The most challenging aspect to the Trachtenberg method is that in researching this topic, I have seen the method explained at least in three uniquely, different ways. Thus, this suggests that either there is a misunderstanding about the method itself, or that the way that people learn about it or how they're being taught about it, is inconsistent.
thank you for your comments. I did my best to understand this method exactly, but it didn't seem to be enough. furthermore I had to adjust the way during making this video. I mentioned about this point in the video. sorry for that😪
Trachtenberg is an interesting method but... way too long for this kind of simple calculation problem. Since the age of 10 I do the mental calculation like this, keeping the most possible rounded numbers: 57 x 135 = 60 x 135 - 3 x 135 60 x 135 = (6 x 100 + 6 x 30 + 6 x 5) x 10 = (and I hope everybody has it) (600 + 180 + 30) x 10 = 8100 3 x 135 = 3 x 100 + 3 x 30 + 3 x 5 = 300 + 30 + 15 = 405 or 3 x 135 = 3 x 130 + 3 x 5 = 390 + 15 = 405 (just because I love multiplying 13) 8110 - 405 = 8100 - 100 - 300 + 10 - 5 = 8000 - 300 - 5 = 7700 - 5 = 7695 This one is quite simple so I only need 4 steps to solve the problem but the trick resides in always decomposing the operands. Believe me or not, I'm very fast 😉
whenever I see people who is good at mental calculation, I have no choice but to show repects. thank you for showing your process and comment! good day!
The problem is, Mary did NOT use the Trachtenberg System. Her uncle told the teacher that so she wouldn’t learn the truth about Mary. Mary had inherited her mathematical genius from her mother, the uncle’s sister. Having seen how the burden of her advanced mathematical abilities ultimately led to her suicide, the brother wanted a different life for his niece. So, when Mary’s teacher asked him about Mary’s abilities, rather than admit that Mary was a super genius who could multiply in her head faster than a calculator, he lied and said that Mary had learned an old, obscure math-trick system called Trachtenberg.
it makes sense. I was convinced. in fact, I haven't thought like that. is that your excellent guess? or is there any mention about that in the movie? thanks!
@@math_travel It's been a couple of years since I've seen the movie, and it wasn't on any streaming services I have access to, so I borrowed it from the library. Below are relevant dialog snippets: --As Frank and Mary are leaving school Bonnie: Excuse me? Hi. Mary: Oh. Look, it's my teacher. Probably wants to remind me what one plus one is. Frank: Go to the car, OK? --Mary gets in the truck Bonnie: I think ... I think your daughter ... I think Mary might be gifted. Frank: What? Bonnie: Yeah. Today in Math she answered some really ... Frank: Oh, no, no, no. That's ... Bonnie: No, please. Frank: It's not gifted. Bonnie: Really difficult questions. Frank: All right. Bonnie: Just that a seven-year-old would ... Frank: It's Trachtenberg. Jakow Trachtenberg. Bonnie: I'm sorry? Frank: Spent seven years in a Nazi concentration camp. Developed a system to rapidly solve problems. It's the Trachtenberg method. Bonnie: But she's ... I mean, she's seven though. Frank: I learned it when I was eight. Do I look gifted to you? It's kind of gone out of vogue since the invention of the calculator ... but, uh, I can still win a drink at a bar using it. --Mary is with Frank as he is fixing a boat Mary: For the record, I didn't want to go to the stupid school in the first place. And the boy in the next row acts inappropriately for someone who's a child. Frank: I'm sorry. I'm still passive-aggressively ignoring you. --They discuss the day a bit Mary: Frank, I'm sorry. I'm really sorry. Frank: Yeah, right. You can't show off like that at school. Mary: I know. Frank: You promised you wouldn't, then on the first day ... Mary: I know, I know. I screwed up. --That Friday, after finding info on the Adlers on Google, Bonnie confronts Frank at the bar Bonnie: You lied to me. Frank: Okay. Can you narrow it down? Bonnie: Mmm-hmm. I'd like to know the truth about Mary, my student ... your niece. The overall vibe is like Jonathan Kent telling Clark to try to blend in at school, and don't play football, and don't go out for track, etc. Frank knows Mary isn't normal. Mary knows she isn't normal. But Frank wanted Mary's life to be as normal as possible. He'd seen what this sort of "gift" had done to his sister and wanted something better for his niece. Frank himself WAS gifted - a philosophy professor with a knack for mechanics. He may not have had his sister's talents, especially in mathematics, but he was no intellectual slouch. HE probably DID learn Trachtenberg at eight. (I looked at it - it's not that easy.) But it seemed obvious to me that "Trachtenberg" was just a clumsy ruse to hide the fact that Mary was a math genius capable of doing college-level work.
Here's a simpler way of doing the calculation, based on understanding that multiplication is actually repeated addition. Fifty-seven times one hundred thirty-five means that 135 is used as an addend (a number that's added) 57 times. We want to change this problem so that we're using simpler numbers. Let's change the 57 to 60. In other words, change the 57 to 57 + 3 = 60 . Multiply 135 by 6: 100 x 6 = 600, 30 x 6 = 180 5 x 6 = 30 and add: 600 + 180 + 30 = 600 + 210 = 810 then multiply by 10 (this is because we're originally multiplying by 60) 810 x 10 = 8100 Now, since we added 3 to 57 to equal 60, we need to subtract three quantities of 135 from that 8100. In other words, 8100 = (57 + 3) x 135 = (57 x 135) + (3 x 135). So we need to subtract 3 x 135 = 405 from 8100 to give us 7695.
Funny, I just realized that 8100 is also the square of 90. (When thinking of of thr hell she could approximate 87,7 that easily) Personally, I try to come up as fast as possible with the most easily usable pieces of informations, for example here I'll need: 135x2 135/2 135x50 135x7 135x57 135*2 = 270 135/2 = 140/2 -5/2 = 70 -2.5 = 67.5 135*50 = 6 750 135*(7= 2+5) = 270+675 = 945 = 1k -55 135*57 = 1k + 6k+750 -55 = 7 695
For a mental math problem, more likely she recognized that 57 = 3 * 19, and 135 * 3 = 405. Then 57 * 135 reduces to 405 * 19, a much simpler mental math problem.
I guess you are good at mental calculation. I have no idea about that. so it seems to me that there is not a big difference between 57*135 and 405*19. sorry. I want you to know that there could be a possibility not to be correct im my explantions. it was not easy for me to understand exactly the Trachtchenberg system. thanks😁
@@math_travel Others may feel differently, but for me it's necessary to reduce carrying in order to reduce memory load. Whereas calculations are easy on their own. Perhaps if I practiced the Trachtenberg system I'd change my mind - but first impression is that it is making the opposite trade-off to what works for me.
@@math_travelI wish I was more trained to come up with refactorizations with actual multiplications, instead of additions, but now that it's there I can easily explain. 57 is 60-3 and it sounds cool because 60 and 3 are likely to have a very good relation, and it's called 20. So now that we know we're using 1x135 x 3x19 factors, we can simply reform them into, 135x3 x 19 to get 405x19. And this also looks very easy because of the 0 and the easier digits. (any multiplication table is more convenient than 7 isn't it 😅) Those two numbers 19 and 405 are very close to be round, 2x10 -1 and 4x10 +5, so the small differences won't cause too much burden. We can simply start with 405*20 = 8100 And then there's a single one to remove to get 19*405 so 8100 -(400+5) = 7700 -5 = 7695 !
@@Aztonio you used the fact that 135 is factored into 3*19. and calculate 135*3, and then 405*19. you calculated 405*19=405*20-405. right? thank you for sharing your smart method. 👌
Personally, I try to come up as fast as possible with the most easily usable pieces of informations, for example here I'll need: 135x2 135/2 135x50 135x7 135x57 135*2 = 200 + 60 + 10 = 270 135/2 = 140/2 -5/2 = 70 -2.5 = 67.5 135*(50= 100 /2) = 100*67.5 = 6k +750 ; k=1000 135*(7= 2+5) = 270+675 = 945 = 1k -55 135*57 = 1k + 6k+750 -55 = 7 695
My Grandfather could speak read and write 7 languages fluently ( English ,French ,Italian,Russian,German ,Polish and Latin) and could do complex engineering calculations in his head to 6 decimal places.He was a metallurgist in france before ww2. He could do these calculation in his head faster than I could do them with a calculator....I suspect he might have been a polymath...very obnoxious and short tempered but could do anything with his hands...couldn't buy a piston for an engine? he cast his own and machined it made the rings gudgeon pin etc etc...
your grandfather amazes me like you were amazed. I am sorry that you couldn't calculate faster than him even with the help of calculator. it's funny. I am wondering how much you are able like him. thanks for your comments!!!
@@math_travel He thought I was the dumbest member of the family...and I have a degree in mining engineering!!! He used to give me engineering problems while he was laying in bed in hospital dying from Prostate cancer...He truly was unique ..and he fought in the free french army then later in the underground..He started my mechanical education when I was 10 .Working on motorcycles clock mechanisms .At eleven I could set the valves set the timing float heights spark plug gaps on halve a dozen different engines
@@mottthehoople693 your grandfather and you were unique. I think you should thank for the relationship with your grandfather.I appreciate you share your story. thanks
The most challenging aspect to the Trachtenberg method is that in researching this topic, I have seen the method explained at least in three uniquely, different ways.
Thus, this suggests that either there is a misunderstanding about the method itself, or that the way that people learn about it or how they're being taught about it, is inconsistent.
thank you for your comments. I did my best to understand this method exactly, but it didn't seem to be enough. furthermore I had to adjust the way during making this video. I mentioned about this point in the video. sorry for that😪
@@math_travel
No worries.
It's all good.
Thank you for your video and the explanation of the Trachtenberg method.
I appreciate it.
True, it's very wonky, as presented. Quite forgettable.
How did Mary calculate 57×135 so quickly? She read the script.
I do hope so! thanks😆
@@math_travel57*135 would basically be the number 7695 via my calculation.
@@byronrobbins8834 congratulations and thaks
Trachtenberg is an interesting method but... way too long for this kind of simple calculation problem.
Since the age of 10 I do the mental calculation like this, keeping the most possible rounded numbers:
57 x 135 = 60 x 135 - 3 x 135
60 x 135 = (6 x 100 + 6 x 30 + 6 x 5) x 10 = (and I hope everybody has it) (600 + 180 + 30) x 10 = 8100
3 x 135 = 3 x 100 + 3 x 30 + 3 x 5 = 300 + 30 + 15 = 405
or 3 x 135 = 3 x 130 + 3 x 5 = 390 + 15 = 405 (just because I love multiplying 13)
8110 - 405 = 8100 - 100 - 300 + 10 - 5 = 8000 - 300 - 5 = 7700 - 5 = 7695
This one is quite simple so I only need 4 steps to solve the problem but the trick resides in always decomposing the operands.
Believe me or not, I'm very fast 😉
whenever I see people who is good at mental calculation, I have no choice but to show repects. thank you for showing your process and comment! good day!
I love reading all the comments on here. Very educational.
I am happy to hear that. let’s enjoy math-travel. thanks😏
The problem is, Mary did NOT use the Trachtenberg System. Her uncle told the teacher that so she wouldn’t learn the truth about Mary. Mary had inherited her mathematical genius from her mother, the uncle’s sister. Having seen how the burden of her advanced mathematical abilities ultimately led to her suicide, the brother wanted a different life for his niece. So, when Mary’s teacher asked him about Mary’s abilities, rather than admit that Mary was a super genius who could multiply in her head faster than a calculator, he lied and said that Mary had learned an old, obscure math-trick system called Trachtenberg.
it makes sense. I was convinced. in fact, I haven't thought like that. is that your excellent guess? or is there any mention about that in the movie? thanks!
@@math_travel It's been a couple of years since I've seen the movie, and it wasn't on any streaming services I have access to, so I borrowed it from the library.
Below are relevant dialog snippets:
--As Frank and Mary are leaving school
Bonnie: Excuse me? Hi.
Mary: Oh. Look, it's my teacher. Probably wants to remind me what one plus one is.
Frank: Go to the car, OK?
--Mary gets in the truck
Bonnie: I think ... I think your daughter ... I think Mary might be gifted.
Frank: What?
Bonnie: Yeah. Today in Math she answered some really ...
Frank: Oh, no, no, no. That's ...
Bonnie: No, please.
Frank: It's not gifted.
Bonnie: Really difficult questions.
Frank: All right.
Bonnie: Just that a seven-year-old would ...
Frank: It's Trachtenberg. Jakow Trachtenberg.
Bonnie: I'm sorry?
Frank: Spent seven years in a Nazi concentration camp. Developed a system to rapidly solve problems. It's the Trachtenberg method.
Bonnie: But she's ... I mean, she's seven though.
Frank: I learned it when I was eight. Do I look gifted to you? It's kind of gone out of vogue since the invention of the calculator ... but, uh, I can still win a drink at a bar using it.
--Mary is with Frank as he is fixing a boat
Mary: For the record, I didn't want to go to the stupid school in the first place. And the boy in the next row acts inappropriately for someone who's a child.
Frank: I'm sorry. I'm still passive-aggressively ignoring you.
--They discuss the day a bit
Mary: Frank, I'm sorry. I'm really sorry.
Frank: Yeah, right. You can't show off like that at school.
Mary: I know.
Frank: You promised you wouldn't, then on the first day ...
Mary: I know, I know. I screwed up.
--That Friday, after finding info on the Adlers on Google, Bonnie confronts Frank at the bar
Bonnie: You lied to me.
Frank: Okay. Can you narrow it down?
Bonnie: Mmm-hmm. I'd like to know the truth about Mary, my student ... your niece.
The overall vibe is like Jonathan Kent telling Clark to try to blend in at school, and don't play football, and don't go out for track, etc. Frank knows Mary isn't normal. Mary knows she isn't normal. But Frank wanted Mary's life to be as normal as possible. He'd seen what this sort of "gift" had done to his sister and wanted something better for his niece.
Frank himself WAS gifted - a philosophy professor with a knack for mechanics. He may not have had his sister's talents, especially in mathematics, but he was no intellectual slouch. HE probably DID learn Trachtenberg at eight. (I looked at it - it's not that easy.)
But it seemed obvious to me that "Trachtenberg" was just a clumsy ruse to hide the fact that Mary was a math genius capable of doing college-level work.
@@garywagner2616 I really appreciated your comment. after I read, I was sure what you said is right.
Here's a simpler way of doing the calculation, based on understanding that multiplication is actually repeated addition.
Fifty-seven times one hundred thirty-five means that 135 is used as an addend (a number that's added) 57 times. We want to change this problem so that we're using simpler numbers. Let's change the 57 to 60. In other words, change the 57 to 57 + 3 = 60 .
Multiply 135 by 6:
100 x 6 = 600,
30 x 6 = 180
5 x 6 = 30
and add:
600 + 180 + 30 = 600 + 210 = 810
then multiply by 10 (this is because we're originally multiplying by 60)
810 x 10 = 8100
Now, since we added 3 to 57 to equal 60, we need to subtract three quantities of 135 from that 8100. In other words, 8100 = (57 + 3) x 135 = (57 x 135) + (3 x 135). So we need to subtract 3 x 135 = 405 from 8100 to give us 7695.
thank you for sharing. it is simpler way!
Funny, I just realized that 8100 is also the square of 90. (When thinking of of thr hell she could approximate 87,7 that easily)
Personally, I try to come up as fast as possible with the most easily usable pieces of informations, for example here I'll need:
135x2
135/2
135x50
135x7
135x57
135*2 = 270
135/2 = 140/2 -5/2 = 70 -2.5 = 67.5
135*50 = 6 750
135*(7= 2+5) = 270+675 = 945 = 1k -55
135*57 = 1k + 6k+750 -55 = 7 695
@@Aztonio I see😄
or (60 x 135) - (3*135) = 7695. She does not need anything but 2 grade division.
For a mental math problem, more likely she recognized that 57 = 3 * 19, and 135 * 3 = 405.
Then 57 * 135 reduces to 405 * 19, a much simpler mental math problem.
I guess you are good at mental calculation. I have no idea about that. so it seems to me that there is not a big difference between 57*135 and 405*19. sorry. I want you to know that there could be a possibility not to be correct im my explantions. it was not easy for me to understand exactly the Trachtchenberg system. thanks😁
@@math_travel Others may feel differently, but for me it's necessary to reduce carrying in order to reduce memory load. Whereas calculations are easy on their own.
Perhaps if I practiced the Trachtenberg system I'd change my mind - but first impression is that it is making the opposite trade-off to what works for me.
@@pietergeerkens6324 'to reduce carrying~~' it is a interesting mention.it gives me some feeling about your way. thanks!
@@math_travelI wish I was more trained to come up with refactorizations with actual multiplications, instead of additions, but now that it's there I can easily explain.
57 is 60-3 and it sounds cool because 60 and 3 are likely to have a very good relation, and it's called 20.
So now that we know we're using 1x135 x 3x19 factors, we can simply reform them into, 135x3 x 19 to get 405x19. And this also looks very easy because of the 0 and the easier digits. (any multiplication table is more convenient than 7 isn't it 😅)
Those two numbers 19 and 405 are very close to be round, 2x10 -1 and 4x10 +5, so the small differences won't cause too much burden.
We can simply start with 405*20 = 8100
And then there's a single one to remove to get 19*405 so 8100 -(400+5) = 7700 -5 = 7695 !
@@Aztonio you used the fact that 135 is factored into 3*19. and calculate 135*3, and then 405*19. you calculated 405*19=405*20-405. right? thank you for sharing your smart method. 👌
Reminds me of the writing method called chiasm. Typically seen a lot in biblical writing .
I'm sorry I have no idea about chiasm. thanks for a comment. good day😁
my school classic way was faster
you were so lucky. you just follow that way. thanks😁
Personally, I try to come up as fast as possible with the most easily usable pieces of informations, for example here I'll need:
135x2
135/2
135x50
135x7
135x57
135*2 = 200 + 60 + 10
= 270
135/2 = 140/2 -5/2 = 70 -2.5
= 67.5
135*(50= 100 /2) = 100*67.5
= 6k +750 ; k=1000
135*(7= 2+5) = 270+675 = 945
= 1k -55
135*57 = 1k + 6k+750 -55
= 7 695
My Grandfather could speak read and write 7 languages fluently ( English ,French ,Italian,Russian,German ,Polish and Latin) and could do complex engineering calculations in his head to 6 decimal places.He was a metallurgist in france before ww2. He could do these calculation in his head faster than I could do them with a calculator....I suspect he might have been a polymath...very obnoxious and short tempered but could do anything with his hands...couldn't buy a piston for an engine? he cast his own and machined it made the rings gudgeon pin etc etc...
your grandfather amazes me like you were amazed. I am sorry that you couldn't calculate faster than him even with the help of calculator. it's funny. I am wondering how much you are able like him. thanks for your comments!!!
@@math_travel He thought I was the dumbest member of the family...and I have a degree in mining engineering!!! He used to give me engineering problems while he was laying in bed in hospital dying from Prostate cancer...He truly was unique ..and he fought in the free french army then later in the underground..He started my mechanical education when I was 10 .Working on motorcycles clock mechanisms .At eleven I could set the valves set the timing float heights spark plug gaps on halve a dozen different engines
@@mottthehoople693 your grandfather and you were unique. I think you should thank for the relationship with your grandfather.I appreciate you share your story. thanks
@@math_travel thank you for your kind words