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hello and welcome to another engine math 7 lesson by emathinstruction my name is Kirk Weiler and today we're going to be doing unit 2 lesson 3 more work adding signed numbers so in the last lesson we learned about 0 some pairs and how to visualize the idea that when you add negative numbers to positive numbers portions of those numbers pair up as 0 some pairs and effectively cancel each other out right today we're going to be working with more adding of signed numbers and we want to get to a point where you certainly don't want to be drawing diagrams or anything like that but where you want to have some methods for figuring out what their sums are so that you know you're not drawn pictures basically alright so let's get into it with doing a little bit of review in terms of these zero sum pairs in the first exercise alright now let's recall another name for zero sum pairs additive inverses the word inverse is used in math all the time it kind of drives me nuts generally speaking an inverse is an opposite for lack of a better term and additive inverses are opposites so numbers that when added together give you a sum of 0 now kind of testing this in the first problem right away exercise number one fill in each of the following to make the equation true so here's what I would like you to do I'd like you to take a moment I'd like you to look at each one of these equations and fill in what should go into the blank in order to make the equation a true equation pause the video now and see if you can do this all right well in letter A we literally have a good example of a zero-sum pair or a pair of additive inverses if I add negative 5 to 5 that gives me 0 on the other hand and letter B I have 10 Plus what it gives me 0 well again the whole idea is that there's only one number you can add to 10 and get 0 and that's going to be negative 10 and finally letter C right negative 7 plus what is equal to 0 just like in B there's only one number that we can add to negative 7 and get 0 and that's positive 7 all right now again this is a very very important idea right this idea that we could end up adding to additive inverses together and getting the number 0 and then of course 0 plus anything that doesn't change a single thing so let's keep moving on and use this idea in the next few exercises here we go breaking up a number into a sum all right now it's kind of funny because in previous years you've looked at breaking up a number into a product that's called factoring but it's actually also important to be able to take a number and break it up into the sum of other numbers this is pretty easy let's take a look at exercise number to fill in the missing blanks with either a positive or a negative number to make the equation true is is simple enough literally 14 equals 8 plus what well 14 equals 8 plus 6 it's simple enough now 14 is also equal to 10 plus 4 and 13 plus 1 and 2 plus 12 right but here I'm just looking to break 14 up into 18 plus 6 and we're going to eventually see why I've picked these numbers in the next exercise now letter B might be a little bit more confusing I've got negative 11 equals negative 2 plus what pause the video right now and see if you can think about both letter B and letter C and then we'll revisit them all right let's do it so never forget that what negative 11 represents is 11 negative ones so if I have 11 negative ones well that's gonna be two negative ones plus another nine negative ones or the number negative nine all right likewise negative 20 is 20 negative ones right a negative a negative negative negative negative a negative right so I can break that into thirteen negative ones and another seven negative ones I can break up positive and negative numbers into all sorts of different sums right and let's use that now in the next exercise here we go finding sums by canceling zero-sum pairs let's take a look at exercise number three use what you do Dan exercise number two to find each of the following sums rewrite the sum so that it involves a zero sum pair all right this is really cool right so I have negative 8 plus 14 all right and what I can do is I can take the number 14 and I can break it up as follows I can say well 14 is simply the same as 8 plus 6 right that's my 14 now because addition is commutative associative and all of those good things I can actually do negative 8 plus 8 to begin that's a zero sum pair which is just equal to zero and zero plus 6 is just equal to 6 so I literally take that 14 and I break it up into a positive 8 plus a positive 6 that positive 8 and that negative 8 now match up as additive inverses or zero sum pairs right they combine to be zero and zero plus six leaves me with six let's do letter B together and then we'll have you do letter C in letter D on your let her be negative 11 plus 9 alright well again I kind of want to think about breaking up this larger number so to speak and I'm gonna break up negative 11 into negative 2 plus negative 9 now when I do that that negative 9 and that positive 9 are a zero sum pair so I can cross them out make them into 0 and negative 2 plus 0 is just negative 2 all right I'd like you to pause the video now and play around with this idea on CMD and see what you get all right let's go through I've got positive 13 plus negative 20 I'm gonna break up that negative 20 all right and I'm gonna break it into a negative 13 plus a negative 7 sometimes you'll see math teachers put parentheses around negative numbers little side note because different teachers do things in different ways sometimes you'll see teachers including myself put negative numbers inside of parentheses so that we distinguish them from the operation of subtraction I haven't been doing that much in today's lesson because we aren't doing any subtracting in today's lesson we're just adding negatives and positives together anyhow positive 13 and negative 13 those two will cancel to give us 0 + 0 + negative 7 is negative 7/25 plus negative 15 I can take that 25 and I can break it up into 10 plus 15 then I have that plus negative 15 the 15 and negative 15 or a zero sum pair 10 plus 0 then just leaves me with 10 all right it's kind of fun you know we take these numbers the sort of ones that have a larger size to them we break them up so that we can see a zero-sum pair right so we take negative 20 we break it up into negative 13 and negative 7 and specifically we do it this way so that we can have that positive 13 cancel that negative 13 and leave me with negative 7 let's keep moving on let's keep playing around with this all right exercise number four find each of the following sums by rewriting so that it includes zero sum pairs all right so this is exactly what we did before let's do letter A together have your work on letter B on your own and then we'll kind of play around with it all right so letter A we've got 18 plus negative 25 if you will the larger number in terms of size this negative 25 so that's the one I'm gonna be breaking up I'm gonna break it up into negative 18 plus negative 7 right negative 18 plus negative 7 is my negative 25 the 18 and the negative 18 are a zero sum pair so I just have left with zero plus negative seven and that's negative seven this line right here isn't strictly necessary to write down right and a lot of times students teachers etc will just see those opposites those additive inverses and kind of cancel them out and then say ah my final answer is negative 7 but just to kind of be very strict about it 18 plus negative 18 is 0 then 0 plus any number is just that number all right let's take a look at letter B pause the video now and figure out what that sum is all right now this one might be a little bit more challenging to think about right the numbers are a little bit bigger you have to think about it a little bit you're probably looking at 32 and thinking well I'm gonna break that into a negative 14 and something else what's that other thing you might even go onto the side of your paper and go like this you might go ah 32 minus 14 you know a little bit of this business tene so then you might say all right well negative 32 is gonna be negative 18 plus negative 14 plus 14 there it is right here's my zero-sum pair and I'm left with a negative 18 now the reason I kind of went through that is that right you do this a little subtraction over here to sort of figure out well okay I've got I've got 32 negatives right so how many more negatives do I have than positive so I have 18 more right and that allows me to break down the negative 32 into a negative 18 plus negative 14 plus 14 those two cancel and I'm left with negative 18 but we can actually use this idea of subtraction in order to help us figure out what the result of adding a negative and a positive is together you know without having to break them up into zero sum pairs or draw out diagrams or anything like that now before that we do that I want to review what's known as absolute value let's do that in the next exercise absolute value is literally how big a number is now back when you were dealing with just positive numbers and zero there was no need for absolute value because the size of the number seven is seven right it's only once you get into negative numbers that you want to talk about absolute value and literally the absolute value of the number is its size or its distance away from zero on the number line the absolute value of negative nine is not right the absolute value of 13 well that's 13 okay so literally negative nine has a size of nine units or it's nine units away from the origin just like positive nine is the absolute value of negative eight point two five is eight point two five maybe the trickiest one of all is the absolute value of zero but it's just zero right the number zero is zero units from zero on a number line it's a lot of zeros there you know at the end of the day what a lot of students learn is when you're taking the absolute you have a negative number it makes it positive when you take the absolute value of a positive number it doesn't do anything to it etc but I want to be able to talk about absolute values of numbers especially in the next exercise so let's take a look using subtraction to add signed numbers exercise number six consider the sum negative one hundred and fifty six plus seventy nine letter a will the result be positive or negative how can you tell alright well I'd like you to pause the video right now answer whether or not you think the sum is going to be positive or negative and then how you know all right let's go through it well the answer is going to be negative all right and it's going to be negative for a very simple reason the absolute value of negative 156 is greater than the absolute value of 79 okay and if you will you know if you had said instead of using this kind of mathematical jargon if you had literally just said ah the larger number is negative that would be correct you wouldn't want to say that the lot that that negative 156 is greater than 79 you'd want to sort of stay away from that because all positive numbers are greater than all negative numbers but in terms of its size there are simply more negatives than there are positives so how do we figure out then in letter B using subtraction what our answer is well we literally go back to thinking about how many more negatives do we have than positives right and the way that we calculate how many more one number is then another number is by subtraction so we literally say okay forget about negatives and positives if I just went over here and I did 156 minus 79 right we'll have to do a little bit of borrowing here 16 minus 9 is 7 and then 14 minus 7 is 7 right then literally what this subtraction tells me is that there are 77 more negatives than there are positives now I know my overall answer is negative and it's got to be negative 77 and that is by all for all practical purposes the way most addition without using a calculator most addition of a negative and a positive works a person looks at it and they say oh there's more negative than positive or there's more positive than negative how much more of one than the other is there you subtract their absolute values right we do 156 minus 79 ah that tells us there are 77 more negatives than there are positives and therefore the answer must be negative 77 right 79 positives would cancel out with 79 negatives leaving us with negative 77 let's use this idea now to do a lot of addition of negatives and positives in the upcoming exercise here we go find each of the following sums show the work that leads to your answers all right so here's what I want you to do every time in a situation like this I want you to look and I want you to say ah we have 274 positives negative 319 negatives right that means that must be a negative answer right more negatives than positives so my answer is negative right now how do I figure out negative what well that's where my subtraction comes in right I come over here and I say well 319 minus 274 right how'd I do this we get 45 all right not 0.45 things just random so my answer is negative 45 alright think about what we were doing yesterday we're we're drawing about little pluses and little minuses I never we want to draw out two hundred and seventy four pluses 319 minuses and start crossing out zero-sum pairs what I do want to think about is just the fact that well I know my answer is negative because there's more of that than positive how much more I do the subtraction of the two absolute values I get 45 so the answer is negative 45 this even allows us to do some messy stuff with decimals right I have negative eighteen point seven one plus positive twenty six point five four that means right since twenty six point five four is bigger than eighteen point seven one my answers got to be a positive now normally we don't put a positive symbol there to represent a positive number but I want to do it just because I kind of did it over there now positive what well in that case we do our subtraction again this time unfortunately we have to bring decimals into the situation but that's okay how much bigger is that positive number than the eight then the negative number well we'll just subtract we'll get three here I have to do a little bit of borrowing fifteen minus seven is eight and I have 25 minus 18 which is seven and again literally this just means that there are seven point eight three more positive quantities the negative quantities so the answer is positive seven point eight three there is nothing wrong with putting the plus sign out front again we typically don't so much do that because a number is assumed to be positive and less there's a negative sign sitting there but it also doesn't hurt all right let's go through one more exercise to make sure you've got this adding more than two signed numbers all right so this is important because so far all we've been doing is just here's a positive here's a negative add them together but what if we have more than two what if we have three four or five etc let's take a look at that scenario in exercise number eight consider the sum negative eight plus two plus negative 11 plus 14 letter A asks us to find the sum by adding the numbers together from left to right as the order of operations would imply so let's do that together I'm going to rewrite the sum right down here so I've got negative 8 plus positive 2 plus negative 11 plus 14 so order of operations hey there's a little bit of red going on there I'm gonna get rid of that quarter of operations would say well the first thing I should really do is do negative 8 plus 2 well negative 8 plus 2 is negative 6 right because 6 minus 8 or 6 minus 2 aha 8 minus 2 is 6 and the results got to be negative now order of operations would then say I should do negative 6 plus negative 11 well that's just 6 negatives and 11 negative so that gives me 17 negatives and finally the last thing I would want to do is negative 17 plus 14 but of course 17 is bigger than 14 by 3 units and that means my final answer must be negative 3 all right just a little bit of order of operations working from left to right now what a lot of people will do though when they're adding negatives and positives together and there's more than two of them is they'll do what we're gonna do and letter B let's take a look at that find the sum by first summing the positive and negatives separately and then combine the final two signed numbers all right so let's do this literally addition allows us to rewrite it in any order we want so it sometimes makes sense to say well I've got a positive 2 and I've got a positive 14 so I'm going to group those two together and I then have a negative 8 and a negative 11 I'm going to combine those two together all right the reason that this is helpful is that positives when added together well that's simple and negatives when added together although a little bit more strange should also be relatively simple right 2 plus 14 is 16 and negative 8 plus negative 11 is negative 19 right I mean it's just 8 plus 11 but it's negative because everything's negative and now it's pretty easy to then say 16 plus negative 19 must net me overall a negative 3 right and oftentimes you'll see me doing that especially in later on courses like math 8 and algebra 1 and geometry etc I'll just say I've got negative 8 negative 11 that's negative 19 I've got 2 and 14 that's positive 16 negative 19 positive 16 gives me a net negative 3 right I'm combining the positives I'm combining the negatives then I'm taking those 2 and combining them together alright let's wrap it up so today we reinforce the idea of how to think about adding positives and negatives together by cancelling negatives and positives that turn out to be additive inverses or zero-sum pairs right we then saw how to look at a sum of a negative and a positive identify whether the result should be negative or positive based on the sizes or absolute values of the two numbers involved and then using subtraction to actually figure out that absolute value finally we also saw how to add more than two numbers together by grouping the pauses and the negatives separately tongue-twister there and then combining the final results all right we're going to use this a lot as we move on in the next excerpt and the next lesson and we start to do a little bit of subtraction for now I'd like to thank you for joining me for another engine math 7 lesson by emathinstruction my name is Kirk Weiler and until next time keep thinking and keep solving problems