Common Core Algebra I.Unit 2.Lesson 5.Linear Word Problems.by eMathInstruction

Hello and welcome to another common core algebra one lesson. My name is Kirk weiler, and today we're going to be doing unit two lesson 5 on linear word problems. Before we begin this lesson, let me remind you that you can find the worksheet and a homework set for this lesson. By first clicking on the video's description. As well, on the upper right hand corner of every worksheet, we've got a QR code. That can be scanned using your smartphone or a tablet to bring you right to this video. All right, let's jump into everyone's favorite topic, word problems. You know you love them, you know you're excited about them. Let's do them. So modeling and solving linear word problems. It's hard to really give a recipe or anything like that for this. So take these sort of bulleted points more as pieces of advice. Things that are good habits of mind that will help you with solving these types of problems. So first, you always want to clearly define the quantities involved with common sense variables and let statements. Choose good letters, letters that make sense to you. Do really good let statements so that you have no doubt about what things you're actually representing with the variables. Number two, use your let statements to write out expressions for quantities that you're interested in. In other words, when we did that translation lesson, a couple lessons ago, where we were translating English into math. That step number two, you want to be very careful to write out expressions that you're interested in using those LUT statements. Number three, carefully translate the information that you're told in the problem into an equation, right? Ultimately, there's going to be information that allows you to write an equation that you can then solve. Solve that equation, though, and just like we did in the last lesson, make sure that you're mentally justifying every step. Eventually, of course, you're going to be solving equations almost by rote. You know, you just get into it to a point where you're not thinking about the justifications, but right now it's important to be thinking, I just use the commutative property, or I just use the additive property of equality. Things like that. And I'll be going through that a little bit here and there. Finally, check the reasonableness of your answer right if you're trying to figure out how many people get on a bus and you get something like 32.7, you can't have 32.7 people get on a bus. So try your best to make sense of your answers. To me, what math is all about is helping us make sense of the world and the universe that we live in. All right? If you just are doing math and getting answers and those answers really don't make sense. Then I'm not sure really what the point is. So try to make sure that your answers make sense. All right. Well, after that helpful advice, let's jump right into it and solve some problems. All right. Exercise one. The sum of a number and 5 more than the number is 17. What is the number? All right? So we could jump right into this algebraically. But I think it's actually fun to experiment with some answers. So what I don't want to do is randomly grab the right answer because then that'll kind of spoil B, right? So let's try a number. Let's try three. All right, what does it say? The sum of a number three plus and 5 more than the number, see what's 5 more than three. That would be 8. Is 17. Well, nope. That's false. Okay, so clearly three is not the right number. Let's try something else. Let's try. Let's see. I don't know. We'll try 5. All right? So the sum of the number 5 plus a number 5 more than 5 here. Let me actually even do this so that we really get it. That's 17. Let's see. That's going to be 5 plus ten equals 17. 15 equals 17. Closer, right? Almost true, but still not. So 5 is not the right answer. So all right, let's do it. Let's actually just go through the algebra, left with guess and check. Let's let, let's let the number be equal to N okay. Let N plus 5 B 5 more. Then N all right, that makes sense, right? Now, we've got our careful let's statements. What we want to do is set up an equation. So we're going to sum, right? We're going to sum N and N plus 5, and that's going to be 17. Now, a lot of people wonder why I put the N plus 5 in parentheses. Number one, it's safe. But number two, it's correct. In other words, the two numbers are N and N plus 5, right? So N plus 5 is its own thing. It's its own quantity. But now I'm going to use some properties to solve this equation, right? For instance, I'm going to use the associative property of addition to simply remove the parentheses. Because I know that when I add three things together, I can add any two of them I want. So in other words, I'm going to actually add these two ends first. Before I add the 5. Now I'm going to do a little additive property of equality. By adding negative 5 to both sides, that's going to leave me with two N equals 12. That almost looks like zn. Then I'm going to use the multiplicative property of equality to multiply both sides by one half or divide both sides by two. So I find out that the number is 6. Now, of course, I can check this pretty easily because this one at least I can take my number 6. I can add it to 5 more than the number, which is 11 now at almost looks like a 12. And I can find that that is, in fact, now a true statement, right? That actually makes the statement true. Okay? So careful let's statements, careful translating of the let statements into an equation, and then solving the equation using techniques we already know. Now this one was pretty easy. We'll get into some more complicated ones in a bit, but take a look at that one. Pause the video, write down anything you need, and then we'll do some more. All right. Now ultimately, what we're always doing here is we're modeling. We're modeling a real world physical situation. All right, maybe exercise one wasn't so much, but we will be. But we're modeling it with equations with symbols and then we're allowing that modeling process to play out to give us the correct solutions. All right? Let's go on and look at the next problem. All right, nice multiple choice problem. The difference between twice a number and a number that is 5 more than it is three. Which of the following equations could be used to find the value of the number N explain how you arrived at your choice. So what I'd like you to do is pause the video right now and see if you can figure out which of the four multiple choice answers is the correct one. And how you went about thinking about it. All right, let's go through it. So the difference. Difference means subtraction. Now we have to be careful with subtraction, right? Because subtraction is not commutative. We have to make sure we do the subtraction in the right order. So they'll always give us the first number first, if you will. So the difference between twice a number. Well, if the number is N, then twice the number is two N, right? This is twice the number. Twice the number. And that does not look at all, like the word twice. It looks like tua. There we go. Twice the number. So the difference between twice the number and the number that is 5 more than it. So what's the number 5 more than it? N plus 5, right? And then it's equal to three. Now, if you put down choice one, well, that's a reasonable choice. The problem is it's wrong. And it's wrong because right now what I'm doing is I'm doing two N minus N and then I'm adding 5 to the result. What I need to do is I need to do two N minus a number that is 5 more. And because I'm subtracting the entire number, I have to have it in parentheses. All right? Maybe in the last problem having those parentheses around N plus 5 is somewhat irrelevant. I don't think it was. But maybe it was in terms of just solving the problem. But here it's absolutely critical because it's the difference between choice one and the correct choice, which is choice for. All right, I want you to think long and hard about that. Very important to have parentheses, especially when you're dealing with subtraction. Okay? So I'm going to clear this out, pause the video if you need to. I'm going to apparently move my arms around a lot. All right, here we go. Let's move on. What can they throw at me? Evie and her father are comparing their ages. All right, so Evie and her father are comparing their ages. At the current time, Evie's father is 36 years older than her. Three years from now, EV's father will be 5 times her age at that point. How old is he now? Whoo. All right. Well, why don't we try some guests and check? Okay? This is important. I often will tell students that is a really, really good idea to do some guessing and checking before you launch into the algebra. Because it'll help you. All right? So let's do this. Let's keep track of their ages. Let's say that EV is ten. Okay? Then we're going to say dad instead of father because it's a shorter word. What does it say? It says Evie's father is 36 years older than her. All right, so her dad will be ten plus 36. All right, which is obviously equal to 46. All right, what is it, three years from now? Three years. Three years from now, let's see what's going on. Well, three years from now, EV will be ten plus three now I know that many of you would just write down 13, but again, I want to establish some algebraic background before we go on to let her be. So Evie is going to be 13 and her dad would be, well, what? It would be ten plus 36 plus three. Which would be 49. And it says EV's father will be 5 times her age at that point. So the question was, were we right? Was Evie actually ten? Well, we can check it by saying, well, is 5 times 13 equal to 49. Right? Well, 5 times 13 is 15, 65. And is 65 equal to 49? No, that's false. So what does that really tell us? What it tells us is that EV isn't ten years old. All right. So let's now try to solve for EV's age. All right, now what we're going to do is we're going to essentially do exactly what we did before, except instead of saying that EV is ten, we're going to let EVs age. Age be equal to, I don't like using E or a I'm going to go with X, all right? If I don't like the variable that begins the word like E or a for age. Then I tend to go with X or Y they're nice letters. Okay. So what's her dad? Well, her dad's age, dad's age. He's going to be 36 years older than her. So X plus 36. So three years from now, what do we have? Well, three years from now, obviously, Evie's age is going to be X plus three. By the way, notice what I'm doing. Right? Everywhere that there was a ten, I'm basically now just putting in an X and watch me do that. So three years from now, eve is going to be X plus three. An EV's dad is going to be X plus 36. Plus three. Which is going to be X plus 39. Now that final statement we're going to actually use to set up our equation, right? What we know is that EV's dad is going to be 5 times her age. So Evie's dad's age is X plus 39. And EV's age is 5 times X plus three. Now please note, I kind of flip flop the order of the equation here. That kind of switched it a little bit. But we can now solve this. So X plus 39, I can now distribute, use the distributive property. 5 X plus 15. I can now subtract an X from both sides using the additive property of equality. So I'll get 39 is equal to four X plus 15. Likewise, I can subtract 15 from both sides again using the additive property of equality. So I'll get 24 is equal to four X and finally I'll divide both sides by four using the multiplicative property of equality. And I'll find out that EV's age is 6. All right. But notice how I was able to use what I did in part a to walk myself through part B whatever I was doing to the number that I was sort of guessing and checking. I'm doing to the variable X. Certainly seems reasonable. Just for a moment, talk about some things that wouldn't be reasonable. Number one, it wouldn't be reasonable if Evie's age came out to be negative. Number two, it wouldn't be reasonable if EV's age even turned out to be a non integer fraction. Like if I'd gotten 6.3. We don't report ages that way, right? When you ask me what my age is, you know, I'll say that my age is 43, let's say. Okay? I won't say that I'm 43.8. Of course some kids talk about their ages in terms of halves. But generally speaking, we report ages and whole numbers. So pause the video now. If you need to, copy down what you need, and then I'm going to scrub out the text. Okay, here we go. Let's move on. Okay, exercise four. Kirk has $12 less than Jim. If Jim spends half of his money and Kirk spends none, then Kirk will have $2 more than Jim. How much money did they both start with? Wow. Okay, well again, let's do a little, let's do a little guess and check over here. Guess and check over here and then we'll play around with the algebra over here. So notice that Kirk's money is based on Jim's money. That's actually kind of important in how we how we attack this problem. Kirk has $12 less than Jim. So why don't we start off with Jim having let's say that Jim has $50. Okay? Then, of course, Kirk will have 50 -12. Which will equal $38. All right? Then if Jim spends half his money, so after we spend money, right? Jim has one half of 50, which is 25, then Kirk will have two more dollars than Jim. Well, Kirk has $38 right now. Right? And specifically, Kirk is supposed to have 30 supposed to have $2 more than Jim. In other words, if I take $2 and I add it to Jim's money, then I should get Kirk's money. But 38 doesn't equal 27, right? So Jim didn't start with $50. That's all that tells us. Okay. But watch as we manipulate. So let's let M be equal to the amount. Of money mummy, hopefully no mummies today. The amount of money. Jim has. All right? Kirk, then M -12 equals the amount Kirk has. All right, fair enough, right? Then after they spend money, because that was originally, Jim now has one half M and Kirk still has M -12. Now what does it say though? It says that if James spends half his money, Kirk will have $2 more than Jim. This is the hardest part, right? Kirk's amount of money, which is M -12, will be Jim's amount of money, which is one half M plus another two, right? And that's tricky because it would be very understandable to think, oh, I'm supposed to take Kirk's money and add two to it to get Jim's money. But then that would mean that Jim has two more dollars than Kirk. Kirk, right? Which is M -12 has two more than Jim. All right, let's do some solving. We're good to go, right? So I'm going to use the additive property of the quality to add 12 to both sides. I'll have M equals one half M plus 14, I'll use the additive property of equality to subtract one half M from both sides. Don't get scared of that fractions. One M minus one half M will be one half M that'll be 14. And now I can use the multiplicative property of equality to multiply both sides by two to find out that Jim had $28. So, Jim, originally had $28. And Kirk had 12 less than that, so 16. Now, what's interesting is that this one's a real easy one to check, watch, watch as we check for reasonableness. Right? Because what did it say? It said when Jim spent half his money, so Jim has $14 left, Kirk will have $2 more than Jim. Well, Kirk has $16 and notice that that is two more than what Jim has. So it makes sense given what we were told in the problem. All right. I'm going to clear out that text. Think about think hard about what we did with the example and then how we translated that into the algebra. All right, here goes the text. Cleared up. All right, let's wrap it up. So we did some modeling today. We took some real-life scenarios, we tested them with just numerical values, then we used that process to develop equations that modeled the scenario. And then solve those equations to find out what we wanted. This is a tough process. It really is, because it combines equation solving techniques along with translating English into math. And that can be a struggle for students. I guarantee, though, that if you stick with it, if you keep working at it, you will be able to do it. And modeling with mathematics is where it's at. Just solving equations and manipulating numbers eh, we don't do much with that in the real world, but modeling, setting up real world problems and having the math solve them for us, that's power. All right. Let me thank you for joining me for another common core algebra one lesson by eMac instruction. My name is Kurt weiser. And until next time, keep thinking and keep solving problems.