all right we're going to talk about chapter six division okay so first thing we want to talk about with division is that we have different ways of signaling division we have a divided by B that division sign we also have A/B or a fraction a over B and we also have the a divided by B but it's in that little like division symbol that little house so to speak right where we have this this way we can set up a division problem that's quite a bit for students to understand especially the lower level that you know with addition or subtraction we really have one symbol for that with multiplication we have many symbols for that right we have the little X you have the little dot you have parentheses and division now we have other multip representations of division so we've got to be aware that we need to make sure we differentiate and differ what kind of symbols we use throughout our division units so interpr ations of division a divided by B the result is called the quotient a is called the dividend and B is called the divisor okay and I have a little picture there using our definition of multiplication we can then go down into what's happening with our division okay and in with division we have whoops different types of problems we have a how many groups division problem and how many units in one division in one group division problem so how many groups inter interpretation of exact Division if a and b are non- negative numbers and B is not zero then according to the how many groups interpretation a divided by me means a number of groups of B units that make a units okay so the how many groups so in this instance we have a lot of examples of this actually like in your activity that we do um in just a minute uh that I have a video for but you're ideally asking um like you have a certain amount of candies and you want a certain amount of candies in each bag how many bags how many groups do you need we also have how many units in one group interpretation if a and b are non- negative numbers and B is not zero then according then according to the how many groups how many units in one group interpretation of division a divided by b means the number of units that equals one group when a units are divided among B groups okay so how many units are one group so this is like a flip side so you have a certain amount of candy and you have a certain amount of bags how much candy goes in each bag now what type are these so it if it take it takes seven cups of snod grass to make three pints of potion how many cups of snod grass does it take to make one pint of potion and then the next question bezel bugs cost $3 per pound you have $7 to spend on beesel bugs how many pounds of beesel bugs can you buy so I want you just thinking about what these types of problems would be okay I get more into that whenever we talk about the act in the activity we go in depth about that so like give your guesses as of what type of problems these would be Now activity 6B you have a video link there please watch that video because it goes into what's going on with those word problems and the different types once you've completed activity 6B you can come back here and double check to make sure that's what you should have gotten with your guesses of which type of problems they were all right now division word problems there are 54 cans arranged in an array of six rows there's a rectangular room with an area of 54 square meters one side of the room is 6 meters long there are 54 Penguins on Shore and there are six times as many penguins as were in the water or in a game you can roll six di six-sided d a number cube sorry and then pick a card there are 54 different possible outcomes how many different cards are there for you to pick from so these are all situations that are essentially 54 divided by 6 so these are all different ways we can write problems where our goal is to get 54 divided by six okay so I want you thinking about you can do an array type situation you can do an area situation you can do an application where you have like penguins or something going on you also have the numbers of candies that go in bags or how many um pieces of candy you put in each bag how many bags do you have have you have all different ways you can set up division word problems now you can never divide by zero ever whenever you divide by zero that is Technic is called undefined so 8 divided by 0 10 divided by 0 you can never ever ever divide by zero so make sure you understand that 0 divided by three is zero but a number divided by zero is undefined you can never divide by zero activity 6D I have a link for you there we talk about how division relates to fractions now Division and remainders so division with remainders how many units in one group we want to talk about with the remainder is the number of objects left over okay because I'm talking about right I have um I'm trying to figure out how many units go into one group so I have a piece I have all these candies I have certain amount of bags how many pieces of candy go in each bag my remainder would be the leftover candy okay that maybe wouldn't go into each one of the bags like let's say I have eight bags but I don't have enough leftover candy to go into all eight so I would just have some leftover setting out what if I have how many groups type of problem so if a and b are whole numbers and B is not zero then a divided by B is the largest whole number of groups of B objects that can made out of a objects the remainder is the number of objects left over okay that can't be placed in a group okay now play close attention to page 248 to 2 59 okay they talk about all different types of situations you need to make sure you're paying close attention to activity 6f is what you need to have a reflection on on canvas now the division algorithm so we have over on the left this common method for division this is what you've done before right you think about seven divided into four that doesn't work seven go how many times is s divide into 45 6 * 6 * 7 is 40 two you subtract you get three you drop the eight down and so on now the scaffold method is a different way to do it okay so the idea here is you think about it just like what we did for um the partial sums method with multiplication we think about it in a similar way so I think how many time seven times what can give me really close to four four 4500 okay so 600 * 7 gives me 4200 6007 okay so I have 600 up here and then I subtract I get 380 okay so what's left over after subtracting those 67s I have 380 and then I think to myself okay well 57s will go into 380 that's 350 okay that gives me 31 left over so then 31 what's left over after subtracting 57s um and then what happen What can I do there well how many sevens go into 31 well I can have four sevens go to 31 that would give me 28 okay and then if I subtract that I get three and that's going to be my leftover or my remainder okay now you can use the scaffold method and get a little bit more flexible a little bit more shorthand so here hundreds and tens are an easy way to work through but it's not very efficient so here it's just being broken down into hundreds so if I don't know what is going to go into 572 or how many eights are going to go into that I can say well I know for sure 100 go into it and that gives I must subtract 800 and then another 100's going to go into it and then another hundred's going to go into it and I can keep doing that until I get to where I'm like okay now I'm at um 272 so I'm going to subtract 10 * 80 okay because 800's not going to go into that and then or 10 * 80 so that gives me 80 then I'm going to subtract another 10 * 8 and another 10 * 8 until I get down to the bottom okay there's fewer steps that's more efficient fives and twos would will work a little bit better so whoops so here I have um 500 time 8 gives me 4,000 then I have another 100 of that then another 20 then another 10 then another four so it's just kind of merging saying well I know 500 of these go into that right they're just kind of merging a littleit a few of the steps and then an efficient way of using the scaffold method is to implement the standard algorithm somewhat right how many 8s go into 572 well you can think about like 4,800 that's 600 right 6008 would go be 4,800 you have a remainder of 272 so then you think okay well I know 38s go into that so that's 240 so 32 is left over and 32 8 goes into 32 four times and so on so why does the standard algorithm work so here's just a breakdown of why the standard algorithm works for us okay um we essentially are pulling out bundles of hundreds or thousands and things like that so it's breaking it down into our base 10 um but keep in mind though that we are not dropping like this is says here it's 42 well yes but that's 4200 right so we got to be careful with that our place values really matter whenever we're dividing um and using our standard algorithm that's why we keep everything kind of vertically aligned as we move through because that's kind of cons it is essentially conserving our place value so we don't have to think about the zeros being there but they are all right so be careful with that now what about decimals in the standard algorithm okay so let's say I have 2 143.000 divided by 7 now I can go in and I can break it down normal process as you would go through this but instead of having our remainder um be part of a dollar or something like that well essentially this was going to end up happening think about it with money okay so I have five dimes which will be the tenths and then each person gets seven dimes then one dime is a tenth so I have seven pen 10 pennies left so each person gets a penny and then I have three pennies left over right so here we're kind of breaking it down into money as an example but we're just using the same standard algorithm we just have to make sure that we preserve those place values by lining up vertically also we're preserving by putting the decimal place right above where our decimal was in the number we were dividing okay now what if we want to exess express fractions as decimals so here I have a fourth that's the same thing as 21 + 5 hundreds which is 0.25 okay so here's a picture for that so a fourth is the same thing as 2/10 plus 500s which is 0.25 or 1 divided four okay 100 divided 4 which is 25 and that and so on now how many groups division problems so this is what we're going to talk about with the how many groups division with fractions though so let's say I have this is how many groups both of these are how many groups so you ate five eight cups of popcorn one serving of popcorn is two cups how many servings did you eat or you ate 8/3 cups of cheesy popcorn one serving of popcorn is 2/3 of a cup how many servings of cheesy popcorn did you eat so even though we're dealing with fractions now it's the same kind of setup and you can see here you can uh possibly solve it using a table a strip diagram or a double number line activity 6m is a reflection that you have on canvas as well now dividing fractions okay so let's look at this example this example is showing well what happens if I have 2/3 divided by a half so how many halves are in 2/3 so the idea here is that oh what if we got a common denominator first well that would be 4 six divided by 36 okay and then all we really have to look at is well if I have a common denominator I just divide my numerator four divided 3 okay so that would be four divided 3 or 1 and A3 now we can also divide fractions by dividing numerators and dividing denominators which is an interesting way to do this because it's something that you might not have seen before so if I have 18 divided by 50 18 55s divided by 6 11s I can take my take my 18 and divide by six and my 55 and divide by 11 and I'm going to get 3 fths now becomes a little bit more complicated though if I want to do four fifths divided 37th because three does not divide into four evenly and seven does not divide by into five evenly so what you have to do here is you got to say okay well that doesn't divide evenly so I've got to get a common numerator common denominator essenti sort of okay so what we would do is we look at factors and we' say well in order for three to divide into four I need to have four * three but I can't just do that to the numerator I have to do that to the denominator as well otherwise I'm changing that numbers completely okay or the value of this fraction now seven doesn't divide into five but I can say well okay what if I did 5 time 7 so I have and if I do it in the denominator I have to do that in the numerator otherwise I'm changing the value of that fraction so now I can do 4 * 3 * 7 Divi over 5 * 3 * 7 divided by 37 okay and now here I can divide by three and I can divide by seven I'll have four time 7 divided 5 * 3 which is 28 divided 15 it's an interesting way to think about dividing fractions now what you probably remember with dividing fractions is well I guess I'm going to get to that in the next slide but this is a how many units in one group division problem with fractions so six lers of uh of a marathons Runner Sports string gets two servings how many liters are in one serving or four fths of a liter of a sports drink is 2third of a serving how many liters are in one serving okay um so again you can do this with a table you can do this with a strip diagram or a double number line now this is probably the way you remember learning how to uh divide fractions why does invert and multiply work or taking the reciprocal and multiplying or flip and multiply keep stay flip Keep Change Flip there's all these different ways you can think about this so A over B divid C over D um uh one way that we can do this is take the reciprocal that's flipping over the second fraction and changing our U operation to multiplication and then we multiply straight across just like you do with um multiplying fractions so to get to why that works there's many different examples and so I wanted you to read 379 through 3 sorry 279 through 281 just to see all the different examples as to why one ways you can look at it okay and then finally we have dividing decimals our last unit so whenever I'm dividing decimals what I essentially do is I try in our standard algorithm we try to get rid of the decimal of what we're dividing by right here so what we do is we multiply by 100 well I can't just multiply the outside by 100 I've got to multiply whatever I'm dividing into by 100 as well and then whenever I divide those since I've multiplied by 100 and to both of these and I'm dividing them I've kind of undone it before I needed to worry about it so I'm essentially I'm G to get the same answer if I have 371.50 as if I would have 3 and 74,000 divided by 2 and 3500s