Common Core Algebra I.Unit 4.Lesson 6.Modeling with Linear Functions
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Learning the Common Core Algebra I.Unit 4.Lesson 6.Modeling with Linear Functions by eMathIstruction
Hello and welcome to another E math instruction common core algebra one lesson. I'm Kirk Weiler. And today, we'll be looking at unit number four lesson number 6 modeling with linear functions. As always, you can download the worksheet and a homework that go along with this lesson by clicking on the videos, description. Or by visiting WWW dot E math instruction dot com. We've added something new to our worksheets lately that we think you'll like. It's called a QR code, and I know that many of you already know what that means. But for those of you that don't, there'll be a small little two dimensional barcode right below the date field on every worksheet. What you'll be able to do is scan this code and it will take you directly to this video, so if you're working late at night and you're trying to remember how to do an exercise, but it just isn't coming to you. You can scan that code, bring it right to this video and we'll give you a hand, okay? Let's move on to the lesson.
So today what we're going to be looking at is modeling real world phenomenon with the equations of lines, right? And every linear model has two parameters. It's slope and its Y intercept. So let's talk about those two a little bit before we move on to the problems. Right, the slope is all about how fast the Y coordinate is changing compared to the X coordinate, right? So we kind of visualize it as well. All right, the Y coordinate changes for a certain change in the X and we simply call this ratio the average rate of change, and for a line we give it the letter N I don't know. I always think of it as the movement of the line and therefore the letter M we also have what's known as the Y intercept, right? And in real world problems, the Y intercept almost always talks about the amount of whatever you have population savings, distance, whatever your modeling, how much of it you have when the input variable is zero right at the beginning, okay? So watch for these two parameters as we move throughout the problems. The slope, the rate at which the Y coordinate is changing compared to the X coordinate, and where the Y coordinate begins. All right, at what value it begins at. I'm going to scrub out the blue.
Let's go on and do some modeling. Holy cow, look at the size of this problem. All right, modeling problems are wordy. And probably the most important thing for us all is to make sure that we're comfortable with what's going on in the physical situation. So what I want you to do right now is pause the video. Take as long as you need, but read the problem over once twice thrice. I don't care. Read it over as many times as you need to to understand what is physically happening. Pause the video now. All right, let's take a look. Janine has $450 in her savings account at the beginning of the year. All right, no problem. Got that. She places money into the account at a rate of $5 per week. All right, great at the end of the week. Puts in 5 bucks. We want to model the amount that she hasn't savings. Oh, here comes the algebra. The amount she has in savings with the letter S as a function of the number of weeks she's been saving W now, that phrase, which I did a terrible job underlining a function of the number, right? It means that S is the output and the input is weak, so the amount of savings we have, that's what comes out of our gumball machine, but what goes in is how many weeks we've been saying. All right? But I still have it. She starts with $450, and she saves up $5 per week. So take a look at letter 8. In letter a, I'd like you to fill out that table.
Tell me how much jeanine has. After she's been saving for zero weeks after she's been saving for one week, 5 weeks, ten weeks. Go ahead and do that. And please write down the calculation that you use. In order to find your answers. All right, let's go through it. So well, okay, gotta admit. There's really no calculation I need for zero weeks, right? But I do know that after a zero weeks, she's got $450. I don't want to forget that. After one week, what does she have? Well, after one week, she saved $5, right? And she adds that on to the 450 that she has, and she gets obviously a savings of $455. All right, but after 5 weeks, after 5 weeks, she saved an additional $25, right? 5 times 5. I'm gonna actually write that calculation down, okay? 5 times 5, 25, and we add that on to the four 50, of course that would give us 25 plus four 50. I'm gonna skip right down that intermediate step. And we have $475. All right. After ten weeks, right, she's saved what? Ten times 5 $50, right? So let me take ten, and we multiply it by 5. We get that $50, and we add it onto the four 50. And we get $500.
So what we now want to be able to do is turn what we just did with just normal numbers into an equation and letter B I don't want you to worry about equations of lines. I literally want you to write down an equation that gives us the savings, right? What we did here, the savings, right? Gives us that, here I'll even start it off for you. S equals. S equals what? How is that calculated if you know the number of weeks W? And by the way, I thought I'm going to do something really quick in red. I'm going to circle the number of weeks. There's a W value, there's a W value. Where is it here? Well, it's really right here. We never wrote it down. One week, 5 weeks, ten weeks. So what's that formula over here? Pause the video if you need to and see if you can write it down. All right, I'm going back to blue. I'm gonna try and go back to blue. Here we go. So what did we do each time? Well, we took $5 and we multiplied it by the number of weeks. I know the W was here in the 5 was here, but that's what we did, right? We multiplied 5 and W and then we took the result and we added on to 450. Now, I don't need the equation of a line. I don't need Y equals MX plus B to do that.
I just know that that's how the calculation got done, but please note, right? That was the slope. That 5. And that was our Y intercept. That was where our linear model started. We started at $450. Now literacy says, if jeanine saves for exactly one year, what is the range in her savings over the year? Well, pause the video for a second. And see if you can figure out the range of her savings. I think the thing about what that term means range, all right? And then we'll come back to it. All right, this is a little bit of a tricky problem, because of course, what do we know? Well, we know that her savings starts at 450, but I don't really know where it ends. I'm going to just put a little question mark there. In order to do that, I really need to use this formula, and I also need to know that there are 52 weeks in a year. Right. I need to know that. So what I'm going to do is I'm going to figure out I'm going to use a little function notation here, all right? I'm going to figure out S of 52. Not S times 52, S of 52. So I'm going to do 5 times 52, right? And then I'm going to add that onto my four 50. 5 times 52 is 260. Of course, we might use a calculator to do a calculation like that. And then we'll add on the 450. And that gives us 7 ten. So I know at the end of the year, she saved up $710. So tentatively speaking, let me write this down as my rage.
I'll explain why I say it's tentatively speaking. You know, in a certain respect, when we talk about the range and statistics, saying, well, hey, her savings ranges from 450 to 710. That's actually perfectly acceptable. When we talk about functions though, the range is literally the outputs to the function. So that four 50 to 7 ten is a little leaves a little bit to be desired, but we'll come back to it. Take a look at letter D, read it for yourself. All right. Letter T says or asks, why would it not make sense to evaluate S of 6.5? In other words, why would it not make sense to do this calculation to do S equals 5 times 6.5? Plus four 50. Why does that not make sense? All right? Well, it gets into something very technical. You see, we can't really put 6.5 in as an input because the inputs to this function are all whole numbers, right? We don't get savings halfway through the week. The idea is, you know, we have $450. We wait a week. We're given a $5 bill. We have $455. We wait a week. We're given a $5 bill, we're up to $460, et cetera. We don't get money at the half week point. So in fact, what this leads us to talk about is what's known as the domains.
Remember, the domain is the set of all inputs that make sense. So what really makes sense are what are known as the whole numbers. All right? This is the domain. The whole networks. And I suppose if we were going for a year, maybe we would just say up to 52. Let's see if I can get that in there. Just barely. Now that brings us back to the range. Remember how I said the range I kind of left it a little bit to be desired up here. The range of the outputs, and we really don't have every number from four 50 to 7 ten. In fact, what do we have? Well, our range is really four 50. Then it's four 55. Then it's four 60. Et cetera, all the way up to 7 ten. That is actually our range. The domain is the inputs, the range is the outputs. Let's take a look at letter E letter is interesting. I just wanted to make sure that we really understood this rate of change idea. In letter I ask, use two points from the table to verify that the rate of change of the function is 5. How do the units show up in this calculation? So remember, rate of change is just slope. So I actually like to use the .5 comma four 75 and the .10 comma 500 and I'd like to show you how this calculation works out, right? If I just calculate the slope, the change in Y divided by the change in X, what will that be? 500 minus four 75 divided by ten -5. The calculations easy enough, the numerator ends up being 25.
The denominator is 5 and look at that. That 5. We knew that, by the way, that was the slope of our model, right? Up here, the slope of our model. Now, it also says, how do the units show up in the calculation? Well, this is kind of cool. Units especially in linear modeling can be tracked. In other words, the numerator here is not unilateral. $500 $-475. Giving us a result of $25. The denominator is not unitless either. This is ten weeks -5 weeks, giving us our abbreviate it 5 weeks. So this isn't just a unitless 5. This is $5 per week, right? Which is exactly what we knew, right? The slope was dollars per week, specifically $5 per week. But what's really kind of cool about these types of calculations is that you can really see the units come through the calculation itself. Units are very important. Don't brush them off. Don't think that they're just that thing that gets you that last point of credit. All right, I'm going to scrub the text so pause the video now if you need to. All right, here it goes. Okay, let's go on to the next problem. Exercise number two, again, I'd like you to pause the video and take as much time as you need to understand the physical scenario going on here.
All right, Kirk. I don't know why I can't come up with enough names not to use my own, but hey, why not? Kirk is driving along at road a long road at a constant speed. He's driving directly toward Denver. He knows that after two hours of driving he's 272 miles from Denver, and after three and a half hours he's a 176 miles from Denver. All right, the first thing I'm going to ask you to do in letter a is to summarize the information that's been given in coordinate points. Where H, the amount of hours that Kirk's been driving is the input, it's the X coordinate. And D, the distance from Denver is the output, the Y coordinate. So in other words, if I get to take this piece of information and I'd like you to turn it into a coordinate pair and take this piece of information. Turn it into a coordinate pair. Pause the video for a minute. All right, that should be pretty quick, right? Because what do we have? We have after two hours, we're 272 miles from Denver. And after a watch out, 3.5 hours were 176 miles from Denver. All right, that's simple enough. Letter B calculate delta D divided by delta H all right, so that's the average rate of change, include proper units in our answer.
Okay, now again, don't freak out by this type of notation. I know this looks like D times 3.5, but the way that we interpret it is here is the distance at 3.5. That's 176. Now watch this. I'm going to track the units. That's a 176 miles. Minus D two, right? That's the distance after two hours. That's 272. Miles. Divided by, and let's say it is be very specific. This is 3.5 hours. Minus two hours. Now, if we do that subtraction in the numerator, we get negative 90, what do we get? Negative 96. Yeah. Negative 96 miles. And then of course we get divided by 1.5 hours. Don't worry. And then when we do that division, we get. Negative 64 miles per hour. Isn't that nice? Look at how those units just kind of come through with us. A lot of students, of course, say, oh, how do I have to have the units here and here? And I often say, well, no, you don't. Often they can become laborious to write down, it can be messy, it can be kind of confusing. So it's understandable if students don't want to write them there, but they're always there. Always there.
Now take a look at letter C it says you should have found that the rate of change was negative. Why is it? Explain what is physically happening to result in this negative rate of change. So what I'd like you to do is pause the video and really think about those questions. This is important. All right, so why is it that we get negative 64 mph, right? Well, the reason that it's negative is because. The distance. And this is very important. The distance is decreasing. See what rate of change tells us really. At the end of the day is whether the output is increasing or decreasing as time. Increases. We always allow the input to increase. All was allowed the input to increase and very often that's because the input is most importantly time. Not always, but very often the input is time. And then what happens is we just ask, well, is the output increasing or decreasing when we have a negative rate of change like we did in B, it means that the distance is decreasing.
Now we knew that the distance was decreasing. We're driving towards Denver, but had we gotten a positive rate of change, it would have been a problem. Okay, letter D this one's a little bit harder than an exercise one. Letter D it says, assuming the relationship is linear, which it would be if we're driving at a constant speed, right an equation for the distance D as a linear function of the number of hours H all right. Now in this case, what I'm going to actually do is I'm going to write down Y equals MX plus B okay. One of our two parameters, either M or B, we already know. What I'd like you to do is I'd like you to pause the video and think about which one of those two we know. Did you get it? Did you realize we already know the slope? The rate that the distance is changing, right? As compared to time, is right here. This is our slope. Our slope is that negative 64 miles per hour. So I'm going to check that off. In fact, now I'm going to write it in terms of D, what we know is we know that the distance. Is negative 64 mph, times the number of hours we've been driving, plus the Y intercept. But that's the real whammy, right? Actually, let me forget about blue. Let's go red, right? That is. That is hugely important. I do not know this. And that's very often the case.
You know, especially when we're given two pieces of information, we can calculate the slope of the model by simply doing what we didn't be, but then we don't know what the Y intercept is. But we can find it the same way every time. Think about this for a moment. You've done this before. You've found equations of lines. What did you do in the past to find B? That's right. You substitute in one of the two points you know. Which one doesn't matter. All right? You substitute one of these in. All right? I'm going to go back to blue. So I don't know, let's do the two, two 72, right? So I take my two to 72, be careful the two 72 is D and I put in the two for H and what that leaves me is only the B to solve for, right? Look at what happens. Of course, at this point, we're very good at solving equations, so this is no problem. We do negative 64 times two. We get negative one 28. And then we can get rid of that negative one 28 by adding a 128 on both sides. So this should look very familiar to you. You should have done this quite often. What do we end up getting 400? Yeah. So what's our final model? D equals negative 64, H plus 400.
Now it's a great idea is always to check something like this by taking a point from up here and substituting it in. I'm going to let you do that yourself so that we don't make the video too long. Let's take a look down at letter E how did how far did Kirk start from Denver? Show the work that leads to your answer. So how far was I from Denver right when I started driving? Pause the video if you need to. Did you figure it out? It's actually quite easy. Oh my. What happened there? What we, what we, how far I started from Denver is 400 miles. Right? Well, that's just the Y intercept. You know, one thing, of course, that you could do is you could just say, well, I'm assuming that time works the normal way, right? I started from Denver at H equals zero. But if I evaluate, let's use some function notation. If evaluate D of zero and I get negative 64 times zero plus 400, right? Well, negative 64 times zero zero, right? Plus 400. It gives me 400 miles. That is a long ways away from Denver, long drive. Let her have, I love this one. Let's use the model finally.
After how many hours will Kirk arrive in Denver show the work that leads to your answer? All right, so I'd like you to try to do this. How long will it take to get to Denver now? Pause the video and see if you can figure this out. All right. Well, the real key here is understanding that when Kirk gets to Denver, the distance is zero, right? That's the key, right? The distance away from Denver is zero when we get to Denver. So what we end up solving is this equation, negative 64 H plus 400 equals zero. That's not so bad. We'll subtract 400 from both sides. Just undo what's been done to H negative 64 H equals negative 400. Divide both sides by negative 64. Run in that room. And what we have. Oh, nice. 6.25 hours. 6 and a quarter hours, long drive. Long drive. So this one was a little bit more challenging and exercise number one, we were literally given pretty much the slope $5 per week, and the Y intercept $450, they were just there. So writing the model was pretty easy.
When we're given a linear situation where we simply know two points that lie on the graph, well, then what we have to do is calculate the slope, like what we did in letter B, find the Y intercept like we did in letter D, and then we have our model. All right? So it's a little bit more challenging. I'm going to scrub the text. We're going to look at one last multiple choice problem, so pause the video now if you need to write anything down. All right, here we go. Clear screen. Last problem. All right, little multiple choice problem. This would be relatively simple modeling problem, but in a multiple choice context. Amanda is walking away from a light pole at a rate of four feet per second. If she starts at a distance of 6 feet from the light pole, which of the following gives her distance D from the light pole after walking for T seconds. All right? So we want the distance she is from the light pole after T seconds. And to kind of help us visualize, take a look. We've got her at zero seconds. She's 6 feet away.
After one second, if she's walking at four feet per second, she's got to be ten feet away. Let's do that one more time, right? Got it? So we got the two parameters. Where do we start? What is our starting output value? If we kind of think about it as Y equals MX plus B, right? This is where we start. And this is the change in Y over the change in X, right? The rate. Well, they tell us two things in this problem. Four feet per second, and we start 6 feet. One of them's got to be one of them. One of them's got to be M, one of them's got to be B which one's which. Did you get it? Right? This is most certainly the slope. M is four feet. Per second. Look at those units, right? Change in Y divided by a change in X or a change in T and B, well, that's got to be the 6 feet, right? It's where we start. It's the value we have when the input is zero. So our model Y equals MX plus B becomes D equals M, T plus B, right? Four T plus 6. Choice one. All right. Got it? So oftentimes it's very easy when we're trying to extract the slope and the Y intercept from a problem. Because very often in a real world situation, we know them.
You know, we know what our starting value is, and we know what our rate is. We just have to put them together in terms of M and B all right. One last scrub, pause the video now if you need to. All right, and it's gone. Hey, she's back. All right. Well, thank you for joining me again for another E math instruction. Common core algebra one lesson. I'm Kirk weiler. Remember that you can download the worksheet and the homework for this lesson by clicking on the video's description. As well, try out that QR code and please leave us a comment to tell us how you think it works. Obviously, we could see it being a big advantage, but if photocopied it could get a little bit messy. But hopefully it'll work and help you get to the videos quicker. Okay, until next time, keep thinking and keep solving problems. I'm Kirk Weiler.