Common Core Algebra I.Unit 1.Lesson 6.Seeing Structure in Expressions.By eMathInstruction

See, very often when you do algebra, you end up applying a lot of properties, doing a lot of manipulations, you're distributing numbers, you're combining like terms, you're solving for X, your graphing, Y, et cetera, et cetera. And many times people kind of look at algebra is just a collection of mindless algorithms, mindless recipes. So what we want to do each and every time we're playing around with this community of associative and distributive properties, we want to be using them for a reason. We want to be mindful about it. Now there's an argument to be made that once you become really fluent at a skill, you actually start doing it almost mindlessly. But when your first playing around with things, you definitely want to be mindful so that you don't do things that are a little bit silly. Let's take a look at the first exercise. 

All right, an exercise number one, we're asked to consider two expressions. Two X plus one and 6 X plus three, and we're supposed to find the value of both expressions when X is two. This is a skill you've done before. So what I'd like you to do is pause the video now and come up with the value of both of these two expressions when X is equal to two. All right, let's go ahead and go through it. Let's activate our pen. Let's kind of lay it out like this. We have the expression two X plus one, and we want to evaluate it when X is two. So we know our order of operation says that we have to do that multiplication by two first. So we're going to get two times two, which is obviously four plus one, and that gives us a result of 5. On the other hand, the expression 6 X plus three, well, for that, we'll put two in. 

Again, we have to remember our order of operations and we have to do that multiplying first. But simple enough, we get a result of 15. All right. Fine. Well, letter B asks us, what is the ratio of the larger outcome to the smaller outcome? All right, now you should understand ratios. And then it asks, why did the ratio turn out this way? And what property could we use to justify all of this? This is a little bit tricky, but if you think you have a good idea about what the ratio is, you should certainly be able to do that. But why it turned out that way pause the video right now and think about it. All right, let's take a look, because this is all about mindful manipulations. First, let's take a look at the ratio. Remember when we have something that says a ratio of something to something, right? The larger to the smaller. We sometimes write it with colon notation. Sometimes we write it as a division problem. But we get three or three to one. In other words, this how you always want to interpret ratios. The larger. Is three times the smaller. 

Now, the real question then becomes, why did it turn out that way? Or is that just a complete and utter coincidence? Maybe for other numbers, it wouldn't turn out to be a three to one ratio. Or maybe it would. So look at the two expressions now. Look at the expression, 6 X plus three. And look at the expression two X plus one. Can you think of any reason why this would be three times more than this? Well, I can. In fact, if I take three and I multiply it by two X plus one, using, of course, the distributive property. And remember what the distributive property says it says I multiply this by each part of the sum inside, I'll get two times three, which is 6 X plus three. So in fact, 6 X plus three gives us three times two X plus one because 6 X plus three is always three times two X plus one. In fact, watch this. Let's pick a different X let's say we picked X equals, I don't know, 5, 5 is kind of a nice number to work with. If we put it into two X plus one, we get two times 5, which is ten plus one and 11. 

If we put it into our 6 X plus three, we'd get 6 times 5. Plus one gives me oops, not plus one. That's a mistake. I'm like wait a second. I'm not getting the right ratio. I'd get 6 X plus three, and that would be 30 plus three, or 33. And notice that 33 is 11 times three. So no matter what value of X we picked, 6 X plus three will always give us three times the value of two X plus one. And we should be able to see that by thinking about the distributive property. Let me scrub out my text. And why don't we go on to the next exercise? Okay, exercise number two gets into kind of the heart of the lesson. So I'd like to take a little bit of time on it. 

All right. I'm going to tell you, I'm going to give you an expression, and I'm going to tell you what it's equal to, but I'm not going to tell you the value of X so in this problem, I'm telling you that we have the expression three X plus two. And for some value of X that I would substitute in here, the expression would end up equaling 7. All right, now I don't want you solving for X, even if you know how. Because that will miss the point of the exercise. Don't get me wrong. It's great if you can solve for X but don't do it. All right, what I want to do is I want to determine the value of this expression and this expression for the same value of X that gave me 7. 

All right. And I want to do this by using a mindful manipulation. So take a look at 6 X plus four and compare it. To three X plus two. How do these two compare? Right? Think about that for a moment. Pause the video if you need to. Get to figure it out. 6 X plus four. Is twice three X plus two, right? That's by using the distributive property. I'm not going to write it all the way out. But the distributive property. We can see that. But three X plus two is equal to 7. Right? That's what it's equal to. So 6 X plus four must be 14. Got it? That might be a little bit tricky. We're going to be looking at this more. Now let's take a look at three X plus 5. Well, how does three X plus 5 compare it to? Three X plus two. Think about that for a minute. 

Can you mindfully manipulate this M and M? Not any of the candy, right? Can you mindfully manipulate this to try to get the three X plus two involved? Pause the video and see if you can get the right answer. All right, let's take a look. Here, we're actually going to use a different property. No distribution here. I'm going to recognize that 5 is the same as two plus three. Now this might seem like a funny step, but I think you will have to agree that 5 is the same as two plus three. And now watch what I'm going to pull. I'm going to decide to add the three X and the two and then the three. Think about it, that for a moment. What property did I just use? Did you get it? I used the associative property, right? The associative property says if I'm adding one, two, three things, I can add any two that I want first before I add the third one. But ding, ding, ding, we have a winner. 

Three X plus two is 7. And then I just have another three more. So three X plus 5 must always equal ten when three X plus two equals 7. These are kind of fun, aren't they? But you have to be able to play around especially with two properties. The distributive property and to a certain extent the associative property. I don't know how much the community is going to come into play today, but we'll see. Let's scrub that out. Let's move on to the next problem. All right, exercise number three. Here's a big one. Here's where we want to start to develop a little bit of fluency with mindful manipulation. So same idea is the last exercise. So I'm going to kind of turn you loose on this one in a second. We've got an expression two X plus 5, right? That's the thing that we're basing everything on. And it's got a value of ten. Again, don't solve for X kudos if you can. But don't do it. 

All right, what we want to do is mindful manipulations now on each of these expressions to try to figure out what they're equal to. Again, this is kind of tricky, especially letter G but I'm sure that many of you will be up to the challenge there. Pause the video now and take some time, okay? See what you can do, what we'll do is when we start the video again, I'll go through a couple of them that I'll let you have another chance to pause in case you're having a rough time. But pause the video now, take as much time as you need to see what you can do on this lesson. All right, let's take a look. So again, what we always want to be doing in these mindful manipulation problems is we want to be looking at the expression that we're trying to figure out the value of versus the expression that we're starting with. Hopefully this one was pretty easy because it was similar to the one that we did last time, right? I can say that four X plus ten is simply twice two X plus 5. 

Again, all that is is the distributive property, right? And since I know that two X plus 5 has a value of ten, four X plus ten must have a value of 20. Got it? Let's take a look at B as well. So now I'm looking at two X plus 20 and I want to compare that to two X plus 5. All right, well, what can I say? This is actually very similar to the last problem as well. I can say that two X plus 20 would be the same as two X plus 5 plus 15. I would imagine many of you might not put this step in. All right, many of you might skip a directly to this. Directly to this associative step. Where we can say that two X plus 20 is the same as two X plus 5. 

Our winner, plus another 15. Well, since I know two X plus 5 is ten, I can now just add 15 to it and get 25. All right, let's pause the video again and see if you can do the rest of them. If you haven't already. All right, let's take a look at the next problem. Let's take a look at C two X plus one again versus two X plus 5. This one might be a little bit trickier, right? How do you take two X plus one and make it into two X or involve two X plus 5? Well, again, kind of skipping over some steps. Hopefully, we could see this, right? Two X plus one will be the same as two X plus 5 minus four. Now let me just kind of group it so that you can really see the structure. We want to think about two X -5 as being a single quantity. That's why I put it in parentheses. Oftentimes, people think that parentheses tell you to do whatever's inside them first. And that's one way to interpret them. 

But I think it's most helpful to interpret parentheses as, hey, I want to treat everything inside of that as one number. So I can now say that this must be ten minus four, and that's 6. All right, let's take a look at some more of them. Negative two X -5. This is another situation of the distributive property. So negative two X -5 can be thought of as negative one times two X plus 5. Again, just the distributive property. But then that would be negative one times ten. And we would get negative ten. Ten X plus 25. Let me put my two X plus 5 up here. How do we get there, right? Well, hopefully you'd see. Again, using the distributive property, see how often we use the distributive property. Probably formally more often than the other two. I know that ten X plus 25 can be thought of as 5 times two X plus 5. And again, because two X plus 5 can be treated as a single quantity. IE ten. I can say that's 5 times ten. And that expression must be 50. All right, let's do letter F letter F is actually very similar to letter C, maybe the subtraction is a little bit trickier. But two X -5 can be thought of as two X plus 5, but then I have to subtract ten. 

So again, using the associative property, which says I can group this in any way I want. I get a result of zero. You got to like that. Okay, I hope my head wasn't bobbing and weaving too much across the page. Let's take a look at the challenge. Now, in the challenge, what we're looking at down here is we're trying to manipulate 6 X plus 20, but that is a challenge. You know, you kind of think to yourself what, I mean, if I multiply two X plus 5, let me get my two X plus 5, which is right here. If I take my two X plus 5 and I multiply it by three, right? Well, that doesn't quite get me there. It gets me 6 X plus 15. But I need 6 X plus 20. So that's not big enough. Oh, but now maybe I combine a couple operations. Maybe I say this. Well, let me do three times two X plus 5. Because I know that's going to get me. 6 X plus 15. But then I have to add on another 5, right? In order to get me my 6 X plus 20. But now remember we look at two X plus 5 as a single quantity. 

So I'm going to put a ten in there. Add 5. I now have 30 plus 5 for 35. All right, wow. That is a lot of blue on the sheet. Let's scrub that away. And let's take a look at the next problem. All right, last problem of the lesson. I wanted to get something going other than just multiplication and division and subtraction in addition. So take a look at exercise number four. Multiple choice problem, but even in multiple choice problems, we're always going to look to justify and show our work, okay? We just have four choices now. Anyway, it exercise number four. We're told that we have the expression three X minus four, and it's equal to negative three for some value of X and it is. All right, we want to figure out what the value of this expression is. So why don't we write that down here so that we can kind of play around with it? We have three X minus four squared plus 6. I don't know what happened there. 

Come back to my pen plus 6 X -8. Now obviously somehow we've got to get three X minus four involved, right? Well, it's clearly right here. So that's not a problem. The real question is, what do we do with this? So if you think, you know, pause the video and try to work through this problem. All right, let's take a look at it. Well, nothing that I'm going to do with this. I'm going to just leave it as three X minus four. Squared. But then I'm going to write this as two times three X minus four. So again, I'm looking at this three X 9 is four, not as a binomial, which it is. It's got two terms. But it's a single quantity because I know that single quantity, I know this single quantity is equal to negative three. 

So now I can simply say this expression must be negative three squared. Plus two times negative three. Remember a negative times a negative is always a positive. Folks don't use your calculator if you don't need to, two times negative three is a negative and 9 plus negative 6 is positive three. There we go. That might help you a little bit when something comes up like this on a standardized exam. Anyway, let's scrub this out. Get rid of my text. Come on. There we go. All right. So that was our lesson on seeing structure. Thank you for joining me for another common core algebra one lesson from E math instruction. I'm Kirk weiler, remember you can access the worksheet and a homework for this lesson by clicking into the video's description. Until next time, keep thinking and keep solving problems. Bye for now.