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Math 7 - Module 2A - Lesson 1

Added Feb 17, 2021 •

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How to Use a Number Line to: a) Describe Positive and Negative Numbers b) Find the Distance a Number is from Zero aka The Absolute Value c) What are opposite numbers d) A number is combined with its opposite number also called its Additive Inverse, the result is zero e) How to and which direction to move from one number to the next (ADDING)

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positive numbers so okay let's go with this positive numbers are above 0 negative numbers and I'll take this a good one be Lo 0 and to know a negative number besides being below zero you have a sign as a minus sign in front of number positive number so mine is still the one I say minus sign is the minus sign I don't want to just put a dash on there but that is totally correct what you put on so it's a subtraction sign but the one is negative sign -4 now but will I bet you just called the negative sign and then positive number so I got two different answers and both of you are correct so optional optional optional mean you don't need it you could have optional Plus sign in front number but no sign needed in front you need a Janita pause you need a plus that while you don't need it you could plus sign in front of any number to make it negative but you don't need it no sign needed so what else is there someone so pretty much were talking about number line earlier so where would a negative number Rihanna number line where would a positive number be on a number line think about how a number line to write a number line this person good I'm not going to wait for your and I'll wait to say I want some other p what time in first on a number line where's the positive number where's the negative number on a number line all right that's true but is there another situation for where it is so I'll put that on there so lot of people have done the same yeah okay one person add it on to what you guys are said I'm going to say what most of you said so far so negative left leave a little space the word or left side or leaving my son wine of number line play right here right side or leave a space before of number line and when I say space so we always think of a number line like a ruler and we have for some reason we we only go left and right a numberline take about a thermometer narwhal if you're looking outside how cold it is but it wasn't as cold as like a couple days ago I was freezing my toes are cold so left side or bottom or below Avana Westside bottoms go bottom cuz I can go top down up and down think about like a swimming pool if you I've never climbed up on a diving board cuz I'm you know not that good of a samurai freak out but anyways just go right here and then what's the opposite of the bottom the top good cuz if you go above or below think about think about the swimming pool the water itself is 0 will you climb up before you jump in the water you're going up on the number life if you had to jump and dive into the water under the water below the water in my positive and negative up in the up in the sky he can breathe the air below the water you're drowning swimmer Destiny practice other things looks like we got most of all I did was put a couple more so looks like and I'm just these are just example looks like negative 7 negative 8 so on and so forth and just to look like looks likes does not like look looks like 7/8 goes on on on but you don't need those plus sign that's how I'mma put an oar + 7 yes... So this is say you can write it without the sign or with the side in a some of you guys need that extra visual aid to say alright I can see the difference I got to sign a plus sign versus the negative sign for some people are like I already know that I don't need to put the sign there cuz if I don't have a sinus accepted to be positive so you fall into one of these but there's only one way to write negative and then negative numbers opposite have positive and then if you think about a positive numbers it's the opposite of negative I'm in the middle there's a couple different things we can say but both negative and positive numbers on number lines so no matter what it wants you to think of it how do we connect to the best way to understand positive negative numbers is on the number line not just a ruler a ruler is a number line with only positive numbers negative numbers negative numbers don't exist in real life so that's yes I'm going there right now the negative numbers are are in majan Arie and we don't know where they are we we just have an idea where's a positive we can get the bird 1 2 3 4 5 6 7 8 9 10 11 12 bird and out of those bird sitting there three of them are ducks the rest are dirty pigeons so understanding we can count in real life we can see positive things in real life as long as we get, it's considered positive but what is a negative and that's what we're going to do there and we don't have the cards for the number card game the energy game so it's going to be annoying in module 2 you're always going to see this this a card game they talk about we're going to fake it in the card game is like this I'm trying to stay so I think there were cuz I did it last year and annoyed me sucked up too much time and few people played it I would give you guys couple sheets of cards to a couple sheets of paper thick cardstock paper with numbers on it I think it was one through 12 on one of them and on the other one is -1 to 12 and you were also given for card with 0 do the numbers 0 on their 0izza number and I'm going to go ahead and tell you zero is not negative or positive so that's where this is right here it's B closer O negative numbers it's above 03 pause armor but what if it's zero it's not positive or negative cuz it's not on the side it's in the middle but it's not possible next that's worth gun someone sign up so let's go to the next page and just think about rebuilding we're building here the deputy let's do this one together right there so you the number line below here's a number line below to practice counting up and down counting up starting at 0 cenotes counting up starting at 0 corresponds to blank counting down starting at 0 so look at this here is zero I'm going to highlight zero in green that's where we start 1000 on the number line so sick right here use your fingers and it does help use your finger so counting up notice up coming up is in what direction there's no up here Tommy up is in which direction thank you Tommy up you going to the right so put that on there that will help up what two colors use purple and red shining up to the right cuz it can mean up and down also I'm going to give you another number line right here just to visualize that should be aligned you know it's curved it's cuz I know you know my hands are next day me drawing a straight line but let's look at this number line right here and the middle let's put that at zero and just like we said earlier you count so this is counting by ones and you don't have to count by 1 we did it when we were graphing we make member how we made a little grass start to grab her stuff and what was that module 1A and 1B will be grasped there were two number lines X and Y axes those are number lines so we don't have to count by ones cuz running don't have too much so 1 2 3 liquor store to 3 and then we'll go down -1 negative 2 and -3 so we have a number line that goes up and down and a number line goes left and right so this one here counting up but we don't have to count up counting to the right starting at 0 if we're coming to the right area corresponds matching look at that we're using vocabulary word matches with which numbers think about it type it in if you know counting down and down is another way of saying to the left but if it's a number line up and do look at this counting down corresponds to blank numbers thank you so counting down matches two negative numbers counting up or to the right matches with positive numbers and notice they want us to start at 0 in a couple weeks when we get lazy I'll show you we don't have to start at zero but for now for now start counting at 0 alright so we'll get lazy and we will not have to do start at zero but eventually we will have to do for now let's just started so starting at 0 if I go up let's go up I'm going to 123 I'm coming up 123 going to the right and Time 1 2 3 4 5 6 7 9 10 counting down from 0 negative 1 3 all the way tell me the negative numbers look at this though so look at this going this way I start at 1 week at the end same thing it's just going office dress you start counting further away the further away it is the smaller this is one thing that I'm jumping ahead if you're always comparing two numbers here this how I remember left and right when you put your fingers and make a shape of an l d l the correct side of Elvis your left side and then you're right so left finger right finger if you're comparing two numbers put your finger on the two numbers the number on the left the number on the left is always less than left and less is always less than the number on the right so it's easy to see us a positive number is bigger than a if number a negative number is less than a positive number and then look at here we compare two positive remember the left finger is always the less the smaller number 4 is less than you can use all that but when we get two negative numbers we want you to eventually not use your fingers on a number line but it helps right now to imagine so we want you to eventually notice of my fingers off to have to be able to bid Plies picture this number line in your head and position the left and right but look at this right here I'mma put my finger on -3 and let's go negative in the left side is less than right side playlist -10 is less than -3 but let's go here 3 is less than 10 so you need to verse your way of thinking is what I'm getting it I'm jumping ahead we can see that has a two digit number that's bigger than a one digit number but when you go to the negative it's the other way around negative 3 is bigger than -10 on the number line negative 10 you would take us 10 10 is greater than 3 but it's negative so it's the opposite the bigger the digits the smaller it is smallest amount of digits the greater it is if you're comparing negative numbers only for comparing negative numbers and then you might compare 9 and -9 which ones bigger the right numbers bigger the left number is smaller so 9 is greater than -10 even though it's something but they're not equal so basic I get more different ways of thinking so let's go to the where do you begin when locating a number on the number line so what did I say where do we always start at 0 thank you so what are we give start at 0 to always start at 0 for now will going to do a lazy thing after module 1 it after module 2A 2B we should be lazy I'll show you how to be lazy I'm sorry Mom but always start at 0 haircuts that they're always I put it in quotation marks cuz like I'm telling you right now we're going to eventually be lazy and we don't have to do that but for now start at zero always started that's not funny right there all right what do you call the distance between a number and a 0 on a number line so this right here is some of you guys have learned what do you call the distance between zero any number look at this right here sick of this like a ruler here's 0 + 7 how many units is how many spaces away is 027 thank you good good 7 all right I'm going to do the other way how many spaces away is 0 to -7 how many spaces Google I got two answers to answers so when you count spaces that's why you got that's what we use a ruler think about it like a ruler how many spaces away I'm going to give you a chance to fix that some of you guys how many spaces summer half of your correct half of your wrong but I understand what you want to listen to me we are 7 spaces from 0 to 7 how many spaces away 0 to negative 7 it's still seven if we use a ruler I don't know where my ruler went cuz I got all this paper here stacked up for all the glasses if we use a ruler ruler ruler to get a second hard enough Sun all that that you get like on everyone should have got this in the mail is the stupidest thing for a water heater but was great about it is that it came with the ruler Plumbing draining service I'm not advertising for them we're just letting you know there's a ruler so it's not cardboard it's like plastic so I'm saving this for a real cheap ruler and it's a nice little thing of right on so look at this when we measure something we never measure o u r -5 ft tall your -4 ft tall we always measure in positive numbers so even though I said how many units is there a negative 7 I don't care that it's -7 you still count and measure by a distance so I don't know if it's inches or not but whatever it is the unit's it's 7 spaces away still so 02702 -7 both are 7 spaces so that's why this is important what do you call the distance between a number and 0 on a Timberline there's a special name to call that distance yes so if you look at it you just put the ruler under that measure so you know I don't know there was was it sending me to see if it's if it's bigger than centimeters but you could put the ruler under you know and line it up from the number to Zuri could still find good it's like you cut it and shipped it over measured that's a great way of thinking about but there's a name for it some of you guys know and hopefully if you don't that's okay so the distance what I just did the measurement between doesn't matter 0 and a positive 0 and a negative it's called the absolute value of solute value there's a shorter way of writing the apps value ABS stands for absolute v a l stands for Valley so read it as absolute value don't read it is ABS valve read it as absolute value absolute value but sad on nice short way of describing the definition distance from zero and I'll see you later this what is the relationship between 7 and -7 we just went over both and let's not use the word spaces but use the word units when we don't know what the measurement is I don't know if that's centimeters what's not centimeters it's not even inches but when we know we got some type of measure call it the units both are 7 units from zero and here's a night here's how you write an absolute value for it's a special parentheses and imma show you the Lay's the better way to do it so we don't want it to look like the number one so you need to make it look bigger but here's how you write the absolute value of a number you put the numbers between parentheses straight lines I know that I made it bigger see how it's above and below the 7 when people get lazy like you will get lazy you're going to this for an absolute don't do it don't do not write this down see how they're kind of the same some of you are going to say oh I have a 171 even though those are supposed to be absolute value parentheses look at that look at it looks like 171 but if you make it bigger you can see it's an absolute value so it's a special parentheses so and here's the thing do not write it like this cuz he's our regular parentheses do anything that just means to send a parenthesis these even though that's a messed up looking at like 171 assume it's absolute value this one right here that's an absolute value parentheses so absolute value absolute value of -7 so always make your absolute value parentheses I'm calling them parentheses for now but eventually your absolute value signs so the signs for absolute value is just almost like a princess but straight line so absolute value of 7 is 7 absolute value of -7 is 7 what it means is it 7 units from 0 7 units from 0 good observation this right here so we'll get to this we're going to write this down eventually absolute value is always positive you put that absolute value is always a positive number that's a good observation your your cracked that's a great thing and we say it's always a positive I don't like it but it's true I don't like I do not want you guys to think about this as a definition I'd rather you guys think distance from zero everything always positive that you stuck in changing signs of them know it's always positive because again we use a ruler and good pretty much I like what you said Rodrigo you just put a measuring the distance from the negative side we just imagine this cutting it up and lining it up with zero what how long is it from 0 and you can count all we got a positive number so we always measure distance we measure distance in positive numbers we never use a ruler and measure in negative numbers but we know it exists all right simple concept hate this today with you this is very important all this all right if we have the the stupid integer card game we would have this would have been it so you had four cards let's say we took four card and we got a 5 + -5 + -4 + 8 what is some of the card values shown use the counting count use the counting on method on the provided number line to justify your answer so that's the thing that assumed that you knew this but you don't and we're going to do this part together we hold on while I use on the number 7 numbers add negatives no one likes negative unless you play golf unless you play golf is only time you like negative is the only to us the only sport where the lower your score is the bigger netted with a little smaller your negative number is what's the best but we don't like negatives to think about this when you owe somebody money you're you're you're in the negative when you have money in your pocket you're in the positive so that's the way I always think my positive negative I don't want to any money and they are about 20 bucks and then he like he's stupid enough to let me borrow twenty bucks he's shaking his head no but if you listened about 20 bucks I have 20 bucks now but cuz I owe him so understand what owing unless unless someone owes me money sure but then that's never and have it I'm not going to let you borrow Penny no look up the ground for a penny don't buy from somebody else cuz I don't trust you to pay me back so the first number you always star 0 so that's one thing right there so you'd only counting money made sure counting method is hops I don't like hops but we're going to do Hobbs and will eventually we'll get lazier all right so start at 0 remember always start at 0 always start at 0 and our first number we go to 500 from there what's next is what it's saying and so you never so once you start at zero this is where you end up and then the next number this is how many spaces negative. we count left but we don't go to -5 this is how many spaces you move start at 0 someone started zero to go to 5 3 from 5 go to well move that many spit let's move move spaces the only on the first time do you go straight to that number from zero cuz after that we're counting spaces if I go back left if I go left by space what number were back at 0 so let's make bigger jump so we're back at zero and then from where we ended up from let's see for this is for from 0 move this many spaces and because it's negative we're going left so she's another Grant with migraines there it is I'm here we go negative Force bases and then finally from -4 move 8 spaces to the right what number do we end up at for good cuz now we moved eight space is over we end up all the way back at or you just got to think about like an airplane we start at 0 we fly we flutophone we flew to 5 and then we return we had a round trip and we flew back to zero does remove five spaces back and then this plane flew to Nega 4 and then we flew eight space is over so we kind of a space we ended up back at 4 without thinking without thinking while you guys using the number line is moving like that quick you add and subtract negative numbers but for some reason when we just write it if if I'm not going to write it here I'll write it on a separate piece of paper if for some reason we wrote this 5 + -5 5 + -4 + 8 for some reason we write this you guys have a brain fart and you guys mess up cuz you look at the size I got negative numbers I do this this mess but when you use a number line I have a lot of guys participate and yes move back and forth so much you got to think about it what if you're driving flying an airplane think about how he's facing the only time I pretty much go straight to the number is when you I doubt you start at zero we go straight to 5 and then I got to move five spaces back 1 2 3 4 5 and end up at zero then from zero whatever I ended up with a go to -4 and then from -4 I move 8 space is over at 4 the number line in your brain the number line you imagine that's what you need to start thinking about when we do all this stuff all right what is the final position on the number line we just said that for Now read this question type it in the chat if you know so if we had to pull another card out what number would that card have to be to end up back at zero so we're at 4 right now what card do we need to get back to zero I've been in if you know you are correct your cost correct what card would you need to get back so we are right now currently at 4 this is where we are at right now what car do we need to end up back at zero think about which way these things moved 8 + 5 move this way -5 and 4 movie is that way so good correction you would need the negative for card positive numbers move to the right negative numbers moved to the West use the number line to answer each of the following questions alright let's see what we know so far how far is 7 from 0 and in which direction you got to mention Direction Left Right positive negative so right here how far is 7 from 0 thank you for answering seven spaces 7 units 7 units to the right and let's go right but don't say positive direction let's go right and understand right left up and but the number line is right to left so we're going to go right if it was the number line was up and down 7 units up what is the opposite of 7 -7 to understand the opposite opposite is just the number that's the same distance so we're we did in the beginning 027 is 7 units from 0 the opposite is the number that's also 7 units from 0 so that means it's negative 7 how far is -7 from 0 and in which direction do seven units but it's to the left and that's one way to understand what opposites are the same amount of units from 0 but in opposite directions so 7 is 7 units to the right negative 7 is 7 units to the left opposites are the same units away and just a different sign and their opposites of each other who's annoying me up fix that later nice and simple I need you is Augustus for all of you guys right here that aren't taking this stuff down or listening or following long I can't help you if you don't get the easy stuff right here but wait no more questions so we'll still use this Saudi thinking back on the previous work that we just did explain how it is how you would use the counting on method to represent the following so if we reply integer card game which is 1 through 12 and will negative 1 through -12 + 4 0 cards just imagine what is that 24 cards you would have no. 12 and 12 is 24 you would have 28 cards so while playing the integer card game the first card you selected is 7 and the second card selected is -7 what's a question here oh explain how you would use the counting so what we just did think about it like an airplane if the first card you got was 7 and the second card that was -7 how would you describe the sequence of moves if you were to describe what we did pretty much like right here how is he described it if we grabbed the first car was a seven and the second card was -7 what would it look like and how would you describe it I'll give you 30 seconds to write it down yeah good so someone just said someone so much just said writing God is so much more than just doing it erect the writing it out help you understand all right so I'll copy yours but I'm going to add something to this one that you just typed out so starting a 0 member always start at zero starting at zero move seven units to the right and I'm going to be lazy and I want to use improper English, speak I can move 7 units right then move seven units to the left and here I'm going to put them move 7 units to the right I'll put a parentheses to say that's for 7 move 7 units to the left that's parentheses -7 that's a negative 7 right here and you will end up we're back at zero very good so what does it tell us about the sum the sum means to add what is a tell us about adding 7 and its opposite -7 what is the Tellus yeah good you get zero 7 + -7 equals 0 the sum 7 + -7 is 0 that's a sentence that's an equation description so understanding math you need to know how to do two things you need to be able to write an expression that describes what you said and you need to buy does a sentence so I jumped back and forth because it helps you understand this is why this class is connected to English and I'll say it again say it again very important very important look at the curb Rose yuju for 7 and -7 what relationship exists between these two arrows that would support your claim about 7 + -7 so here's the thing about this so here's our number line here 0 ears 7 so the curved Arrow so here's our first Arrow we go down here that's a plus seven and then we drink back right here - I don't care about the numbers in between cuz it doesn't matter to me what's special about those arrows yes that's that samples the arrows are the same distance from zero it's just a nice way to describe the arrows are the same distance from zero it's another describe bringing it back what an absolute value means the only difference is the arrows are the same distance from zero but in different and I'm going to use the word opposites 2 directions I had to give this to his most you going to see that what is a ray my first one plus 7 the arrow goes to the right but it's the same distance going the other way except the negative 7 is an are going to the left 7 units so Arrow direction will come into play and it helps you understand positive and negative little thing like arrows don't mess that up where I'm going to keep emphasizing until you guys understand arrows draw the arrows left and right so you don't mess up so when you start think about in your mental state you'll be able to get this all right do you think this will hold true the sum of any number and its opposite so this right here is the rhetorical question you're going to ask yourself use any number someone type in a number just type in an isomer quick 12 so we have let's say we have the card 12 what is the opposite of 12 what's opposite of 12 Meagan Good let's say we have the card 12 and -12 if you drew the same thing like this what number do we end up at if ya zero yeah we end up back at zero if we did this instead of 7 and negatives that we got 12in they do we end up back at zero so why is he that's the whole the wise what we're going to work on so do you think the truth yes so yes and it's on any number and its opposite so yes we know it's yes but why this is what we're going to do next and there's a property for that this will describe this is a y Somerset can we start with a negative number good so let's look at that number line quickly before I move on let's use purple instead so we can cover the purple and brown so let's look at this right here I'm going to give you the card I'm going to keep it simple let's go 5 + negative let's go -5 + 5 -5 5 instead of starting while we start going towards the negative side so here 0 we go to -5 + 4 + -5 we end up back at 5 so -5 + 5 the only difference is the number line started on the left side instead of the positive side instead of the right side so we started at but we still ended up back at 0 so look at this song 0 we went to the right and then we went back so it's on the right side of the number line on this one if we start with a negative we go to the left side but we end up back at zero so it doesn't matter that part so we still end up back at zero so good question it doesn't matter if we start with a positive or negative we still end up back at 0 so this is called the property right for every number a don't know if it's a positive or negative there is an opposite look at this for every number that you have there's an opposite A and negative a so that a plus -8 equals 0 and -8 plus a equals 0 so it doesn't matter which one comes first you have a number and its negative version if you add them to get 0 but if you start with a negative and the positive you still end up with zero and that's called the additive inverse of a number the additive inverse of a number is a number such that the sum of the two numbers is 0 so another way of saying the opposite or the negative version is the additive inverse the opposite the number satisfies this definition for example the opposite of 3 is -3 + 3 + -3 + 0 + -3 is the additive inverse of 3 the property above is usually called the existence of additive inverses just call it additive inverse now week we call it additive inverses cuz we don't want you guys to think I'm going to tell you why now and it's going to look for some reason you already think that a lot of guys the opposite of a number is always negative not true what if the number starts out negative so that's why I wanted use the phrase additive inverse I always end up going back and forth between this and the word opposite so what is the additive inverse of 5 we did that -5 what is the additive inverse of 9 negative 97 what is the additive inverse of -97 positive 97 they're opposites of each other so know that you could start with just the number the additive inverse is just the opposite and the opposite to be positive or negative depending on what you start with all right so we're not going to play the integer game with your group cuz we don't have the card and that's why we are done but here one last and I want to do as much good example fine that's pretty much it so understand I'm going to do a quick pull back right here if we ever for now for now we're building up just make your brain mush remember start at zero and go to the first number after that from wherever you end up with moving left moving right count spaces do hops if some of you guys want to do little baby hops cuz you want to count that you can also do that someone said to someone said Can I Do by I didn't understand you what you were talking about when you type that in now I remember all right I'm going to show you quickly what up bunnyhop is you don't have to use it but you can so look at this person right here we start at 5 go to 5 you can do bunny hops 1 2 3 4 5 or I can just take a normal step right here we end up there and then you take -5 bunny hops 1 3452 the last we end up right here and then what color does it have a head Green from there I do negative I go bunny hops To the Left 4 1 2 3 4 5 -4 and then we end up with what color we use black then negative H to the right so from here 1 2 3 4 5 6 7 8 where we end up I hate this method and you should too I want you guys to think we'll use this little spot yes I know it's annoying but it doesn't help you think adding subtracting do please if you need to use the bunny hops there's no other way to describe it or little hops but try to use the big one try to do big hops because it gets you to start thinking adding subtracting but what if I asked you to add a hundred are you going to do a hundred little bunny hops no see that's why I don't do these little small ones if you could avoid it if you're forced to do it but try not to do the problems

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