + \underbrace{5}_{\text{constant}}$

Answer

The statement $2(w + 4) = 3w + 5$ is an equation. Its left side is the product of $2$ and $w + 4$ . Its right side is the sum of the terms $3w$ and $5$ , where $3$ is the coefficient of $w$ and $5$ is the constant.

Takeaway. An expression has a structure, and each part of that structure has a name. Terms are the parts that are added, and a coefficient is the number that multiplies a variable. The operation you would do last (adding, multiplying, or dividing by a number that is not zero) makes the whole expression a sum, a product or a quotient. An expression names a quantity. An equation states that two expressions are equal.

Next, you will name the parts of expressions that come from formulas, and you will learn to treat a whole part of an expression as one single thing.