# Properties of Numbers

The lesson below is still images taken from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. If the PowerPoint is not one of my freebies, you may also head over to my YouTube channel to see the slideshow fully animated and can pause as needed to be sure you grasp each concept before moving forward (each lesson will always be free there!).

This lesson will familiarize you with the commutative property of addition, commutative property of multiplication, associative property of addition, associative property of multiplication, distributive property, identity property of addition, identity property of multiplication, and the concept of inverses.

So without further ado, read through the slides below to get a feel for how these properties of numbers work and can be used to make some calculations easier!                Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Order of Operations

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

• Properties
• Of numbers and operations
• Objectives
• By the end of this lesson you should feel comfortable:
• Identifying and utilizing the commutative property of both addition and multiplication.
• Identifying and utilizing the associative property of both addition and multiplication.
• Identifying additive and multiplicative inverses in identity properties.
• Identifying and utilizing the distributive property.
• Commutative Property
• In our lesson with integers, we introduced the commutative property.
• The commutative property is shown to the right:
• Here, “a” and “b” just represent real numbers.
• To make this a little more simple, think of the numbers commuting from one place to another, just like some people commute to work.
• So the commutative property of addition says you may add two numbers in any order and arrive at the same solution.
• And the commutative property of multiplication says you may multiply two numbers in any order and arrive at the same solution.
• Associative Property
• We have already seen how the commutative property can be useful, so let’s move on to another.
• The associative property is demonstrated to the right:
• Yet again, our letters “a” “b” and “c” all represent real numbers.
• In simpler terms
• The associative property of addition states that the order you add numbers does not matter.
• The associative property of multiplication states that the order you multiply numbers does not matter.
• Associative property
• Notice here that in both addition and multiplication that the numbers remain in the same order, where as in the commutative property, the numbers moved around.
• These numbers don’t commute.
• Instead, they associate with one another.
• In our first example, 3 and 4 are hanging out on the left. But we have a close group of friendly numbers, so on the right, 4 and 5 decide to hang out.
• Both sides are the same, though.
• Associative property
• Just to be sure, let’s dig deeper as we did with the commutative property back in our integers lesson.
• In our first example, 3 and 4 are in the parentheses together, which means we perform that operation first if you remember our order of operations PEMDAS.
• Yet on the other side, 4 and 5 are associated together, so we add those first.
• Both sides are equal to 12, so the property holds true.
• Very quickly, what about the multiplication example?
• Inverses
• Before we can discuss the third property, first we will need to understand what inverses are.
• The easiest way to think of an inverse is as the opposite of a number, though this isn’t exactly the definition.
• But haven’t we heard of opposites before?
• Negatives! That’s right, our additive inverses are simply the opposite sign of a number.
• These are additive inverses because we can do this:
• Inverses
• Ok…..So what exactly is the point?
• Well, zero is the additive identity.
• Think about what the word “identity” and where you have heard it before.
• Perhaps the secret identity of a superhero or how you identify.
• An identity in math refers to a number that does not change our first number.
• Identity Property
• Alright, why don’t we write out a problem to get a better idea what’s going on?
• As you can see, we add our identity to our starting number, 5, and arrive back at the same number as the solution.
• While this may seem trivial now, in the coming lessons, knowing what number will not change the value you are starting with can be very important!
• Thus we can see the additive identity property to the right.
• Where “a” is of course representing any real number.
• Identity property
• Hold on. Aren’t we forgetting something?
• So it stands to reason there must be multiplicative forms, too.
• Observe:
• Remembering what we said about additive inverses, let’s come up with a formal definition for multiplicative inverses now.
• When two multiplicative inverses are multiplied together, the product will be the multiplicative identity.
• Identity property
• Again, let’s write out exactly what a multiplicative identity does so we can see one in action:
• Hopefully you notice the similarities here to our additive identity.
• While before we could add 0 to get the same number back, here we must multiply by 1 to return to the same number.
• Thus our multiplicative identity property is:
• Note: the additive inverse and multiplicative inverse are not the same number!
• Distributive property
• Finally we have the distributive property:
• Lots of letters floating around in there.
• Ok, we know by now that these letters are just standing in for whatever real numbers we want to plug in, so let’s do just that:
• Well that looks…better…
• Ok what just happened?
• Distributive Property
• In a class, the teacher sometimes hands out stuff. Stuff like cookies!
• By doing so, the teacher is distributing a cookie to each person in the class.
• The same thing is happening in our example.
• 5 is being distributed to both the 3 and 2 inside the parentheses.
• Distributive property
• Remember back in our order of operations lesson where we mentioned that a number outside a set of parentheses means to multiply?
• This is why the right side of the equal sign looks how it does.
• With this in mind, let’s check our solutions on both sides, shall we?
• Using order of operations on the left and right:
• Both are equal to 25, so both sides are equal!
• Mental Math
• Many of these properties are useful primarily in theories and algebra, but there are also some cool math tricks you can do in your head.
• Let’s look at one example just to bring it all home:
• Normally, this problem could be pretty tricky, but what if we rewrote it.
• Why would we do this? Doesn’t this make more work and a longer problem?
• Well, piece by piece we can do this mentally now. (with a little practice)
• Mental math
• First distribute 23 on the right side:
• Now we have two slightly easier problems.
• 23 x 20 is like 23 x 2 with a zero added on the end.
• So 23 x 20 is 460
• 23 x 6 can be a little trickier, but with practice you can think of it as 20 x 6 + 3 x 6 (much like we are doing now).
• Lastly, we just add the two results together.
• Certainly this is not a requirement in math, but for those who wish to perform mental calculations quickly, this is one of many tricks!
• conclusion
• The commutative properties state you may reverse the order of addition or multiplication to arrive at the same answer. (The moving one.)
• The associative properties state you may add or multiply in any order to arrive at the same answer. (The parentheses one.)
• Adding inverses will always result in a sum that is the additive identity.
• Multiplying inverses will always result in a product that is the multiplicative identity.
• Adding a number to the additive identity (0) does not change that number’s value.
• Multiplying a number by the multiplicative identity (1) does not change that number’s value.
• The distributive property multiplies a number to every number inside a set of parentheses.