Back To Course

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

Are you a student or a teacher?

Try Study.com, risk-free

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.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

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

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did 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 109: Introduction to Programming
- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Progressive Politics & American Imperialism
- Reconstruction, Westward Expansion, Industrialization & Urbanization
- North America & the 13 Colonies
- The Renaissance & The Age of Exploration
- Algorithmic Analysis, Sorting & Searching
- CEOE Test Cost
- PHR Exam Registration Information
- Claiming a Tax Deduction for Your Study.com Teacher Edition
- What is the PHR Exam?
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP

- Convergent Sequence: Definition, Formula & Examples
- Mauryan Empire Art & Culture
- Multi-Dimensional Arrays in C Programming: Definition & Example
- Tests for Identifying Common Gases
- Singing Lesson Plan
- Arrays & Strings in JavaScript: Conversion & String Methods
- Heuristic Methods in AI: Definition, Uses & Examples
- Quiz & Worksheet - Average & Instantaneous Rates of Change
- Quiz & Worksheet - Speed, Velocity & Acceleration
- Quiz & Worksheet - Functions & Parameters Overview
- Quiz & Worksheet - Incremental & Radical Change
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Parts of Speech Worksheets
- Music Lesson Plans

- Calculus: Help and Review
- GACE Mathematics (522): Practice & Study Guide
- College Chemistry: Tutoring Solution
- Introduction to Computing: Certificate Program
- NMTA Essential Academic Skills Subtest Reading (001): Practice & Study Guide
- AP Physics 2: Thermal Equilibrium & Entropy
- Sedimentary Rocks in Geology: Help and Review
- Quiz & Worksheet - Culture & Technology in the 1990s
- Quiz & Worksheet - Casualties During the Civil War
- Quiz & Worksheet - Battle of Cold Harbor
- Quiz & Worksheet - Expatriate Staffing Issues
- Quiz & Worksheet - Conflict in India & Partition with Pakistan

- Historical Change: Causes and Effects
- What is the Praetorian Guard?
- How to Find Study.com Courses for College Credit
- NYS Earth Science Regents Exam Information
- Finding Traveling Teacher Jobs
- What is Professional Development for Teachers?
- How to Pass the CCRN Exam
- Poetry for 2nd Grade
- Following Directions Activities & Games
- Harlem Renaissance Lesson Plan
- Free Online High School Math Courses
- French Revolution Lesson Plan

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

Browse by subject