When Tigger was in public school, they taught him some math shortcut and he kept using it wrong because he really did not understand how to do it the long way, so he really did not understand how to apply the shortcut. He would use it inappropriately and tell me "It is a Rule. My teacher said so."
Prime Factoring
Alternately, I factor using prime numbers. If you factor with primes, you cover everything quickly. For example, when prime factored, the number 8 breaks down to '2x2x2' and 12 breaks down to '2x2x3'. I can readily see that 4 is a factor of both of these numbers because both of them have '2x2' in them. Such information can be useful when working with fractions, looking for the least common denominator, and similar.
Note: I learned to factor primes out by starting with dividing by 2 and keep dividing by 2 until you get an odd number. For example, I would write 24 at the top left of the page and put an upside down 'V' under it. I would write 2 beneath the first 'leg' of the inverted V and 12 underneath the second leg. Then, under the 12, I would write another inverted V and write 2 again and 6. Etc. I think this is the 'standard' method for factoring with primes, and it looks like this: 24
I was also shown this alternate method of listing prime factors.
I had to go back and show him the long way and explain WHY the shortcut Usually works but would not work for this specific problem. So, when a concept is initially introduced, the student probably needs to do it the long way a few times in order to understand it. But, AFTER they get the basic concept, I hope my thoughts on factoring are useful to some people:
Factoring
I handle factoring differently from most people. This is not something I was ever taught. It was something I discovered by observation: It is half as much work (or sometimes way less than half as much work) to stop factoring when the numbers 'meet in the middle'. If I were factoring for 8, once I got to 2x4, I would stop because listing 4x2 and 8x1 is just repeating the numbers already listed as 1x8 and 2x4. So my factors lists go down one side and up the other. I know that 3 is not a factor of 8, so once I got to 2x4, I would know there were no more factors to cover.
Like so:
1 x 8
2 x 4
3
It just so happens that 3 is not a factor, and there is no point in going further: 4 has already been listed and I am sure that there are no more factors to be found.
To prove it, let's factor 24 and walk you through my reasoning:
1 x 24
2 x 12
3 x 8
4 x 6
Logically, next you would check if 5 factors into 24. It doesn't because 24 does not end in 0 or 5. So, then you would go on to 6, 7, 8, 9, 10, etc. But, since we have covered 1 through 4 and 5 isn't a factor and the next factor is 6 (and we already listed it, next to 4), there are NO whole numbers by which you can multiply 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, or 23 in order to get 24.
Since 6 and 8 are both factors of 24, you can see that 7 is not because, in order for it to be a factor, you would need a whole number between 3 and 4 (the factors by which you multiply 6 and 8) and there is no such thing. By the same token, any numbers between 8 and 12 would have to multiply by a number between 2 and 3 to get 24. And any numbers between 12 and 24 would have to be multiplied by something between 1 and 2.
Therefore, there is no need to spend time examining any of those numbers as possible factors of 24. For 24, 4 x 6 is where the list of small numbers going up from 1 and the list of large numbers coming down from 24 'meet in the middle' -- and I am finished.
The entire list of factors of 24 is: 1, 2, 3, 4, 6, 8, 12, 24.
Factoring
I handle factoring differently from most people. This is not something I was ever taught. It was something I discovered by observation: It is half as much work (or sometimes way less than half as much work) to stop factoring when the numbers 'meet in the middle'. If I were factoring for 8, once I got to 2x4, I would stop because listing 4x2 and 8x1 is just repeating the numbers already listed as 1x8 and 2x4. So my factors lists go down one side and up the other. I know that 3 is not a factor of 8, so once I got to 2x4, I would know there were no more factors to cover.
Like so:
1 x 8
2 x 4
3
It just so happens that 3 is not a factor, and there is no point in going further: 4 has already been listed and I am sure that there are no more factors to be found.
To prove it, let's factor 24 and walk you through my reasoning:
1 x 24
2 x 12
3 x 8
4 x 6
Logically, next you would check if 5 factors into 24. It doesn't because 24 does not end in 0 or 5. So, then you would go on to 6, 7, 8, 9, 10, etc. But, since we have covered 1 through 4 and 5 isn't a factor and the next factor is 6 (and we already listed it, next to 4), there are NO whole numbers by which you can multiply 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, or 23 in order to get 24.
Since 6 and 8 are both factors of 24, you can see that 7 is not because, in order for it to be a factor, you would need a whole number between 3 and 4 (the factors by which you multiply 6 and 8) and there is no such thing. By the same token, any numbers between 8 and 12 would have to multiply by a number between 2 and 3 to get 24. And any numbers between 12 and 24 would have to be multiplied by something between 1 and 2.
Therefore, there is no need to spend time examining any of those numbers as possible factors of 24. For 24, 4 x 6 is where the list of small numbers going up from 1 and the list of large numbers coming down from 24 'meet in the middle' -- and I am finished.
The entire list of factors of 24 is: 1, 2, 3, 4, 6, 8, 12, 24.
Prime Factoring
Alternately, I factor using prime numbers. If you factor with primes, you cover everything quickly. For example, when prime factored, the number 8 breaks down to '2x2x2' and 12 breaks down to '2x2x3'. I can readily see that 4 is a factor of both of these numbers because both of them have '2x2' in them. Such information can be useful when working with fractions, looking for the least common denominator, and similar.
Note: I learned to factor primes out by starting with dividing by 2 and keep dividing by 2 until you get an odd number. For example, I would write 24 at the top left of the page and put an upside down 'V' under it. I would write 2 beneath the first 'leg' of the inverted V and 12 underneath the second leg. Then, under the 12, I would write another inverted V and write 2 again and 6. Etc. I think this is the 'standard' method for factoring with primes, and it looks like this: 24
/\
2 12
/\
2 6
/\
2 3
I was also shown this alternate method of listing prime factors.
24 | 2
12 | 2
6 | 2
3 | 3
1
If I factor out all the primes, I am nearly done. I can 'construct' any other factors of a number by multiplying combinations of the prime numbers that it breaks down to.
For example:
24 = 2 x 2 x 2 x 3
From here I can 'see' (or construct) all of the factors of 24:
2 x 2 x 2 x 3 = 2 x 12 (because 12 is 2 x 2 x 3)
2 x 2 x 2 x 3 = 3 x 8 (because 8 is 2 x 2 x 2)
2 x 2 x 2 x 3 = 4 x 6 (because 4 is 2 x 2 and 6 is 2 x 3)
And then you are done. I see no reason to keep going and re-list all the factors in reverse order of how they were listed the first time.
Since multiplication is commutative, the order of operation does not matter and I now have the entire list of factors, if I include both columns of numbers. I promise you, once you find where they 'meet in the middle', if you didn't miss any going down, you haven't missed any on the 'other side'.
Using Prime Factors to quickly find the Least Common Multiple
If you prime factor 2 numbers, you can readily find the least common multiple (something handy when working with fractions) without doing hardly any more math. Just remove all of the identical prime factors on a one-for-one basis (like 'canceling') and whatever is left is what you need to multiply the other number by. This gives you the LCM every time.
As a simple example, the prime factors for 8 are 2, 2, 2 and the prime factors for 12 are 2, 2, 3. When you remove the first two '2's, you have 2 as the remaining prime factor for 8 and 3 as the remaining prime factor for 12. If you 'cross multiply' -- 2x12 or 3x8 -- you get 24 for both answers, which is your LCM for 8 and 12.
I don't think I have ever been shown another straight forward, guaranteed method for finding the LCM, which can be a confusing hassle (for me anyway, and, presumably for other people as well). But this one works very well.
As a simple example, the prime factors for 8 are 2, 2, 2 and the prime factors for 12 are 2, 2, 3. When you remove the first two '2's, you have 2 as the remaining prime factor for 8 and 3 as the remaining prime factor for 12. If you 'cross multiply' -- 2x12 or 3x8 -- you get 24 for both answers, which is your LCM for 8 and 12.
I don't think I have ever been shown another straight forward, guaranteed method for finding the LCM, which can be a confusing hassle (for me anyway, and, presumably for other people as well). But this one works very well.