Common Core Algebra I.Unit 1.Lesson 3.The Commutative and Associative Properties.by eMathInstruction

Hello and welcome to another common core algebra one lesson by E math instruction. My name is Kirk weiler, and today we're going to be doing unit one lesson number three on the commutative and associative properties. I'd like to remind you before we begin that the worksheet for this lesson and a homework set can be found by clicking on the lessons or on the video's description. Also, you can find it on our website WWW dot E math instruction dot com. As well, let me remind you that at the top of each worksheet, there's a QR code. You can scan this code with a smartphone or with a tablet, and it will take you directly to this video. But let's begin. In order to really understand algebra, algebraic expressions, equations, manipulating them. Without thinking that it's just a bunch of symbol manipulation symbols flying here or there. What you really have to understand is the way that numbers behave and the way that they combine with the various operations of addition subtraction, multiplication and division. In the last lesson, we saw this a little bit in terms of the order of operations. But in this lesson, we're going to get it two very important properties of numbers. The associative property and the commutative property of addition and multiplication. But let's take a look at the first exercise because I think this really gets to the heart of the matter. It says add the following numbers without using a calculator. It says hint. Although order of operations tells us that we should add from left to right. In other words, we should do three plus 9, get an answer, then do plus four, get an answer, then do plus two, get an answer, et cetera. Think about an easier way to sum these numbers. Some meaning to add up, right? Some means to add up. All right. Show how you sum them. So take a moment. And see if you can figure out a very, very easy way to add these numbers together. All right, let's go through it. I don't know if you caught this. But what we can do is we can see, well, you know, I mean, I've got three plus 7. Right? I've also got 9 plus one. I've also got four plus 6. And I've got two plus 8. Now what I can do is I could really think of it like this. I could add those two together. Those two together. Those two together. Those two together, and of course the reason that that's an advantage is that's ten, that's ten. That's ten. That's ten. So I add them all together and I get 40. Yeah, that just seems crazy. I mean, we just spent an entire lesson talking about order of operations and how you have to do things in a particular particular order. And yet for some reason, I decided I would flip flop this edition all over the place. And I would just decide which two numbers to add together in whichever order I decide. So I'm going to add the three to the 7, the 9 to the one, the four to the 6th, the two to the 8, and then I'm going to add those all together. My ability to do this gets into these properties that I was talking about. And we're going to review those in the next exercise. So write down anything you need to right here before I scrub out all the text. All right, here we go. Let's take a look at the next exercise. All right. So all of these properties have important names, and there are names that we want you to know, terminology is important. So the first property is called the commutative property of addition. 8 plus four gives the same sum as four plus 8. In other words, it doesn't matter whether I do a plus four or four plus 8, I get the same answer, which is of course 12. The commutative property of addition when I'm adding two numbers, just two, I can add them in either order that I want, right? And this is helpful because if you're thinking about what three plus 17 is, it's kind of hard to think of being at three and adding 17 to get up to 20. It's much easier to think about, ah, I got 17, I add three more. I met 20. And it really doesn't matter. Whether I have 8 baseball cards and I add four baseball cards or whether I have four baseball cards and I add 8 baseball cards. Now the same thing works with multiplication. The commutative property of multiplication, 6 times three gives me the same product as three times 6. Both products are equal to 18. Right? Now again, a lot of students like to use this, especially in helping them remember their multiplication table because that means you only have to remember kind of half of it, if you will. If you know that three times 7 is 21, well then later on when you have to do 7 times three, you know it's 21. Right? Now, the associative property. This is interesting. So commutative property works when you have two things that are either being added or multiplied and you can think about them in either way. The associative property always involves three numbers or three expressions or three variables, right? And it basically says the following. Let's take a look at addition. If I look at this three plus 5 plus 9, what that implies is I'm really supposed to do three plus 5 first and then add 9. The associative property says, hey, look, I got three numbers. I can add any two of them I want first. Think about this for a moment. If I do this, I get three plus 5, which is 8. Then I add 9, and I get 17. On the other hand, if I do this and I add 5 plus 9 first, I'll get 14, add three. I also get 17. So the associative property of addition says that if I have three things added together, ten plus 8. Plus four. I can add any two of these I want. Any two I want before I add the third one. So I could do ten plus 8. Or I could do 8 plus four. Technically speaking in order to do ten plus four, I'd have to use the commutative property to flip flop this edition. But we'll get into that a little bit more later. Associativity, associativity, that makes me sound smart. Also works with multiplication. In other words, if I'm multiplying three things together, two times 5 first, then times 7, that's going to give me the same thing as if I do two times 5 times 7. Let's look at that for a second. Two times 5 is ten, and then if I multiply by 7, I get 70. On the other hand, if I do 5 times 7 first, I get 35. Now I know that's a little bit harder. But two times 35 is also 70. All right. These two properties the commutative property of addition to multiplication and the associative property of addition and multiplication are very, very important properties. I'm going to use them a lot. I'm going to cite them a lot. So that you get used to their names and use to think about them. Although hopefully this isn't the first time you've seen them. I'm going to scrub out the text so copy down what you need to. All right, here I go. And it's gone. Let's go on to the next problem. Exercise three says give an example that shows that subtraction is not commutative. So this is very important. We've talked about the commutative property of addition and multiplication. And the associative properties of addition to multiplication. But not of subtraction and not of division. Right. So addition is commutative because three plus 5 is a and 5 plus three. Is 8. But watch what happens when we do subtraction, right? Let's go with the same two numbers. 5 minus three is two. But three -5 is negative two. So not commutative. Now, why do I even bring this up? How do I bring it up at all? I bring it up because a lot of times we're going to want to manipulate algebraic expressions. Remember an expression is just a combination of numbers that we know and numbers that we don't know. Maybe we know them all, maybe don't know any of them. But it's just a combination of them using addition subtraction multiplication division and other operations, right? And we're going to want to be able to manipulate those using the rules of numbers and operations, like order of operations and the commutative property and the associative property. So subtraction presents us with a problem. We can't rearrange it. But subtraction can be commutative if you turn it into addition. And that's what exercise four is all about. Exercise four says change the following expression involving addition and subtraction into one only involving addition, then use the commutative and associative properties to quickly determine the value of this expression. Through this, please review some properties of negative numbers. So first, let me review how to change subtraction into addition. Why don't we stick with that this 5 minus three business? 5 minus three can be changed into an equivalent addition by simply adding the opposite of the number we're subtracting. So 5 minus three is the same as 5 plus negative three, either way we get two. Let me do another one of these. Let's say I had even something like this, let's say I had a minus 12. That would be the same as a plus negative 12, which would be then negative four. Got it? So let's do this one. Let's change everything into addition. So that's going to be 7 plus negative three. Plus 8 plus negative two. Oh, here's a subtraction. Plus, oh, that does not look like a plus at all. Let's get rid of that. Plus negative 6. Plus one. Oh, here comes the hard one. This is tricky for most people. Subtracting a negative three would be the same as adding a positive three. Remember, we always add the opposite so the opposite of negative three would be positive three. Now the reason that this is so nice is now using associative and commutative properties of addition. I can write this any way I want. Often what I like to do is I like to group all the positives together so watch this. Let me go in red. That'll help us see it. Positive, positive, positive, positive. So 7, 8, one, three. And let's go negative. Go different color. Negative, negative, negative. So plus negative three plus negative two plus negative 6. Watch what I'm going to do next. I'm going to actually use the associative property. So I can I can group these addition in almost any way that I want. So 7 plus 8 is 15 plus one is 16. Plus three is 19, okay? Here negative three and negative two is negative 5 plus negative 6 is negative 11 and now 19 plus negative 11. Is 8. I'm going to be doing that a lot. I'm going to be changing subtraction into an adding opposites quite often. I suggest you do the same thing because addition is way more flexible than subtraction. Subtraction you have a problem. But addition, you can flip flop, you can group in various ways, commutative and assertion of properties. All right, I'm going to be clearing this out. Here we go. All right. Next problem. So one thing that you've done in previous courses is you've combine like terms. And we're going to be working with that quite a bit this year. All right, so some real basic review. Let's take a look at exercise 5. In fact, my bet is you could probably do this on your own. So I'd like you to pause the video and see what you remember about this. If you don't remember it at all, don't worry. We're going to review it. Let's see what you can do, okay? Pause the video now. All right, let's go through it. Again, we're going to be looking at sort of a more formal basis for how to do this eventually. But really, it boils down to this. 5 X's really means that you've got X plus X plus X plus X plus X I think I did enough. And two X means that you've got two X's. So obviously if I've got 5 X's and I add two X's, I end up having 7 X's. Right? Here it's a little bit trickier if I have 7 positive X's and three negative X's than I have a grand total of four positive X's. Likewise, if I have negative 8 X's and negative two X's, then in total, I have negative ten X's. Let me just a little combining light terms. No different than combining oranges. If I have 5 oranges and two oranges, I have 7 oranges. If I have 7 oranges and I subtract three oranges, I have four inches. I don't know about letter C anyway. One thing that the common core is going to challenge us to do is to explain our math, you know? Previous curriculums and Alex grew up really just had you manipulating symbols, solving equations, doing a lot of what we're going to do this year. But it was pretty rare that you had to explain things justify things, list properties. Common core is going to demand that of you. You're going to have to raise your game. So let's take a look at exercise 6. It says in the following exercise, we show how two linear expressions will get into this word more. How to linear expressions are combined using various properties. List what those properties are. So here, we want to understand why we can rewrite three X plus 7 and two X plus 8 without the parentheses, right? Why can we do that? Because with the parentheses, it means that we're adding these two first, adding these two first, then adding the totals together. Without parentheses, it really means I'm doing three X plus 7 plus two X plus 8. So what gives us permission to add in any order we want, that's going to be the associative. We're going to abbreviate property and this is important. Of addition. Because we didn't do anything with multiplication there. It was all addition. All right. Now, notice that this line and this line are the same. Okay? So now, when we take this line and compare it to this, what happened? And what happened was we essentially said that 7 plus two X was the same as two X plus 7. We just took that addition and we reversed its order. That is the commutative property. And again, it's of addition by F's are not looking very much like S. All right, again, notice this line and this line are the same. And look what we're doing. Now we're saying even though the addition normally would be three X plus two X get a result add 7, get a result at 8, get a result. We're now going to group the three X in the two X together. And the 7 and the 8 together. And that. Is, again, the associative property. Completely okay to abbreviate property as prop. Nothing here. That's just combining light terms. All right? So make sure you can not only recognize these properties. But that you, as you manipulate things, you're kind of constantly reminding yourself in your head of what property you're using. Really, really important. All right? We'll use them a lot more in lessons to come. But right now, I'm going to scrub out the text, so pause the video if you need to. All right, here we go. And it's gone. Let's do a wrap. Actually, no, we've got one more problem to go. What was I thinking? Exercise 7, combine the expressions below, replace subtraction by addition of opposites if needed. So take a little bit of time and try to write these expressions in their simplest form by combining what you can and leaving what you can't. Okay? Pause the video now. All right, so you don't have to write down the properties, but I'm going to talk about them as I go. So what I'm going to do is I'm going to first rewrite this. Subtraction by 9 with an addition of negative 9. Then what I'm going to do is I'm going to use the commutative property to flip flop this addition to make it into this addition. Finally, what I'm going to do is I'm going to use the associative property to group these two particular additions together. Four X plus negative two X will be a positive two X 6 plus negative 9 will be negative three. And we normally don't leave things this way, we would normally then write it as two X minus three. All right? Now, pause the video again and try letter B and letter C if you got letter a wrong or you were confused about how to do it. All right, let's go through it. Letter B, again, what I'm going to do is I'm going to flip flop this edition using the commutative property of addition. So it's going to become ten X plus 9 instead of 9 plus ten X then I'm going to use the associative property of addition to group those together. Negative 6 X plus ten X would be a positive four X and 9 plus three is a positive 12. That's it. Oh, let her see it's got a lot of subtraction. So I'm going to rewrite this as four Y plus negative ten plus negative 7 Y plus negative three. Now I do that simply because subtraction is not commutative for associative, but addition is. So now I can play around with this the way I did before. I'm going to use the commutative property of addition to rewrite that as negative 7 Y plus negative ten plus negative three. Again, using the associative property, I'm going to group these two additions together for Y plus negative 7 Y is negative three Y negative ten plus negative three is negative 13. And again, although that's a perfectly good answer. And I'll likelihood we would want to leave it like this. Negative three Y -13. All right, now that is the last exercise. So pause the video now, write down anything you need to. Because we are scrubbing the text bye bye text. Okay, let's finish up. We're going to be doing a lot of manipulation of algebraic expressions in order to do a lot of different things to see structure to play around with ideas to play around with equations and variables. Knowing the properties of the operations addition and multiplication, along with the numbers, these commutative properties and associative properties will be key in doing these manipulations. So be sure to know them. All right. Well, I want to thank you again for joining us for another common core algebra one lesson by E math instruction. My name is Kurt weill. And until the next lesson, don't forget, keep thinking. It keeps solving problems.