Two-Operation Phrases and Grouping | Grade 6 Math | LevelKite

You can already turn a one-operation phrase into an algebraic expression: a math phrase made of numbers, operation signs and at least one variable, which is a letter that stands for a number. For example, “7 more than a number $n$ ” becomes $n + 7$ .

You also know the operation words. A sum is the result of adding, a difference is the result of subtracting, a product is the result of multiplying, and a quotient is the result of dividing. “Twice” means 2 times.

In this lesson each phrase has two operations, such as “7 more than 3 times a number $n$ .” The new question is: which operation do you do first, and how do you show that order in symbols?

Start with plain numbers. Take the number 4 and use the same two operations, multiplying by 2 and adding 5, in two different orders.

“5 more than twice 4”: multiply first to get 8, then add 5 to get 13.
“Twice the sum of 4 and 5”: add first to get 9, then multiply by 2 to get 18.

The order changed the result, so an expression has to show the order. Grouping symbols are marks that show which part you work out first. Parentheses ( ) are the most common: you work out whatever is inside them first. When there are no grouping symbols, you multiply and divide before you add and subtract. Here are the two orders in symbols:

2 \cdot 4 + 5 = 8 + 5 = 13 \qquad \text{but} \qquad 2 \cdot (4 + 5) = 2 \cdot 9 = 18

Key concept

Inner quantity first, then the outer operation

Each two-operation phrase in this lesson has a one-operation phrase inside it, such as “the sum of $n$ and 4” or “twice $n$ .” That inside part is the inner quantity: you work it out first. The other operation is the outer operation: it acts on the whole inner quantity.

Write the inner quantity first. Then apply the outer operation to all of it. If the inner quantity is a sum or a difference and the outer operation multiplies or divides it, you must group it: use parentheses, or a fraction bar when you divide. Parentheses around the inner quantity never change the meaning, so you may add them whenever you are unsure.

Rows 1 and 2 repeat the number example with n in place of 4. In every row, n stands for the unknown number.

Word phrase	Inner quantity (work out first)	Outer operation	Expression
5 more than twice a number $n$	twice $n$ , which is $2n$	add 5 to it	$2n + 5$
twice the sum of a number $n$ and 5	the sum $n + 5$	multiply the whole sum by 2	$2(n + 5)$
4 less than the quotient of a number $n$ and 3	the quotient $\dfrac{n}{3}$	subtract 4 from it	$\dfrac{n}{3} - 4$
the difference of a number $n$ and 4, divided by 3	the difference $n - 4$	divide the whole difference by 3	$\dfrac{n - 4}{3}$

Worked example

Multiply first, then add

Write an algebraic expression for “7 more than the product of 3 and a number $n$ .” Then test it with $n = 2$ .

Name the variable. The phrase calls the unknown number n, so n stands for that number in every step.
Find the inner quantity. The words “the product of 3 and a number n” are a one-operation phrase inside the bigger phrase. They mean 3 multiplied by n.
$3 \cdot n = 3n$
Apply the outer operation. “7 more than” means add 7 to the whole product. You do not need parentheses here, because with no grouping symbols you already multiply before you add.
$3n + 7$
Test with n = 2. Follow the words: the product of 3 and 2 is 6, and 7 more than 6 is 13. The expression also gives 13, so the test finds no mistake.
$3 \cdot 2 + 7 = 6 + 7 = 13$

Answer

The phrase “7 more than the product of 3 and a number $n$ ” is written as $3n + 7$ . When $n = 2$ , the words and the expression both give $13$ .

Worked example

Add first, then multiply

Write an algebraic expression for “5 times the sum of a number $n$ and 2.” Then test it with $n = 4$ .

Find the inner quantity. Here n stands for the unknown number. The words “the sum of a number n and 2” are the one-operation phrase inside, so the inner quantity is n plus 2.
$n + 2$
Apply the outer operation. “5 times” multiplies the whole sum, not just n. A sum that you multiply as a whole must be grouped, so put it in parentheses. Now you add before you multiply.
$5(n + 2)$
Test with n = 4. Follow the words: the sum of 4 and 2 is 6, and 5 times 6 is 30. The expression also gives 30, so the test finds no mistake.
$5(4 + 2) = 5 \cdot 6 = 30$

Answer

The phrase “5 times the sum of a number $n$ and 2” is written as $5(n + 2)$ . When $n = 4$ , the words and the expression both give $30$ .

Now go the other way: you start with an expression and must match it to a word phrase. Every two-operation expression in this lesson has one addition or subtraction and one multiplication or division. Read it with two questions.

What do you work out first? If the sum or difference is inside parentheses, or fills the top or bottom of a fraction, you work it out first. Otherwise you work out the multiplication or division first. This is the inner quantity.
What happens to that result? This is the outer operation.

The matching phrase names the same inner quantity and does the same thing to it. The try-a-number test helps too: a pair that gives two different results is not a match, so the test can rule out wrong pairs.

Worked example

Match three expressions to three phrases

Let $n$ stand for a number. Match each expression to its word phrase. Expressions: $4n + 8$ , $4(n + 8)$ and $\dfrac{n + 8}{4}$ . Phrases: (A) 4 times the sum of the number and 8. (B) The sum of the number and 8, divided by 4. (C) 8 more than 4 times the number.

Read 4n + 8. It has no parentheses and no fraction bar, so you multiply first: the inner quantity is 4 times n. Then you add 8 to it. Phrase C names the same inner quantity and the same outer operation.
$4n + 8 \;\rightarrow\; \text{phrase C}$
Read 4(n + 8). The sum is inside parentheses, so it is the inner quantity. Then you multiply the whole sum by 4. Phrase A says the same.
$4(n + 8) \;\rightarrow\; \text{phrase A}$
Read the fraction. The sum fills the top, and the fraction bar groups it, so the sum is the inner quantity. Then you divide the whole sum by 4. Phrase B says the same.
$\dfrac{n + 8}{4} \;\rightarrow\; \text{phrase B}$
Test the first pair with n = 4. Phrase C: 4 times 4 is 16, and 8 more than 16 is 24. The expression also gives 24, so this pair passes.
$4 \cdot 4 + 8 = 16 + 8 = 24$
Test the second pair with n = 4. Phrase A: the sum of 4 and 8 is 12, and 4 times 12 is 48. The expression also gives 48, so this pair passes.
$4(4 + 8) = 4 \cdot 12 = 48$
Test the third pair with n = 4. Phrase B: the sum of 4 and 8 is 12, and 12 divided by 4 is 3. The expression also gives 3. The results 24, 48 and 3 are all different, so any other pairing would give two different results. The test rules those pairings out.
$\dfrac{4 + 8}{4} = \dfrac{12}{4} = 3$

Answer

The expression $4n + 8$ matches phrase C, $4(n + 8)$ matches phrase A, and $\dfrac{n + 8}{4}$ matches phrase B.

The big idea: each two-operation phrase you met here has an inner quantity and an outer operation. Write the inner quantity first. Then make the outer operation act on all of it. If the inner quantity is a sum or a difference that gets multiplied or divided, group it with parentheses, or with a fraction bar for division. To match an expression to a phrase, read it the same way: what do you work out first, and what happens to that result?

Next, you will use these skills to write expressions for word problems and to find mistakes in written expressions.