Common Core Algebra I.Unit 1.Lesson 5.Equivalent Expressions

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 5 on equivalent expressions. Before we get into that, let me remind you that the worksheet for this lesson, along with a homework set, can be found by clicking on the video's description below it. As well, don't forget that on the top of every worksheet we've got a QR code. That's going to allow you to use a smartphone or a tablet to scan that QR code and come right to this video. Let's get into it though. It's important to understand terminology, right? We already know what an expression is and expression is a combination of numbers we know with possibly numbers that we don't know. Two expressions, though, will be known as equivalent. If they have the same value for every value of the substitution variable, or variables. In other words, no matter what we stick in for the variable X or Y or Z, no matter what numbers we put in, the two expressions have exactly the same value. All right? Really important. Equivalent expressions have the same values. No matter what you put in for X, Y, or Z so we're going to play around with this for the entire lesson. It's an amazingly amazingly important idea in algebra. All right, let's take a look at exercise one. It says consider the three expressions below. All right, and by the three expressions what I mean is this one, 5 times X minus three. 5 times X, then minus three. And 5 times X -15. It says by substituting the values by substituting the values of X given, determine which of the two expressions are equivalent. Show your calculations of the expressions values and circle your final answers. So let's take a look. All right, in other words, what we're going to do is we're going to take 7 and we're going to put it in here for X, put it in here for X, put it in here for X, and we're going to figure out what their values are. So let's do it. All right? Now remember in this calculation, what's going to happen first is the subtraction. So we're going to have 7 minus three, which is four, then we're going to do 5 times four, and we're going to get 20. So this expression is equal to 20 when X is equal to 7. Let's try this one. Here we have to do 5 times 7 first, remember your order of operations. That's going to give me 35. Minus three, which is 32. Now right away, by the way, this tells us that these two expressions are not equivalent. Equivalent expressions will have the same value no matter what you put in for X, Y, Z, VM, it doesn't matter. All right? Let's try this last one. This last one says that we're going to do 5 times 7. And then we're going to subtract 15. So 5 times 7 is 35. -15 gives me 20. Hey, that one's the same. Maybe I'll put it in a square or rectangular, whatever that is. So this gives us evidence that this expression in this expression or equivalent, but they may not be. That's just one value of X let's try some other ones. Let's do 5 times zero minus three. I'm going to speed this up a little bit. So zero minus three, watch out. That's going to be negative three. Then a positive times a negative is a negative, so we get negative 15. If I substitute zero into this expression, 5 times zero minus three, I first have to do 5 times zero, which is zero. Then subtract off three. And that gives me negative three. Again, evidence that these two expressions, or sorry, these two expressions are not equivalent. But let's see if it keeps working out that the first and the third are. Here, I'm going to do 5 times zero, then I'm going to subtract 15. That's going to be zero -15, and that's going to be negative 15. Ah, it is the same. More evidence that these two are equivalent. All right. We can't prove equivalency by just substituting in values, but we can sure get a good idea of it. Let's put one in. Remember, do as many of these as you can without your calculator, even tricky ones that involve negative numbers, like one minus three is negative two. 5 times negative two is negative ten. All right, here we'll do 5 times one minus three. It's going to give me 5 minus three. That's going to give me two. Again, not the same. Finally, 5 times one -15 gives me 5 -15. And that is negative ten. All right, so what we see in this table is that at least for these three X values, the first expression, and the third expression give exactly the same values for those inputs, giving us a really good sense that the first and the third expressions are equivalent. But exercise two asks us, well, which property, the community of associative or distributive, justifies the equivalency of these two expressions that you determined to be equivalent above. So think about that. Which of those things can justify that those two expressions are the same. Well, hopefully you said the distributive property. All right, because the distributive property says that if we've got 5 times the quantity X minus three, that will be the same as 5 times X -5 times three, right? So we distribute that multiplication, sometimes teachers will show that with like little arrows like this. I know I love the span of this course. 5 times X, not much I can do with that, just 5 X and then 5 times three, 15. So in fact, 5 times X minus three, and 5 X -15 are the same, and that's justified. By the distributive property. So what is the moral of the story? The moral of the story in this problem and pretty much all the problems that we're going to be doing today. Is that we can use these three properties commutative associative and distributive. To rewrite expressions in equivalent forms, okay? The look different. And that's one of the keys in algebra is that often to expressions will look very different and yet will have exactly the same values no matter what you put in for X or T or Y or Z it won't matter. And if it doesn't matter, then the two expressions are called equivalent. All right, I'm going to clear the text out and pause the video now if you need to. All right, here we go. And they're gone. Moving on. Nice little multiple choice question. Which of the following expressions is equivalent to blah, right? Which of the following is equivalent to this? It says show your work to justify your response and then test at least one value of X to check your answer. So here's what I'd like you to do. I'd like you to pause the video and see if you can use things like the distributive property. The commutative property, the associative property, to simplify or to rewrite this expression as one of those four choices. All right? Pause the video now and see what you can do. All right, let's go through it. Well, there's not much we can do with the commutative and associative properties, not yet. But what we could certainly do is distribute the multiplication. In other words, when I multiply two X plus one by 5, I can multiply the two X by 5. And I can multiply the one by 5. Now what happens to the subtraction by four? Well, it just kind of hangs out. Now, of course, here I can use the associative property we talked about this in another lesson. I can use the associative property to actually multiply the 5 and the two first. Here, I'm just going to do 5 times one and get 5. And now, of course, I can get ten X plus 5 minus four. I can also use the associative property of addition and subtraction to group those two together and say, well, 5 minus four, that's just going to be one. So it appears that ten X plus one is my equivalent expression. Now, how can we test this? This is kind of cool where should I put the test? Who knows? Anyway, let's put it up here. Now, I want to grab an X file. What X value should I grab? I don't know. Let's grab X equals three. Grab something that you feel confident to work with. And what I'm going to do is I'm going to test the two expressions. Let me put a little line here to separate it. Let me test 5 times two X plus one minus four. Let's see what this expression is equal to. Oh, this is a doozy, isn't it? So we have to put two times three. I'm going to use the dot instead of the parentheses, otherwise I'll have a lot of parentheses. So two times three is 6. Plus one is where a calculator actually might be quite helpful. Register or of operations. 35 minus four gives me 31. Now let's test ten X plus one. Right? So ten times three is 30. 30 plus one is 31. Look at those two. They're the same. So that is an excellent check. In fact, some would argue it's one of the ways that you could do the problem. You could take a value of X, you could substitute it into the original expression, and then substitute it into the four choices. Any of the four choices that don't come out to be the same are clearly not equivalent. Now on the other hand, you could get to expressions to come out the same and not be equivalent for a value of X they just might have the same value for that particular value of X so watch out a little bit. I would still suggest using properties like distributive associative commutative to simplify or to rewrite the expression, and then testify. All right, I'm going to clear out the text, so write down anything you need to. All right, here we go. Let's move on to the next page. A little bit worse, a little bit worse, but the same type of problem, right? Which of the following expressions is equivalent to, again, I'm not going to read it all off. We'll just work on it a little bit. Again, show your work by thinking carefully about order of operations in the properties we've learned about. Finally, check your answer by substituting a value of X show this check. Pause the video. Take as much time as you need. Really think about order of operations, think about the distributive associative commutative properties and see if you can figure out which of those four choices is the same as the one setting up there above. Okay? All right, let's go through it. Very often, the first property that people will choose to use is the distributive property. If you see a multiplication by a sum or a difference, then it's good to distribute. So I'm going to do four times three X plus four times one, then that minus two is just hanging on. All divided by two, and then we can't subtract. Forget the subtraction by 5. I'm going to speed this up a little bit four times three X. I hope you're comfortable now that I can do the four times three and get 12 X four times one is four. Minus two divided by two -5. I'm now going to do the associative property here, group those two together. So I'm going to get 12 X plus two divided by two -5. Now I'm going to distribute division. Now we have to be very careful about that. It would be very easy to divide the 12 X by two. But not the two divided by two, so I'm distributing the division that distributive property of division, not for the 5. The 5 is not being divided. Here I'll do 12 divided by two, and I'll get 6 X here, okay, two divided by two, which is one. And -5. And again, I'm going to group these two together associative property. And I'm going to get 6 X minus four. There we go. Let's do a little test. We'll put the test over here right now. Now again, picking a value of X, that's almost like a like an art. I won't go with negative because I don't want to confuse people too much, but one thing that's a nice check sometimes is just a nice simple one like X equals one. So let's put that into the more complicated expression. Let's see, we have four times three times one plus one minus two, all divided by two -5. We'll put the other expression over here. But let's clean this up a little bit. Three times one is three plus one minus two, all divided by two -5. Wow, a lot of calculations here. The more you do these in your head, the better you'll become at math. Four times four is 16. Minus two divided by two -5. I'm running out of room. 14 divided by two -5. 7 -5 gives me two. Oh my. Okay, let's try it. This one much shorter. 6 times one minus four. So now we're just testing this one. That's 6 minus four, and that gives me two. Yay. Yay. All right, wow. And again, what's critical here is that that 6 X minus four and that four times three X plus one minus two divided by two -5. They're the same thing. Isn't that remarkable? They're the same. It wouldn't matter what value of X we chose. We chose X equals one to test it, but we could have chosen any value of X and they would have given us exactly the same output. All right? So I'm going to clean this up, pause the video. All right, here we go. Let's keep going. Okay. Now, this is interesting. Exercise 5. Which of the following expressions is equivalent to ten X plus 15? Explain how you made your choice in the split in the space provided. All right, so this one's a little bit different, but I think that you all have the ability to do it. It definitely has something to do with the distributive property. So pause the video now and see if you can figure out which one of these is the same as ten X plus 15. All right, let's go through it. Well, probably the most straightforward way of doing it is to actually apply the distributive property to each of these four. I'm going to do it very quickly. If I applied the distributive property in number one, I'd get 16 X plus 26. Well, those aren't the same. So no. If I distributed the 5 here, I'd get 5 times 5, which is 25 X plus 15. Closer, but still not the same. No. If I distribute the 5 here, I'll get 5 times two X, which is ten X 5 times three is 15. Ah, yes. There's my choice. If you'd like, I could do four, but all I would get is ten X plus 50, which is again a big fat no. All right, so just choice two. We could test with values of X but the plain fact is here what we're doing is we're if you will kind of reverse it the distributive property. And trust me, we're not really reversing it. We're just using the distributive property. All right? And what we're doing is we're getting our first look at what's known as factory. All right, factoring is amazingly important. So we're going to talk about it a little more in a second. Let me clear out the text first, though. Okay. Here it goes. All right, factoring expressions. You are going to factor so much this year. Yay. Factoring is the process of writing an equivalent expression as purely the product of other expressions. So for instance, if I told you to factor 18 and you said, oh, well, that's two times 9. You'd be correct. You just factored it. You didn't do what was called complete factoring or complete prime factorization, but you factored it because 18 and two times 9 are equivalent. 18 and two times 9 are equivalent. In the last problem, ten X plus 15 was factored as 5. Times two X plus three. So these two expressions are equivalent. Okay? But this one is written as a product of the product of 5 and two X plus three. Okay? So factoring is a amazingly important. It's a process that we're going to practice a lot over the span of the year. It has many different uses. We'll just get a taste for it today. So let's do some easy ones. I'm going to clear this out. And let's do one final problem. All right, factor each of the following expressions by writing an equivalent expression that is in the form of a product. Check your work by using the distributive property. All right. So here's how we do this. All right. We look at 6 X plus 21. And what we want to do is we want to kind of write it like this. A times B plus C, right? Now sometimes the thing out here is a number. Sometimes it's a variable. In this case, it's going to be a number. All right? Now this is a little bit tricky because we want to undo some multiplication that we don't really know about. So here's what I'm going to do. Take a look at the 6. Take a look at the 21 and find the largest number that divides both of them. And that happens to be three. So that's going to be my a all right. Then the question is, well, what would I have to multiply three by to get 6 X? Well, I'd have to multiply by two to get the 6 and X to get the X plus now what do I have to multiply three by to get 21? That's going to be 7. So that is an equivalent expression that's been factored. We can check it by simply multiplying back through to see if we get that. And we do. So take a look at the letter B, right? Take a look at the two. Don't worry about the negative and the ten. What number divides both of them? What number goes into both? And that would be two. Or negative two, whichever one, maybe we'll do it with both. If I write it as a two, then the question is, well, what would I have to multiply two by to get negative two X? And I'd have to have a negative one X the ones actually not mandatory. And then I'd have to multiply it by 5 to get the ten. On the other hand, if I had factored out a negative two, then I would just simply have to multiply it by X to get negative two X this is the tricky part. And then I'd have to multiply it by negative 5 to get positive ten, either one of these would be completely okay. All right, let's look at letter C 14 and 14. Oh, you know, I know what to buy some both. 14. So if I pull the 14 now, then I'd have to multiply it by X to get 14 X and then I'd have to multiply it by one to get 14. So 14 times X plus one. All right. So that's how we factor. Well, that's one way that we factor. There's going to be many different types of factory. But ultimately, at the end of the day, all factoring is is reversing the distributive property. Every time it's reversing the distributive property and writing an expression as something that's equivalent in the form of a product. Okay? So pause the video now and I'm going to scrub out the text. All right, here we go. All right, let's finish up. The idea of equivalent expressions could be one of the most important in all of algebra. Because every time we manipulate one side of an equation or another, we're manipulating an expression. And we're always manipulating it to get an equivalent expression. All right? Once we lose equivalency, then we really change a problem. It's just changed entirely. So it's very important to understand that we use these laws of numbers and operations specifically the commutative property of addition and multiplication is associative properties of addition and multiplication and the distributive property to rearrange algebraic expressions to get ones that are exactly the same, but might look different. All right? So I want to thank you for joining me for another common core algebra one lesson by eMac instruction. My name's Kirk weiser. And until next time, keep thinking. And keep solving problems.