"8*7 is 56 because 7-6=1 and 50-1 is 49 and everyone knows 49 is a perfect square, and it must be the square of 7 since 8 is 64 squared and 5 is 25"
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@futurebird this kind of thing is great if you want to find simple roots of polynomials in your head, but not great if you just want to multiply.
@futurebird I've been thinking about this for an hour now, could you explain what the logic here is? I can see how it works as a "soft proof" that 56 is correct once you have it, but how does it work forwards?
If you already know 7²=49, why not just add 7, where does the 1 come from?
If you want to subtract 7 from 56 you can think of 7 = 1 + 6
Then subtract the 6 first then the 1.
@futurebird Sure, but for me that's what I would do to get 56 / 7 = ?, as in "subtract 7 (in parts), ah neat, thats a number I know". How does that help for the multiplication (if I haven't half-remembered 56 and want to know if that could be it)?
@futurebird For some reason "seven eights are fifty-six" in words is one of the ones I just know (maybe I practised it a lot or something). But I think of that as 7×8, so remember 8×7 by "turning it round". (I remember as a child thinking it was silly to learn them all twice, and smaller-first made most sense to me).
Now I'm not sure whether I do "turn it round" or simply see there's an 8 and a 7 and translate them straight into "seven eights", but the words have to be that way round.
@futurebird didnt remember, had to calculate 8*5+8*2 = 40+16
8*7 is the most frequent error elementary students make when learning their times tables.
If you have a younger child, and it causes them no distress have them learn their multiplication tables up to 13. Like learning a language there is a window where this is easier.
It is, however to survive with only the perfect squares (essential) and numbers that have a "trick" to remember them.
@futurebird This is cool, I didn't know this. When I was a kid this was also the one multiplication I struggled with (I was just slower, but always remembered 7x7=49 and 7x7+7=7x8=56).
I asked my students how they remember 8*7 and the above was my favorite answer. Distressing but understandable.
How do you remember 8*7?
@futurebird I have 56 memorised, but covid's made my memory unreliable. So, after a pause, 56 emerged, but felt fuzzy ... so I double checked it by doing 4 * 7 and doubling it!
@futurebird I remember it because I'm Hungarian and my dad told me it was a bit of a thing for a while back when he was in school in the seventies, older kids would come up and ask "7*8?" and you just say 56 and they'd nod and walk away. Or something like that. 56 being a reference to the 1956 revolution.
@futurebird now it’s just something I know. But when I was learning I’d get confused with knowing 6x9 and 7x8 were some combination of 54 and 56. I would disambiguate by recognizing 54 is as a multiple of 9 by the rule that the digits sum to 9.
@futurebird What’s funny is, just now, 7*8 came from memory pretty fast, but then later, when uou said, 8*7, I had to do 8*8-8.
I’m mostly a memorizer for multiplication/division, but with addition/subtraction, I find myself pulling out mental unit sticks, like the ones we used in early elementry school. There’s a 10 stick, a one cube, and a five stick for each decimal place.
@futurebird when I was a kid I just remembered numbers effortlessly. I remember being so far ahead I did the tables upto 23 because the teacher couldn't think of anything else for me to do. Btw, people find this kind of parlor trick annoying I learned 😅
Now that I'm a bit older and my brain is fried I just go from the perfect squares if the number doesn't pop up immediately.
As a kid I stored the numbers in a 9x9x9 3d visual. No idea why I figured that out, but that visual is still hard coded in there somewhere, and it's useful somehow.
Multiplication of numbers >9 doesn't fit neatly in that array, but the answer for 12*x is in the (3,4,x) slot or is just one addition away. Of course the cube can expand beyond 9 if convenient for some repetitive task.
I tried to go 4d, and also tried remembering numbers by prime factors, but my brain never liked that for some reason (probably a few hundred million years of evolution in approximately 3d space screwed that up )
@futurebird 5678 just stuck in my head early
Fascinating discussion. I hadn't thought about how much/little of multiplication tables I still remember.
I'm of the age when endless drilling was the preferred teaching technique. So mostly I still just pop the answer. But, given how little arithmetic I do the tables are certainly fading with age and non-use.
Generally I go to posers of 10, so 8 * 7 becomes
(10 * 7) - (10 - 8 ) * 7 =
70 - 2 * 7 =
70 - 14 =
56
Yes I know this is weird and complicated
@futurebird my elementary school teacher told us it was the hardest to remember, which I then always remembered. I'm more likely to have to think about 6*8. (Which I get because 6*7 is 42 is locked in there, and then I add six)
@futurebird I don't, but I am old enough to have had the multiplication tables driven in until they became innate.
@futurebird I memorised it and 6x7.
@futurebird somebody shouting "EIGHT EIGHTS ARE SIXTY FOUR!" is burned into my brain for some reason, so I then figure out 64-8
@futurebird I learned it by it being an example in the American Girl book "Molly Learns a Lesson," which my grandma bought me because she thought Molly looked like me (brown hair and glasses?) and my deadname sounds kinda like her name.
My God it's so strange how human memory works. An arithmetic fact can be a door to a million memories.
@futurebird my parents taught me to chant "five six seven eight. Fifty- six-is seven-times-eight" until I remembered it
@futurebird I feel fortunate that my grade 6 teacher had us start each day with times table drills. I can remember being very slow at it at the start of the year, and being quite fast by the end.
So I don't reason it. It's just something I memorised.
@futurebird "7*7 is 42, right, so... wait, that's wrong 'cause it's not meant to be an even number, that's 7*6? So 42 + 7 is 49, I think, so then 49 + 7 is... 45, no, 46, NO, 56! Maybe next time I should try 8*8 - 8 but subtraction is harder than addition... also 8*8 is 48 right? uh... no, it's 64 because reasons....wait isn't 8*8 meant to be 10*8 - 2*8 so shouldn't it be 62?? oh wait 8+8 is 16, not 18, so 64 is right"
It's generally something like that.
I really don't get why no one flagged either the dyscalculia or the ADHD before university. And I am very glad calculators exist so I can double check my weird number flipping behaviour.
It's generally something like that.
I really don't get why no one flagged either the dyscalculia or the ADHD before university. And I am very glad calculators exist so I can double check my weird number flipping behaviour.
I have a masters degree in mathematics and can end up in such rabbit holes with arithmetic. The only bit I don't relate to is not getting 7*7 right. It's a perfect square and therefore "Important"
@futurebird also very glad that I can just go "no, forcing me to do calculations in my head is evil, I am going to write it all down and have someone else check if it needs to be right for whatever reason"
@futurebird This was one I memorized. Had a certain color or feel and just put it there. One that came to mind, though was something like what got you to 7 then 8? 5 then 6. Doesn't help the underlying math, or the order it happens in. Sometimes it doesn't have to make sense, I guess
@futurebird
This is one of the few I remember how I learnt as a kid, 56=7*8, just the digit progression for the trickiest one.
@futurebird I was thinking "I don't", then did 7*7=49, 49+7=56, then realised that 56=7*8 is a lot easier to remember (5,6,7,8)
@futurebird I now just know, but for many years it was 7x7+7.
@futurebird a simple
49+7
@futurebird Blink if you need help
@futurebird I love this line of thinking. It's so clearly individual to that person as the obvious path, but still explicable to others in ways the rote "8*7=56 because it just is" explanation could never be