Out of all the comments this one made me the most depressed. 11x12 is something you need to think about?
At my high school we had these expensive graphing calculators that were supposed to last for four years. I of course lost mine in early grade 10. Best thing by far that happened to me.
I was always that the top of my class for mathematics, so I was usually pretty bored with the homework, but all of a sudden I thought of a new challenge: Do all the homework without any calculator whatsoever. Yes, even things like 3^4 + 5^(1/4). At first of course it was painful, but at least I was doing something. I had a much better understanding of actual graphing than anyone else as well, because I would boil the functions down to their key points and play with them until I saw what they did at infinity.
I ended up getting my calculator back in late grade 11 (funny story, I had to sub in for a math teacher at the last minute because there was a wave of sickness at the school, so I was teaching people one year younger than me) and I saw my trusty calculator on the desk in front of me (I had engraved my name with a knife). I picked up and said "hey thanks, I've been looking for this."
Since then I've always been able to do math in my head and it is unbelievably useful. When people say "oh so that is log(800), what is that" and I respond with "roughly 2.9"* they are flabbergasted, but I know how the curves actually work, I haven't outsourced that knowledge to a machine.
Programming is great and we should have graphing calculators in some part of the math curriculum, but students should be able to do this type of math by hand/in their head.
*I checked this after answering it in my head and was mildly amused. Here was my thought process: log(1000) = 3, log(100) = 2 a log graph has a consistently degrading slope, but one that will still take you to infinity, so the last 200 units in the 100-1000 range will be worth less than the first 200 units. And since halfway there is roughly log(300) and we assume a kinda-semi linear path form there (rounding down because 800 is higher on the scale) it comes out to roughly 2.9.
Out of all the comments this one made me the most depressed. 11x12 is something you need to think about?
I don't find that especially depressing. If you have to sit down with a pencil, yes, you need to play catch-up a bit, but is it so terrible not to have it memorized? I personally always get by with things like "12x12 = 144, so 144 - 12" or "12x10 = 120, so 120 + 12". I just don't have a use for memorizing the complete table, so while I knew it 2 decades ago, I have lost pieces of it.
To multiply 11 by 12 you just "split" the 12 and put a zero between the digits: 1_0_2. Then you add the digits in the 12: 1+2 = 3. Then you add the result of the 2nd operation to the zero in the first split: 1_0+3_2 = 1_3_2 = 132.
This technique works for any 2 digit number multiplied by 11.
What IS especially depressing is that the slowest and least efficient arithmetic techniques keep being passed on for generations. Mainly because it's easier to teach the dumb way and teachers are too lazy to learn how to teach anything new. The result of this is that most students end up hating arithmetic and calculation, and subsequently any further math, because they are never taught the fast and fun way to do things.
If you ever have a job interview at a hedge fund or trading firm, you will not get past the phone screen if you can't do this sort of arithmetic in your head.
Well, to each his own. I always hated the kind of math where you have to remember a basketful of little tricks, like your method of computing products of 11. I have always much preferred "the-method-I-invented-on-the-spot", my approximate method using nearby known numbers being one of them.
The most exciting math tests for me were always the ones where I couldn't remember the 'trick' for half the problems, and would re-invent the solution. It didn't always go well, but the GREAT SCOTT!!! moments were some of the best in all of college.
Approximate estimation is what you need to multiply other two digit numbers quickly. It just happens that there's a trick for multiples of 11. There's actually probably less than 20 other "tricks" that help with rapid calculations.
I don't understand why this trick is necessary. For 11x<number> you can just do 10x<number> + <number> which is easy until you get to 3 digit numbers, but even then continues to work.
Speed. The "trick" breaks down the problem so there's always a simpler summation to perform. In half of the possible products, one only has to add the sum of the digits to zero. For the rest, just slot the 2nd digit of the sum where the zero goes, and add 1 to the first digit. Most people can do either of those two simple additions faster than they can do sums like 210+21 or 390+39. If one is already quick with the latter kind of addition, then the trick is unnecessary.
At my high school we had these expensive graphing calculators that were supposed to last for four years. I of course lost mine in early grade 10. Best thing by far that happened to me.
I was always that the top of my class for mathematics, so I was usually pretty bored with the homework, but all of a sudden I thought of a new challenge: Do all the homework without any calculator whatsoever. Yes, even things like 3^4 + 5^(1/4). At first of course it was painful, but at least I was doing something. I had a much better understanding of actual graphing than anyone else as well, because I would boil the functions down to their key points and play with them until I saw what they did at infinity.
I ended up getting my calculator back in late grade 11 (funny story, I had to sub in for a math teacher at the last minute because there was a wave of sickness at the school, so I was teaching people one year younger than me) and I saw my trusty calculator on the desk in front of me (I had engraved my name with a knife). I picked up and said "hey thanks, I've been looking for this."
Since then I've always been able to do math in my head and it is unbelievably useful. When people say "oh so that is log(800), what is that" and I respond with "roughly 2.9"* they are flabbergasted, but I know how the curves actually work, I haven't outsourced that knowledge to a machine.
Programming is great and we should have graphing calculators in some part of the math curriculum, but students should be able to do this type of math by hand/in their head.
*I checked this after answering it in my head and was mildly amused. Here was my thought process: log(1000) = 3, log(100) = 2 a log graph has a consistently degrading slope, but one that will still take you to infinity, so the last 200 units in the 100-1000 range will be worth less than the first 200 units. And since halfway there is roughly log(300) and we assume a kinda-semi linear path form there (rounding down because 800 is higher on the scale) it comes out to roughly 2.9.
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