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Math 7 - Module 2B - Lesson 11

Added Apr 17, 2021 •

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The Rules for Multiplication Note: I misspoke at the end. I boofed. -1 x -1 = (positive) 1 I don't know why I said = 0 (ACTUALLY I DO - BUT....)

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11 develop rules for multiplying signed numbers this is going to take most of the. So let's try and get this so let's go read it work granted Chromebook if you need Amazon package right there I'm in the package right here next to the paper towels all right so part a complete quadrant 1 and 4 1 Quadra to Quadra 3 Quadrant 4 show no one ever messes up quadrant one no one will ever mess of quadrant 3 it's 2 and 4 so let's do a quick thing about quadrants so this is the x-axis the y-axis so forms for questions we've only been doing so far quadrant 1 you know your x axis or Y axis on a graph we're going to put in the other quadriceps like so lot of people mess up cuz most of us are you used to looking at the clock so when we go around we go this way that's called clockwise but when we talked about quadrants in math its back counter clock run Alger clockwise so 1 2 3 4 1 2 3 3 is here but notice if you mess up 123 South one and three will never be messed up its test where you go counter-clockwise the correct way vs. clockwise the incorrect way so watch out but right now it only wants us to do one in ignore ignore complete quadrant 1 and 4 of the table below to show how sets of matching integer card so are playing the card game again where am I vaginal a card game that will affect a player's score in the integer game for example remember we're doing repeat it's a liquor I hear the x-axis is the number of May card so you know like I said I would have brought in a spinner so you choose a number this is the integer card you would choose so we throw away 6 to 12:06 and doesn't count pick a car you would get one of these options you get what to do for 5 or negative one negative to negative so you would pick one of these 10 cards and then I would flip the spin and it would tell you how many times you should repeat that card you would clone that car so right here for example 3 too so let's say you chose the card number to put the spinner and land on three what would happen to your score for example three twos would increase a player's score by 0 plus 2 plus 2 plus 2 equals 6 points but that's the whole thing repeated addition right there repeating three times so I would flip the spinner is how many times you would repeat whatever card you pick and you would have pick one of these cards so I put the spinner lands on 3 you pick a card it's too what would your score what happened to your score you increase by 6 what's go down here it's like the spinner lands on for you pick negative so that means your cards get a repeat four times and you pick negative 5 the car value choose what's happens to your score right here so let's look at I'm going to put the example up here we're doing this one together right here ice like the spinner lands on for you pick the car it's a negative five let's look at what happened let's say you start start II start at 0 starting at 0 starting at 0 you picked which car you pick negative 5 so we're going to add negative 5 but how many times are you going to eat that 4 time plus -5 plus -5 plus -5 notice how I drag that last one out even repeating something only four times is exhausting its why do you want to say it over and over and over and over again that's why we need to understand what multiplication is so let's look at this I know we're going to have 0 + -5 and we are repeating it four times -5 + -5 + -5 + negative Magic on a number line we're adding the same number so we're at zero we're going to -5 and then we're going another negative 5 and then we're going another -5 for x she going to think going backwards Let's ignore the negative sign we know all we're doing is adding on cuz if we're going in the same direction it's addition and we're going in the negative Direction how many spaces he died 5 5 10 15 20 and since we're on the negative side -20 but that's the whole thing why do I want to do addition when I can just see this as and again we don't need the zero but we're doing it for now eventually we're going to get rid of it will get rid of him in a couple steps 0 plus what level I just do this why why do I want to do this one at a time when I could just do 0 Plus 4 * -5 back to last lesson let's review this the first number is how many times your number is repeated we are repeating our number four times what are number -5 so4 -5 combined thank you for some person that jumped way ahead thank you let me know you're paying attention when you put in the comment put in the chats that participation are there if you want to sneak out thank you so or -5 is -20 so let's just complete let's just do a couple of these let's just finish this one right here so 1 one set choose is to two sets of two's for three sets of two six yes somebody someone said aren't we but are we just doing a multiplication filling in a mug let's do finishes four sets of 285 sets of 210 correctamundo let's go down here though let's finish this row and you can do the rest one set of 251 -5x and -5 if I have to -5 to -5 would be / -10 do you see a pattern -5 -10 3 -5 negative 15 find a five- 25 so let's see which one right let's go do roll number for right here One X 4 to X to one set of 2 is to one set of negative Tuesday is -2 two sets of -2 that means negative two plus negative to negative for look at this right here look how they think about three sets of negative negative 6 -8 five sets of naked -10 based on this right here correct so fill in the top fill in the bottom I'm going to give you 45 second because a minute is too much filling these two and we're going to answer these two together so do these right here fill it in and we're going to think about what we're doing with those numbers 45 seconds not a minute go race writing helps writing lets me know you're falling along cuz some of you guys I see a flickering lights in your eyeballs that means you're you're watching Caillou right now Caillou you know that right, yeah so sad for you I'm sorry Philadelphia Illidan another 30 seconds a ride lot of people are sending me your answer is fine for typing it out that's fine but I just want you to fill in it all right so if you're not done all along so listen to this all this row is look at this one x 1 1 7 1 1 2 6 1 2 3 4 five first role in counting by ones what did he provide understand by that number the natural it's look at this one set of three three two sets of three three six three sets of three nine stop for a second you see how I'm going back each time Emerson is where logic begins it's like I'm going back way old school there's two games that I can think of the cat, there's what doctor lighten on the Nintendo BS was that detective game where you were supposed to figure out Clues and connect things together looking at patterns and there was that lawyer game Ace East something is Ventura that's the Pet Detective DS game where you were playing that lawyer or something lawyer is a Capcom when the people make Street Fighter Megaman which is Bizarre Adventure fighting game I forget it a something lawyer but logic number sense is logic look at this we're counting by ones were counting by twos look at this I stop cuz I gave my three choices math is always about Ace right yes Ace Attorney so those games were about logic it's like why wait till you you get all these Clues you get less points if you figure out in the very beginning it's like it's like a wheel of fortune if you're able to figure it out right in the beginning you win but you could get greedy and get more money but the chassis Uline on the back of it but once you see a pattern know what the blacks are why go back and forth that's just slowing you down I'll do it in another County Mi once I'm counting by twos I'm counting by threes three 6 9 12 15 look how quick that is I want to go back to the slow way because this I got one set of four four two sets of 4 x 4 + 4 eight three sets of four 4 + 4 + 4 + x 12 notice I'm doing each little step over and over and over again I'm literally doing each step four sets of 4 4 + 4 + 4 + 4 for which is 16 five sets of four for a 12:20 but that's the whole thing I literally didn't care about looking this as a pet Twisted the one by one you can do that you're still going to get it right what we're trying to get you guys to do as a student and this is where stem comes in the play why do people's eyes with STEM Science technology engineering and math all that stuff so you nice rich job on Rittenhouse Rich honest people anyways if you want to think about that one 5 is 5 to 5 and he looks let me double check this out three sets of 5:15 look I'm counting by 5 5 10 15 20 25 you see that process of Jesse has a pattern will just fill it in why wait why go one step what I just did reading everything over and starting over that right there stem is about seeing that there's a pattern and I'm just going to repeat it it's called The recursive concept you do it over you see a pattern you do. We already saw the hey look at this negative to negative 4 negative 6 negative 8 it's just the opposite of up and someone said that I just put a negative sign down here for the same stuff up here correct that was pattern recognition right away cuz we did it twice doing it one time you're not sure do it two times that's a pretty good then you should assume and that's one of the Assumption is that's what lawyers do this with detectives do we make it a substantially prove it right unless just fill in the rest later this if I'm going to fall at school with negative that would be disruptive for 8 12 14 for 8 12 16 20 how about I just write that here for 8 12 16 20 yes I know I was holding off so I'm just going to write these numbers now but this person said I messed up their supposed to be Navy you are correct I was waiting to do that so I know it's going to match up with this one for 8 12 16 20 except down here negative notice I didn't do work I've just filled it in the work I did was actually up here and help me see down here so I could just fill it in super quick go back with these right here this is going to be matching with this or 12345 accept negative so let me just write down the numbers Force 1 2 3 4 5 6 7 look how quick that is once you recognize a pattern what's a good this that should Match 3 6 9 12 15 3 6 9 12 15 and we can put the negative sign but let me double check one of these I'm going to check this one and this one I got five sets of negative three negative three negative three that's two of them negative six three of them negative nine four of them -12 V -15 that totally works nice patterns make things easier that's what multiplication is recognizing that there's a pattern of repeated addition to take meds that I don't want to add I'm going to multiply all right so what song is this what does this quadrant represents ignoring it we're playing the game right now I flicked the spinner lands on one of these you pick the card look at this dude what happens to your score good we're picking what does a squadron represent we're picking up a positive card and repeated This Man X let's keep it simple picking A+ card so taking a positive car what happens your score increases so when you repeat a number and you pick a positive card year your just increases down here look what happens I flick a spinner lands on 5 you picked -4 so you got to repeat naked for up this is the total of you -20 you pick the negative card cuz this site is picking so picking a negative hard and repeating that negative card your score doesn't go up because it gives you a giant negative number that means you know it goes down so another way. score decreases some of the guys play Pokemon Yu-Gi-Oh or Magic the Gathering or you got the digital thing I would have been a Hearthstone you guys know you can make money playing Hearthstone is a heart forget I'm not going to get in the stone I hate that game cuz I always get beat it's unfair you always play Advanced people why they pick on noobs like sad that when I was so if you played any of those hard games sometimes there's a multiplier a multiplier and then you have to when you get a x you pick a second card it multiplies together city of that next card so look at this we picked while they started at 6 so you picked your multiplier of 3. Okay you flip that card whatever next card to do you do it three times you get two of them you know you can you repeat also I get a new score of 6 it's like in Yu-Gi-Oh when you're when you're polymerize your characters and then right here also that's that's a great thing I ride but what is in your deck you had a card that hurt you cuz that's look up the side or the Texas near the mall the party got to pick from the deck out there then not in your hand deck and then all right all you got to get pulled your multiplier of five hand parts and then you go for a bit over sorry you've been poisoned for next five rounds in oh and you flipped you hear toys in your every time you play you lose 3 hit points but you got to do it five times until you get a cure or you get tired so you're going to lose a total of 15 points so your score goes down so yeah think about it like that stuff played with like I said only come pick up his card games right now or role-playing games Kingdom Hearts and what not realized just not game what's not just games if you get a credit you get credit we had a credit card credit card isn't free money you're borrowing money from the bank that's what it's credit card is your borrowing if you pay it off in your first month to forget the first if you pay it off when the bill comes they don't charge you interest they're hoping that you don't pay it off in that month they hate you when you do that that's good that's a good time to get some hate but when you don't pay it off in the first month they like you because whatever money is over they charge interest on the juice which means they take a percentage so let's say you you every month every month they take away let's say $5 if you don't pay it off so each month so let's say you bought this is just a general idea to charge something for 500 bucks but you only paid $250 on it months the bank takes away $5 that you don't pay it all off but it's nothing worth its percentage would do something like that later you spend the bank says alright 5 months first month I'm going to take away $5 second month I'm going to take away $10 third month, take away 15 until you pay it all back next month you're taking which one so you know they're going to keep on taking away whereas I don't know how it works on the positive side cuz if you pay it all off you don't even move anywhere you don't get money that's all that banks will always take money from you but they don't like giving money to you there's a savings account which still give you interest also another insult say you borrow money and they charge you this much it take a big chunk from you but if you leave money in a bank account that builds interest you think I'm getting paid you're not really getting paid you're getting on small percentage yep so so when you have a savings account you get interest every month but it's like pennies what banks do banks taking so anybody who has money in a in an account at a bank the bank actually borrows your money they're using your money as credit for them so they can invest that's how Banks make money let's say there's two people with $1,000 they are saying $1,000 will give you a little chunk of what we make so they use that thousand dollars and banks have people that know how to invest in the stock market like GameStop stocks to hear about that but they have people that like to buy oil stocks I'm going to buy a Tesla stock and they get profit every month off of find stocks because but they didn't use their money that used your money those people $2,000 and they were able to buy $2,000 worth of stock and if they get a percentage of it that's their your money stays there here's their profit now and here's the thing it's a huge chunk the prophets huge and all they give you you gave it you let them use your thousand dollars they're giving you $0.25 what are they get to keep the other $599.75 for profit that that's that's unfair but that's what they do and that's why they want more people have in savings account cuz the more people that have savings account leaving money untouched in a bag that's more they can use to buy stocks and brothers just buy a safe yeah but then people find out cuz you just let everybody know you know we'll do that thing they're going to stalk you now not everyone's going to stop become the school to self and then don't forget it when I do what happened right in front of my eyes to my corn man around the corner I can do anything they just have them sad alright you killed it again this is we caught your new nickname is killer bee kill the moment all right so get I'm going back to what someone said this is not a graphing chart it but it's like a multiplication table but that's what you guys don't realize a graphing chart animals application table is very similar look at this right here I'm going back this right here is the x axis we always do horizontal Left Right line first so this is the x axis which means the up-and-down vertical is the y-axis and look at this it's a number we got zero notice there's no rule no lie no row or column 400 times anything is 0 we don't move anywhere is this 01 2 3 4 5 0 negative 1 19-18 it's a number line it's the X & Y axis graphing and a multiplication table is pretty much very similar let's go read this what patterns do you see in the right half of the table you guys already told me the pattern but let's go right down at first quadrant 1 and quadrant or are opposites opposites of each I just took you know this but here's the thing that I just take this and move it down as if I just shifted it down five would be on this first row or opposites but there's a term when you flip it over Lotus right here let's go. I got four fingers but five little baby fingers and that means I had each finger that would line up on each one of these roads 12345 noticed this is a this is front of my finger beside market and when I flip it over and noticed that one that's human that's pointing to five notice when I flip it over it says I got my fingernails does the top of my finger now but I'm still at buy but it's now the negative one so I flipped it over that's called a reflection when you flip it over it reflects you don't just shift it down you don't just screwed it down you flip it and my flip didn't decide was negative so quadrants 1 and 4 are opposites and it's called reflecting on the middle line in the middle line is the axis are opposites reflected I'm teaching you this term right here it is a geometry term but you guys got to learn we'll get to that eventually but I'm giving it to you early and then let's go quadrant 1 to summon set it up in the thank you quadrant one is positive Quadrant 4 is negative I know that's what I said earlier this right here it's quadrant but I wrote it as quad but notice I read it as quadrant in math and in life when anytime you write in shorthand you still read it as a complete even though I shortened it out I wrote it at Squad but you read it at I rate as quadrant quadrant one quadrant for so always read your shorthand as a complete border but it math lot of things you can always write it but do not shorthand in an essay English class no science and math yes in the left side of the middle row and then enter the missing integers left side of the middle row and describe what they represent all right so the mineral here's the middle row feel that in I'm giving you a 10 seconds you could you should be able to super quick looking at the pattern what goes here Lynette middle row do that in right here that's the first part right there and I swear I wouldn't represent in terms of this I'm ignoring you bring up good alright everyone got it so this right here the Middle Road 81 negative to negative three negative for -5 this is a number line this is your original number line based everything on 0.0 in the middle your origin someone said earlier like a draft correct this is the origin 0 1 0 1 2 3 4 5 looking at this 1 0 1 2 3 4 5 you go up notice when you go down going to the left is the same thing to the negative good jobs but what the represent in this number game so this right here is how many times we're repeating the card to add to our score we're adding a positive or adding a negative but it's the opposite times where remember someone got it and just starting throwing away your card good job so it's the number of matching not only are you doing matching cards again but instead of adding it to your score or combining it you're removing it or discarding it you're throwing away at hard so think about this right here we're going to fill in the next page that's going to be right here do this in a second so we're going to fill it till this in right here so let's look at it this means how many times were going to repeat it also but instead of putting it into words for we're going to throw it away let's go choose this I flick the thing my spinner Atlanta -4 you chose the card 3 so we're going to repeat three four times but we're going to discard to think about it three three 3 that's 12 but your score instead of being added to your score we're removing it good someone said so the answer is negative so this is the multiplier -4 this is what's being multiplied 3 you think I got I'm going to not great but I'm going to ask 3 but no the x o I'm going to take away from my my hit points instead so -12 huh look at this how it matches with here it's the opposite pattern recognition again let's go look at this right here I'm going to go straight here -1 so I'm going to just -1 I'm going to discard one card discard one time your card is three I have three points but imma throw it away look at this how it matches instead of this world reflects down here because this and reflects that way let's finish this road just to check it out I got you got a three card but the multiplier says you got to throw way twice if I have three points I throw away three points I throw another two points I throw away 6 points -6 look at that that matches up that totally fits our criteria that science right there we are Fairmont did with numbers with spermatid once we experimented twice yes we experimented three times that's how science works we usually do an experiment 3 times a test stroock that means I'm not why bother we already know all right I got a 3 I got a 6:00 this should be at 9 why waste time and this should be 15 accept negative so fill in quadrant 2 right now go do that fill in the rest fill in the rest I'm going to give you 20 seconds selling it in some of you guys already did it thank you but check your work once we get it done I'm going to try the biggest mistake for people that are being played right now super easy done here's the mistake someone shutting someone already asking so they put the five here to put a 10 here 15 the mirror that no remember looking at this I need to be on the edge here where we are afflicted down here and change negative here you go flip it this way instead of flipping it up and down you flip it left right left that this thing is on the 25 it flips over that should be the 25 so and it's just the opposite so just go in out with a butterfly like wings one size of positive Wing one side the negative wing so let's go right here 5 10 15 20 25 except they're all negative thank you I'm recognizing you. how's yours someone use butterfly wings Start on the inside and spread out that's a good way of describing it will give this 510 just spread out like wings so we're going to have to go in Idaho for spreading out that or 8 12 16 20 but don't forget the negatives these are all negative and just fill in the rest 2468 and this is how you could move faster and work someone has Matt your fast no I just recognize a pattern that's how you can move fast when someone says slow down it's cuz you're doing work one step at a time recognize a pattern just a man I can just throw it in all right then remember this side means throw away let's put that their number of cards thrown away I didn't write that down was put that right here this right here means number matching removed so we're moving in five times moving and four times he get that idea look at this right here let's say when you pick a card you picked -3 man I'm going to lose three points but look at this you're going to remove your negative 3 card what if I remove my negative 3 car I don't go down in place I get rid of it I'm not going down under if you chose in a hand the card game whatever it is I just use like I said you yeah you pick the card and it damages you yourself Emmanuel yeah maybe we did fine cuz it would be better so when we get if light has come back here you can bring like Pokemon that you guys choose Pokemon Yu-Gi-Oh will play I haven't played in so long I forgot teach me but I still have my blue eyes white dragon somewhere no I don't have the head so that I don't have a complete collection so I don't even know where they are you chose a card that says you you got a spell reflect on something you lose three points but at the same time you also get to choose a community card but you take that card away with just put down oh I take away a car that hits myself so you actually don't lose points and what's the opposite of not losing gaining so if you throw away a negative card your score goes up so that right here is just three will come back to this in a second we'll come back to this think about that let's go answer these questions right here we're going to come back to the first page just look at it so Part B complete quadrant to which we just did what relationships or patterns do you notice between the products values in quadrant 2 and the products in quadrant 1 so let's compare 2 and 1 and will fill it elegant two in one we just talked about that right there we just said it would be this kind of these compare their opposites quadrant one is positive to is negative but that's the thing they're not shifted over it's the same thing we just wrote down here so let's do that for quadrant one or two so quadrant one is positive and quadrant 2 is negative they are opposite reflections their opposite Reflections listen to the next thing we wrote down the same thing about 1 + 4 + 1 + 2 are you a sweet wrote the same thing about this to this with this. I just chose different words what relationship or patterns do you notice between the product values of quadrant 2 and 4 2 + 4 2 + 4 except it's it's not a reflection per se but they're the same values except they're in a different order but in a it's almost like a mirror image but on a diagonal but if we had a Atlanta right here and we made the butterfly this way because this it's a reflection going this way so that's a little bit different so we have another reflection again but I don't want to write that they're both neg they're both negative and let's just say they are reflections quadrant 2 is the reflection notice I didn't say opposite it's just the reflection so opposite reflection let's go that the opposite reflection just the reflection of quadrant for it's not opposite cuz we're not changing the side it's literally the same yeah it is so now listen to this till this is just pattern recognition use what you know about the product that is a 1/2 + 4 to describe what do you think three will happen if we look at the pattern who is the opposite of two one is the opposite of 4 2 and 4 are the same good yes looking at this pattern just base this look at this based on this one quadrant 3 is the opposite of 4 next to it cuz look there's the opposite because that that's the opposite and it's the same as one look at that it's a reflection of 1 so what type of numbers do we have right here good someone said all positive let's fill it in fill it in watch out just paying attention remember it's a reflection we want to go 3 this should be to the should be 145 let's just do what you do for 5 1 2 3 4 5 8 goes inside out it's the reflection to 46810 it is when you follow your pattern recognition 3 6 9 12 15 for a 1216 auntie look at that 5 10 15 20 25 and what does this represent this side represented us removing cards removing removing positive card and what happens to our score when we remove a positive card score decreases look at that that's the opposite of this side look at this it's the same as this side removing negative and when we remove a negative a positive score increases look at this right here it's the opposite of the one next to it but the one diagonal to it or score increases I'll let you read them that time 8 minutes the rest is going to go super fast so looking at this look at this positive times a positive positive negative times a negative live positive times a negative negative answers a negative times a positive a negative answer patterns that's a sign we're going to get there using what you know what what do you think describe quadrant 3 where he said that quadrant 3 is a reflection quadrant 1 so quadrant 3 is positive is it possible to know the sign of a product two integers by knowing in which quadrant each figure is located yes this what what it's saying is it if you are if you know this but if you have the numbers do you know which sign in which were the two numbers you multiply to get ya it's a good choice you have a good chance of guessing is it possible yes because 1 + 3 r+ you know that if it's in quadrant one you're at if they put the number in quadrant 1 you know you have a positive and a positive I wanted to write the whole thing I could have just short-handed it if you know what's in quadrant 3 you know what the number has to be negative and negative the two number you * if it's in quadrant 2 or Quadrant 4 are negative you know your aunt if it is the two number at the number itself was in quadrant 2 you know your first number has to be negative or second number has to be positive or if you have a positive number and then a negative number look at this right here you got a positive number and a positive number you got that right here switch to cut them some number in here look negative number then a negative number doesn't matter which one if we go in quadrant II got some numbers are we North first number has to be negative why has to be positive if we're going in here or first number is positive second number has to be negative that's what it's saying that's like Google making which quadrant contains which value describe an integer game scenario let's just use rules quadrant 1 + 3 raise your score quadrant 2 and 4 lower your score and we'll just finish with this next thing and we'll be done we'll skip those promise down there let's just do that quickly I'll leave this right here pretty much we got positive times positive negative times negative same signs we know the answer equals a positive positive times a negative and a negative times a positive that equals a negative it's a sign for the same thing that's so similar to an addition sounding to an addition rule same sign in multiplication if it's the same silence positive notice I didn't say keep the sign this is multiplication mole tips vacation not Edition so don't get those mixed up all right but a lot of people were asking at why is negative times negative positive that was using the graph wait here is the short way of explaining proving that negative times negative is pause that's what I could use is most people so let's look at this we know that how wide is -1 * -1 equal a positive one that's the part people get stuck on all right let's start with this what is anything times 0 what is -1 * 0 + 18 * 0 is thank you thank you thank you three people that's always through for some reason all right so instead of writing zero I'm going to rewrite it as this -1 * I'm going to go 1 + -1 bestie additive inverse right there cuz what is 1 + -1 0 so I'm lining yeah it's doing the opposite opposite I'm going to use this to show you so this right here is the additive inverse when you have when you have offices that zero so I change 0 into that so I substituted that in there oh look at this what happens to a number in front of a parentheses you multiply so let's distributes choose a different color distribute so we're going to distribute the negative 1 times 1 negative 1 this one we got right here and then we're going to distribute this negative one times this negative one so we got a negative one * this -1 and what's connecting them this little addition right here some put parentheses around both of these look at this right here what is 1 * anything negative 1 and all of these are still equal to zero cuz I didn't change anything it should be would have 0 negative 1 plus what number equals 0 -1 plus what equals 0 1 look at this -1 * -1 equals 0 the only way

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