posted
The recent thread on pedagogy got me thinking. I've got a niece who's going to be going to school in a couple of years and I've promised to try to shield here from the stuff I hated about school.
So one of the things I hated most in early education was the stupid way they went about teaching math. Rote memorization was just plain dumb in my opinion. It may be different now. To be honest, I really don't know.
Anyway, I got really bored with the math stuff, so I started playing with the numbers, and I pretty much taught myself basic number theory. This made math intereseting and gave me the luxury of ignoring nearly all the math instruction I got up till at least 8th grade. So it's something that I'd like to pass on.
However, while I've got a pretty good semantic memory, my episodic one is for crap. So I'm having problems remembering what I learned about number theory and how I got there. So I figured I'd throw the topic out there to see what people thought would be important and how to get there and also as a way of exploring how I actually came up with this stuff.
One of the biggest things I remember is understanding the implications of base 10 math. Once I grasped this, the world of cyclic patterns, scaling, and 10s complement (the number added to a given number that would give you 10, so the 10s complement of 4 is 6, etc.) opened up.
So for example, if you're subtracting a number from another with a lower ones digit, say 7 from 24, all you need to do is take the tens complement of 7 - 4 = 3 -> tens complement = 7, so the answer is 17.
More interesting for me is when you combined it with the cyclical patterns in something like multiplication. So, like the 3 times tables have a specific pattern. 0-3-6-9-2-5-8-1-4-7-0-and so on. This pattern exists because of how 3 fits into base 10 math. And the multiples of 3 go up in the 10s column at 0-4-7-10, or (adding one for the original leap part of the cycle) 0 + 4 + 3 + 3. And that pattern holds. So by understanding where you are on the cycle, you can easily come up with the multiple of 3 without directly calculating it.
Now, if you take the 10s complement of 3 (i.e. 7) you can reverse this pattern. seven goes 0-7-4-1-8-5-2-9-6-3-0, which is just the 3 pattern subtracted from 10 each time. Also, 7 is the opposite of 3 in that it goes up in the 10s except at 4 and 7. So knowing the 3 cycle also gives you the 7 cycle because of the 10 complement. This is true for each of the 10s complement pairs. Also, you could just subtract the 10s complement multiple from 10 times the multiplying number to get the answer, so 7 * 4 = 40 - 12 = 28 (knowing this made certain stuff in algebra easier later on).
Oh and scaling. There's a famous story about somebody (I don't remember who exactly, might be Einstein) where the teacher assigned them to add the first 100 integers together as busy work and this guy came up with the correct answer in like 15 seconds. Now this is an easy thing because any consecutive series of numbers follows a pattern around the central pivot. In this case, the pivot is 50 and the pattern is one of 100 complements (e.g. 100 and 0, 99 and 1, 98 and 2, etc.) There's actually 101 numbers if you include 0, so your answer is 101/2 * 100 = 5050 or 50 pairs of 100 complements (50*100) plus your pivot (+50). But scaling lets you do this with just the first 10 integers, because 100 is one scale above 10, so in that case, you've got 5 10s complement pairs plus the pivot of 5 for 55. If you realize that the scaling kicks in twice for the first part (instead of 10s complements it's 100s and instead of 5 pairs it's 50) and once for the second part (the pivot is 50 instead of 5), you can easily figure out that it's got to be 50 * the two scaling effects (10*10=100) + 5 * the one scaling effect or the 50*100 + 5 * 10 we said above. If you understand the scaling effects you can by using the numbers from 1 to 10 figure out the answer to adding the first 1,000 integers or 10,000 or 1,000,000,000 if you wanted. You could also do the same for the first 600,000 or the numbers between 300 and 600 or even 222 and 743. Not a particularly useful application, but the principle holds for any operations that have scale effects.
One other thing I remember about multiplication is that it's just repeated addition. This can be very useful when you want to figure out the answer to numbers you don't know but which are close to numbers that you do know. For example, say I know that 10 * 10 = 100. To figure out what 11 * 10, all I need to do is figure out how the adding is different. In the first case, what I really did was add 10 10s together. In the second, all I need to do is add anthoer 10, so 10 * 11 = 110. Pretty simple right?
Well, consider squares of numbers, which is where this really came into play for me. 10*10 = 100 is a known square. Based on the same thinking as above, I could use this to figure out any other square (although for ones not around it, that involes a heck of a lot more addition than I'd want to do, unless I used the pivot method from above, which is possible). So, say I don't know what the square of 11 is. Well the difference between 10 * 10 and 11 * 11 is one is adding 10 10s and the othe 11 11s. So, let's take it a piece at a time. We've already figured out that 11*10 = 110, beacuse all we're doing is adding another 10. Now the difference between 11*10 and 11*11 is that we need to add another 11, so 11*11 = 10*10 + 10 + 11 or 121. We can do 12 * 12 using the same stuff. 100 + 10 + 11 + 11 + 12 = 144. Or say 15 * 15 is a known square with a value of 225. So 19 times 19 is 225 + 15 + 16 + 16 + 17 + 17 + 18 + 18 + 19. Here, using the pivot method (pivot 17 + 17 = 34 with 4 pairs which is 34 * 4 or the 4th instance of the 4th cycle of 4 (although 4 is trickier because it's even, and thus can't cycle through the odd numbers and so actually has a double cycle 0-4-8-2-6-0-4-8-2-6-0), thus 40 * 3 + 10 + 6 = 136) we can figure out that 19 * 19 = 225 + 136 = 361. Of course, it would have been much easier to use the known 20*20 square = 400 - 20 - 19, which also gives us 361.
Anyway, that's a bit of the number theory that I can remember coming up with. I'm sure other people have come up with tricks that I've either forgotten or never figured out. What do you lot have?
Posts: 10177 | Registered: Apr 2001
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posted
Okay - you've totally lost me. I can balance my checkbook, handle very large corporate budgets, and know my multiplication tables. What more do I need???
quote:Oh and scaling. There's a famous story about somebody (I don't remember who exactly, might be Einstein) where the teacher assigned them to add the first 100 integers together as busy work and this guy came up with the correct answer in like 15 seconds.
Gauss.
As to your topic, I'll come back to it after giving it some thought. Posts: 13680 | Registered: Mar 2002
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quote:There's a famous story about somebody (I don't remember who exactly, might be Einstein) where the teacher assigned them to add the first 100 integers together as busy work and this guy came up with the correct answer in like 15 seconds.
I love finding underlying number theory. My friend went to a university with a class called Math Munchies where each lesson was dedicated to some aspect of number theory the professor liked.
Edit: Foiled by Icarus because I wanted to find a link.
posted
Shan, Part of my point is that it's likely partially due to the way that you were taught that you are limited in the math place. I'm trying to make sure that this doesn't happen to my niece and I think that one of the main reasons why it didn't happen with me is that I realized that math is actually a game. I'm somewhat limited by the way we communicate here into explaining this stuff in a way that makes it look cool to non-math people, but as a kid it was and it made things so much easier.
edit: I'm reasonably sure that the stuff I'm talking about is actually really easy to get if it's explained well and approached in the right way. I'm not sure how to do that on an online bulletin board with people who may have math anxiety, but I'm pretty sure it's there.
posted
I know you're trying to do a good thing, Mr. S.
My orientation to math and mathematic principles has never been . . . anything to brag about. Sometimes funny -
In early elementary school I have very clear memories of being totally baffled by story problems along the lines of say:
Jane has three red marbles, Joe has five black marbles, Bill has 10 rainbow marbles and Danny has no marbles. How many marbles would Danny have if everyone shared two marbles withn him?
My mind leaped to the following:
First, if they were all nice kids, they'd share anyhow, and how many marbles anyone had wouldn't matter. Second, the only girl in the crowd would only have one marble left if they all shared two.
You can see how my parents and teachers might have despaired, and just required some basic memorization and utilization skills.
Obviously, my interests and talents lay elsewhere. (Chuckles)
posted
There's a great book, published by Little Brown I think, called Math for Smarty Pants. I got a hold of this book when I was about 7, and it took me through 6th grade math in a matter of a couple months. Not only that, it was a fun book. I seem to recall learning a lot of basic number theory using that book. I'd recommend trying to find it... heck, I can probably find it.
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