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I'm studying for eight hours a day, and it seems that no matter how hard I try I just can't make progress. For example, I can't figure out how to simplify fractions. I know that you have to find the greatest common factor , then divide the numerator and denominator by it. Problem is, I can't figure out what the GCF is. It seems people can just look at two numbers and know intuitively with the GCF is, because they've been doing it since they were very young children. It depresses me reading tutorials and having them talk about it like "oh yes this is obviously the conclusion here and it comes second nature" when that doesn't happen for me.
Where did he get 3 from? Those numbers aren't in my times tables. How did he know intuitively that 3 was the GCF? It seems that I just can't process mathematics.
I'm studying for eight hours a day, and it seems that no matter how hard I try I just can't make progress. For example, I can't figure out how to simplify fractions. I know that you have to find the greatest common factor , then divide the numerator and denominator by it. Problem is, I can't figure out what the GCF is. It seems people can just look at two numbers and know intuitively with the GCF is, because they've been doing it since they were very young children. It depresses me reading tutorials and having them talk about it like "oh yes this is obviously the conclusion here and it comes second nature" when that doesn't happen for me.
Where did he get 3 from? Those numbers aren't in my times tables. How did he know intuitively that 3 was the GCF? It seems that I just can't process mathematics.
It really helps to know your multiplication tables here but if you don't, just start dividing the numbers by smaller numbers and see which factors are common.
Neither 57 or 63 are even so 2 and 4 are out, they don't end in 0 or 5 so 5 is out at first glance so try 3, which works. 19 is prime so you're done even though 21 can be divided by 3 and 7.
I would set this up as two problems. 57/3 = 19 and you can't go any further and 63/3 = 21 21/3 = 7 so you really have 19*3 in the numerator and 7*3*3 in the denominator. Cancel one 3 and you get 19/(7*3) or 19/21
What you really need to do is practice factoring numbers. With enough practice, you start seeing the common factors on inspection. The greatest common factor is simply all the factors that are common multiplied together. In this case they only have one which is 3 so the least common factor is also the greatest common factor in this case.
I am going to add to your explanation. I hope you don't mind. -- davidv
Quote:
Originally Posted by Ivorytickler
It really helps to know your multiplication tables here but if you don't, just start dividing the numbers by smaller numbers and see which factors are common.
Neither 57 or 63 are even so 2 and 4 are out,
Not just 2 and 4, but all even numbers are out. So you are left looking at odd numbers only.
Quote:
Originally Posted by Ivorytickler
they don't end in 0 or 5 so 5 is out at first glance
Exactly right.
Quote:
Originally Posted by Ivorytickler
so try 3
Don't just "try" 3. You can determine if a number is divisible by three by adding the numerals in the number together and then dividing that answer by 3. If it can be done, then the number is divisible by 3.
In the example above: 57/63
For the numerator, 57, add 5 and 7 together and you get 12 (5 + 7 = 12) Divide 12 by 3 (12/3) and you get a whole number (4) which means that 57 is divisible by 3.
For the denominator, 63, Add 6 and 3 together and you get 9 (6 + 3 = 9) Divide 9 by 3 (9/3) and you get a whole number (3) which means that 63 is also divisible by 3.
Taking a random number like 457, we find that 4 + 5 + 7 = 16. Since 16 is not divisible by 3, that means 457 is not divisible by 3.
Another example: 5,946. Add the digits together (5+9+4+6) to get 24. Since 24 is divisible by 3 so is 5,946 (5946/3 = 1982)
Quote:
Originally Posted by Ivorytickler
19 is prime so you're done even though 21 can be divided by 3 and 7.
I would set this up as two problems. 57/3 = 19 and you can't go any further and 63/3 = 21 21/3 = 7 so you really have 19*3 in the numerator and 7*3*3 in the denominator. Cancel one 3 and you get 19/(7*3) or 19/21
What you really need to do is practice factoring numbers. With enough practice, you start seeing the common factors on inspection. The greatest common factor is simply all the factors that are common multiplied together. In this case they only have one which is 3 so the least common factor is also the greatest common factor in this case.
I'm studying for eight hours a day, and it seems that no matter how hard I try I just can't make progress. For example, I can't figure out how to simplify fractions. I know that you have to find the greatest common factor , then divide the numerator and denominator by it. Problem is, I can't figure out what the GCF is. It seems people can just look at two numbers and know intuitively with the GCF is, because they've been doing it since they were very young children. It depresses me reading tutorials and having them talk about it like "oh yes this is obviously the conclusion here and it comes second nature" when that doesn't happen for me.
Where did he get 3 from? Those numbers aren't in my times tables. How did he know intuitively that 3 was the GCF? It seems that I just can't process mathematics.
Try solving the equations based on common sense - thinking it through and trying to understand it in your mind - instead of just following rules from a math book. No matter how long it takes, if you can solve it using your own logic, you will develop that intuition you are talking about. It does not come from experience - it comes from understanding. You've probably been taught math rules your whole life and never developed that understanding - it's our lousy system of math instruction. You are better off figuring it out on your own.
For example:
20/10 can be simplified at 2/1 - or 10/20 as 1/2 - try to figure out why that is true and why it makes sense before you move on to complex numbers. Then you will have the basic understanding. It might even help to take some physical objects and try making simple fractions with them. Hershey bars work well - easy to break into different size pieces, and also delicious study snack.
Thanks for the advice folks. I'm still struggling to understand. I tried to find the GCF of 315 over 405. Following your advice, I added 3 + 1 + 5 and got 9. I divided 9 by 3 and got 3. I then added 4 + 0 + 5 and got 9. I divided 9 by 3 and got 3. So that means they're both divisible by 3. But it turns out 3 isn't the GREATEST common factor, 45 is. How was I supposed to come to the conclusion that 45 is the GCF?
Math is one of those things that some people have knack for--but you don't have to be a math whiz to be considered an educated person as an adult. Don't worry about it if you struggle with some high school math topics; unless you choose certain careers you won't have to know much math as an adult.
It is also something that is much easier to learn if you have a good teacher--but there aren't many good math teachers out there.
Math is one of those things that some people have knack for--but you don't have to be a math whiz to be considered an educated person as an adult. Don't worry about it if you struggle with some high school math topics; unless you choose certain careers you won't have to know much math as an adult.
It is also something that is much easier to learn if you have a good teacher--but there aren't many good math teachers out there.
Well, I have to learn high school algebra now that the GED test was made more difficult. And if I can't even simplify fractions reliably there's no way I'll ever learn that.
I am going to add to your explanation. I hope you don't mind. -- davidv
Not just 2 and 4, but all even numbers are out. So you are left looking at odd numbers only.
Exactly right.
Don't just "try" 3. You can determine if a number is divisible by three by adding the numerals in the number together and then dividing that answer by 3. If it can be done, then the number is divisible by 3.
In the example above: 57/63
For the numerator, 57, add 5 and 7 together and you get 12 (5 + 7 = 12) Divide 12 by 3 (12/3) and you get a whole number (4) which means that 57 is divisible by 3.
For the denominator, 63, Add 6 and 3 together and you get 9 (6 + 3 = 9) Divide 9 by 3 (9/3) and you get a whole number (3) which means that 63 is also divisible by 3.
Taking a random number like 457, we find that 4 + 5 + 7 = 16. Since 16 is not divisible by 3, that means 457 is not divisible by 3.
Another example: 5,946. Add the digits together (5+9+4+6) to get 24. Since 24 is divisible by 3 so is 5,946 (5946/3 = 1982)
Everything else here is spot on.
I didn't go higher with even numbers because 2 covers them all (I only included 4 because I skipped it when I went to 5 and find that confuses my students if I don't point out it's even and divisible by 2). I originally had the trick of adding digits to see if they're divisible by 3 in my post but chose to delete it because this one confuses my students if the number turns out to be higher than 10 so I left it out. If you don't know your 3 times tables it's easier to just use a calculator and divide by 3 and see if you get a whole number. If I were doing the division manually, I'd definitely apply the 3 test.
I'll agree on practice though. There's no replacement for practice when learning how to factor. You just keep practicing until you start seeing the factors and even if you don't, if you approach it systematically, you will still find the answer. I teach my students to divide by small numbers instead of looking for the GCM. A number that can be divided by 16 can be divided by two 4 times. They still get the same answer but smaller numbers don't scare them as much. They don't see the 16 but they'll see the factor of 2 when they realize the number is even. 3's they don't see even with the trick of adding digits unless they happen to add 3, 6 or 9. If they get 21, they don't see it and, as I said, it's just as fast to test divisibility by 3 on a calculator and see if you get a whole number.
I tell my kids to see if it's even and if it is divide by 2, if not, does it end in 5, if so, it's divisible by 5. If neither of these work, try 3 then 7 then 11, 13,17, 19 and 23. We stop there. I don't expect them to test every prime number. Most of my kids can't see the factors but they can do successive divisions so that's what they do. I do it like upside down division.
2 | 210
..5|105
...3 |21
........7
Ignore the leading periods, the software keeps left justifying when I try to post. They're just there to position the numbers.
So the factors of 210 are 2, 5, 3 and 7.
Last edited by Ivorytickler; 12-26-2013 at 04:49 PM..
Well, I have to learn high school algebra now that the GED test was made more difficult. And if I can't even simplify fractions reliably there's no way I'll ever learn that.
OP, I can't help much with the math, because honestly I don't remember it. I took that several semesters ago. But just wanted to give you some encouragement. Math is difficult for me as well, extremely difficult. I even had trouble adding positive and negative integers in Pre-Algebra. I've been enrolled in CC for 3 years and am graduating in May with an Associates in Science and I took 5 math courses, 2 of which were remedial. I only needed to go up to Pre-Calc which is apparently even worse than Calc. I would say if I can pass all this you can pass the GED test. I am awful and math and just finished my last semester of it with all passing grades. I even got a B in a few of my college level math courses. I assure you if you put the time in it'll all be worth it.
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