Chapter 8 · Expressions

Read a Formula as a Structured Expression

Read the expression in a formula part by part. Name its terms, factors and coefficients, and see a grouped part as one single quantity.

13 min read

A formula is a rule written as an equation. It tells you how to find one quantity from other quantities. In each formula in this lesson, the right side is an expression, and the parts of that expression stand for real quantities.

You can already name an expression as a sum, a product or a quotient. Now you will read the expression in a formula the same way. You will name its parts, say what each part stands for, and learn to see a whole part as one single thing.

Start with a rectangle. Let $l$ be its length in centimeters (cm) and $w$ be its width in centimeters. Both are positive numbers. Its perimeter $P$ is the distance around it, also in centimeters. A rectangle has two sides of length $l$ and two sides of length $w$ , so one formula for its perimeter is $P = 2l + 2w$ . Here $2l$ means $2 \times l$ .

Keep four ideas in mind as you read the expression $2l + 2w$ :

The terms of a sum are the parts being added.
The factors of a product are the parts being multiplied.
A coefficient is the number factor in a term that has a variable, such as the $2$ in $2l$ .
The operation you do last, when you follow the order of operations, names the whole expression. Add last and it is a sum. Multiply last and it is a product. Divide last, by a quantity that is not zero, and it is a quotient.

Worked example

Read a sum from a formula

A rectangle has length $l = 9$ cm and width $w = 4$ cm. Its perimeter in centimeters is $P = 2l + 2w$ . Name the kind of expression $2l + 2w$ is. Identify its terms, factors and coefficients. Then find $P$ .

Find the last operation. There are no parentheses here, so the order of operations puts multiplication before addition. You multiply $2 \times l$ and $2 \times w$ first, and you add last. Adding last makes the whole expression a sum.
$2l + 2w = (2 \times l) + (2 \times w)$
Name the terms. The terms of a sum are the parts being added, so this sum has two terms. The term $2l$ stands for the two sides of length $l$ together. The term $2w$ stands for the two sides of length $w$ together.
Look inside each term. Each term is a product of two factors. The term $2l$ has the factors $2$ and $l$ , and the term $2w$ has the factors $2$ and $w$ . The number factor $2$ is the coefficient in each term. It counts a pair of equal sides.
$2l = 2 \times l \qquad 2w = 2 \times w$
Find the value of each term. Replace $l$ with $9$ and $w$ with $4$ , because these are the side lengths in centimeters.
$2 \times 9 = 18 \qquad 2 \times 4 = 8$
Add the two terms, because the expression is a sum. The sides of length $l$ give $18$ cm and the sides of length $w$ give $8$ cm.
$P = 18 + 8 = 26$

Answer

The expression $2l + 2w$ is a sum of two terms, $2l$ and $2w$ . Each term is a product with the coefficient $2$ , and the perimeter is $26$ cm.

Now walk around the same rectangle in another way. Go along one length and then one width. You have covered $l + w$ centimeters, and you are halfway around. Make the same trip again and you are back where you started. So a second formula for the perimeter is $P = 2(l + w)$ . Both formulas work for every rectangle, because both count the same four sides.

The parentheses matter. They bundle $l + w$ into one quantity: the distance halfway around. The $2$ then multiplies that whole quantity. When you treat a part of an expression as one quantity like this, you are viewing it as a single entity.

Key concept

One part, two views

A part of an expression that is built from smaller pieces can be viewed in two ways at once.

Zoomed out, it is a single entity: one quantity doing one job in the whole expression, such as one term or one factor.
Zoomed in, it is an expression of its own, with its own terms or factors.

In $2l + 2w$ , the part $2l$ is one term, and it is also a product of two factors. In $2(l + w)$ , the part $(l + w)$ is one factor, and it is also a sum of two terms. When an entity is written in parentheses, the parentheses show where it begins and ends.

Worked example

Read a product that has a sum inside

Use the same rectangle, with $l = 9$ cm and $w = 4$ cm, and the formula $P = 2(l + w)$ , where $P$ is in centimeters. Name the kind of expression $2(l + w)$ is. Describe the part $(l + w)$ in two ways. Then find $P$ .

Find the last operation. Parentheses come first in the order of operations, so you add $l + w$ first and multiply by $2$ last. Multiplying last makes the whole expression a product.
$2(l + w) = 2 \times (l + w)$
Zoom out and name the factors. The factors of a product are the parts being multiplied. There are two: the number $2$ and the single entity $(l + w)$ , which stands for the distance halfway around.
Zoom in on $(l + w)$ . Inside the parentheses the only operation is addition, so this part is a sum of the two terms $l$ and $w$ .
Find the value of the entity. Replace $l$ with $9$ and $w$ with $4$ , and add inside the parentheses first so that the entity becomes one number.
$9 + 4 = 13$
Multiply the two factors, because the expression is a product.
$P = 2 \times 13 = 26$

Answer

The expression $2(l + w)$ is a product of the two factors $2$ and $(l + w)$ . The part $(l + w)$ is a single entity and also a sum of two terms, and the perimeter is $26$ cm.

Watch the plus sign. A common mistake is to see the $+$ in $2(l + w)$ and call the whole expression a sum with the terms $2l$ and $w$ . That reads it as $2l + w$ , so the $2$ multiplies only $l$ .

Test it with $l = 9$ and $w = 4$ . The mistaken reading gives $2 \times 9 + 4 = 18 + 4 = 22$ . The formula gives $2 \times (9 + 4) = 2 \times 13 = 26$ . The values differ. Equivalent expressions agree for every allowed value of the variables, here every positive $l$ and $w$ . So this one mismatch shows the two expressions are not equivalent. One match would never prove that two expressions are equivalent.

The mistake fails because it counts both lengths but only one width. The parentheses make $(l + w)$ one entity, and the $2$ multiplies all of it.

Viewing $(l + w)$ as a single entity also helps you reason. Rectangle A is $9$ cm by $4$ cm. Rectangle B is $10$ cm by $3$ cm. The entity has the same value for both, because $9 + 4 = 13$ and $10 + 3 = 13$ . So $9 + 4 = 10 + 3$ .

Now multiply both sides by $2$ . Applying the same valid operation to both sides keeps them equal, so $2(9 + 4) = 2(10 + 3)$ . Each side is $2 \times 13 = 26$ , so both rectangles have a perimeter of $26$ cm. You did not need to treat $l$ and $w$ separately. The value of the single entity decided the perimeter.

A fraction bar groups what is written above it and what is written below it, just as parentheses do. Think about a bike trip. Let $a$ be the distance you ride before lunch, in kilometers (km). Let $b$ be the distance you ride after lunch, in kilometers. Let $t$ be your total riding time, in hours.

The time $t$ must not be zero, because you can divide only by a nonzero quantity. A real ride always takes some time, so $t$ is greater than $0$ .

Your average speed $s$ , in kilometers per hour, is your total distance divided by your total riding time:

s = \frac{a + b}{t}, \quad t \neq 0

Worked example

Read a quotient with a sum on top

You ride $a = 24$ km before lunch and $b = 36$ km after lunch. Your total riding time is $t = 4$ hours. Your average speed in kilometers per hour is $s = \frac{a + b}{t}$ . Name the kind of expression $\frac{a + b}{t}$ is. Describe its top part in two ways. Then find $s$ .

Find the last operation. The fraction bar groups the top part, so you add $a + b$ first and divide by $t$ last. Dividing last makes the whole expression a quotient.
$\frac{a + b}{t} = (a + b) \div t$
Zoom out. The dividend, the amount being divided, is the single entity $(a + b)$ . It stands for your total distance. The divisor, the amount you divide by, is $t$ . Here $t = 4$ , which is not zero, so you may divide by it.
Zoom in on the dividend. It is a sum of the two terms $a$ and $b$ . Replace $a$ with $24$ and $b$ with $36$ , and add to turn the entity into one number of kilometers.
$24 + 36 = 60$
Divide the total distance by the riding time, because the expression is a quotient.
$s = 60 \div 4 = 15$

Answer

The expression $\frac{a + b}{t}$ is a quotient. Its dividend $(a + b)$ is a single entity and also a sum of two terms, and your average speed is $15$ kilometers per hour.

Three formula expressions, read zoomed out and zoomed in

Expression	Last operation	Zoomed out	Zoomed in
$2l + 2w$	Addition, so it is a sum	Two terms: $2l$ and $2w$	Each term is a product of the coefficient $2$ and a variable
$2(l + w)$	Multiplication, so it is a product	Two factors: $2$ and $(l + w)$	$(l + w)$ is a sum of the terms $l$ and $w$
$\frac{a + b}{t}$ with $t \neq 0$	Division, so it is a quotient	Dividend $(a + b)$ and divisor $t$	$(a + b)$ is a sum of the terms $a$ and $b$

Takeaway. The expression in a formula has a structure you can read. Find the operation you do last, following the order of operations, to name the whole expression as a sum, a product or a quotient. Then name its parts, such as terms, factors, coefficients, or a dividend and a divisor, and say what each one stands for. A part that is built from smaller pieces can be viewed in two ways: as a single entity doing one job in the whole expression, and as an expression with parts of its own.

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Evaluate Expressions by Substituting Numbers

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