Commutative, Associative, and Distributive Property
This video gives examples of the Commutative, Associative, and Distributive property. The associative property states that when adding or multiplying a series of numbers, it does not matter how the terms are ordered. Simplifying within the parentheses is the first step using the order of operations.
Free Associative Property Fact Sheet
Use the associative property fact sheet below to help you get a better understanding of how associative property work. You are encouraged to print or download the associative property fact sheet with the PDF link at the bottom of the page.
Commutative, Associative, and Distributive Property
As you may have already realized through the years of math classes and homework, math is sequential in nature, meaning that each concept is built upon prior work. Arithmetic skills are necessary to conquer algebraic concepts, which are then developed further to be used in calculus, and so on. As you’re building these concepts over time, the math process may become automatic, but the reason, or justification for the work, may be long forgotten. In this video, we will go back to the basics to review the Commutative, Associative, and Distributive Properties of real numbers, which allow for the math mechanics of algebra and beyond.
The names of the properties that we’re going to be looking at are helpful in decoding their meanings. Consider the word, “commutative.” What do you think of when you see this word? When I look at this word, I see the word commute. That word reminds me of move, which is pretty much what the commutative property allows you to do when adding or multiplying algebraic terms. The Commutative property looks like this, mathematically:
Let’s take a minute to remember that the definition of an algebraic term: it is a number, variable, or product of coefficients and variables. Examples of algebraic terms are 3, 3x, 3xy, 3xy2, and so on. To prove that moving, or rearranging, terms is acceptable, let’s look at a few examples of the commutative property being used in addition problems.
If we add 5 plus 3, we get 8. But if we switch our terms and make it 3 + 5, we still get 8. So 5 + 3 is equal to 3 + 5. Let’s alter one of our terms a bit for this next example.
Five plus (-3) equals 2, and (-3) plus 5 still equals 2. So 5 plus (-3) equals (-3)plus 5. Note that there is a very important distinction between the addition of a negative integer and the operation of subtraction. It is important to note this distinction because the commutative property does not apply to the operation of subtraction.
For instance, 5 minus 3 does not yield the same as 3 minus 5. This property also does not apply to division. One hundred divided by 2 does not equal 2 divided by 100.
The commutative property does, however, apply to multiplication. For example, 4 times 3 times 5 is equal to 5 times 3 times 4. Let’s do the math just to make sure. Four times 3 is 12, times five is 60. Five times 3 is 15, times 4 is also 60. Even though we switched around the terms, we got the same result.
Our final example involves the use of variables. Simply substitute values for the variables to show that rearranging terms is acceptable when adding and multiplying.
Let x=1, y=2, and z=3. You could substitute with any values, but we’ll use these for now. This gives us 3(1)2 + 5 × 1 × 2 + 3 is equal to 5 × 1 × 2 + 3 + 3(1)2. After evaluating our exponants, we get 3 + 10 + 3 is equal to 10 + 3 + 3, which gives you 16 equals 16. Once we add each side, we are left with 16 on both sides, which is true, 16 = 16.
The next property that we will look at is the Associative property. Again, the name provides a helpful hint to its meaning. What comes to mind when you hear the word associative? For me, the word associate stands out, which could also bring to mind the word group. Accordingly, the associative property allows us to group terms that are joined by addition or multiplication in various ways. Parentheses are used to group the terms, and they establish the order of operations. Work inside the parentheses is always done first. Mathematically, the property looks like this:
Let’s look at an example of this property used in an addition problem. This example will show that adding the last two terms first or adding the first two terms first simply does not matter.
3 plus (4+5) is equal (3+4) plus 5. If we solve the parentheses first, we get 3 plus 9 equals 7 plus 5, which leaves us with 12 equals 12.
Likewise, the order that we perform multiplication does not matter either.
Let’s say we have (3 times 4) times 5 equals 3 times (4 times 5). We solve the parentheses to get 12 times 5 equals 3 times 20, which gives us 60 equals 60. The commutative property of multiplication shows that it is acceptable to rearrange terms when multiplying. In contrast, the Associative Property of Multiplication moves parentheses to order the multiplication.
And finally, the last property we’ll be looking at is the Distributive Property, which looks like this:
The notation, once again, dictates that this property applies only to the operations of multiplication and addition. Specifically, if a term is being multiplied by an expression in parentheses, then the multiplication is performed on each of the terms. Here is an example to prove that this algebraic move is justified.
The parentheses on the left tells us to first add 3+7, then multiply that sum by 2. The sum of products on the right side of the equation gives the same result.
Two times 10 equals 6 plus 14, which gives us 20 equals 20.
Ok, now that we’ve gone over the three properties, let’s test your memory. For each problem, state the property, Commutative, Associative, or Distributive, that justifies the statement. Go ahead and pause the video if you need more time.
Think you got it? Let’s see!
The answer for number 1 is the associative property, because the parentheses are moved to order the multiplication. The answer for number two is the distributive property, because 3 is multiplied by both terms in the parentheses. That leaves us with the answer to number three being the commutative property, because we’ve simply rearranged the terms.
As you can see from our work in this video, you have been using the Commutative, Associative and Distributive properties for quite some time without even giving the “WHY” much thought. You will be asked to think about these concepts again in higher-level math courses when some of these properties simply do not hold up! Until then, keep using these rules with confidence to guide your work and thought processes.
I hope this review was helpful! Thanks for watching, and happy studying!