Learning Linear Equations Through Kai's Journey

Well, hello. Today we are going to do co-requisite activity 6a. And this is, it's not a review of linear equations, it's more of looking at linear equations in a different way. I know you've probably all been taught about equations of lines, y equals mx plus b. Kind of want you to put all of that aside. And we're going to approach it from a different angle. And we're going to use, really, I find it a pretty touching story. And it's the story of a beautiful dog named Kai. So here goes. I hope you can hear this. So meet Kai. And Kai is going to help us. retain information about linear equations in a whole different way so keep an open mind he looks like a table with like stuff way under it even the pictures don't do justice of how overwhelmingly large he truly was I took him to our vet and he just said it's the fattest dog he'd ever seen and he thought he should lose roughly 100 pounds. The previous owners had taken him in to actually put it down because he was having a hard time getting up and the vet just felt that he still had a lot of life in him. When he got to the house, it took us probably 20 minutes to get him up the three stairs into the house. I remember saying to the vet, like, I literally don't know what to do. And the vet was like, anything you do is going to help. He literally couldn't do anything. He could walk a few steps and then he'd... and lay down. It would just bring me to tears. I remember going to the dog park at first and he'd just sit there, like he'd watch everyone play and he'd just kind of look at me and I'd be like, buddy, I don't know if we'll ever get there, but we're going to try. I'm a nurse and I used to work cardiac rehab and we'll just start with the simplest thing, which is walking. So I walked him three times a day, no treats. His food was monitored and every day we went out. And at first he could just walk maybe five or ten steps and then he'd stop and lay down. Then I thought maybe... If I could get him in water, it would be way better just for the buoyancy and for him to be able to have full range of motion. There was little things every day that you would just think. wow he couldn't do that before. One day he like licked his back legs and then sure enough he started you know prancing along and his speed got faster and faster and then he'd run or he'd jump something or he'd climb up he clawed his way up onto the bed. It would overwhelm me. We kind of started getting a little fan club at our local dog park of people cheering him on and telling him he could do it. He's showing me how to remain positive and happy and take really big goals and just break them down into little tiny steps. I worked harder than any person, animal I have ever known in my life. to where he could enjoy life again. All he wanted was to have fun and be part of a family. He just never gave up. People might have given up on him, but he never gave up on himself. By the time we were done, I just couldn't give him up. There was no way. I'm so proud of him. Like, comment, and subscribe. Okay, so you know how I feel about dogs, and that dog, that really resonates with me. So we're going to use this little puppy, this not so little puppy, as a kind of a story to help us with learning equations. But before we do that, there's another lesson. I mean, regardless of how you're doing in this class, I'm sure there are some other classes that maybe... are struggling are a struggle for you or being in school or maybe something entirely different so um we watched the video and now for number one um i want you to reflect on how this video made you feel and is there something in your life that you'd like to change so i know you have to upload these notes so you don't have to you can just think about it you don't need to write it down though oftentimes data shows that if someone can write down a goal and kind of just the act of writing it down helps you solidify in your mind. So I'm going to write something down and I'm sure my goal is different than yours. Is there something you would like to change? So you're not required, but I'm going to go ahead. I would really like to get in better shape. So Bronwyn. wants to become more active, more healthy. So for example, it is now 2 30 in the morning, so maybe improve my sleep would be a good thing. and exercise. Kind of excited about the next chapter of my life. I've been working for many, many years and my kids, my daughter has an awesome partner and I want to be a really active grandma one day. So I got to start planning that now. So that's what this video made me think about. So listen phrases. or ideas from this video that might help you meet your life goal or like your goal or life challenge. So some of the things that I, that jumped out at me and this, so write down your own for a minute, pause and write down your own. We'll see if ours are the same. So one of the things that jumped out at me is we'll start with just the simplest things. So it doesn't for you, maybe it's not that you need to be active. Maybe you need to be better about how you, how you spend money. Maybe you need to improve your relationship with somebody else in your life. Maybe you need to get out of a bad situation that you're in. So one of the things she said is we'll start with the simplest things. Well, I think she said walking. So, and I say that a lot, just one foot in front of the other. It's one of the reasons why I have so many assignments every day. You just chip away at them. Don't think about what's way ahead of you. Just do one piece at a time. And you'll go far if you do that. Another one, little things every day. um and i really liked this one take big goals um and just break them down into tiny steps So that's just a few. I wish we were in class together so that I could hear the things that you came up with that are different and the things that are the same. Both having common ground and having differences, I find kind of inspiring. So I do want to point out that I don't, so Kai the dog was successful in losing his weight. He started at 173 pounds and he lost the 100 pounds that he needed to lose, which is pretty amazing, which was more than half the weight he had. But he couldn't do any of this without Pam. Pam was his ally. and that's a beautiful thing and um uh you know if you're struggling in this class or if you hit a roadblock later on i'd like you to think of me and the tutors so this semester mark and alec i hope they stay with me but um whatever tutors i have i always pick the best tutors and you we really are rooting for you. And I wouldn't be where I am today if I didn't have people who helped me out. It's almost impossible to find people who are self-made people. And I certainly am not one of them. So if you need help, I really hope you feel comfortable reaching out to us. Maybe not now, maybe in a while. Okay. So when Kai, so now we're getting into the math. So that was kind of the growth mindset thing. When... Kai met Pam. He was almost a hundred pounds overweight. I mean, that's, I don't think I'm, I mean, I just can't imagine that would be like us being like two or 300 pounds overweight. And his previous owners want to euthanize because of that. It was just, it had gotten out of hand and they didn't see any hope for him. And I'm sure there might be people in your life that think you're not going to change and they're wrong. People can change. um maybe you don't want to change but if you do and you probably will later on it's not now um so initial weight for kai was 173 pounds when when when pam brought him into the office that's how much he weighed and that was his initial weight okay um so she decided to set out a goal of just two pounds each week. So she was going to put him on a diet. She was going to make him exercise and it was going to be two pounds a week. And that kind of goes into the take a big goal, which was losing a hundred pounds and break it into tiny steps. And one way or the other, you're going to get there. So I want you to hold onto this. for linear equations. So here we go. Assuming Kai meets his goal of losing two pounds a week, so two pounds a week, let's track his weight from week to week for the first six weeks of his time with Pam. So day zero, when the vet, when Pam decides to adopt him, he weighs, or she's fostering him at this point. I've got two foster dogs. There's Oliver right there. Foster fails because now they're my dogs. So after a week, Pam was successful in helping Kai lose two pounds. I'm going to write a minus two and it is two pounds, but I'm not going to put the units in there. They're all pounds. So I'll be just to remind me. So Kai loses two. And I'm going to put the minus sign in there to indicate that he lost two. So if it's a constant, if he's going to lose two every week, we call that a constant rate of change. So from here, oops, I'm going to blur everything. Let's see why. So constant rate of change. He's going to lose two pounds again. Well, so I just need to figure this out. 71 take away two is going to be 69. So he will be. 169 pounds, which is still massive. And so I just want you to go ahead and fill out. So for the first six weeks, fill out this chart and just track what's happening. And this is a constant rate of change. In this case, a constant loss. So minus two, minus two, minus two, and minus two. I'll put that in there. 2, 4, 6, 8, 10, 12. So after six weeks, he's 12 pounds lighter. He's getting there, right? So how much did he weigh? 167, 165, and then 163, and then 161. Okay, so we get the pattern there. So what I'd like you to do now. create a graph that keeps track of Kai's weight loss for the first six weeks, assuming he manages to consistently lose two pounds a week. Label your axes accordingly. So in math, you probably were not taught to do this, but in this class, because it's statistics, we're going to do, I'm doing that little zigzag there, which indicates that we're not, we can't keep track of everything. So this is going to be your y-axis, always is your y-axis, and this is going to be your x-axis, and the x, so we want to label this, I think I have one there, so I'm going to say x equals, and we're keeping track, if you look here, it's the weeks. So week zero, week one, week two. So this is going to be number of weeks since the diet started. Number of weeks since diet began. So that's the X variable. What's the Y variable? The Y variable is going to be Pi's weight. And it's a really good idea to put the units. So we've got weeks and pounds. So we're going to start at week zero, one, two, and so on. And so. I want the scale to be as easy as possible. So the zigzag means the next one isn't one. I'm going to start from the top and we'll just start here. Zero is going to be 173 because that was his initial weight. That's his initial weight, which is right here. Okay. So after a week. He's going to lose two pounds. And this is his ordered pair right here. X comma Y. At one week, he's 171 pounds. So you go over one and you go up to 171. Well, if this is 173, 171 is going to be two down from that. So I've literally fallen. two pounds and I'm going to put here one comma 171. So there's my ordered care. This is my X and that's my Y. This is my week and that's my pounds, not my pounds. Well, these are Kai's pounds. So for a dog, that's massive. So after, if we're going to go over two weeks. We're going to drop another two pounds and you can kind of see this now for, so for two, it's going to be 169. So you literally drop, so you, you're moving over two weeks, one week, and you're dropping two pounds. And I just want you to fill that out. So you go down another, so it's a week later. One week, two pounds. Another week goes by. That's going to put him at three weeks. And he's going to lose another two pounds. Bam. And so this is the third week. And he's losing two pounds. So that's 167. So he's starting probably to feel a little better. Now, if we keep, so we'll just keep going because he's certainly going to keep going until he gets to his ideal weight. So another week gone by, losing another two pounds, bam. And what you're seeing, so you can do that, but I'm going to cheat. So let's back up. There's our graph. What do the horizontal and vertical axes represent? So the horizontal axis right here, horizontal axes. horizontal is weak on diet and the vertical axis is weight. Explain the ordered pair 5,163. So 5,163 is going to be this one right here two three four five whoops it wasn't quite right there let's count a little tiny bit so it's going to be this one right here 16. was it drop two drop two which means drop four 163 okay so um number of weeks number of pounds not how many pounds he lost but what his actual weight is so um if we so that i always kind of take my time on this So I'm going to write it in a sentence in context. So I better talk about Kai and I better talk about the dog and the weight and the whole situation. So after five weeks of dieting, Kai weighs 163 pounds. Okay, and this says that so succinctly. That's why mathematicians really like what we call ordered pairs. And they're ordered pairs because you always, always have to write the x first and the y second. That's why we call them ordered pairs. What pattern is made when... you have a constant rate of change. And in this case, it's two pounds lost per week. So if we look at this, what pattern are we making with those blue dots? And I hope you can see it, that what we're making, I'll pick a, it would be an amazing color. I'll do a turquoise. So. you know, we're tracking what's happening every week, but everything in between is also, so if I connect these, so we could figure out what happened in the middle of week one and week two. And since it's a continuous rate of change, there'll be something in between, like halfway through he's lost one pound. So what I see, the pattern I see is a line is being made. So it's a linear pattern. And that's just a fact of rate of change being constant. So we talked about in the last section, in chapter five, we talked about linear equations and we talked about scatter plots and that kind of thing. There's two different kinds of it. So the way you describe them is direction, form, strength, anything unusual. That's what you talk about. So the direction could be positive or negative. The form is linear or nonlinear. And you could come up, there's, we're going to learn something called regression and we're going to learn linear regression and there's quadratic regression. exponential regression. There's all different kinds, but linear is the most common. And a lot of times you can see lines in your data. So that's a line comes when the change is pretty constant. At this constant rate of weight loss, how much will Kai lose after 10 weeks and after 25 weeks? So we want to know what's happening after 10 weeks and what's happening after 25 weeks. So, gosh, I ran out of graph paper here. So unless you want to go buy a bigger piece of graph paper and just keep counting down the two, lose two, lose two, lose two, it would be better to look at the pattern. It might be less work. So, so let me just go through this and I'm not using any algebra. I'm not using any Y equals MX plus B, whatever that is. So if he loses, so just reasoning this out, if he, Kai, loses two pounds per week, well, so it's going to be. two the first week, two the second week, two the third week. So I could, I'm looking at this pattern here. It's two every time. So after 10 weeks, he's literally lost 10 twos. So he's lost two 10 times. so that's going to be two 10 times is going to be 20 pounds so um how much will kai weigh we we're not done though we we don't just want to know how much he loses we want to know how much he weighs so the way you calculate the weight is his new weight you His new weight is old weight minus loss. So what was his old weight? Well, if we go back to his initial weight, that was 173 minus the loss. And the loss was 20 pounds. So his new weight is going to be 153. So we can do that without the graph once we see the pattern. And I asked for 25 weeks. For 25 weeks, well, he's losing two every time. So for 25 weeks, I could just subtract that two 25 times. For 25 weeks, I'm going to, if I want his new weight, his new weight. is going to be initial minus two is how much he loses each time and now it's going to be oh 25 times so i'm seeing a pattern here initial so it's the initial value And this right here is the rate of change. And this right here is the number of weeks. So we're kind of building a linear equation here. So it'll be 173. That was his initial value. We're losing weight. So it's going to be minus 2 times 25. And I hope you can visualize that you're just taking... Every week, you're taking away two pounds. two pounds. And you did it how many times? In this case, you did it 25 times. So that'll be 50 pounds. So 123. And probably Kai feels so much better. 123 pounds. After 25 weeks is his new weight. So he's not done yet. He really needs to keep going until the, I think the doctor said 75 pounds. So that is a sense of what linear equation is without any memorization. We want it all to be just common sense here. So number nine, describe how you arrived at your answers to eight and nine. Use mathematical operation in your description. So I want you to describe. So we have an initial weight. 173. We have a constant rate of change. I'm going to do something a little bit different here. I'm going to say it's negative two. If it were positive two, it means I'm gaining two pounds a week, which that can happen, especially in a pandemic. And then we know the number of weeks it's given to us. In this case, it was 25 or 10 or whatever. Okay. And so I kind of put that all together. where I saw the pattern was that new weight equals initial weight in this case minus two times the number of weeks. Okay, so what we have is we have two variables here, and this one can be thought of as the explanatory variable, and this one You calculate this, that can be thought of as the response variable. So as the weeks go by, you get a new weight. The weeks determine the weight rather than the weight determining the weeks. Okay, use your description in mind to write a linear equation that describes Kai's situation. So make sure to explain what the variables mean. So you could use X and Y, but I strongly recommend that you use letters that help you remember what the heck you're doing. So instead of a Y, I am going to use a W. W equals. And then we know the initial weight is a constant 173. That's the initial weight. And we know that we're losing two pounds, not once, not twice, but every week. Oh, W and W, we can't have the same thing. So I'm going to say it's time. It's a time unit. So I'm going to say T and I'm going to put parentheses around it to kind of emphasize, oh my gosh, we've got a linear equation there. So this is time in weeks. That's what T is. And T gets plugged in. And then what gets plugged out is weight. and it's chi's weight always new weight and presto bingo we have a linear equation and um so there we go that was what most of the hard work was um in a hundred weeks here so You're used to y equals mx plus b, but I'm doing instead of the y and instead of the x, I'm doing t and w to remind me of what the heck I'm trying to do. All right, so x is the explanatory variable, and it always goes along the horizontal axis. We've talked about that in the last discussion section. Y is the response variable. In math classes, you said independent and dependent. We're not using those terminologies because independent means something else in statistics. The Y-intercept. is the point in line where the y-axis intercepts the line. But I am going to say that the y-intercept, and it's always true, this is the initial value of y. And I'll just put in parentheses here when x is zero. So it's also, you can think of it as your... start point. So when you started the study, where were you? So that's this right here, initial value. That's our base. That's our baseline. And then B, you you've learned in another class that b is your constant rate of change and we have this word called slope so right there that's your slope or constant rate of change and we know it to be negative two you it's helpful to me to put a one under it. So I can see these are the pounds and these are the weeks. So it literally reads this line in the middle is a per pounds per week is exactly what it translates to. the rate of change is you lose, you lose two pounds per one week. So every week, so that's, and, and you always want your slope to be a ratio of. the change in the response divided by the change in the explanatory. So in this situation, and so you're much more used to, and I'll just write it here, kind of in boring black, y equals a plus bx. that's what you're used to seeing. And it's the same thing, initial value. rate of change, explanatory and response variable. So what was Kai's explanatory variable? So if we look at the situation here, so I'll pause it and you write, fill out number 11. And you're looking at, this is your equation right here. And tell me what's the word for the explanatory and the response variables. So. the explanatory, what does she look at first? She's keeping track. And it's also, by the way, the first column of data. So the explanatory is going to be the number of weeks, the color I used for that number of weeks is used to determine um weight and the slope is going to be my negative two and that negative sign is really important and I'm going to say negative two over one change in weight over change in weight So for every one week, Kai loses two pounds. That's how that's read off. And the Y intercept is going to be 173 pounds. That's what poor Kai was when he came into the hospital or the doctor's office. Rewrite your equation in question 10 using the new naming convention. So the... I don't know which one's new, but you could say Y equals 173 minus 2X, or the new naming convention is weight equals 173, and it's a minus 2 because you're losing two pounds a week. and then this would be t for time. And I like to put parentheses around because it reminds me time goes in, weight comes out. I think that's what I wanted to tell you. So that's kind of a crash course in linear equations. This is not an optional assignment. This is a required assignment because I know that I need to take the time to go over and learn some new values. So go ahead and upload these notes. They are required. And I'll meet you in the next Zoom, which will be not too, now what you want to do now is you want to do, it's, if there is a correct, do that first, then go ahead and do the preview assignment for 6A. and then I'll meet you back here for the zooming on the in-class exam. All right, take care.

Kind of want you to put all of that aside. And we're going to approach it from a different angle. And we're going to use, really, I find it a pretty touching story. And it's the story of a beautiful dog named Kai. So here goes.

I hope you can hear this. So meet Kai. And Kai is going to help us.

retain information about linear equations in a whole different way so keep an open mind he looks like a table with like stuff way under it even the pictures don't do justice of how overwhelmingly large he truly was I took him to our vet and he just said it's the fattest dog he'd ever seen and he thought he should lose roughly 100 pounds. The previous owners had taken him in to actually put it down because he was having a hard time getting up and the vet just felt that he still had a lot of life in him. When he got to the house, it took us probably 20 minutes to get him up the three stairs into the house. I remember saying to the vet, like, I literally don't know what to do.

And the vet was like, anything you do is going to help. He literally couldn't do anything. He could walk a few steps and then he'd... and lay down. It would just bring me to tears.

I remember going to the dog park at first and he'd just sit there, like he'd watch everyone play and he'd just kind of look at me and I'd be like, buddy, I don't know if we'll ever get there, but we're going to try. I'm a nurse and I used to work cardiac rehab and we'll just start with the simplest thing, which is walking. So I walked him three times a day, no treats. His food was monitored and every day we went out.

And at first he could just walk maybe five or ten steps and then he'd stop and lay down. Then I thought maybe... If I could get him in water, it would be way better just for the buoyancy and for him to be able to have full range of motion. There was little things every day that you would just think. wow he couldn't do that before.

One day he like licked his back legs and then sure enough he started you know prancing along and his speed got faster and faster and then he'd run or he'd jump something or he'd climb up he clawed his way up onto the bed. It would overwhelm me. We kind of started getting a little fan club at our local dog park of people cheering him on and telling him he could do it.

He's showing me how to remain positive and happy and take really big goals and just break them down into little tiny steps. I worked harder than any person, animal I have ever known in my life. to where he could enjoy life again. All he wanted was to have fun and be part of a family.

He just never gave up. People might have given up on him, but he never gave up on himself. By the time we were done, I just couldn't give him up.

There was no way. I'm so proud of him. Like, comment, and subscribe. Okay, so you know how I feel about dogs, and that dog, that really resonates with me. So we're going to use this little puppy, this not so little puppy, as a kind of a story to help us with learning equations.

But before we do that, there's another lesson. I mean, regardless of how you're doing in this class, I'm sure there are some other classes that maybe... are struggling are a struggle for you or being in school or maybe something entirely different so um we watched the video and now for number one um i want you to reflect on how this video made you feel and is there something in your life that you'd like to change so i know you have to upload these notes so you don't have to you can just think about it you don't need to write it down though oftentimes data shows that if someone can write down a goal and kind of just the act of writing it down helps you solidify in your mind.

So I'm going to write something down and I'm sure my goal is different than yours. Is there something you would like to change? So you're not required, but I'm going to go ahead.

I would really like to get in better shape. So Bronwyn. wants to become more active, more healthy. So for example, it is now 2 30 in the morning, so maybe improve my sleep would be a good thing. and exercise.

Kind of excited about the next chapter of my life. I've been working for many, many years and my kids, my daughter has an awesome partner and I want to be a really active grandma one day. So I got to start planning that now. So that's what this video made me think about.

So listen phrases. or ideas from this video that might help you meet your life goal or like your goal or life challenge. So some of the things that I, that jumped out at me and this, so write down your own for a minute, pause and write down your own.

We'll see if ours are the same. So one of the things that jumped out at me is we'll start with just the simplest things. So it doesn't for you, maybe it's not that you need to be active. Maybe you need to be better about how you, how you spend money.

Maybe you need to improve your relationship with somebody else in your life. Maybe you need to get out of a bad situation that you're in. So one of the things she said is we'll start with the simplest things. Well, I think she said walking.

So, and I say that a lot, just one foot in front of the other. It's one of the reasons why I have so many assignments every day. You just chip away at them. Don't think about what's way ahead of you. Just do one piece at a time.

And you'll go far if you do that. Another one, little things every day. um and i really liked this one take big goals um and just break them down into tiny steps So that's just a few.

I wish we were in class together so that I could hear the things that you came up with that are different and the things that are the same. Both having common ground and having differences, I find kind of inspiring. So I do want to point out that I don't, so Kai the dog was successful in losing his weight.

He started at 173 pounds and he lost the 100 pounds that he needed to lose, which is pretty amazing, which was more than half the weight he had. But he couldn't do any of this without Pam. Pam was his ally.

and that's a beautiful thing and um uh you know if you're struggling in this class or if you hit a roadblock later on i'd like you to think of me and the tutors so this semester mark and alec i hope they stay with me but um whatever tutors i have i always pick the best tutors and you we really are rooting for you. And I wouldn't be where I am today if I didn't have people who helped me out. It's almost impossible to find people who are self-made people. And I certainly am not one of them.

So if you need help, I really hope you feel comfortable reaching out to us. Maybe not now, maybe in a while. Okay.

So when Kai, so now we're getting into the math. So that was kind of the growth mindset thing. When...

Kai met Pam. He was almost a hundred pounds overweight. I mean, that's, I don't think I'm, I mean, I just can't imagine that would be like us being like two or 300 pounds overweight. And his previous owners want to euthanize because of that. It was just, it had gotten out of hand and they didn't see any hope for him.

And I'm sure there might be people in your life that think you're not going to change and they're wrong. People can change. um maybe you don't want to change but if you do and you probably will later on it's not now um so initial weight for kai was 173 pounds when when when pam brought him into the office that's how much he weighed and that was his initial weight okay um so she decided to set out a goal of just two pounds each week. So she was going to put him on a diet.

She was going to make him exercise and it was going to be two pounds a week. And that kind of goes into the take a big goal, which was losing a hundred pounds and break it into tiny steps. And one way or the other, you're going to get there.

So I want you to hold onto this. for linear equations. So here we go.

Assuming Kai meets his goal of losing two pounds a week, so two pounds a week, let's track his weight from week to week for the first six weeks of his time with Pam. So day zero, when the vet, when Pam decides to adopt him, he weighs, or she's fostering him at this point. I've got two foster dogs.

There's Oliver right there. Foster fails because now they're my dogs. So after a week, Pam was successful in helping Kai lose two pounds. I'm going to write a minus two and it is two pounds, but I'm not going to put the units in there. They're all pounds.

So I'll be just to remind me. So Kai loses two. And I'm going to put the minus sign in there to indicate that he lost two.

So if it's a constant, if he's going to lose two every week, we call that a constant rate of change. So from here, oops, I'm going to blur everything. Let's see why.

So constant rate of change. He's going to lose two pounds again. Well, so I just need to figure this out.

71 take away two is going to be 69. So he will be. 169 pounds, which is still massive. And so I just want you to go ahead and fill out. So for the first six weeks, fill out this chart and just track what's happening. And this is a constant rate of change.

In this case, a constant loss. So minus two, minus two, minus two, and minus two. I'll put that in there.

2, 4, 6, 8, 10, 12. So after six weeks, he's 12 pounds lighter. He's getting there, right? So how much did he weigh?

167, 165, and then 163, and then 161. Okay, so we get the pattern there. So what I'd like you to do now. create a graph that keeps track of Kai's weight loss for the first six weeks, assuming he manages to consistently lose two pounds a week.

Label your axes accordingly. So in math, you probably were not taught to do this, but in this class, because it's statistics, we're going to do, I'm doing that little zigzag there, which indicates that we're not, we can't keep track of everything. So this is going to be your y-axis, always is your y-axis, and this is going to be your x-axis, and the x, so we want to label this, I think I have one there, so I'm going to say x equals, and we're keeping track, if you look here, it's the weeks. So week zero, week one, week two.

So this is going to be number of weeks since the diet started. Number of weeks since diet began. So that's the X variable. What's the Y variable?

The Y variable is going to be Pi's weight. And it's a really good idea to put the units. So we've got weeks and pounds.

So we're going to start at week zero, one, two, and so on. And so. I want the scale to be as easy as possible.

So the zigzag means the next one isn't one. I'm going to start from the top and we'll just start here. Zero is going to be 173 because that was his initial weight. That's his initial weight, which is right here. Okay.

So after a week. He's going to lose two pounds. And this is his ordered pair right here.

X comma Y. At one week, he's 171 pounds. So you go over one and you go up to 171. Well, if this is 173, 171 is going to be two down from that. So I've literally fallen.

two pounds and I'm going to put here one comma 171. So there's my ordered care. This is my X and that's my Y. This is my week and that's my pounds, not my pounds.

Well, these are Kai's pounds. So for a dog, that's massive. So after, if we're going to go over two weeks.

We're going to drop another two pounds and you can kind of see this now for, so for two, it's going to be 169. So you literally drop, so you, you're moving over two weeks, one week, and you're dropping two pounds. And I just want you to fill that out. So you go down another, so it's a week later.

One week, two pounds. Another week goes by. That's going to put him at three weeks. And he's going to lose another two pounds. Bam.

And so this is the third week. And he's losing two pounds. So that's 167. So he's starting probably to feel a little better.

Now, if we keep, so we'll just keep going because he's certainly going to keep going until he gets to his ideal weight. So another week gone by, losing another two pounds, bam. And what you're seeing, so you can do that, but I'm going to cheat.

So let's back up. There's our graph. What do the horizontal and vertical axes represent?

So the horizontal axis right here, horizontal axes. horizontal is weak on diet and the vertical axis is weight. Explain the ordered pair 5,163. So 5,163 is going to be this one right here two three four five whoops it wasn't quite right there let's count a little tiny bit so it's going to be this one right here 16. was it drop two drop two which means drop four 163 okay so um number of weeks number of pounds not how many pounds he lost but what his actual weight is so um if we so that i always kind of take my time on this So I'm going to write it in a sentence in context.

So I better talk about Kai and I better talk about the dog and the weight and the whole situation. So after five weeks of dieting, Kai weighs 163 pounds. Okay, and this says that so succinctly. That's why mathematicians really like what we call ordered pairs. And they're ordered pairs because you always, always have to write the x first and the y second.

That's why we call them ordered pairs. What pattern is made when... you have a constant rate of change. And in this case, it's two pounds lost per week.

So if we look at this, what pattern are we making with those blue dots? And I hope you can see it, that what we're making, I'll pick a, it would be an amazing color. I'll do a turquoise. So. you know, we're tracking what's happening every week, but everything in between is also, so if I connect these, so we could figure out what happened in the middle of week one and week two.

And since it's a continuous rate of change, there'll be something in between, like halfway through he's lost one pound. So what I see, the pattern I see is a line is being made. So it's a linear pattern.

And that's just a fact of rate of change being constant. So we talked about in the last section, in chapter five, we talked about linear equations and we talked about scatter plots and that kind of thing. There's two different kinds of it.

So the way you describe them is direction, form, strength, anything unusual. That's what you talk about. So the direction could be positive or negative.

The form is linear or nonlinear. And you could come up, there's, we're going to learn something called regression and we're going to learn linear regression and there's quadratic regression. exponential regression. There's all different kinds, but linear is the most common. And a lot of times you can see lines in your data.

So that's a line comes when the change is pretty constant. At this constant rate of weight loss, how much will Kai lose after 10 weeks and after 25 weeks? So we want to know what's happening after 10 weeks and what's happening after 25 weeks.

So, gosh, I ran out of graph paper here. So unless you want to go buy a bigger piece of graph paper and just keep counting down the two, lose two, lose two, lose two, it would be better to look at the pattern. It might be less work. So, so let me just go through this and I'm not using any algebra.

I'm not using any Y equals MX plus B, whatever that is. So if he loses, so just reasoning this out, if he, Kai, loses two pounds per week, well, so it's going to be. two the first week, two the second week, two the third week.

So I could, I'm looking at this pattern here. It's two every time. So after 10 weeks, he's literally lost 10 twos.

So he's lost two 10 times. so that's going to be two 10 times is going to be 20 pounds so um how much will kai weigh we we're not done though we we don't just want to know how much he loses we want to know how much he weighs so the way you calculate the weight is his new weight you His new weight is old weight minus loss. So what was his old weight?

Well, if we go back to his initial weight, that was 173 minus the loss. And the loss was 20 pounds. So his new weight is going to be 153. So we can do that without the graph once we see the pattern.

And I asked for 25 weeks. For 25 weeks, well, he's losing two every time. So for 25 weeks, I could just subtract that two 25 times.

For 25 weeks, I'm going to, if I want his new weight, his new weight. is going to be initial minus two is how much he loses each time and now it's going to be oh 25 times so i'm seeing a pattern here initial so it's the initial value And this right here is the rate of change. And this right here is the number of weeks.

So we're kind of building a linear equation here. So it'll be 173. That was his initial value. We're losing weight.

So it's going to be minus 2 times 25. And I hope you can visualize that you're just taking... Every week, you're taking away two pounds. two pounds.

And you did it how many times? In this case, you did it 25 times. So that'll be 50 pounds.

So 123. And probably Kai feels so much better. 123 pounds. After 25 weeks is his new weight. So he's not done yet.

He really needs to keep going until the, I think the doctor said 75 pounds. So that is a sense of what linear equation is without any memorization. We want it all to be just common sense here. So number nine, describe how you arrived at your answers to eight and nine. Use mathematical operation in your description.

So I want you to describe. So we have an initial weight. 173. We have a constant rate of change.

I'm going to do something a little bit different here. I'm going to say it's negative two. If it were positive two, it means I'm gaining two pounds a week, which that can happen, especially in a pandemic. And then we know the number of weeks it's given to us. In this case, it was 25 or 10 or whatever.

Okay. And so I kind of put that all together. where I saw the pattern was that new weight equals initial weight in this case minus two times the number of weeks.

Okay, so what we have is we have two variables here, and this one can be thought of as the explanatory variable, and this one You calculate this, that can be thought of as the response variable. So as the weeks go by, you get a new weight. The weeks determine the weight rather than the weight determining the weeks.

Okay, use your description in mind to write a linear equation that describes Kai's situation. So make sure to explain what the variables mean. So you could use X and Y, but I strongly recommend that you use letters that help you remember what the heck you're doing.

So instead of a Y, I am going to use a W. W equals. And then we know the initial weight is a constant 173. That's the initial weight. And we know that we're losing two pounds, not once, not twice, but every week.

Oh, W and W, we can't have the same thing. So I'm going to say it's time. It's a time unit.

So I'm going to say T and I'm going to put parentheses around it to kind of emphasize, oh my gosh, we've got a linear equation there. So this is time in weeks. That's what T is. And T gets plugged in. And then what gets plugged out is weight.

and it's chi's weight always new weight and presto bingo we have a linear equation and um so there we go that was what most of the hard work was um in a hundred weeks here so You're used to y equals mx plus b, but I'm doing instead of the y and instead of the x, I'm doing t and w to remind me of what the heck I'm trying to do. All right, so x is the explanatory variable, and it always goes along the horizontal axis. We've talked about that in the last discussion section. Y is the response variable.

In math classes, you said independent and dependent. We're not using those terminologies because independent means something else in statistics. The Y-intercept. is the point in line where the y-axis intercepts the line.

But I am going to say that the y-intercept, and it's always true, this is the initial value of y. And I'll just put in parentheses here when x is zero. So it's also, you can think of it as your...

start point. So when you started the study, where were you? So that's this right here, initial value.

That's our base. That's our baseline. And then B, you you've learned in another class that b is your constant rate of change and we have this word called slope so right there that's your slope or constant rate of change and we know it to be negative two you it's helpful to me to put a one under it.

So I can see these are the pounds and these are the weeks. So it literally reads this line in the middle is a per pounds per week is exactly what it translates to. the rate of change is you lose, you lose two pounds per one week.

So every week, so that's, and, and you always want your slope to be a ratio of. the change in the response divided by the change in the explanatory. So in this situation, and so you're much more used to, and I'll just write it here, kind of in boring black, y equals a plus bx.

that's what you're used to seeing. And it's the same thing, initial value. rate of change, explanatory and response variable. So what was Kai's explanatory variable?

So if we look at the situation here, so I'll pause it and you write, fill out number 11. And you're looking at, this is your equation right here. And tell me what's the word for the explanatory and the response variables. So.

the explanatory, what does she look at first? She's keeping track. And it's also, by the way, the first column of data.

So the explanatory is going to be the number of weeks, the color I used for that number of weeks is used to determine um weight and the slope is going to be my negative two and that negative sign is really important and I'm going to say negative two over one change in weight over change in weight So for every one week, Kai loses two pounds. That's how that's read off. And the Y intercept is going to be 173 pounds. That's what poor Kai was when he came into the hospital or the doctor's office. Rewrite your equation in question 10 using the new naming convention.

So the... I don't know which one's new, but you could say Y equals 173 minus 2X, or the new naming convention is weight equals 173, and it's a minus 2 because you're losing two pounds a week. and then this would be t for time.

And I like to put parentheses around because it reminds me time goes in, weight comes out. I think that's what I wanted to tell you. So that's kind of a crash course in linear equations. This is not an optional assignment. This is a required assignment because I know that I need to take the time to go over and learn some new values.

So go ahead and upload these notes. They are required. And I'll meet you in the next Zoom, which will be not too, now what you want to do now is you want to do, it's, if there is a correct, do that first, then go ahead and do the preview assignment for 6A.

and then I'll meet you back here for the zooming on the in-class exam. All right, take care.

Transcript for:Learning Linear Equations Through Kai's Journey

Transcript for:
Learning Linear Equations Through Kai's Journey