Back To Course

ELM: CSU Math Study Guide17 chapters | 147 lessons | 7 flashcard sets

Are you a student or a teacher?

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

The order of operations is important and useful, but a few mathematical properties highlight cases where order doesn't matter. In this lesson, we'll learn about the commutative and associative properties, which may save you time and effort.

Math is full of rules. You have to divide before you subtract. 2 + 2 has to equal 4. You can only eat pie after you finish your vegetables. Working with pi will always make you think of pie.

Fortunately, there are a few rules that actually make math simpler. These are like the Casual Fridays of mathematics. They're rules, yes, but they define how you can loosen up a bit and lose that tie.

Let's say you want to know what 3 + 8 is. Do you have to add 8 to 3? Or, could you add 3 to 8? It doesn't matter, right? And, the same thing is true if you want to know 2 * 5. That's the same as 5 * 2.

These examples illustrate the **commutative property**, which states that the order of the numbers when you add or multiply doesn't affect the sum or product. In other words, *a* + *b* = *b* + *a* and *ab* = *ba*.

The name comes from the word 'commute.' When you commute, what are you doing? You're moving from one place to another. When you get to the end of your commute, you're still the same person. Well, unless you're commuting using a malfunctioning teleporter.

Note that the commutative property doesn't work for subtraction or division. 3 - 8 does not equal 8 - 3. And, 10/5 does not equal 5/10. But, it does apply with addition and multiplication.

Let's practice. Let's say you have 2 * 5 * 3. That's 30. What if we rearrange it to 3 * 5 * 2? Still 30. 5 * 2 * 3? Still 30. 2 * 3 * 5? Yep, still 30.

Here's one with addition: 10 + 7 + 9. That's 26. If we rearrange it to 9 + 10 + 7? Still 26.

You could use the commutative property to justify eating your dessert first at dinner. After all, no matter which order you eat the food, it all ends up in your stomach. So, why not polish off that ice cream before getting to the broccoli? Well, that's sort of the same thing, but not exactly.

When you add 10 and 7 and 9, you're always dealing with constant numbers. If you had all the parts of your meal laid out, and you were sure to have room for all of them in your stomach, then order really doesn't matter. Granted, parents everywhere may still not approve.

There's another law that's similar to the commutative property. To understand this one, let's imagine the world's saddest yard sale. You're selling three things: a broken hair dryer for $1, a three-legged chair for $4 and a box of old VHS tapes for $2.

Let's say your neighbor Mrs. Lake buys the hair dryer. Then your other neighbor, Mr. Rivers, buys the chair and tapes. You just made $1 from Mrs. Lake and $4 + $2 from Mr. Rivers - that's $7. While that won't buy you nicer stuff, it will buy you a burrito with guacamole.

But what if Mrs. Lake bought the hair dryer and the chair? And then, Mr. Rivers bought just the tapes? You'd then make $1 + $4 from Mrs. Lake and $2 from Mr. Rivers. You'd still get $7. And, you'd still get that burrito.

This sad yard sale illustrates the **associative property**, which states that the way you group numbers when you add or multiply doesn't affect the sum or product. In other words (*a* + *b*) + *c* = *a* + (*b* + *c*) and *a*(*bc*) = (*ab*)*c*.

Whether Mrs. Lake buys two items and Mr. Rivers buys one or Mrs. Lake buys one and Mr. Rivers buys two, you still get $7.

That was an addition example, but it works the same with multiplication. Let's say you have this: (5 * 2) * 3. If you remember the order of operations, you need to handle the stuff inside the parentheses first. That gets you 10 * 3, which is 30. But, the associative property says that (5 * 2) * 3 is the same as 5 * (2 * 3). That latter format gets you 5 * 6, which is, yep, also 30.

Note that I said the property works for addition and multiplication. The associative property doesn't work for subtraction and division.

(7 - 4) - 2 does not equal 7 - (4 - 2). With (7 - 4) - 2, you first subtract 7 - 4 to get 3. Then you do 3 - 2, to get 1. In 7 - (4 - 2), you start with 4 - 2, which is 2. You then do 7 - 2, which is 5.

Let's try a few of these. Here's one: (5 + 10) + 7. Again, the order of operations says we need to do that 5 + 10 first. But, since everything here is addition, the grouping doesn't matter. So, you could add the 10 and 7 first. In other words (5 + 10) + 7 = 5 + (10 + 7). No matter how you group it, you get 22.

How about this one: 6 * (2 * 5)? If you do 2 * 5 first, you get 6 * 10, which is 60. But, the associative property tells us that we could go (6 * 2), which is 12, then multiply that by 5, which still gets us 60.

In summary, the commutative property states that order doesn't matter when you are performing only addition or only multiplication. *a* + *b* = *b* + *a* and *ab* = *ba*. Your terms can commute around without changing the result.

The associative property states that the way you group numbers when you add or multiply doesn't matter. (*a* + *b*) + *c* = *a* + (*b* + *c*) and *a*(*bc*) = (*ab*)*c*. Your terms can associate with whomever they'd like.

After watching this lesson, you'll be able to describe what commutative and associative properties are and work problems with these properties.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackDid you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
4 in chapter 6 of the course:

Back To Course

ELM: CSU Math Study Guide17 chapters | 147 lessons | 7 flashcard sets

- What is a Variable in Algebra? 5:26
- Expressing Relationships as Algebraic Expressions 5:12
- Evaluating Simple Algebraic Expressions 7:27
- The Commutative and Associative Properties and Algebraic Expressions 6:06
- Combining Like Terms in Algebraic Expressions 7:04
- Practice Simplifying Algebraic Expressions 8:27
- Negative Signs and Simplifying Algebraic Expressions 9:38
- Go to ELM Test - Algebra: Basic Expressions

- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Computer Science 203: Defensive Security
- GRE Information Guide
- Computer Science 310: Current Trends in Computer Science & IT
- FTCE: Equations and Inequalities
- FTCE: Analyzing Data and Drawing Conclusions
- FTCE: Data Analysis & Visualization
- The Cybersecurity Threat Landscape
- Cybersecurity Policy, Governance & Management
- What is the ASCP Exam?
- ASCPI vs ASCP
- MEGA Exam Registration Information
- MEGA & MoGEA Prep Product Comparison
- PERT Prep Product Comparison
- MTLE Prep Product Comparison
- What is the MTLE Test?

- How to Determine the Number of Main Ideas in a Text
- Sequence of Events in a Narrative: Lesson for Kids
- The Square Root Property
- Number Theory: Divisibility & Division Algorithm
- Guided Reading Lesson Plan Template & Example
- Practical Application for Introduction to SQL: Views
- Computer Security Risk Assessment Computations: SLE, ALE & ARO
- Quiz & Worksheet - Slang Words in The Outsiders
- Quiz & Worksheet - Othello's Soliloquy
- Quiz & Worksheet - Adjugate Matrices
- Quiz & Worksheet - Double Angle Properties & Rules
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Common Core ELA Standards
- 2nd Grade Math Worksheets

- College Algebra Textbook
- Critical Thinking, Problem Solving & Decision Making
- Nutrition 101 Curriculum Resource & Lesson Plans
- Business Law for Teachers: Professional Development
- Introduction to Counseling: Certificate Program
- Invertebrates for High School Biology Lesson Plans
- Chapter 10: Nuclear Chemistry
- Quiz & Worksheet - Club Drug Dependence Symptoms
- Quiz & Worksheet - Drug Tolerance & Overdose
- Quiz & Worksheet - Middle Imperial China's Imperial State & Scholar-Official Class
- Quiz & Worksheet - Call & Response in Music
- Quiz & Worksheet - How Christianity Spread in Medieval Europe

- Found and Prepared Instruments in Modern Music
- Verifying Trigonometric Identities with Unit Circles
- How to Pass the California Bar Exam
- Life Cycle of a Frog Lesson Plan
- Common Core State Standards in Colorado
- Is Studying Computer Science Hard?
- Is AP Environmental Science Hard?
- Creative Writing Exercises for Kids
- Texas Physical Education Requirements
- Civil Rights Activities for Kids
- How to Write a Personal Statement for Law School
- AP Chemistry Exam Scoring Information

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject