 This is going to be partly review because I snuck in some slope earlier But this says rate of change less than three and it's saying investigate the relationship between a slope and a rate part one Says Robin is preparing to make taquitos for supper. She preheats the oven to 400 degrees. This graph shows the heat of oven of the oven increasing over time Take a look at it. Here's your oven temperature in Fahrenheit because oven temperatures are still traditional in Fahrenheit time in minutes Look at the numbers and then let's see if we can answer some of this. What does the point zero comma fifty on the graph represent? You're right. You're right. What? Starting time starting temperature original temperature of The oven at time zero the oven was at 50 degrees. Wait a minute. Shouldn't it be room temperature? Actually ovens the insides are metal. They're often a little cooler inside because it radiates heat out faster Also for what it's worth. This is the physics nerd within me. Don't write this down. Actually the graph should look like this If you haven't noticed ovens turn off then they turn on then they turn off then they turn on then they turn they cycle But for the purposes of easy math Tell us and you just don't talk or I'll move you. Thank you For the purposes of easy math straight line What does five comma four hundred represent? Okay? final temp 400 degrees and How long did it take to get that hot? Five minutes clearly a newer oven line takes 15 drives me crazy see See says calculate the slope of the line Joining these two points. How do we find the slope slope is what over what? Okay, I find it helpful To draw a little triangle like that because this is going to be my rise How high am I careful when you're reading? How high am I right here? Think about it. I heard it How high am I right here? It's not meant to be a tough question folks. How high am I right here? No Read how high am I right there? 400 good. How high am I right here? 50 what's the rise do the math in your head preferably? Tanner Okay, the rise is 350 400 is the top one 50 is my starting height. What's the change in height? What's the rise? 350 so when they want me to find the slope Slope is rise over run. It's 350 is my rise over. What's my run? Well, let's see What value is this right here as an x-coordinate how far to the right? Five where I start out nice easy number to do math with what's the run? The run is five. So the slope the rise over the run is that and put it in lowest terms for me, please 70 70 even 70 what What are the units? Well The rise is my y-axis. Look at your y-axis. What are the units on your y-axis read? degrees per The run is my x-axis. What are the units on my x-axis? And call me silly, but that sounds like a rate from last lesson What we want you to realize today is actually most slopes if they're from real-life data Not from the stupid graphs that they gave you last year that were just x's and y's and have nothing to do with science Most slopes if they're from real-life data the slope is a rate and you can figure out the units calling by simply looking the y units over the x units and those You that are in physics 11 and some of you are or especially those you take physics 12 You've just learned a great trick in physics 11 and physics 12 We actually ask you to recognize some graphs Some students try to memorize all the different graphs that are out there I tell my kids don't if they ask you what the slope is just look at the units and you can almost always figure it out The slope is the y units over the x units So part D says the slope represents a rate of change a Change in temperature divided by a change in time what units are used degrees per minute You guys know the symbol for degrees is a little tiny Zero hanging in mid-air apply You can write degrees, but that's too much writing for me so Says complete the following in this oven the temperature is First of all increasing or decreasing How do you know? So now we're trying to go from last year math 10 where we simply said slopes were positive or negative based on uphill or downhill To now we're trying to start to apply them a little bit more to actual data We'll say increasing or decreasing so increasing at a rate of what's the rate? 70 What per what? degrees per minute You did a little bit of this in science 10 last year I think if your teacher did the physics unit you learned that the slope of a distance time graph was the velocity and It was actually distance meters time seconds meters per Second you actually didn't need to memorize that you could have just told it by looking at figured it out by looking at the units Let's bring the trade Part two Often you're going to be asked to freehand rough sketch a graph and what I mean by freehand rough sketch a graph is just label Your points clear. I'm not going to make you measure or anything like that Says this water is leaking out of the bottom of a barrel at a constant rate So if it's leaking out at a constant rate, it means it's a straight line not a curve When do we look at curves calculus? After two minutes the water is level is 62 centimeters. Here's how I would show that I'd put a two right there That's after two minutes write that down and how much water is there at two minutes. What does the question say? Now really quickly check the next number of centimeters is the next number of centimeters that the question mentions bigger or smaller So I'm going to put the 62 nice and high because I think that's my biggest. There's my first point right there Two comma 62. This is what I mean by a rough sketch. You'll notice Tatiana. I haven't counted not that interested in counting That's a waste of time Then Tatiana if I continue reading it says after how many minutes Look up kiddo after how many minutes You say seven so I'm just going to move over. I'm going to say how about seven right there ish Maybe well, I know seven's not over here. I know I'm running out of graph paper there. I Know I don't want to put seven Right here because it's too tough for me to do this. I'll make a dumb mistake because it's such a cramped drawing That honestly I can't give you a math answer all I can give you is I've done enough of these to know It's much easier if things just spread out, but you heard me. Let's try that again Come back wrong button Okay, here we go scares everybody at home on the internet, but you did hear me ask Hey, look check your next centimeters number if the next centimeters have been higher than 62 I would have put the 62 down here and the next number up here somewhere. It's a rough guess I just want it to be an accurate picture sort of so I put seven over here And what was the next Joe? What was the next number centimeters? Hey, that's 62 22 is less than half. You know what about there ish You know what if they want me to do a scale later. I'll fit that in if I have to I'll do most of this in pencil But usually they don't and I'm kind of lazy I'll take a short approach first and if I really got to make it prettier later fine I'll do that later if I have to I'm gonna I'm gonna argue with you that I think we're smart enough now in grade 11 That we can probably get away without it most often and be lazy watch Connected connect the two dots right This is why Cheyenne I always said to you I draw a triangle because I think I can go rise over run here What's the lowest this graph gets can you see it? What's the highest it gets? What's my run? Now I'm gonna be fussy uphill or downhill So you know what I'm gonna call the rise not positive 40 You know what I'm gonna write in my little triangle. Why not just so I don't forget Negative 40 did I need to put hash marks in there or were you able to figure it out just by looking at the numbers? See, I'm gonna argue that the extra hash marks sometimes are more confusing They're prettier and if I'm in science and I want accurate data those you're in my physics class. Yes Here we're just doing rough sketches. In fact Really quickly right now. What's the wrong? See I would argue without the extra numbers in it's much more obvious that it's five if I put a four and a six I got to then think which numbers am I using I got to do more thinking I would argue this is cleaner In fact, I'm gonna argue that the slope is negative 40 Over five and once again, this one works out evenly in lowest terms negative eight units on The y-axis what are the units of these numbers and we're gonna write that in in just a second to figure that out I got a look at the original question. What was the 22 and the 62 your hint is look for the units right next to those numbers in the question Okay, I'm gonna put a little centimeters and I'm gonna draw mine vertically like sideways so the units are centimeters per and what units what I put on the x-axis here Your hint is find the two find the seven and then find the units that appear right next to that once again Which is why I don't want extra numbers on my graph minutes This is not a graph that I would accept as a lab graph in a science class, but this is not a lab I just want to get a rough idea of what's going on and I think that it's eight centimeters per minute and The negative tells me are we gaining or losing? We're losing It tells us an awful lot This slope represents a rate of change a Change in water level divided by a change in time what units? centimeters Per minute, so it says complete the following to explain what the rate of change of this particular scenario Represents water level is changing at a rate of blank blank per blank. What's the rate? negative eight centimeters per keep going Rashaun minute What we're trying to do today is to get you to recognize that unlike last year when you simply said slope is rise over run This year if we're giving you actual data slope is a rate of change It's a rate it tells you something if you're using actual data. Oh and Decreasing how do I know it's decreasing? Downhill now that means something at a rate of it's decreasing at eight centimeters per Minute you don't usually say that it's decreasing at negative eight because then you have a Decreasing which means negative and a negative eight which means negative I wouldn't take marks off if you said that but usually the word decreasing tells people downhill Turn the page Going back to Cheyenne's earlier question the other reason I knew not to bother trying to graph more accurately Is the piece of graph paper that they gave me in that question? Look at the next question. Did they give me an actual grid in the next quest? I'll probably graph that one a little more accurately because I can count So there's rough sketches, which honestly I like more than actual grid paper because they're less cluttered But grid paper has its uses as well. I Don't know. Let's find out first of all once again Here's a lovely summary when you're studying you might want to put a little dog ear on your book Or you might want to put a little post-it note right there or something says slope represents a rate of change Positive indicates increase a positive rate of change negative Indicates decreasing negative rate of change and then from science 10 Distance versus time in science. What do you measure distance in what units? Meters, what do you measure time in in science almost always? In science usually Okay, so if you memorize in science 10 that the slope of a distance versus time graph was velocity Great, but you didn't need to because the slope would be the rise meters over the run Seconds and hopefully you learned that meters per second is a velocity way easier to learn the units and have that tell you What the graph is then memorize what the heck the graph is Uh-huh. You're not sorry yet after school example one now I'm guessing they want me to be more accurate on this grid because they gave me actual grid paper But let's see Dave entered his car in a long-distance car race He traveled the first hundred and fifty kilometers in two hours after six hours. He had traveled six hundred and fifty kilometers Now Here's how you know Tasha, what's the first word in part a right here? What's the first word? Plot means graph properly with x's and y's and counting You'll notice on the previous page. Oh, they did say plot. I would have said sketch. I Disagree with them. I think they should have said sketch Look at the data that they gave me What's the maximum number of hours that we're talking about in this question? read We're gonna put time on the x-axis and we'd better be able to fit six hours on the x-axis Count how many squares you have on the x-axis? What would be a good scale that would use up most of the graph paper? Do I want each square to be one? I don't think yeah, I think I want to go like this one. You know what? Oh, no, I'll go like this two three four five six and I'm gonna call this time T I'm gonna graph Distance in kilometers on the y-axis What's the biggest number of kilometers this question talks about read? 650 I don't think I'm going up by ones here. How many squares have they given me vertically? You're gonna have to count I Got 13 squares vertically. What would be a good scale here? I think what I'm gonna try first and this is where I use pencil I'll try every other square be a hundred. I'll go by squares of 50 But I don't want to label that much because that's gonna be really really hard to read I'm gonna try going like this. First of all, I'm gonna be clever because I can I'm gonna move that over out of the way I'm gonna go 100 200 300 400 500 600 and that works pretty good if not I would just get out my eraser and I would try a different scale But I do try and think about it a bit ahead of time Every two squares is 100 every one square is 50. I had to write small, but hopefully you can see that now I've zoomed in Okay, by the way, if this looks really messy to you because it does to me I Would had no problem if you did this If you went 100 200 300 400 500 600 in fact, I would almost say that's easier to read by putting hash marks But not putting actual numbers there figure out what works best for you in terms of not making dumb mistakes when counting It wants us to graph this so what's the first point gonna be? What comma what read? Okay, it says it wants me to include zero zero Why is zero zero included because before I start how far have I gone? And as a number that means at time zero my distance is as a number Zero, okay. What's the next point that they gave me now? I got to look at the data Read the sentence he traveled. Yeah So what comma what can you give it to me as x comma y in the proper order, please I? Don't think it's 150 comma two So what comma what? Two comma 150 this is a point here and even though this appeared first This is my x coordinate because I put time on the x-axis To comma 150 I got a point right about there in fact exactly there. Yes What's the next point? What went with 650 I can't remember six so six comma 650 Right there, so it says Plot the ordered pair as explained why zero is zero now take out your rulers and If we lay our ruler down will it go through all three points? Now I'm wondering is there a glitch in my counting. I'm a little worried here Didn't make a glitch it's six hours and 650 is that right and it's two hours and 150 and that is 150 two three four five that is 600 That is 100 that is 650 I Am having a problem here after six hours you traveled Really quickly. Let me see if this is gonna work here 150 divided by two 650 divided by six ah It's not gonna work out evenly. I think they glitched on this question a bit Here's what I would like you to do I Would like you to connect the two points that they gave you these two right here with a nice straight line Then what I would like you to do is connect these two points right here with a straight line at a different end Okay, so the second line It's gonna be Like that not quite a steep and I'm gonna make the graph small again, okay So first of all question a says explain why zero zero is included Why is zero zero included at? Time zero Haven't started haven't gone anywhere at time zero. We haven't gone anywhere yet B says Calculate Dave's average speed between two hours and six hours How would I calculate the average speed when they say find the average speed? What are they actually asking me to find in math terms slope? so Go rise over run For this section right here, what's the rise? What's the highest we get on that line? 650 what's the lowest we get? We're not going down to you were right here. What was that? What was it? Okay? What's the rise? 650 minus 150 I can do that in my head 550 500 isn't it? Yes, what's the run from two to six? What's the run? Four what's the average speed? What's the average speed reduce that to lowest terms, please? 500 divided by 425 or not 125 125 units kilometers per hour between two hours and six hours Dave was traveling freeway speed probably on the coca-cola fast freeway speed Then it says this by looking at the grid and without doing any Calculations how can we tell that the average speed during the first two hours was less than the average speed during the next four hours? How can you tell that he was going slower here than here? Yep, we're gonna use a math word though. We're gonna say Less Steep and that way I can relate it to slope one more. We're done. His distance was increasing But his fuel was decreasing. It's not a drumstick added Or a lightsaber or a sword or a helicopter or a backstrap So as the graph shown here Represents the amount of fuel in Dave's gas tank as a function of the distance traveled and this Cheyenne goes back to your question I would argue this graph is way less cluttered and easy to read Than this graph. I like graphs with fewer numbers myself. That's my preference, but I find them way easier to read calculate the slope of the line Calculate the slope of the line slope is what over what rise over run First of all, what's my rise positive or negative and how can you tell just by glancing at this? Which one I heard both negative How high are we right here? 50 by the way most common mistake kids go run over rise I think I told you I had a kid in physics 12 about eight years ago Who got perfect on the provincial exam and on the last question because he was so relieved He was almost done. He went run over rise because he was in a rush So 50 what's my height right here? What's the run? Sorry, what's the rise as I do it, too? What's the rise 50 40? What's the rise? 50 high 40 high we've dropped 10 dropped negative 50 high 40 high Okay need you back here Okay, deep breaths calm blue ocean and all that stuff Nicole. What's my x-coordinate right here 20? What's my x-coordinate right here? What's the run? Yeah, usually they'll pick nice enough numbers. You can do them in your head run is a hundred What's the slope in lowest terms? It's negative and you can either write that as a fraction Or you can write that as a decimal It depends on the situation to be honest sometimes you want it as a decimal Especially if you're dealing with money money is usually dollars and cents sometimes you'll want it as a fraction So it says complete the following statements The amount of fuel in the tank is increasing or decreasing How can you tell it by glancing at the graph? decreasing decreasing The rate of change of fuel in the fuel tank is how much? negative point one or one over ten I'll use the decimal here because there's not much room to write a fraction What are the units? Number of what what's this being measured in on the y-axis? Leaders abbreviation for leaders is the written letter L lowercase L right leaders per What are the units? kilometers That's his well. We call it mileage because we used to measure it in miles, but now we measure it That's his numbers of meters per kilometer. That's how much gas mileage he's getting The amount of fuel in the fuel tank is decreasing at a rate of Point one Leaders per kilometer Why didn't you put a negative in front here because it said the word decreasing and then word decreasing implies negative Sam what's the whole point of this what I want you to realize is slopes mean something What do they mean look at the units and Almost all the time if you look at the y-axis units and you look at the x-axis units and you realize that's your rate You'll be able to figure out what it is because you're old You've seen things Turn the page. This is why you have the rulers Says assignment Consider the following circles each with center. Oh what you're going to do is you're going to use your rulers and you're going to measure each of those Diameters I can't do it up here because the size of the circle on my tablet is different from the actual size on your Workbook so my answers will be totally different from you, but fill in the chart here Then once you filled in the chart They would like you to do a rough graph Diameter versus circumference. They've told you the circumference for circle a circle b circle c circle d circle e You should find if you measure carefully and you record the data on a chart that it forms roughly a straight line So measure the diameter of each circle to the nearest centimeter record each diameter on the table above Plot the points then it says calculate the slopes of the following line segments I'm not that interested in that one cross it out or that one cross it out or that one cross it out Or that one cross it out. I want you to do the slope from the smallest to the biggest and then complete part D If you're stuck, we'll talk about it next class turn the page, please number two question two Here they're giving you one two three four different points So if they want the average speed when they're asking you for the average speed What are you actually finding on the graph the slope? Okay Three we're gonna skip am I gonna Know I'm gonna skip three Try four you have to think about a little bit five is good Here in number five I wouldn't do an accurate graph. I would do a rough sketch like we did for that first graph You're graphing sales and earnings rough guess Six is good skip seven eight a and b number nine very quickly Okay So we're going to pause there. Oh don't want that one. I Want mr. Dewey