Common Core Algebra I.Unit 1.Lesson 2.Variables and Expressions

Hello and welcome back to another common core algebra one lesson by eMath instruction. My name is Kirk Weiler. And today we're going to be doing unit one lesson number two variables and expressions. Before we get started, let me remind you that you can find the worksheet that goes with this lesson, along with a homework set by clicking on the video's description or by visiting our website WWW dot E math instruction dot com. As well, let me remind you of that on the corner of each worksheet is a QR code. That code can be scanned with either a smartphone or a tablet to take you right to this video. Let's begin. All right, today we're going to be talking about variables, and we're also going to be talking about expressions. Now, the definition of an expression is very, very important. It's any combination of numbers that we know and or numbers we don't know. In other words, variables. So we can have expressions like two plus 7. We can have expressions like 12 divided by two plus four. We can have something that involves a variable, like two times X plus 7. Or something that involves more than one variable. M times V squared. These are all expressions. Please note they're not equations because they don't establish any kind of equality between two things. There's simply a combination using things like multiplication, division, addition, and subtraction, of numbers that we know and numbers that we don't know. Now, a lot of times we want to evaluate expressions. In other words, figure out what their final value is when we combine all the numbers that we know. Okay. Assuming that we know all the numbers. Now, in order to do this properly, what we really have to know is our order of operations. And what we're going to do is we're going to review our order of operations through the first exercise. What I'd like you to do is test yourself. I'd like you to do problems a through C by pausing the video now, and then we'll go through them. This will be a good test for you to see if you know your order of operations. I would suggest if you're watching this video at home and you don't remember them. Do a little Google search or something else in order to figure out what they are. But pause the video now and take as much time as you need. All right, let's go through a through C in letter a, we've got three times two plus 7. And if you remember your order of operations, multiplication always comes before addition. So when we look at this calculation, the first thing that we need to do is three times two, which obviously is 6, and then we add on the 7, 6 plus 7, then is 13. All right? What you shouldn't be doing is adding the two and the 7 first and then multiplying by three. So remember, multiplication and division come before addition and subtraction. So let's take a look at letter B this one's a little more complicated. A minus one half times 6 plus 24 divided by 6. Notice that we've got a multiplication here and a division here. We should really do them in the order that they appear in the expression, so that a, well, at 8 is not going anywhere. And then we have to find one half of 6. Well, that's three. And then we have to do 24 divided by 6, which is four. Now all we have left is subtraction and addition. And what order of operations tells us is that we need to do this from left to right. So we'll do 8 minus three, and we'll get 5. And then we'll do 5 plus four, and we'll get 9. So hopefully that was your final expression. Please note that if you're doing these strictly on the calculator and getting the right answer, don't be patting yourself on the back. It's not about getting the 13 or the 9 or any of the other new other numerical values that we're going to get. It's about knowing what operations come first. So I would prefer that you do these by hand and get the wrong answer because you've made a simple arithmetic mistake. We all make simple arithmetic mistakes, all of us. It's okay. But it's better that you do than that way than get the right answer by typing them into your calculator, but having no idea what you're doing. So let's take a look at letter C here we've got some parentheses involved. And if you remember order of operations from previous years, parentheses insists that we do what's inside of them first. So in fact, even though subtraction tends to come later than multiplication, we're going to do 8 -6 first, and we're going to get two, and then we're going to do 5 minus three, and we're also going to get two. Now we have a multiplication and a multiplication, then a subtraction. So four times two gives me 8. 7 times two gives me 14. Be careful here. Whenever we subtract a larger integer or a larger number from a smaller number, we always end up getting a negative. So we'll get negative 6 there. All right. Now what I'd like you to do is I'd also like you to do letters D, E, and F these are even more complex expressions, but I'm confident you can do them. Pause the video now and see what you can do. All right, let's go through them. Again, remember, don't use that calculator unless it is to check something. So letter D, look at this, this is complicated, right? First, that overall fraction bar is division. Hopefully everybody remembers that fraction bars represent division. All right, but we also have exponents. And in our order of operations, which is often given by the acronym, pemdas, whoops, I can't seem to write. Sometimes people remember it as please excuse my dear aunt Sally, parentheses come first, exponents become second multiplication division tie addition subtraction come last. So in fact, we have to do that these exponents first, 5 squared means 5 times 5, which is 25, four squared is four times four, which is 16, then plus three. And one -5, watch out again it's a negative. This negative four. Now what do we do next? Well, this large division bar actually acts as what's called an implied parentheses, meaning that we really have to evaluate the numerator first. So we're going to do this traction in addition as we move our way over. 25 -16 is 9. Then we've got the plus three. We'll take care of that division by negative four eventually. Then we have 9 plus three, which is 12. Divided by negative four. And remember, a positive divided by a negative always gives us a negative. So 12 divided by negative four is negative three. How's that? Let's tackle letter E here we've got some parentheses. Remember, they come first. So two -7, that's negative 5. I'm just going to keep it in parentheses so that I know I can multiply. 5 minus three, that's positive two. Plus three squared. All right, now, our order of operations says that we've got to do the three times three, which gives me 9. Exponents first. Then we're going to do the multiplication negative times positive as a negative, negative 5 times two is negative ten. Plus 9. And if I have ten negatives and 9 positives, a bunch of those are going to cancel out, and they're going to lead me with negative one. I'm trying to review some basic number faxes we go through this exercise as well. Last one, man, is this vocabulary? Look what we have. In the numerator, we've got this division in this multiplication in the denominator we've got the exponents. We can actually work the numerator and the denominator very separate. So we can do negative 16 divided by two, which is negative 8, we can do 5 times two, which is positive ten. And then we can do two times two times two, right? This is two times two times two. Two times two is four. Times two is 8. All right? Negative 8 plus ten gives me a positive two. Divided by 8. Now watch out two divided by 8 is not four, right? When we have a smaller number divided by a larger number, we're going to get a non integer fraction. So that's going to become one fourth. If you remember how to reduce fractions. Okay? So there we go, a little review of order of operations. Please excuse my dear and Sally. Parentheses, exponents, multiplication, division, addition, subtraction. And I am going to clear this away. So pause the video if you need to. All right. Scrubbed out. Moving on. Okay. One thing I can say for certain, with a completely and utterly blank screen. Is that order of operations is extremely important. I mean, look, I made the entire phrase just blow up or burn with fire. Whatever it was. You need to know order of operations. Okay? That's kind of funny. Order of operation. How about a quarter of operations? That looks a little bit better. It really is quite important, right? You need to be able to look at things and understand exactly what is going on in an expression. In terms of the calculation. I think I'm going to scrub out that S or it's going to hang out there. So let's move on. All right, take a look at exercise two. Now we're going to bring some variables in. A variable is simply a letter that represents some unknown quantity. Now that unknown quantity might be, it's changing in which case we would call it a variable. Or it may be something that's consistent or constant, in which case it's a constant. But we oftentimes let their letters represent unknown quantities. Okay? So if X represents an unknown quantity, we're supposed to explain the calculation that each of the following expressions involves involving X represents. Okay? So for instance, let's take a look at the first one. This gets into how do we read algebra? How do we read it, right? So when I look at that, for me, I see three X -8. And what that means is that we've done two things. We've multiplied. X by three, right? And then we subtracted. 8 from the result. All right. What we don't want to do is we don't want to look at that and think, oh yeah, it's a subtract, got an answer multiplied by three. Order of operations tells me that the multiplication comes first, and that subtraction comes second. So what I'd like you to do is see if you can describe what's happened to X in B and C pause the video and see what you can do. All right, let's talk about these calculations. So in letter B, the first thing that we did was we subtracted. Four from X and then we. Divided the result. By two. And that's it, right? So we have to do this numerator first. Subtracting four from X and then we divide the result by Q problem with the hardest one to describe is letter C keep in mind, what do we have here? We've got an exponent. We've got a multiplication, and we've got to subtraction. Please excuse my dear aunt Sally. Hopefully I'm not writing right on top of my head. Right? So what we know is that exponents got to come first. So we're squaring X. Or squaring X, meaning X times X, right? Then we're multiplying. Multiplying the result. By four, and then we're subtracting it. Apparently I like subtracting 8. Oh, subtracting. Great handwriting says, tracking 8, some the result. All right. So it's important. When you see things like a B or C and I give you a value of X or we think about what's happening to the variable X or T or whatever. We have to be able to read it. We have to be able to say, ah, of course, we took the X like in letter C, we squared it, then we multiplied by four, then we subtracted a. It's critical that you're able to do that. All right? So I'm going to clear out the text, pause the video right now if you need to. All right, here we go. And it's scrubbed. Let's go on to the next problem. All right, so we also sometimes need to evaluate expressions. So we evaluated expressions in exercise one, right? An exercise one, we have these expressions that were completely numerical, meaning that there were no variables. And we figured out what the final values are. Many times though, we'll have an expression that involves a variable. Then we'll be told what the value of the variable is, and we'll have to figure out what the expression is equal to. So an exercise three, it says, for each given expression, explain the steps. That the calculation is doing, and then that should be that, not what. Explain the steps that the calculation is doing, then find its value for the given variable values. All right? So we're going to evaluate four X -7 when X is 5. First, we want to explain what's going on. So take a look at this expression, right? What's going on is that we're first. Multiplying X by four, and then we're subtracting 7. All right. Now often when people show that, especially for an input value, like 5, they'll put that in parentheses, notice how I'm putting a parentheses around that, right? So we can then look at this and go, okay, four times 5 is 20. And then 20 -7 is 13. So 13 is the value of this expression when X is equal to 5. All right? Simple enough. Because we're going to do a number of these. So I'm going to clear out this text. And then we're going to move on. Okay. So let's keep doing it. We're just got a bunch of these things. So for each given expression, we're going to explain the steps that the calculation is doing, and then we're going to find the value for the given variable values. All right, so let her be. The expression 8 minus two X squared when X is negative three. All right. So, well, maybe I should have the explanation first, then the calculation. I don't know. But let's think about what's going on. We're going to square first. We're going to square the X, which will be negative three. Then we're going to multiply. By two, we're going to square the X, then we're going to multiply by two, and then we're going to subtract. From a. From a let's take a look at that. That'll be 8 minus two times negative three squared. So the first thing I'm going to do is square that negative three. Remember a negative times a negative is a positive, so negative three times negative three is positive 9. Now we have to do that multiplication two times 9 is 18. And now we'll do 8 -18, which is negative ten. Again, try to do as many of these calculations as you can without your calculator. We try to keep the numbers relatively small relatively easy to work with, especially early on in the course. So that you can use gesture own brain, your own memory and understanding of multiplication division addition subtraction to get the job done. All right, let her see. Let's talk this one out. Maybe we won't write out the explanation. Well, we'll just talk it out. So what do we have? We've got these parentheses in the numerator. That means we're going to be adding 8 first. Then we're going to be multiplying by two, then we're going to be dividing by three and the last step we'll be adding the one. So let's see how that looks. Let's be careful. Put in that negative two for X all right. My next step, negative two plus 8, evaluate what's in parentheses. That'll be positive 6. I can put an equals between here. Now we'll do this multiplication two times 6 is positive 12. Divided by three plus one. 12 divided by three is four. Plus one and finally that lonely edition gets to happen. So that expression is equal to 5 when X is equal to negative two. Again, I'm going to scrub out the text so pause the video, take as much time as you need to think about what we just did and to write down anything you may have missed. Okay, here we go. Scrubbed. Let's go on to the next page. All right, one final problem, a little multiple choice practice. Every once in a while, I'll be sprinkling multiple choice problems into our lessons to give us a little bit of, I don't know, standardized test prep, if you will. So what is the value of this expression? All right, let's kind of decompose it. X shows up more than once here, but that's okay. Remember, these exponents are going to come first, then the multiplication here and here will come second, and then those subtractions will come last. But let's substitute that value in. Take a look at what we have. All right, so again, order of operations tells me I really should do that squaring first. So four times four is 16. All right. Now I should do this multiplication. One half times 16 is 8. Two times four is 8. And then that minus three. Finally, I should work the subtraction from left to right, 8 -8 is zero. Minus three and zero minus three. This negative three. So choice one. Right on our little scan Tron, we would be bubbling it in. Make sure to get a good bubble. Gotta always get a good bubble. All right, so that's it. Pause the video now if you need to write any of that now. Not that. Filling in at the bubble. All right, let's scrub out the text. It's gone. Let's wrap up this lesson. It's all right. So today we talked about variables and we talked about expressions. We talked about a lot about order of operations. Make sure you know those. It's really, really fundamentally important that you've got your order of operations down. So until next time, let me just say thank you for joining me for another common core algebra one lesson by email instruction. My name is Kirk weiler. Until the next lesson, remember to keep thinking. It keeps solving problems.