Using Parentheses in Math: Rules & Examples

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  • 0:01 Parentheses in Math
  • 0:29 Separating Numbers for Clarity
  • 0:51 Indicating Multiplication
  • 1:07 Order of Operations
  • 2:25 Practice Examples
  • 3:32 Lesson Summary
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Lesson Transcript
Instructor
Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

In this lesson, we will explore how to use parentheses to clarify numbers, indicate multiplication, and group numbers together for the order of operations. After exploring the rules of parentheses, review your understanding of this lesson with a short quiz.

Parentheses in Math

In English, we use parentheses to indicate clarification for a concept (we add extra information using parentheses). The following rules will help you understand how parentheses are used in mathematics for different purposes. Parentheses are used in mathematics for three main purposes: to separate numbers for clarity, to indicate multiplication, and to group numbers together.

Separating Numbers for Clarity

Parentheses can be used to separate numbers for clarity. For example, if you had an addition problem with a negative number, parentheses would be used to separate the two signs. Parentheses can also be used to separate a number from its exponent. This usually happens if you are raising a negative number to a power.

Indicating Multiplication

When a number appears next to another number with parentheses, you need to multiply the two numbers. For example, when you see 2(3), you multiply 2 and 3.

Order of Operations

Think about when you tie your shoes or make a sandwich. You can't put the mustard on your sandwich if you don't have any bread, right? Or, maybe you like to have a routine in the morning: brush your teeth first before you wash your face. Mathematics is the same way. You have to solve mathematical problems in a certain order, called the order of operations.

The primary use of parentheses is to group numbers together in mathematical problems. When you see multiple numbers and operations inside parentheses, use the order of operations to solve the problem. The order of operations is:

  1. Parentheses
  2. Exponents
  3. Multiplication and/or division (whichever comes first from left to right)
  4. Addition and/or subtraction (whichever comes first from left to right)

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Additional Activities

When to Use Parentheses with Calculators

This extension to the lesson will help students understand that on some problems parentheses need to be used to enter the problem into a calculator, even if there weren't parentheses to begin with. Consider the following problem 2/(7 + 3)

We know to simplify the denominator first and then simplify the fraction (and convert to a decimal if we prefer)

2/(7 + 3) = 2/10 = 1/5 = 0.2

Students know to group the terms in the denominator together because they have been taught to recognize the fraction bar as separating the numerator and the denominator into groups. But what if we try typing this into a calculator without simplifying on our own first?


Have the student try to type the problem into a calculator exactly as it is written: 2/(7 + 3) and hit enter. The calculator will say 3.2857142857142..., perhaps with more or less decimal places. This is not the correct answer . What happened?

Calculators and computers are programmed to follow the order of operations. When the student typed in 2/(7 + 3), the calculator evaluated the expression as it was typed: it performed division before addition, as the order of operations states to do. The calculator evaluated 2/7 +3, which is not what we intended. To fix the problem, have the student try using parentheses around the denominator to tell the calculator to simplify the 7 + 3 first. Have the student type 2/(7+3) and hit enter. Now the calculator will say 0.2 which is correct.


Practice Problems

There are other situations where we would need to type parentheses in a calculator when they are not explicitly written in the problem. Have the student try to solve the problems by typing into a calculator. Parentheses will need to be added to get the correct answer. The answer, if typed correctly, will be the answer stated. Have the student try to find where parentheses should be used to get the correct result.

1. (2 + 5)/5 Answer should be 1.4

2. (3 - 7)/(2 - 1) Answer should be 4

3. 2(3 + 1) Answer should be 16

Solutions for How to Type

To get the correct answers listed above, the problems should be typed as follows:

1. (2 + 5)/5

Group the numerator terms with parentheses

2. (3 - 7)/(2 - 1)

Group the numerator terms with parentheses and group the denominator terms with parentheses.

3. 2(3 + 1)

Group the entire exponent in parentheses.

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