 Hey, we're live quiz number one solve the following leave your final answer as a fraction if it doesn't work out evenly I'm going to show all my steps if you didn't do every single one of these steps. That's fine I've already told you I kind of like you to learn to do some stuff in your head But if I was doing this one here, I see variables on both sides. I would probably go Minus 3x from both sides if you went minus 5x from both sides, that's fine If you used I've seen a couple of my international students use a slightly different method. That's fine I'm concerned. Do we get the same answer? So minus 3x from both sides the negative 6 would drop down. I'd have a 2x plus 7 I would minus 7 from both sides if you didn't do the quiz you can decide one of two things You can fill the quiz out as we're doing it right now as kind of a learning exercise Or you can say I'm going to watch and then I'll try the quiz on my own And at least I'll get the benefit of trying to practice the questions because your test is going to be really similar To this but you decide what you think will help you learn the best minus 7 from both sides minus 7 from both sides and I get negative 13 equals 2x you get x equals negative 13 over 2 is that right? So if you get that 2 out of 2 What if you didn't get that? Yeah, so I can't hear you It was a positive I take a half mark off if you lost a negative would that be a fair way to mark I think on a two mark question. I think so Pardon me As long as one of them is negative, but you know what the rule for fractions is if the fraction is negative You put the negative either on the top or in the front It's not wrong leaving a negative in the bottom Just like it's not wrong to walk around with stuff hanging from your nose But if you want friends, you probably don't want stuff hanging from your nose If you want math friends, you don't want to leave the negative in the bottom Specifically for me if I'm marking really fast. I won't look for the negative down there Which means I might miss it when I'm marking But the rule is look the fraction is negative. There's a single negative put on top or in front But I won't take marks off. Okay, and I answered your question as well Otherwise, I would probably say something like this if you subtracted the variables to both sides I would give you a half mark for that if you added seven or subtracted seven or if you did the other way Plus six of both sides if you did that step I give you a half mark for that and if you somewhere along the way Clued in you had to divide by a two or a negative two depending on what you did. I give you a half mark for that To number two three marks. Why is question number two three marks one extra step gathering like terms Three X take away four X. I'm pretty sure that that's actually negative X and the minus eight drops down Here I don't have X's for like terms I would just drop the negative five X down here. I have a negative twelve plus a seven I have a negative five. I think when I gather like terms there. I would give you One mark for that if you got that far now, I've turned it into the previous question You could either go you know what I'm gonna go I think plus five X to both sides here because that gets rid of more negatives When I do that I'll get a four X a minus eight and a negative five. I Would plus eight to both sides. I'll get a four X equals negative five plus eight three positive three X ends up being How'd I get the X by itself divide by four divide by four X ends up being three Over four if you got three quarters and you didn't show every one of those steps. I'm good with that You get three out of three Otherwise I would if I was marking this give you one mark for that line I probably give you one mark for this line and I'd give you one mark for divide half a mark for dividing and half mark for the Final answer. Yes First question. Did you get same answer as me when all said and done? Okay, then if you're not sure later on when I start to collect these I'm gonna start out by saying before I collect them Anybody want to come a lawyer with me now is the time bring it up and I can tell you instantly how I give you Okay, and that way I can let you know where I'd give up the part marks that way, you know for the test as well Number three Yeah, what is a negative divided by negative? I'll give you full marks for that But I'll also give you the frowning of a lifetime because you wrote extra stuff that you don't need to It's like putting the ones up there as exponents. Yes, there's always an invisible one up there It's like putting a one in front of an X. Yes, if the X is by so there is an invisible one there come on We don't need to Number three. What am I gonna do first? Yo, oh, sorry didn't see that. So I probably give you two out of three On a test I probably give you two and a half out of three, but I'll mark the quiz tougher because I want you to find a test What am I doing first here? Get ready. Yeah, I'm gonna say get rid of brackets I'm worried if you say front door bomber if people haven't had Mr. Rock Are they what you terror? What the heck? What's huh? So get rid of brackets and we'll all know what we all mean. Okay fair enough to get rid of brackets I told you I always draw the little loops because it gets rid of most of my dumbest rakes Let's see. I'll get a negative 10x and a plus 15 equals negative 8x and plus six and Now it looks exactly like number one sort of yeah Absolutely Go with red you could also plus 8x, but I noticed if I plus 10x I'll get rid of negatives I'll take that The 15 will drop down like a domino that becomes a 2x Plus six and now we're on pretty familiar territory minus six from both sides I'll get 15 take away six is nine Equals 2x Nicole. How would I get the x by itself divide both sides by? Absolutely x is gonna be nine over two got that three out of three Otherwise one mark for that line if you got rid of brackets properly one mark for getting the variables to one side Half mark for dividing half mark for the answer number four Brackets and like terms and variables on both sides so I'm gonna get rid of brackets boom boom Careful, that's not a two. It's a negative two. I'll make sure when I multiply I treat it as a negative to Chiana and oh I'm not gonna write it, but what number is after the negative in front of the brackets sitting right here. It's invisible Yeah, I'm gonna assume there's a one there you can write it if you want to I won't take marks off I just never will oh We have a visitor Sorry, but that let's continue where we left off You're gonna get and now I'm gonna write a bit bigger because this is gonna take a bit more work a 5x a positive 4x negative 2 times positive 5 and negative 10 equals a 10 a Minus 2x because it's negative 1 times 2x and a negative 1 times a negative 1 a positive 1 Now I'm gonna gather any like terms that I can I got a 5x plus a 4x. Hey, that's a 9x The minus 10 will drop down like a domino I've got a 10 plus a 1 That's an 11 and the minus 2x will drop down like a domino And now hopefully I'm on familiar territory. I'm gonna plus 2x to both sides That's gonna give me 11x minus 10 equals 11 I'm gonna plus 10 to both sides That's going to give me 11x equals 21 Is x 21 over 11 is that correct people are nodding woohoo So four marks if you got that Otherwise, I'd go one mark for getting rid of brackets one mark for gathering like terms properly One mark for getting the variables to one side half mark for dividing half up for the answer. Yes You just forgot the 10. I'll give you a three out of four and the frowning of a lifetime By the way, the the sloppy mistakes like the forgetting ones Those are the toughest ones to get out of our system my math 12 students After every test are always furious because getting rid of what I call the mistakes Those ones where you look at it afterwards. You're like, what was I doing? Those are very tough to get out of our system very tough any success Simplify the following completely. Okay I'm gonna get rid of breath. I'm gonna get rid of brackets When I do that I get okay numbers times numbers three times five 15 I have two x's. I got one more x. It's gonna be an x cubed and a single y Numbers times numbers. I'll have a negative six. I got an x cubed and a y squared Numbers by numbers all the positive nine. I got an x squared and a y squared I think that's correct 15 x cubed y minus 6 x cubed y squared plus 9 x squared y squared. How many terms are there in this answer? So since it's worth two marks a half mark off for each wrong term if you get all three terms wrong But you knew you were supposed to multiply you at least get a half mark That's how I'll break this up into two number six foil 15 x squared minus six plus 20 minus six plus 20 is Plus 14 x minus eight that was in my head. Is that correct? Ain't so bad. How many terms are there? Three, what's this out of two? You know what I do? I give you a half mark off for each term that was wrong But if you put in that you were supposed to foil if I saw that somewhere I give you an extra half mark number seven Okay, I'm not doing this in my head I'm gonna get when I multiply the first factor 6 x cubed plus 8 x squared minus 4 x I'm gonna get when I multiply the second term Minus 9 x squared Minus 12 x plus 6 I Don't stop here now. I gather like terms I Got a 6 x cubed took care of that one I got 8 x squared minus 9 x squared 8 x squared minus 9 x squared is negative 1 x squared Yes, you can put the one there Colleen. I'm not gonna put the one there because I'm lazy I Got a negative 4 x Minus a 12 x negative 4 x minus a 12 x negative 16 x's And I got a plus 6 that drops down like a domino. How many terms in this particular? Answer for for three marks half mark first of all if you got it right three out of three Otherwise half mark off for each term that's wrong But if you started to do this you get at least one mark for starting including in that it's multiple one Can you give yourself a score out of three? nine First page out of 19 But we'll score out of 19 in the bottom makes it easier when you're doing your added Turn the page Says graph the following equations on the grid clearly label each line with the appropriate letter one mark each This guy here has a y intercept of negative 6 and it's up 1 over 3 up 1 over 3 up 1 over 3 Ruler here. We go here. We go here. We go There's line a No, but I would say try and do three like don't just do two and quit because your ruler my the more you do the easier It is to line up your ruler if you just do two all your ruler has to do a slide just a tiny bit You won't notice it because your dots are so close, but you'll be way off further on right? I'll be okay with it Do and again Tanner on the quiz if you didn't put arrows, I won't be fussy But in real life we should clue in hey, it goes forever in math 10 you would have said the domain is all reels B Y intercept of positive 5 and the slope is negative 2 which means down 2 over 1 down 2 over 1 down 2 over 1 down 2 Over 1 Also Tanner. I have to be honest. This is me again I have enough of the 6-year-old within me that I still enjoy connecting the dots which is lame But honestly, I'll freehand this one. So there's line B By the way, you know how I mark these for one mark If your y intercept is in the right place that gets you a half mark and if you did slope correctly that gets you your second half mark make sense See that's not in slope intercept form. I have to get the y by itself I'll do that over here. I would go minus 3x from both sides Which would give me negative 2 y equals negative 3x plus 4 and then I would divide by negative 2 divide by negative 2 Divide by negative 2 and I will get y equals Colleen. What's a negative divided by a negative? So I don't need to leave both the negatives there. I can just get rid of them. Absolutely and What's positive 4 divided by negative 2 negative 2? So this equation as a y intercept of negative 2 I'll go green for this one and The slope of 3 over 2 which means up 3 over 2 up 3 over 2 up 3 Over 2 up 3 over 2 just one second folks And it would look An awful lot Like that we'll press pause because we have some visitors here What's the 70s polyester and ugliness? The 60s is I think more of the flower child thing the 70s 70s had bell bottoms for example Oh, those were just up, you know silent Hittiest polyester colors disco was 70s. Oh, you're great It's next Friday a day one or a day two I'd have to go look at my agenda Meanwhile, let's continue on please. We'll talk about the rest of it later The last one y equals negative three. What's weird about this one? What's missing? No x really, but yes, there's no slope. This is a line This is a horizontal line negative three high. This is the line that's going to look like quiet, please This is the line that's going to look like That that's line D Then it is possible to get a bonus question as a bonus this one here This is a line that's going to go through. Well, I know it goes through One comma one I can do that And what's another point this one goes through? Good God, Mr. Duke. You didn't make this the easiest one eight take away three four negative one this line looks like this So the bonus line goes through one comma one Negative two comma positive three positive four comma negative one Looks like that Give yourself a score on this page out of four, please. You could get five out of four and then At the very very top give yourself a score out of 23 if you have a question about your quiz Now is the time to come lawyer with me. You're wondering So we're on page three. We did a quick review of using the cross multiply method to solve Let's finish off today's lesson Come here And again, I'll be honest. I agree with my colleague over here. This is not my wow I really love doing this stuff. Yeah, but it's useful Says this using unit or dimensional analysis to convert. So this is the third method All three of these are valid I won't make you use any specific one you can use whatever you like whether it's the ratio method that we did at beginning Whether it's the cross multiply method that we did at the end of last class or this one here What is unit analysis unit analysis is when you have to build a chain of Fractions in order to get from one type of unit to another type of unit You can do it all in individual steps and that works perfectly fine But if you want to do it all in one fell swoop you can do that as well For example Example four says in class example three we use proportional reasoning to convert point zero five miles to inches How could I do this using what's called unit analysis? First of all it asks what conversions will the student need to consider to solve this problem using unit analysis? We want to go from miles To inches if I look at this chart here do I have anywhere on this chart miles to inches? No What do I have miles to what? Yards or feet which one of those will help me well. I have feet to inches so what we're going to write here as our Plan of attack is we're going to go from miles to feet to inches Write that down and what we're going to do is we're going to build a fraction chain and The way I'm going to build the fraction chain is the first thing that I do is I write the measurement that they gave me in part B. They want me to convert Point zero five Miles I write that down page three point zero five miles Want to end up in inches and we said first we're going to go from miles to feet and the way we're going to show that is We're going to go times From the previous page from that chart. What is the conversion factor? How many miles how many feet are in one mile? Someone want to read it, please turn back a page. I realize 5,280 okay put your pencils down look up The question kids have most often is do I divide or do I multiply? That's really the most common question in unit conversion The nice thing with this method is it tells you put your pencils down suppose you thought you had to go one mile is 5,280 feet This is incorrect. Which is why I said don't write this down This is incorrect and here so I can tell you that it's incorrect What units are on top right here? I? Want the miles to cancel the only way I can get the miles to cancel is if in the next fraction I have miles on the bottom Which tells you whether you're dividing or not so I'm going to erase this Right away. I think to myself. I want the miles to cancel pick your pencils up So I'm going to put the one mile right here and the five thousand two hundred and eighty feet Right there. There's the first link in my chain times and Then I'm going to with my pencil cross out the miles because I've just taken this from Miles to feet apparently I'm going to go point zero five times 5,280 divided by one, but Colleen am I going to bother dividing by one? Will that change my answer at all? No, but if there was a number there, I would divide by whatever number was there Now I need the conversion factor between feet and inches So from that chart how are feet and inches related? What's the conversion factor one foot is what? Okay, and once again Jacob, I want to put the feet on the bottom because I want this feet That's on top to cancel with these feet that are on the bottom and you said it was one foot is 12 inches and when I do it like this whoo feet cancel You can use whatever method you wish I'm much more concerned with can you get the right answer? I almost don't like the fact that the book is showing you three different methods I would usually go with one as a math nerd who is comfortable with fractions I use this one all the time because I don't have to memorize stuff, but you can use whatever you want. Here's the point I Can now see what the answer is the answer on my calculator is going to be 0.05 times 5,280 times 12 and then It would technically be divided by one divided by one am I going to bother dividing by one is that going to change my answer at all I'm just going to hit enter. Yes Proportional reasoning is one fraction equals one fraction you don't analysis you build chains of fractions. Yeah, what's the difference between the first method and this method? Semantics really it's a term technically if you're using proportional reasoning you're going one fraction at a time one step at a time This you're not The answer of three thousand one hundred and sixty-eight is that we got an example three on the previous page to I hope I think unit inches Jacob, where does this method work a little bit better if? You're doing more than one conversion at the same time example five Physics student was required to convert 75 meters per second into kilometers per minute What am I going to have to do here? two things I'm going to have to go from meters to what? kilometers and I'm going to have to go from seconds to what? minutes Here's how you can build a chain I? Start out with 75 meters per Second what do you want to do first meters to kilometers or seconds to a minute doesn't matter We don't do one at a time, but which one do you want to do first? seconds, okay Where is the seconds in this first fraction on the bottom or on the top on the bottom? So for me to get it to cancel I need another seconds on the top, and I'm going to have minutes down here Because I'm going seconds to minutes how are seconds and minutes related So 60 seconds is the same as one minute and What happens right now is my seconds would cancel I got seconds on the top seconds on the bottom This right now is actually in meters per minute, but we're not done. They didn't want us to go to meters per minute What do they want us to go to? kilometer per minute, okay times So now we're going to go from meters to kilometers. Where is the meters in this fraction on the top or on the bottom on? The top which means I want to put it on the bottom I'm going to have kilometers on top Mostly you've memorized this how are kilometers and meters related One kilometer is what? 1000 meters and this is going to give me my equation my equation is going to be 75 times 60 times 1 divided by 1 divided by 1,000 Although Rachel do I need to times by one and divide by what it's going to be 75 times 60 divided by a thousand and by letting the units Line up for me the meters cancels as well. I'm left with kilometers per minute It tells me whether to divide or multiply by just making sure the units cancel each step of the way If you're a little bit, I'm not so sure I like that you can use the previous methods methods I'm just concerned with hey, can you get the answer not use this method or that method? Meanwhile What is the answer? It's going to be 75 times 60 divided by 1000 4.5 4.5 4.5 what units? kilometers per minute if you're traveling 75 meters every second in one minute you'll cover 4.5 kilometers and The answer to your question Jacob this method shines when you're converting both sets of units The top units are changing and the bottom units are changing It's a lot easier to keep track of stuff although. I've seen people that just still do these intuitively in their head no problem Turn the page We reviewed this already, but really quickly a quick review of slope from math 10 So in this unit we're going to be using the concept of slope the steepness of a line Recall that slope symbol m was rise over run and I already included this a couple of days ago It was y2 minus y1 over x2 minus x1 so Consider the line segments a b and cd Find the slope of each line by using rise over run You know what we're going to do the slope of one of the lines using rise over run And we'll do the slope of the other line using the formula What's my rise here? For me I Find if I'm doing slope I find it very helpful to draw that it gets rid of my counting mistakes So you can I would on a test I always do because I find most common mistake with slope. I can't run Three over what's the run? Six although I probably wouldn't write it as three over six what would I write it as? one over two oh Positive or negative and how do I know? uphill This one's going to be negative if I use rise over run I'd have to add the negative But we're actually going to try using now the actual slope formula now the slope formula says Why to you know what is it written right there? It is I'm not going to rewrite it Nope uphill as we said this always to the right uphill is positive always Always to find that way Heading to the right uphill is positive heading to the right downhill is negative okay now Do me a favor. Let's label this point here This point D is what comma what count, please? I'll give you a hint. I think the x-coordinate is 10 What's the y-coordinate? To what's this point right here what comma what what's the x-coordinate six yes What's the y-coordinate? So we could go rise over run, but they also want us to use They also want us to use the formula So let's see if we can get the slope using the formula and we'll check to see if we're right using rise over run Y2 that's the second points y-coordinate. What is the second points y-coordinate? Find the second point tell me the y-coordinate. That's what y2 means. I heard it to minus Y1 is the first points y-coordinate. So here's my first point What's my y-coordinate x2? That's my second points x-coordinate, which is what? 10 minus So the slope of this is 2 take away 5 negative 3 over 10 take away 4 sorry 10 take away 6 whoops 4 slope is negative 3 quarters I'll be honest if they give me the graph I'll go rise over run. It's faster I would have used this method travel triangle if they give me the points though I don't want to go finding some graph paper and graphing it if they give me the points I'll use the slope formula that could give a question or you just No, okay. Sorry. I thought that's not a point in my eye Part C says it does the matter if the slope in part B is calculated using the two end points of the line Or two points found anywhere on the line. You can use any two points to find the slope. Is that to me favorite? Oh He is alive. Okay. Sorry my bad. You know what I'm concerned about learning So don't worry about C. Yo, almost 10 more minutes. Okay Example 7 says complete the following statements Slope is a measure of the blank of a line. What did you learn last year? What a slope measure? Steepness, let's put that word in there, please Slope is the ratio of the vertical change called the blank over the horizontal change called the blank Slope was what over what what did you learn last year rise? over the run Okay Sam this next one's for you part C. It says this a line segment which rises from left to right which goes uphill has Positive slope or negative slope. What do we just say? Yep, what about if it falls from left to right if it goes downhill? negative slope a Horizontal line segment has a slope of so what did we say remember the skier analogy? Horizontal how fast will a skier be going? slope of zero Vertical line skier dies another word for dies and math was undefined or no slope. I'm gonna use the word Undefined I Found when I say no slope kids hear the word no as the number zero. No, it's not a slope of zero. It's undefined and Don't worry about G home pause So can you turn to page seven, please? Page seven, please I crossed out G was not G. I crossed out. I think number four is fair game Number five is fair game. I'm assigning it. I'm circling it look up six four five six no seven eight Skip skip skip skip skip. No, I will pause there four five six seven eight It's math. How could it not be fun?