g'day i'm james and today i'd like to talk about long division and try to see what's really going on with the whole story of long division then i kind of mean that literally to see what's going on with that story now many people like to think of division as the reverse of multiplication so just to get us going let me do a quick multiplication problem and start drawing pictures for that and then see if we can start undoing those pictures for the fun of it but here goes let me try uh 1 302 times 3 why not um actually see what the house is going to be to that one that's actually going to be 3906 okay grand but what's the picture to go with this well i just said a lot of words there 1 302 i'm speaking base 10 so i can represent this number in a place value chart where we have the ones the tens and the hundreds the ones the tens the hundreds thousands i guess i need those i want to go further ten thousands all the rest now people who know me know i've called that a ten one machine but it's just really a place value chart and when i say number like 1302 i'm literally saying one of these one thousand three hundred three of these three hundreds i didn't mention any tens and two two units two ones great and then this multiplication problem being asked please do that to it please triple everything please triple what you've got oh i'm going to literally do that i'm going to triple what i see there's one dot i'm being asked to triple i'm going to triple the dot and make it three dots there's another dot i'm going to triple it and make it three dots there's a dot i have a triplet make it three dots there's another triple and make it three dots there's a yet another dot i'm gonna triple it make it three another dot triplet make it three so now i've got three dots in the thousands place nine dots in the hundreds place none of the tens place and six in the ones place oh i see the number three thousand nine hundred and six there it is i literally see it all right so now can we actually take this process it's already triple don't need that word anymore it's done can i take that process and do it backwards suppose you didn't see the question all you saw was the answer could i undo this now let me do it over here let me draw the answer let me draw the answer 3906 here it is again it's a three dots nine dots and no dots and six dots here they are and let me see if i can undo that multiplication that can be expressed as this problem three thousand nine hundred six divided by three what got tripled to give the answer three thousand nine hundred six so i'm doing like the reverse multiplication thinking here all right well actually that kind of thinking is great for this problem because look i'm asking what got tripled to give me this answer and i look at the picture and say oh well hang on there is a group of three dots there must have been one dot there that got tripled in fact there's a group of three dots there must be one dot there that got tripled and this must have been a dot that got tripled another one there and that's a dot that must be tripled another one there no knots there got tripled but a dot here got tripled and another dot there got tripled so actually you're gonna see that one dot of the thousands level and three dots at the hundreds level and two dollars on the us level got tripled i can see oh it must have been one three zero two they got tripled aha so i kind of undid this i love it that's division now of course a lot of people like to think of division as groups all like how many threes to go into 3906 and i'm actually seeing that too i'm seeing one group of three at the thousands level i'm seeing three groups of three at the hundreds level and two groups of three at the at the units level i'm seeing 1302 groups of three in that picture so in the groups of thinking i'm also seeing it there wow this is actually lovely this is visual this is beautiful this feels like conceptual depth to me um so let's just practice this some more and let's do another example uh i'll keep it straight forward right now let's do something like uh i know 426 divided by 2. what got doubled to give the answer 426. the answer is 213.
i can see that but oh in my mind i can see that i want to literally now see it with my eyes can we actually see the answer 213. what got doubled to make 4 and two and six so now i'm looking for doubles what got doubled well here's a dot that got doubled one there here's a dot they got doubled one there here's a dollar good doubled one there here's a dot that got doubled another one and another one three dots there got tripled indeed two at the hundreds level one of the tens level three at the us level got doubled to give that picture of 426. beautiful now there can be a snafu and let's see a snafu in an example like uh this one 416 divided by four again i can tell what the answer is going to be it's gonna be 104 okay but let's see if we can see what's going on so this time it's 416 400 and 126.
since english is weird we don't actually say 110 and 6. we say 16 all of a sudden curious and now we're looking for groups of four i'll just be really clear four is literally four dots in a box and i see that in only a couple of places there's a dot that got quadrupled one at the hundreds level and there's a dot here that got quadrupled one of the units level and that's all i'm actually seeing right now and it's got a little bit nervous i've got a queasy feeling in my tummy tummy surely this should be more in there but i know the answer is meant to be 104 not 101. what's missing what can i do because i've got this random stuff left over it feels incomplete it's all jarry this is the lovely thing about math class because you've got these human emotional reactions to these things and they're legitimate have a human experience with maths and they seem to take a deep breath and do something what clever thing could we do to push this forward some more and then you realize how do how does place value work well we've always said that ten ones make ten and ten tens make a hundred and ten hundreds make a thousand the reverse would be one thousand is really ten of these and one hundred is really ten of these and one of these one ten is really ten of these 10 ones let's use that to our advantage this dot here is really 10 ones let me draw in 10 ones the reason i'm doing that because look at this it's showing me loads more groups of four that was 10 of them yes 10 of them great 10 ones because look there's another group of four there's a dot that must have been quadrupled there's a dot that must have been quadrupled and there is a dot that must have been quadrupled and indeed that's all the dots and i see the answer is one zero four is what got quadrupled whoa so this idea of undoing a dot i call it unexploding in my exploding dot story i think the curriculum calls it boring or regrouping or ungrouping some words like this whatever it is the visual is make make 110 back to being 10 ones to see more groups very very handy in fact it's particularly handy when it comes to doing multi-digit division so let's go up a notch in difficulty let's do something like 276 divided by 12.
so start by drawing a picture of 276. two hundreds seven tens and uh six beautiful and twelve 12 is going to be sneaky because technically 12 should be 12 dots in a box except society doesn't allow me to have 12 lots in a box they always want me to regroup tens so this is actually oh think about this 12 in a box there's a group of 10 10 ones really makes 110 and leave 2 behind that's annoying all 12 dots are really there but carrying pushes one over to the left or in my language exploding pushes one over the left so to keep in mind what 12 looks like is one dot next to two dots but all 12 dots are really in the right past most part of that picture so 12 do we see the 12s and 276 what got multiplied by 12 to get this picture so we're looking for one dot next to two dots and do i see any one dot next to two dots why yes there's one dot next to two dots right there now let's be really clear in that pink loop where are those 12 dots really they all must have been here on the right part of the loop and then kaboom 10 of them explode to make the next one one place over so all 12 dots were really there at that level so there was one dot there that got multiplied by 12 and spillage pushed one over to the left in fact there's another one dot next to two dots and again all 12 dots must be in the right part of that loop and boom spilled over so there's another dot that dot there that got multiplied by 12. any other one dots next to two dots why yes if you shift over one spot to here there must have been tall loss there kaboom spilled over there must be one group of twelve there and again there's one underneath that one dot next to two dots right there and actually my lips are getting weird but there's another one dot next to two dots right there all 12 dots has been here and kabooms spilled over so actually i see the answer is two of the tens level three at the ones level the answer is 23 which i have to know is correct wow i'm seeing multi-digit division this is so fun let's keep going um let's try multi multi-digit division let's try something crazy like uh four four seven three divided by two one three whoa whoa i'm into it i'm gonna do it let's do it this is four four four thousand four hundred seventy t y means ten in english seven tens and three now i'm looking for 213s now 213 dots in a box never happens in math class and a place value charts give me lots of tens to spill over and spill over spillover actually what 213 really looks like is going to be two hundreds one ten and three look like two dots one dot three dots with all dots really in the right most part of the loop they're right most part of the diagram they must have all been there but an awful lot of carrying or exploding or grouping regrouping happened right from there all right so i'm looking for two one three two dots next to one dot next to three dots i'm spanning over three boxes this time two one three two dots and one dot that's all i need from that box and three dots how's that how's that for a crazy loop two one three 213 dots must be right there and a lot of explosions or group groupings and carryings happen to spill it over to the left bingo and the other two one threes why yes two one and i'm gonna do this and i can do three like that two one three beautiful another one two to three dots must be here and all the spilling over to the left but them all must have been there in the other two one threes oh yes uh my picture's getting crazy but two there and one there and three there all 23 dots supposed to mean here and lots of explosions pulling over to the left one there the answer is 21.
beautiful two at the tens level one of the ones level we can do multi-multi-digital divisions now 21. in fact as practice if you like 4473 divided by 21 well logic tells me you must have the answer 213. you want to try that once can you get 213 from that actually this is actually really interesting um a lot of people ask me right now what if i change the picture to uh oh i just erased what i did it was i just did four four seven three divided by 213 one of it was say four four seven five divided by two hundred thirteen four four seven one two three four five you say oh as you do this you'll do all the same work but i have two dots left over so you say the answer is oh 21 and depending what grade level you're in might just say with a remainder of five a rate of two certainly two dots left over uh some people use a bigger some people use a small r um mathematicians might say is literally 21 plus two more dots still waiting to be divided by 213.
it'd be 21 plus 2 213 bingo beautiful or actually if i made this 4 5 7 5 4 5 7 5 then you'd say the answer is 213 is 21 still with a remainder of oh one dot zero dots two dots remainder of 102 or the math language 21 plus 102 still waiting to be divided by 213. so if there are remainders you will literally see them they'll be right there before your very eyes beautiful stunning grand so i put a couple of practice problems at the end of this video have fun practicing this stuff it's just great if you want to learn more go check out the exploding dots story at globalmathproject.
