back in my school days in australia i was taught a very strange algorithm for doing long division it looked like this suppose i want to do 276 divided by 12. well i was first required to draw a funny symbol that looked like that i had to write 276 underneath the symbol and had to write 12 to the left of the symbol very very strange now i'll keep explaining what i had to do there but let me just for comparison sake let me draw the dots and boxes version of what we're doing on the right hand side as well so 276 looks like 200 seventy seven tens and six and i'll be looking for groups of twelve which i know with explosions will look like one dot next to two dots okay back to australia so let's see i was first told to look at the very first digit here and ask do i see any twelves in two don't be silly there's no 12's and 2. don't do that what's the equivalent of doing that it's saying like oh james look at those two dots do you see any 12s amongst just those two dots no don't do it not not worth looking at back in australia then say okay 2 didn't work out for you now go 2-7 think of these two numbers together and think of it actually do you think of as 27 which i thought was very strange are there any 12's and 27 are thinking okay 2 12 makes 24 yes 2 12 makes 24 i had to write 2 12's 24 there and that was two of them great all right so what am i really doing there so i'm saying okay look at the two and the seven and i'm asking are there any twelves there do i see any one dot next to two dots in this part of the picture and the answer is yes there's one dot next to two dots right there there's another one dot laser shows there's two 12's right there two at this level in fact i even wrote two at that level great all right then in australia i was required to suddenly do a subtraction problem in the middle of all this which i thought was a bit strange 27 take away 24 is three okay what's that three well look over here okay i say okay i've dealt with those dots i've dealt with those dots i guess i've got three more dots to steal to deal with okay that makes sense there's three dots still to deal with back in australia they need to draw something do something really weird that really confused me as a kid we need to draw a little arrow bring the six down and attach it to the three so this was a three but now it's suddenly 36.
whoa whoa i was asked how many 12's are in 36 um three of them write the three up there and actually write the answer 3 12 is 36. what am i doing there well look at this i've got three dots here and now i'm also bringing my attention to six dots over here so three dots and six dots okay think of three dots and six dots great do i see any twelves there why yes i do there's one dot next to two dots there that's one twelve here's the second 12 here's the third 12. yes i did see 3 12s at that level great all right back in australia i was required to do a little subtraction 36 integral 36 is zero apparently getting 0 is good i never understood why well now i see why that means oh there's no more dots to deal with it tells me i'm done the answer really is 23.
oh the answer really is 23. whoa so i was taught this algorithm with no understanding whatsoever i wish someone would draw me a picture like that because once i see that picture it kind of sticks in my head i don't actually need to draw the picture every single time but now i can see the picture in my mind's eyes i do something strange like this i can see what's going on you've got to love these pictures don't have to draw them every single time but once they're in your brain once they come with you forever and that's fabulous makes perfectly good sense there this is great stuff as a bonus here's another way to do division let's do this example seven five three two divided by two but look i set up like i'm doing like a long addition problem or a lot of subtraction or something looks like a standard algorithm here's how it's going to go i'll show you the method first then i'll explain what's going on with the dots and boxes picture all right seven five three two divided by two first look at the first digit how many times does two go to seven it goes in three times with one remaining now think of that five as fifteen all of a sudden how many times does two go to 15 i'm thinking 14 that goes in seven times with one remaining now think of that three as having been changed to 13. how many times does 2 go to 13 6 times is 12 with 1 remaining and 2 into 12 how many times 212 6 times 7 three two divided by two really is three thousand seven hundred sixty six whoa what'd i just do let me show you what i just did literally show you what i did here's a picture we did seven thousand seven thousand uh 532 and we're looking for groups of two all right i first asked how many times does two go into seven well i see three groups of two there they are and i wrote three groups of two there they are with one left over would i do that one well i made that five become fifteen so i unexploded it one two three four five six seven eight nine ten extra ten dots really did make it fifteen how many groups of two do i see amongst fifteen dots well i see seven of them one two three four five six seven with one left over seven of them with one left over but i'll explode that one and make this three suddenly become thirteen there it is thirteen how many times does two go into thirteen six times i believe one two three four five six with one left over six with one left over why don't we do the extra one i'm gonna bring it over to the two and turn that two into a twelve one two three four five six seven eight nine ten beautiful and how many groups of two do i see amongst twelve six of them one two three four five six beautiful beautiful beautiful i even wrote that final six there it is i can see what's going on that crazy method with the dots and boxes picture so let me just show you the method again on the final two of them and your jobs if you like is to draw the dots and boxes picture that goes with that to explain what i just did and draw the dots boxes picture to go with this one to explain what i'm about to do with that one as well here goes six three two divided by four first digit how many times does four go into six one time and there'll be two left over okay twenty three think twenty three now how many times four go to twenty three i'm thinking five times that's our five fours is twenty with three left over and how many times does four go into 32 eight times six three two divided by four is 158 whoa okay two seven eight three divided by twenty three uh look at the first digit two how many dollars does 23 go into two it doesn't don't bother doing it now i think 27 okay how many 23s are in 27 there's gonna be one with four left over now think of that eight has to change to 48.
how many 23s are in 48 um two 23s is 46 that's close two of them with two left over and how many 23s do i see in 23 1 2 7 8 3 divided by 23 is 121.
