Common Core Algebra I.Unit 1.Lesson 4.The Distributive Property

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 four on the distributive property. Before we begin, let me remind you that a worksheet for this lesson and a homework set can be found by clicking on the video's description. As well, don't forget at the top of every video or sorry at the top of every worksheet. We've have a QR code that can be scanned using either a smartphone or a tablet to bring you right to this video. All right, let's begin. In the last lesson, we talked about the associative and commutative properties of addition and multiplication. And they were quite separate. We had the commutative property for addition, and then the commutative property for multiplication. The associative property for addition, and the associative property for multiplication. Today, we look at the last of the big three, the distributive property. All right? So let's take a look at exercise number one. It says consider the product four times 15. Evaluate using the standard algorithm. All right, now depending. It may have been a little while since you've seen this, but let's do four times 15. All right. The standard algorithm is that we first do four times 5, and we get 20. All right? We write down the zero, and we put a little two up here. Then we do four times one, which is four, and then we add the two, and we get 6. So we get a final result of 60. Now you may understand why that algorithm works, or you may not. But it does work. But now what we're going to do is we're going to think about multiplying 15 by four and a little bit of a different way. Obviously, 15 is the same as ten plus 5. So really four times 15 is the same as ten plus 5, another ten plus 5. Another ten plus 5. And another ten plus 5. Right. So obviously what we get is we get 40 here, and we get 20 here, adding them together. We get 60. So what have we really done? We've really said that four times 15 can be thought of as four times ten plus four times 5. Right? And that's known as the distributive property of multiplication over addition. The distributive property of multiplication over addition. Meaning that when you multiply two numbers together, you can break one of those numbers apart and multiply each of those parts and then do the addition. Ten and 5 are just convenient. I mean, we could do four times 15 in a variety of ways. Watch this. We can look at 15 as being 11 plus four, right? That would then say, well, I could do the 11 times four. And the four times four, I like ten and 5 more, but it's still works. Here we get 44. Here we get 16. And 44 plus 16 is again 60. All right. It even works with subtraction. Watch this. Let's say I did four times 20 -5. That's the same as 15. So four times 20 minus four times 5 would give me 80. -20, and that would also give me 60. Isn't that cool? I love the distributive property. It's one of the many ways that I'll do multiplication in my head when push comes to shove. Especially when I'm doing something that I find a little bit hard, like four times 19 in my head or something like that. You know, 19 is an unfortunate number. So I just think about 19 as 20 minus one. And then I can multiply the 20. I can multiply the one, and then I can subtract. So let's play around a little bit more with the distributive property, but purely numerical, no variable set, okay? I'm going to scrub out the text, so write down anything you need to. All right, here we go. Let's move on. Let's talk about the distributive property. All right, the distributive property says, if we're going to multiply a by B plus C and this could also be subtraction. Then what the distributive property says is that I can multiply each part of this particular number or this particular expression by a and then do the addition second. So normal law order of operations would insist that we add these two numbers together first because they're in the parentheses. All right, and then multiply. The distributive property says we can do the multiplication first. Then add each component part. All right? So let's take a look at how that works numerically. Okay. Evaluate each product by using the distributive product to property to make it easier. On V, express 18 is a subtraction. Do not use a calculator. Again, anybody can do this. I have a 9 year old son max. I have a 6 year old daughter Evie, either one of these could either one of them could do these problems using a calculator. So what I'm asking you to do is be better than a fourth grader, right? Don't use the calculator on this. Let's take a look at that array. So, the whole point is that we can think of 23 as 20 plus three. The distributive property now says, while I can do 7 times 20 and 7 times three, hopefully both of these are easier than this one. 7 times 20 is a 140. 7 times three is 21. So I get one 61. 7 times 23 is a 161. Now I can do the same trick with 18. I could write it as ten times 8, that would work fine. But subtraction also works. With distribution. And the reason this attraction works, by the way, is because remember, subtraction can be changed into addition of opposites. But we're going to leave it a subtraction. So 18 can be thought of as 20 minus two, so now I can do 9 times 20. And I can do 9 times two, 9 times 20 is 180. 9 times two is 18, subtract, and I'll get one 62. Coincidence that these two numbers are only one apart. Actually, yes, it is coincidence. I just sometimes randomly do that. So anyway, it's kind of neat, isn't it? I'm going to scrub this out. So write down what you need to. All right, it's gone. Now keep in mind, because this is very easy to become cynical or sarcastic about. You might look at this and go, I know the standard algorithm. I certainly know how to use my calculator. Why would I be doing this? Well, there's a couple of reasons. Number one, we're just trying to understand the distributive property. We're trying to understand how it works. Number two, let's face the facts. At some level, okay? Now maybe it's just me, maybe I'm just a complete and utter math geek, I know I am. But at some level, it's kind of cool. It's cool that you can do 9 times 18 and think of it as either 9 times 20 minus two or 9 times ten plus 8, and you're going to get the same answer either way. All right? That is a neat property that multiplication and addition slash subtraction have with one another. Okay? So let's move on. All right, probably the hardest thing is using the distributive property twice. So let me show you how that's done. We're going to do 12 times 28. And we're going to evaluate it by thinking about the 12 as ten plus two and thinking about the 28 is 20 plus 8. Now at first what I'm going to do is I'm going to treat this as one whole thing. In other words, I'm going to have ten plus two times 20. And then I'm going to have ten plus two times 8. All right, so I'm multiplying the 20 by ten plus two, and I'm multiplying the 8 by ten plus two. And yes, I know ten plus two is 12. But now here's what's cool. I can now distribute these multiplications through these additions. So in other words, I can do ten times 20 plus two times 20. Then I can do ten times E and then I can do two. Times 8. And what do I get? Here, I'll get 200. Here, lock it 40. Here, I'll get 80 and here, I'll get 8. 20 plus 40, sorry, not 20. 200 plus 40 plus 80 plus 8. Gives me 300. Well, how did I get 8 there? That's just weird. Why didn't you say something? That's 16. Like wait a second. That doesn't match my final answer. Let's try that again. And this couldn't have been more inefficient. All right. That took a little while. All right, final answer of 336. So note to self two times 8 is actually 16, not 8. Look at that double distribution. I really want you to try to understand this. These both are what are known as binomials by meaning two. Binomials, all right? Binomial here, binomial there. We're going to be doing a lot of multiplication of binomials together when there's variables involved, and we'll be doing this double distribution a lot. Some teachers call it foiling, first outer inner last, because you're multiplying the first terms together. That's this. Then you're multiplying outer terms together. That's actually this. Then you're multiplying inner terms together. Those are these two. And then you're multiplying the last two terms together, which is the one I botched. Anyhow, I am going to clear out the text so pause the video if you need to. Okay, here it goes. Wonderful. Let's go on. For the next page. All right, so we can use the distributive property when it comes to purely numerical quantities. But we can also use the distributive property when we have variables in our expressions. So I'm going to walk through this very slowly at first. And then pick up the pace. What I'd like you to do though is pause the video right now and see if you can remember how to do this from 8th grade and 7th grade math. Try to just do letters a and B, but if you're fast, you could do all four of them. Pause the video now and take as much time as you need. All right. It's important, by the way, early on in this course. When I give you a chance to try to remember something, or to simply try something out that maybe you've never done before, that you really give it your best shot. Sure, you can listen to me T, you can certainly pick things up. But it's best if you can remember things on your own and figure things out on your own. It will make you a much better problem solver. So let's take a look. This says 5 times two X plus three. Now notice I'm already thinking about this two times X is a single quantity. And what I really have is I've got 5 times two X plus 5 times three. Now, what do I do with that? Remember, multiplication is associative. Meaning that I can actually do 5 times two, then times X and 5 times three. That gives me permission to do 5 times two and get ten, there's really nothing I can do with that. I just have to leave it 5 times three is 15. And that is the best I can do. Okay? Same thing with letter B, I'm going to do negative four times 5 X minus negative four times 8. Oh, that looks confusing, doesn't it? Maybe I should do this. Maybe that helps. Maybe it doesn't. Again, using the associative property of multiplication, which allows me to group this multiplication together, no worries here. We'll just leave that as negative four times 8. All right? Negative four times 5 is negative 20. Nothing I can do with the X here, we're going to have negative four times negative 8, which is negative 32. And remember, when we subtract, we can replace with adding the opposite. So subtracting negative 32 is the same as adding positive 32. For a lot of people, they'll just see a double negative and immediately change it into a positive. That's okay too. Not a problem. All right, let's take a look at this one. This is a little bit confusing. This says we need to do X times X, and then we need to do X times four. Well, we'll work with exponents a little bit more, but X times X is what is known as X squared. And then there's really nothing we can really do with that. Best we can do is write it down as four X X times four, four times X all right, let's do one more. We'll have 5 X times two. Minus 5 X times 7 X all right, watch me use the associative and commutative properties of multiplication all at once. So I'm going to put the 5 times two and we'll bring that multiplication by X here. Watch this. Look at all this multiplication. We actually have four things. 5 X 7 and X so I'm going to do the 5 times the 7. And X times X associative and commutative properties being used there. 5 times two is ten. X, 5 times 7 is 35. X times X is X squared. And that's it. Oh, man, it hurts after writing all that. Okay. I'd like you to pause the video now, really think hard about this. This is a benchmark skill. You've got to be able to distribute through addition and subtraction. All right, I'm going to scrub out the text. Okay, it's gone. Now, one last little piece. Division distributes just like multiplication. And the reason for that is very simple. Division and multiplication are essentially the same thing. When I divide by two, it's the same as multiplying by one half. When I divide by 7, it's the same as multiplying by one 7th. Watch out, I'm going to blow your mind. When I divide by one tenth, it's the same as multiplying by ten. Okay? So anything that multiplication can do, division can do as well. All right? So take a look. If I have B plus C and I divide by a, I can first divide B by a, C by a, and then add the results. Watch out. This is something that a lot of students make mistakes on. They think, well, I'll just divide the first thing by a, because then I've done the division. But now, the division distributes over addition and subtraction, just like multiplication does. Okay? So let's get some practice on that. Express each of the following quotients is binomials and simplest form. Show your calculations some of your final answers will contain fractional coefficients. A coefficient is simply a number that multiplies a variable. Okay? So let's take a look. What the distributive property tells me is that I can take that division by two, and I can divide 8 X by two, and I can divide four by two. Further, I can actually do the associative property, and I can first do 8 divided by two, then multiply by X and then do four divided by two. 8 divided by two is four. Four divided by two is two. And my final answer is four X plus two. All right. Pause the video now and see if you can finish the last view, watch out, let her C and D both involve fractional fractions in your final answer, okay? All right, let's go through them. Letter B all right, I can distribute that division by 5 through this attraction, so 25 X divided by 5. -50 divided by 5. Again, I'm going to use that associative property, which tells me I can do the division by 5, here first, then multiply by X and we'll eventually do this division, but all in good time. 25 divided by 5 is 5. 50 divided by 5 is ten. So we get 5 X minus ten. All right, now for some fractions. All right, same idea. I can distribute the division by four through that subtraction. All right. Then I can rewrite this as two divided by four. Times X -16 divided by four. Watch out two divided by four. It's tempting to say that's two, but the numerator is smaller than the denominator. So I'm actually going to get one half X minus four. Good? All right, let's do this last one. This one's going to get the ugliest. Negative 9 X divided by 12. Plus 18. Divided by 12, again, negative 9 divided by 12. Times X 18 divided by 12. All right, let's have a little talk about reducing fractions really quick. So far, we've only had to reduce easy fractions, like two fourths into one half. Negative 9 12, so remember to reduce a fraction what you'll do is you'll look for the biggest number that goes into both that happens to be in this case three. So negative 9 12 will end up being negative 9 divided by three, 12 divided by three will be negative three fourths X same deal with the 18 divided by 12, the biggest number that goes into both 6 so divide by 6 on both. And we'll have three halves. So negative three fourths X plus three halves. All right. So division distributes over addition and subtraction, just like multiplication does. We'll see that quite a bit this year. All right, pause the video. I'm going to scrub out the text in a moment. All right, here we go. Excellent. Let's finish up the lesson. Okay. So in the last two lessons, we saw important properties of real numbers under addition, multiplication division and subtraction. We saw commutative properties associative properties and today we worked with the distributive property. All of these are going to be key in lessons that are to come. All right? So let me just thank you again for attending another common core algebra one lesson by E math instruction. My name is Kirk wild. And until next time, keep thinking and keep solving problems.