g'day i'm james over 20 years ago then uk minister for school standard stephen buyers was asked in a live interview quickly what's seven times eight now that's a lot of pressure on a live interview out of the blue and he mistakenly said oh seven times uh that's 54. and then became the laughing stock of the media at that time the stats mercy for school standards did not know what seven times eight was everyone's expecting to say cold please give the answer 56 which says a lot about what society expects of mathematics that mathematics all about computation speed computation and being correct the first time to me that doesn't excite me that's not mathematics that's that's speed computation okay i can imagine it could be exciting and fun to do and no but it's not actually mathematics phase he could have asked this question different ways um he may have said oh seven times eight that's the tricky one so he's giving him an emotional response he's being a human being he said something like that and he might pull out something like um oh i remember my teacher telling me that think of five six seven eight um within those four consecutive numbers i've got seven times eight equals 56. okay that's cute um that's kind of a strange little coincidence about how numbers happen to work for seven times eight but it gives me nothing to hold on to for other products that's one down and what a lot more to go what would have really impressed me if he answered this way seven times eight oh my goodness i'm under pressure so um that's always the tricky one everyone gets that one wrong um okay seven times eight sometimes eight but what do i know i do know that seven times seven is forty-nine for some reason i've got the square numbers in my head so seven times eight seven groups of as eight groups of seven must be 49 with an extra seven uh that's uh that's that's 56.
if he went that sort of process i would have been incredibly impressed he was he would demonstrate to the world right then it's okay not to know answers but then you can find out you can do something about you can problem solve and that was beautiful thinking about structure so so if we're working with our students and we expect them to memorize this table and know it called can we at least make it a conversation about structure now because there's other ways to get these things in here you can play fabulous dice games and card games and so forth but at some point we often get to the point uh students need to just have flash cards and get them in their heads all right then please make it a matter conversation let's be very human if our task is to memorize this table please make it about mathematics about thinking your way through a challenge and what can i do to make my life easier because our challenge is right now to memorize everything in this table and right now i have an emotional response to that because that feels like a lot of stuff to get in my head that feels overwhelming in fact i might ask the natural question is how many products do i need to get in my head you look at it and say okay well it's uh what 10 columns and 10 rows it must be 10 times 10 it must be 100 products i have to memorize whoa whoa except except i just did one of them i just said that 10 times 10 was 100. so i know one of them actually there's only 99 things i have to memorize that's not much better not much better and then you look at that okay so i have to memorize 99 things apparently i have 10 times 10 already in my head that's great so how i'm going to memorize the 99 elements in this table that i already have and you notice things about this table let me stare at it for a while there's actually a lot of structure i mean it is the multiplication table there's a lot of structure in the numbers that example you do see that 5 times 4 20 and 5 times 4 this way like row by column or column by row are the same answer uh 5 times 4 is the same as 4 times five uh eight times two eight times two is the same as two times eight or here's two times eight two times eight so actually there's a lot of repeats going on which is great because if everything down here is repeating anything up here you kind of see that uh this lower triangular half matches this upper triangular half so actually that means i have to memorize half of a hundred things or half of 99 things way better oh except i paused half is it half well if you think about it okay you've got these numbers down here which are like one by one two by two three by three four by four five by five on this diagonal so they're all the square numbers ah it's actually the things below that diagonal all these numbers down here match all the numbers up there so it's not quite half oh so how many numbers do i have to memorize now well it's everything in this upper triangular and this diagonal so how many numbers is that okay now i wrote myself in that little puzzle um how many numbers is that well okay so i know there's ten numbers here on this diagonal one squared two squared three squared up to ten squared there's ten numbers which means oh means those two things are 19 numbers together they're the same as those that each of these must be half of 90. so i've now got 45 numbers plus these 10 numbers to memorize i now realize i only have to memorize 55 numbers that's better than 99 or 100.
but i've already got that one that's really 54 numbers actually a lot of people have the square numbers people are fascinated by the square numbers a lot of people have one times one in their head two times two in the head three times three in the head four times four in the head five times five and the head is very common um these upper ones get a bit uh scary i mean six times six is 36. actually i had that one in my head because 36 happens to me by to be my favorite number if everyone has if you have a favorite number why well think about six times six if i've got six rows of six dots that's literally what i mean by square number because they're make take that many pebbles you can make a perfect square out of it is that six rows of six yeah that's 36. so 36 is a squared number it's also a triangular number because look at this one dot then two dots then three dots then four dots then five dots then six dots then seven dots then eight dots if you count up all the dots i just did it's also 36 it's also what mathematicians call a triangular number so it's both square and triangular i like to call it in fact it's the first squangular number i mean i guess one is both square and triangular it's the first interesting scrangular number i love 36 is square angular okay so some reason 36 sticks in my head because it's scrangula by the way now you kind of want to know what's the next square angular number i'll give it away 1225 is the next square angular number these things are rare what's the one after that oh okay that's a different video that's a different video so i had to have i didn't have 36 in my head six times six is 36 because i like the square numbers um a lot of people have 7 times 7 in their heads it's 49 i guess 8 times 8 and 9 times 9 they're always tricky 8 times 8 and 9 times 9 so maybe i won't have those ones in my head that means okay so maybe maybe i shouldn't circle those well i do have 10 10 in my head so i've got in my head right now everything here circled and i don't need all that so actually there's less in the table i need right now what's that of six times eight over 64 81 okay bingo so i've got all i have to memorize now made it much less that stuff and this stuff um i actually have another question is it obvious that 5 times 4 should be the same as four times five and eight times two should be the same as two times eight is it really obvious that five times four and four times five are symmetrical oh my gosh before i even get to my work here i'm actually all these cool questions five times four that's five groups of four think about five groups of four that's a group of four there's a group of four here's a group of four here's a group of four and here's a group of four five groups of four as opposed to this which is different it's four groups of five one two three four five one two three four five one two three four five one two three four five they look like very different pictures actually why should five times four be four times five i know they're both twenty i know they are the same but philosophically why huh okay well i'm just full of distractions here on this little little game well the thing is that people realized if you actually draw your pictures in a more systematic way for five groups of four do it very systematically there's one group of four a second group of four third group of four a fourth group of four fifth group of four there's five groups of four when i look this way and then you realize oh if you turn your head around 90 degrees and let's say from there one group of five two groups of five three groups of five four groups of five the same picture turned around is four groups of five look at this way you get four five groups of four look this way you get four groups of five it's the same picture the answers have to be the same and i only have to say the word 20 to know they have to be the same because it's going to be the same picture by the same token whatever 103 times 107 is i know i could do 103 rows of 170 columns look this way look this way i'll get the same answer how many seven times 103 whatever that turns out to be so without even knowing what the answer is i could argue that ah the bottom half of the multiplication table matches the top half all right so where am i where am i how many things i have to remember i still haven't memorized anything yet i know the square numbers i know except for those two and i know that one um okay i bet i know a lot of stuff everyone knows they're one times tables so i need to memorize those i've got all those known all right so that's easy i don't have to memorize that much um actually everyone seems to know that ten times tables uh one times ten two times three times this people know that so actually i didn't memorize this much as way less have to memorize what else what else do i have to memorize um well a lot of people actually do know that two times tables um so actually i've got those in my head or my two times tables actually there's less i have to memorize wow this is great um what else do people tend to know well here's what i'd do with the student at some point i'd circle all the ones that you kind of know a lot of people some reason seem to know that three times five is fifteen that just seems to stick in people's heads actually three times four is twelve six and people's heads heads a lot actually a lot of the three times table sticks in people's heads so that's usually usually known for a lot of people what else do people know uh four times five is twenty that at that little talk just then about why four times five times five times four i've definitely got twenty in my head uh four times six people get stuck on that one four times seven people get stuck on this one um a lot of people do know that four times eight is for 32 for some reason but maybe one of those are not one of those people that's all fine in fact go through with students which ones do you know you just haven't had in your head for whatever reason and it could be maybe you know seven times nine and sixty-three so right now it looks like for me what's left me to memorize is one two three four five six seven eight nine 10 11 12 13 14 things 14 things i went down from 100 to 14.
that now seems more tractable but let's still make it a meta conversation for example most everyone on this planet gets that one wrong that is i think they even did studies this is like the universal product that goes wrong people don't have seven times eight in their heads in which case then you ask yourself why is seven times eight so hard why do i have 56 in my head and they say okay i don't know it's probably an emotional block or it just seems awkward it just feels clunky and then you say well what could i do to figure it out under pressure and you say to myself okay let me have a little thought process about seven times eight and you do what i wish stephen buyers did you say well okay seven groups of eight oh seven groups of eight that's hard seven groups of eight no no let's think of this as eight groups of seven because eight groups of seven is very close to seven groups of seven which i don't have in my head is 49 and once i know that seven groups of seven is 49 then eight groups of seven would be 49 plus an extra seven and i have to do a little bit of thinking there but that's going to be 56.
