The difference would just be how you think of the process. I sometimes shuffle around the numbers to make math easier, but the shortcut for adding 9s just feels different. Instead of 9+7 = 10 + 6, it’s more like 9+7 = 17-1. It feels less like solving it with math and more like using a cool trick, since you didn’t really use addition to solve the addition problem.
Sort of, same numbers different logic. Its like mixing up the order of operations. You could learn both tricks but it seems redundant if they do the same thing. Like having two of the same hammer.
And it scales with multiplication too. 9*7 is (7-1) and whatever adds to9, so 63. This breaks down for larger numbers, but works really well up to 9*10. I don’t know what “common core” teaches for that, but you can’t change the 9 to a 10 for multiplication (well, you could, but you’d need to subtract 7 from the answer).
Treating 9s special makes math a lot easier. Doing the “adjust numbers until they’re multiples of 10” works for more, but it’s also more mental effort. 9s show up a lot, so learning tricks to deal with them specifically is nice. I just memorized the rest instead of doing “common core” math to adjust things all the time.
That said, I do the rounding thing for large numbers. If I’m working with lots of digits, I’ll round to some clean multiple of 10 that divides by 3 (or whatever operation I need to do) nicely. For example, my kid and I were doing some mental math in the car converting fractional miles to feet (in this case 2/3 miles to feet). I used yards in a mile (1760) because it’s close to a nice multiple of three (1800), and did the math quickly in my head (1800 - 40 yards -> 6002 yards - 40 yards to ft * 2/3 -> 1200 yards - 120 ft2/3 -> 3600 ft - 80 ft -> 3520 ft). I calculated both parts of the rounding differently to make them divisible cleanly by 3. I don’t know what common core math teaches, but I certainly didn’t learn this in school, I just came up with it by combining a few tricks I learned largely on my own (i.e. if the digits add to 3, it’s divisible by 3) through years of trying to get faster at math drills. If I wasn’t driving, I would have done long division in my head, but I needed to be able to pause at stop signs to check for traffic and whatnot, and just remembering two numbers w/ units is much easier than remembering the current state of long division.
I was very competitive in school like this, wanted to finish things first. I think maybe you make a good point about wanting to solve things faster leading to these types of tricks developing. Sort of puts math competitions in a new light.
This is also how it works in my head, but isn’t it the same as the other guy was saying, 10+6?
The difference would just be how you think of the process. I sometimes shuffle around the numbers to make math easier, but the shortcut for adding 9s just feels different. Instead of 9+7 = 10 + 6, it’s more like 9+7 = 17-1. It feels less like solving it with math and more like using a cool trick, since you didn’t really use addition to solve the addition problem.
Sort of, same numbers different logic. Its like mixing up the order of operations. You could learn both tricks but it seems redundant if they do the same thing. Like having two of the same hammer.
And it scales with multiplication too.
9*7is(7-1) and whatever adds to 9, so 63. This breaks down for larger numbers, but works really well up to9*10. I don’t know what “common core” teaches for that, but you can’t change the 9 to a 10 for multiplication (well, you could, but you’d need to subtract 7 from the answer).Treating 9s special makes math a lot easier. Doing the “adjust numbers until they’re multiples of 10” works for more, but it’s also more mental effort. 9s show up a lot, so learning tricks to deal with them specifically is nice. I just memorized the rest instead of doing “common core” math to adjust things all the time.
That said, I do the rounding thing for large numbers. If I’m working with lots of digits, I’ll round to some clean multiple of 10 that divides by 3 (or whatever operation I need to do) nicely. For example, my kid and I were doing some mental math in the car converting fractional miles to feet (in this case 2/3 miles to feet). I used yards in a mile (1760) because it’s close to a nice multiple of three (1800), and did the math quickly in my head (1800 - 40 yards -> 6002 yards - 40 yards to ft * 2/3 -> 1200 yards - 120 ft2/3 -> 3600 ft - 80 ft -> 3520 ft). I calculated both parts of the rounding differently to make them divisible cleanly by 3. I don’t know what common core math teaches, but I certainly didn’t learn this in school, I just came up with it by combining a few tricks I learned largely on my own (i.e. if the digits add to 3, it’s divisible by 3) through years of trying to get faster at math drills. If I wasn’t driving, I would have done long division in my head, but I needed to be able to pause at stop signs to check for traffic and whatnot, and just remembering two numbers w/ units is much easier than remembering the current state of long division.
I was very competitive in school like this, wanted to finish things first. I think maybe you make a good point about wanting to solve things faster leading to these types of tricks developing. Sort of puts math competitions in a new light.