 So now we're live. Quick review of graphing. Whenever I teach graphing to grade 11s, you guys, almost always as soon as I mention graphing, there's a mental groan, sometimes the audible groan. Often I've heard, I hate graphing. I've never quite understood that. But I think the reason is when we introduce the concept of graphing to you in grade 9 and 10, we don't tell you why it's helpful. So the first question I asked is, why graph anything in the first place? What's the point as I cough? Question, Danielle? Okay. Here's why. Because our brains, you're assuming we have a brain. Yes, Jordan. I'm assuming we all have a brain. Are far better at spotting visual patterns than they are at spotting numerical patterns. It's a way to take a bunch of data in a long convoluted form and make it visual. Put your pencils down. There's a company in what they've done, it's US, not Canadian, but what they've done from 1900 until the current year is they've tracked by looking at hospital birth records what the most popular names for babies are. And although initially it was a big, long numerical data, they realized they wanted to make it visual. And so they came up with this wonderful thing here. It's called the baby name wizard. And if you type in a name, it will take that numerical data and graph it, M-A-R-C-U-S. Marcus, as it turns out, was not that popular 100 years ago. It became fairly popular in the 80s. Not quite as popular now, 2010, I think I saw 130th. Who would like to find out how popular their name is? By the way, you can clearly see what's going on, where if I gave you the numerical data, Devin, you'd be, I can't make heads or tails. This is why graphing is useful. Now the problem is in grade 10 and sadly in much of grade 11, we give you this as a graph, which is the most boring possible graph out there. That's a line. I spotted the pattern. Yeah. Thank you, Captain Obvious. Sorry. Ah! I'm willing to bet non-existent and suddenly in the last five years. Like I said, I'm willing to bet non-existent and suddenly in the last five years. You can also see world history based on this graph. What? Yeah, watch. Why did it suddenly, why did the name Adolf suddenly die out there? World War II, right? I mean, you still have all heard of Adolf Hitler. It was a very common name. It's actually not surprising that a dictator was named Adolf because it was a very, very common name. The odds were there was so many Adolfs, one of them would rise to power. Almost vanished completely. In fact, I would argue, I don't think you've ever gone to school here in North America with anybody named Adolf. Okay? Give me another one. Dwight. You have to know your history here. Why did Dwight suddenly have a huge peak in the 1940s and 50s? There was a famous Dwight. You guys don't know your World War II hit? Yeah. Eisenhower, the guy that was in charge of all of the Allied forces and became president twice right after that. It was a name on everyone's consciousness. So when people were thinking of names, Dwight. Very visual, very clear. Now? Not so many. I'll show you a name that's become almost overused. Girls. Sorry. It's almost your last name, but not quite. Madison. Peaked. It was the number three name in 2003. So if they were born in 2003, how old would they be right now? Do the math. Come on. Eight years old. I'm willing to bet in one out of every two grade three classes, there's a Madison. Betcha. Still number three in 2004, wait a minute, how can it be number three in 2006 but not as high on the graph? What does that mean? Can we interpret that? There's more names out there. People are making up names all the time. People are using names that weren't used before. So there's more names out there. You don't have to be as high to be number three when there's way more things to choose from. Now currently, what are we, 2009, number seven. I'll give you one that I never went to school with, but my nephew is named as is one other person in this school, in this class. I never went to school with an Adam. It wasn't a common name at all. So I went to school in the 60s and 70s. Boy, suddenly it entered the consciousness here. I would like to know if in the late 70s or early 80s, if there was a major news event with an Adam. Wouldn't surprise me. Type in Neil for Neil Armstrong. You'll see in the late 60s, early 70s, there's a bump for Neil because that was the name that everyone had heard of for Sky to Walk on the Moon. You can track some negative history as well. This is kind of sad. I apologize in advance. Jamal. In the 1960s, during the civil rights movement in the U.S., many African-American parents proudly gave their children African names. Can you see it? And then when those kids became 21 or 22, and this is true, research has borne this out, they found that they couldn't get a job. You can send an identical resumes. And if you put William as one name and Jamal on the other name, it's almost three to one chance of William getting hired over Jamal. The resumes can be identical. And it's not that people are trying to be racist. You can argue whether they were raised that way or whatever. But you can see people realized, I can't get a job with an ethnic name. And now, they're vanishing again. All this from a graph, which you wouldn't really get by looking at the data, by looking at the numbers. By the way, if you Google, if you're bored, if you want to play with it, it's called Baby Name Wizard. All one word. You can put your own name there. Girls, you can try names that you'd like to give your children someday because we know you think about those things. I like to put my boyfriend's name in there with my last, no, forget it. Okay. So there's the Baby Name Wizard. Our brains are far better at spotting visual patterns than they are spotting numerical patterns. Problem is, we start yet with boring lines. Whoa, Alex, that's really interesting. Alex, right? Yes? Whoo! That's really interesting. No, I think that last graph was far more interesting. I saw all of you like, oh, I can see what's going on. Let's review the lines, though, because it's part of the curriculum. Three concepts for graphing lines. The first concept is slope. Slope is defined as, what's slope defined as? Okay, you memorize this pattern. You define the slope as the rise over the run. I actually had a math teacher who taught it this way. You don't have to write this down, but look up. She always spelled it like that to remind us that rise was the wise on the graph. That's kind of clever. I just remember that rise and wise rhymes is my way of remembering what's what, but whatever works for you. That's if you had the graph in front of you, and you all got very good at drawing a triangle and counting. What if you don't have a graph in front of you? If you knew two points that the graph went through, you could also find the slope by using a slope equation for a candy. Does anybody remember the slope equation? Marcus is going, ah, son of a... First of all, what, not for a candy, what letter did we use for slope? M, why M? Because not all math was discovered by English speaking people. It's from a French or a German word. I can't remember. It's from their word for mountain steepness slope. Ah, so it does make sense if you're not English. Marcus is going to try the slope formula. Go for it. Y2 minus Y1 over X2 minus X1. Yes, it is. That is the slope formula. You didn't have to memorize it. They gave you the equation on the provincial. However, when I teach slope, I tell students that this right here is the Swiss army knife of mathematics. It can get used in all sorts of places that you didn't expect it to be used. You do calculus. It's amazing how much of calculus is using this thing in a way that you never thought you could. I'll get you the candle later. So find the slope between point A, negative 4, comma 6, and point B, 2, comma negative 1. You've already learned that I'm not big on writing stuff uselessly, but I'll tell you one thing that I always do when I'm finding the slope because it takes one second. Whenever they give me the points, I always write above it X1, Y1, X2, Y2, because really the only mistake kids make with this is putting the wrong numbers into the wrong place and I can get rid of almost all of that by doing that. Now it's plug and chug substitution. M equals, I have the slope formula right here in front of me, so I'm not going to rewrite it. Y2, negative 1 minus Y1, 6, all over X2, 2 minus X1, negative 4. By the way, X1, comma Y1, first X, first Y, second X, second Y. That's how you know it's X1, comma Y1, comma, and X2, comma Y2. I have to do a little bit of math in my head. Negative 1 minus 6, negative 7 or negative 5, negative 7. If you're shaky on integers, use the crutch. This is meant to be a crutch, Jordan, not a stretcher. I don't want to see you going 5 plus 3 on these. I'll catch you, I'll call you out and I'll make fun of you, honestly. But if you're shaky on integers or on a test, I use this almost all the time on the test because my brain trips on itself sometimes. But these are slow. Hopefully you've clued in. In your head is almost always faster. The bottom, what's a minus minus the same as? Plus, so Jordan, no calculator. What is 2 plus 4? Did we want to write the slope as a decimal? That was much more useful as a fraction because then when you were graphing, you knew how many squares to count up or down and how many squares to count over. The only time you put the slope in a decimal is we do it sometimes in physics, but that's different. Jog in your memory? Good. Try 2 on your own. I'll freeze the screen, screen frozen. Let's see if we get the same answer. Find the slope between those puppies, telling you, quickly label the points, it gets rid of most of your dumbest rakes. I was making these up before lunch. I wasn't really concentrating. I think this one works out evenly, yes? Like over one, yes? I got a slope of that. You can write 2 over 1. I'm good with that. You can write 2. I'm good with that too. Usually I'll just write the 2 because whenever I see a 2, I assume it's over 1. I've popped my brain to always see that there's an invisible over 1 there as a math nerd. Everything's a fraction. 3 is a fraction. It's 3 over 1. Next page over. There's a quick review of slope. You talked last year about positive slope and negative slope. What was a positive slope on a graph? You're correct. I'm going to just add the key idea. Whenever we look at a graph, we read from left to right like you do in English. Traveling uphill from left to right, that would be a positive slope. Danielle, what would a negative slope be? Downhill. And then there's the two weird slopes. A slope of 0 and an undefined slope. One of them was a vertical line. One of them was a horizontal line. Which one is which? Yeah, 0 horizontal and undefined vertical. Is he right or is he wrong? In our slope formula, if we have a vertical line, these two numbers are the same. And when these two numbers are the same, what's any number divided by itself? Sorry, what's any number take away itself? 0. Can you divide by 0? No. You guys know why you can't divide by 0? Because you can't. Put your pencils down for a pizza. Because math is supposed to make sense, Boston. Don't write this down. Here's why you cannot divide by 0. And I can explain it in three lines. Pick on my friend Boston here. Boston, if I say 6 divided by 3, what's the answer? That also means that 2 times 3 equals 6. If you went 6 divided by 0 and you got an answer, I'll call that answer x. Because I don't know what the answer is. That would also have to mean that x times 0 equals 6. What's wrong with this? Why? We can have our 0 times table or we can divide by 0. You can't possibly have both. It leads to a contradiction. And so dividing by 0 is undefined because it doesn't work going backwards. You can't get the original question that you started out from. It doesn't have an inverse if you want the fancy-schmancy-matter. Nobody showed you guys that? Man, I got to start teaching the junior grades. Makes sense though, yes? Like, oh, yeah. Timesing by 0, I like the 0 times table. That's the easiest one. My favorite. Back to you. How can you remember which is which? Tell you what worked for me. I envision myself as a skier in a tuck position. If I'm on a slope like that as a skier in a tuck position, if I don't use my ski poles, eventually, how fast will I be going? 0. 0. If I'm a skier in a tuck position on a slope like that, I die. Skiier dies, undefined. Skiier becomes undefined. Skiier stops. Speed of 0. That's how I remember. Skiier dies, skier stops. And it's got a nice, violent change to it. My classes are rated PG. OK, turn the page. You learn about slope, and then they give you something called slope intercept form. They have you memorize this algorithm, this pattern. It looks like this. Y equals mx plus b, where the number that's sitting where the m is tells you what. The slope. And the number that's sitting where the b is tells you what. And I'm going to abbreviate y intercept as y int. Can I do that? And also, once again, you can see that this part of math, whoever wrote this first, wasn't an English speaker. Because how you get b from a y intercept, I don't know. I think this one comes from Latin, but I can't read it. So you get something like this. They say, graph the following lines. First thing you want to ask is, is this equation in slope intercept form? Is the y by itself? Yes? You actually don't graph the slope first, though. What you actually did was you looked at the y intercept. What's the y intercept of this particular graph here? So you counted 1, 2, 3, 4, 5 up. Put a dot there. What's the slope? What did that mean? How did you count? Give me directions. Up 1. And when you say over, it's always to the right. Up when it's positive, down when it's, but always to the right. So it's going to be up 1, over 4, up 1, over 4. You can back count now that you see the pattern. So now I'm going to go down 1 back 4. If you have a ruler, use it. Calculator straight edge, use it. You can freehand, but if you've got a ruler sitting right there, try to. I'm going to try. Oh, cool, cool. I think I have one kind of built in here. Draw line. That's way off right now. Oh, look at that. OK, come on. Work with me. You're going to go like right there. And you're going to go right there. Oh, that is just too cool. There's mine. I'm going to be a little bit fussy intellectually. You should put arrows because it does keep going. There's a big difference between that and it's stopping. I probably won't take marks off if you forget to put the arrows, but I'll give you the frowning of a lifetime. OK. B. Almost identical to A. Same Y intercept. I'll change colors. What's the slope here? Negative one quarter, which means count how? Down one over four. Down one over four. Now, I'll be honest. I'm just going to freehand these. In your homework, if you freehand, I don't freak out. On a test, try and be neat, so I'm not trying to guess what your answer is. Because I'll probably just guess that you did it wrong. Try C and D on your own. I'll freeze the screen. You can try E and F as well. If you remember E and F are weird. If you remember how those worked, though, go ahead. I'll be talking about E and F in a second, but I'm going to do C and D. Oh, look up for one second. Let's be clever. Let's put a little A next to that one. Let's put a little B next to that one. Label your lines. Since we're going to have six lines on here, it's nice if we're studying later on if we know what the heck they came from, right? So do that. Now freeze the screen. C. I'll go red. D. I'll go green. Am I right? Yes? No? Am I right? Yeah. E and F. E and F are weird. Why are they weird? How can you glance at them right away and say something different? Each of them only has how many variables? Just one. I'm going to give you a hint. One of these is a skier dies line. One of these is a skier stops line. One of these is horizontal. One of these is vertical. Which one? Emily, not Emily, but Emily, this Emily. Can you do me a favor? Can you read to me? Read the equation out. Louder? You're right. I don't hear that. Here's what I hear. The height is always three. Isn't that what Y means? How can I draw a line that's always three high? Well, what? It's got to be horizontal. The only way I can draw a line that's always three high is if I say, well, I'm three high right there. I'm three high right there. I'm three high right here, right here, right here, right here. Oh, you know what? It's this line right here. This is the line that's always three high. Sierra, can you read to me please? And again, I don't quite hear that. What I hear is you're always negative six to the left or to the right. Well, sorry. I guess you're always negative six to the left, because negative is to the left. How can I draw a line that's always negative six to the left? Got to be vertical. It's going to look like this. This year, we're going to graph lines a little bit, but this year, finally, we're going to start doing some curves. More interesting stuff, none of this. Next page, what about when the graph wasn't in slope intercept form? Like A, what if the Y wasn't by itself? Boston, got to change it around. Can you be more specific? You know what? I left room here on purpose. Let's put a little A right there so we can do the work and still see the graph. How would I get the Y by itself? I think I disagree. I think you thought it right and said it wrong. I can't just move the 3x to there magically. I have to do a math. I have to minus it. So do I have the original question right here, that I'm going to start doing the work right here? I'm going to minus 3x from both sides. I'm going to write right away 2y equals negative 3x and a positive 8. And then how would I get the Y by itself, Boston? It's OK. Still learning names. I blanked it. Liam, come on, brain. Yeah. Oh, no. Shay, you guys switched around on my seating plan. Now you're really confusing me. That's why I was looking at it the wrong way. Shay, more specific, divide by 2. Divide by 2, divide by 2, divide by 2. Here is the equation in slope intercept form. Negative 3x over 2 and a positive 4. I don't mind this as a fraction because I can count the slope very, very easily. There's the equation. Graph it. Slope intercept form. Now, did anybody reach the point last year where they could do the whole thing in their head and just go straight to slope intercept form? None of you did? Because it can be done. And if you can get that good, I have no problem with it. In fact, I think that's great. Can you really do that? Yeah. How would I get the Y by itself? I think I would minus x, minus x, and then divide by negative 2. It can be done. Do you have to? Oh, Adam, what's a negative divided by negative? I'd pull up my trusty pencil eraser because I do my notes in pencil and I'd say, I think that's a 1 half x. There's my slope. And what's 6 divided by negative 2? I don't think it's 3. Negative, which reminds me to put a minus. Always say the sign because it reminds you what to put. In fact, it's going to be this. The Y intercept of negative 3 right there. Slope of 1 half, which means up 1 over 2, up 1 over 2, up 1 over 2, up 1 over 2. That is line B. Is that OK? Try C and D on your own. Try C and D on your own. I'll freeze the screen here. There's C if you're stuck. And let's go green for D. What's weird about D? What's the Y intercept for D? Jordan, it ends up being 0, right? I got, yes? How many of you got C and D right? Oh, no. They're going to skip E because it's just more of the same. And I want to go to F and G and H. Put your pencils down. F and G and H are problems for slope intercept form. This is why I'm actually not a big fan of slope intercept form. For me to get the Y by itself in F, what would I end up dividing by? Can you see it? How would I get the Y by itself? Louder. I hardly know what you're doing. I can't wait to go out. And then 3, here's the problem. Look up, look up, look up. When you divide this 4 by 3, will it work out evenly? You're going to get a fractional Y intercept, which is really tough to eyeball. Same with G. What would I divide both sides by to get the Y by itself in G, Courtney? Will it go into 5 evenly? No. What would I end up dividing everything by on this one? 5. Will it go into 8 evenly? For all three of those, you're going to have a decimal Y intercept. So the only way you can graph those, Boston, is to do what we call a table of values. And what I mean by that is try some numbers in your head. Let's look at F. The first numbers I always try putting in in my head are a 0. If I put a 0 in for Y, what's 3 times 0? 0. What would I have to put right there so that the left side worked out to 4? You can see I'm losing a few of you. Let me write this. So we're looking at F. If I put a 0 in for Y, Y is 0 because it is the easiest time to be able to play around with. And I want the answer to work out for 4 to 4. What do I have to put in right there? Marcus, I know that one of the points on this graph is 2 comma 0. It is. And I just said that by figuring out what numbers could make this left side work out to 4. Can you find me two more numbers that I can stick in here and make this work out to 4? And there's no trick to this. Try some numbers and see if your brain tumbles onto it. Well, what if I put a 1 there? What's 3 times 1? 3 plus what is 4? 1. Can I make that work out to 1? Not without fractions. Yeah. What if I put a 2 there? What's 3 times 2? 6. 6, take away what is 4. Could I make this work out to a negative 2? How, Jordan? I think another point on this is when x is negative 1 and y is 3, I get a 4 as well. If I was graphing this one here, I would say it goes through 2 comma 0, negative 1 comma 3. How about negative 1 comma what? 2. Is that better? Is that what you're going to say? Yes, I was going, wait a minute. That's wrong. Negative 1 comma 2. But now here's the key. Devon, I can spot the slope. Down 2 over 3. You know another point on this graph? Down 2 over 3. You want another one? Down 2 over 3. I can go backwards. Back 3 up 2, back 3 up 2. One more. Back 3 up 2. This line here is graph F. If you can find two points, then Sierra, you can just continue the pattern and connect the dots. So if the y intercept is a decimal, just try to see if you can find some numbers that will make it a true equation. Let's do h. Can you see something I could put here so that this would work out to an 8? Jordan, x is 3. Oh, 3 plus 5. Very nice. So I'm going to make a little note here. I'm running out of colors. I'm going to have to repeat myself. h I'm going to do in row 3. I'm going to repeat myself. h I'm going to do in red. You said when x is 3, y is 1. 3 plus 5. Absolutely. There's one point. See it? 3 plus 5 is 8. Usually you can find one by scratching your brain. Find the second one. It's a bit of a tough one. Yeah. If I put a negative 2 there, I'll get negative 2 plus 10. Negative 2 goes with 2. Is Marcus right? Negative 2 plus 10. Believe it or not, say that's the fastest way to graph. That's how I graph. I don't rewrite things in slope intercept form. I use my time tables. I plug in numbers. I try zeros and ones and twos first. And almost always I can find two points and connect them faster than I can rewrite the equation. If you like slope intercept form, you can use that most of the time. Set Liam for what it's worth. If your x-intercept is decimal, you have to use this. And I think in the homework, there's a couple where the x-intercept is a decimal. Oh, let's graph this. 2 comma 0. Hang on. I'm graphing the wrong one. I scroll down. 3 comma 1. It's silly me. And negative 2 comma 2. Looks like Marcus, this graph goes down 1 over 5, down 1 over 5. Apparently, 8 comma 0 would have worked. 8 plus 0 is 8. And 1, 2, 3, 4, 5 comma 1. Looks like this line looks like that. That's graph. What's your homework? I have two more of those stupid joke review sheets. Hang on for one second.