Last week's question, reapeated for this week's part TWO of our answer:
I'm supost to know a lot about dinomenaters, and even though it's summer and school is over, my teacher made my mother promise her that by the time we go back I'll really know almost everything about dinomenaters, because by then we kind of haf to know EVERYTHING (except maybe like where they came from and how they started and who started them off, and stuff like that. So, before my mother starts going at it with all her ideas about dinominaters and how great they are, could you explain everything you know on dinominaters really fast? And please don't forget about making them really common, because my teacher absolutely loves common dinominaters. And don't worry you don't have to start at the very beginning because I already get the basic point, that the dinominators are always on the bottom of the fraction, and no matter what, they stay down there. And the tops of fractions which I know I'm spost to be calling the noomerators, they never go under the line no matter what's going on down there. So can you just tell me the rest?
I woudn't say you have to rush it, but can you please try to do it pretty fast?
Welcome back! (Or if you've been living in front of your computer to be sure you wouldn't miss this 2nd part of our response to your question about fractions and their denominators, then it's time to WAKE UP!
Okay, last time I left off with 3 promises:
- I'd show you a way to find lowest common denominators that always works;
- I'd explain why - even though lowest common denominator is usually the approach to adding and subtracting fractions with different denominators, if you even forget the way or get stuck, you can work through problems with different denominators without coming up with a common denominator that's the lowest;
- There is a little trick that surprises a lot of people and gives you the answer to a problem that requires you to add or subtract fractions that don't have the same denominator without you having to find such a common denominator.
First: The best, safest method for finding lowest common denominators:
This method uses something called "prime factorization", which means you break down numbers into factors, and then break down each factor until every single factor you have left is a prime number. (Remember that word 'prime'? It means any number that has two different factors and only two different factors.)
So, for example, 7 is a prime number because it can only be factored like this (7x1) or, of course, (1x7) (order doesn't matter in multiplication). But 12 is not a prime number because it can be factored like this: 3x4 or like this: 6x2 or like this: 12x1.
The numbers that cause students (and their parents!) the most confusion are 1 and 2. The number 1(one) a lot of people figure is prime because you can write 1x1 and there are no more sets of factors. However, the definition of a prime number tells you that the number has to have two different factors, and 1 is the same factor as one. (By the way, when you read most math books, they use the word "distinct" instead of different, but it means the same thing. So 1 is not prime.
The number 2 (two) a lot of people guess is not prime because it is EVEN, and if you ask a lot of adults if a number can be even and prime, their first thought is usually no; and they'd be correct -- if it weren't for one and only one exception in the whole world. That exception, of course, is 2. 2x1=2, and there are no other factors. And both 2 and 1 are distinct (different).
So, now we move on to prime factorization and how you use it to get the lowest common denominator. Let's say you have to add 1/12 and 5/18, well these two fractions have different denominators, so if you recall what we said a few days ago, you can't just think of them as the same kind of creatures, like 1/5 and 2/5 giving you three fifths. No. Instead you need to turn them into the same creatures that will have the same denominator. Okay? Here's how:
You draw a line down the center of the paper, or just imagine a line running from the top of the sheet to the bottom where a center-fold would be. Then you write one of the denominators at the top of one side and one of the denominators at the top of the other side, like this:
Next you draw a large upside down V under each, or branching down from each like this:
Then you come up with any 2 factors you can that work together as a pair to get you back to the number at the top. If you find you have a choice, it doesn't matter which you select because any way you begin that is a set of factors will get you to the answer. So you might try this:
Notice each factor for the top number is at the bottom of the branch of the upside down V.
Of course you can and should always check your work. So 4x3 = 12? Yes. And 6x3 = 18? Yes. Good. So far, so good. Next you repeat the upside down V process for each of the factors that can be broken down like this into 2 smaller factors. Like this:
And you'll notice that the 3 on each side cannot be broken down into smaller factors. (Writing 3x1 doesn't help you do anything because the 3 is no smaller than the 3 above it!) And that means that the 3 is a number that is already prime. Remember, the goal in prime factorization is to keep factoring everything down until your factors are prime, so you're done with the 3's! Circle each one so they stand out and you won't get mixed up when things get complicated-looking! (Note: When at a computer, as I, of course, am, it's just as easy to use color to make the prime factors stand out, but the easiest thing when you're working out problems on paper is to circle them. So, don't forget to circle the two 3's that are shown below in red!)
And a little while ago, we said that 2 is prime so circle each 2. Okay, now if all numbers on the bottom are circled, they are your final factors. If there is one or more that can still be factored, do it until you've got it reduced or a collection of only prime numbers, then circle each of those prime factors.
Next, to get the lowest common denominator, you take each of the red (or circled) prime factors one at a time, and see if that factor is on both sides (columns) of your paper. If it is, then you take what I call the 'winning number' of them. (By that, I mean larger quantity of them). So, if the column under the 12 has two 2's (which it does), and the column under the 18 has one 2, which it does, then 2 is the greater number of twos. So you write down two 2's like this: 2 2 (with a little space between them). Then (and this is important) you go back to the 2 columns and cross off every single circled 2 you see (don't forget to do this with both columns). Next you take another circled factor (and here this means the red ones) and repeat the process of comparing both columns to find the "winning number". Here the left side has one circled 3 and the right has 2 circled 3's. So the "winning amount" is 2 three's. Now you write down 2 threes: 3 3 next to the 2 two's (again each number separated by a little space) and then repeat the process of crossing off all the circled 3's. Now find the next circled number that has not been crossed off yet.
Here, I don't see any, so we're done using the columns; they've served their purpose. You take your group: 2 2 3 3 and put little times signs between each like this: 2x2x3x3.
And when you multiply them together, you will have the lowest common denominator for 12 and 18. So one step at a time:
2x2 x 3x3 =
4 x 3 x 3 =
12 x 3 =
So, 36 is the smallest number that both 12 and 18 "go into" evenly. In other words, you
can multiply 12 by something to get 36, and you can multiply 18 by something to get 36.
So 36 is the smallest number you can find that we can say that about. NOTE: It is VERY important that you realize that you are not finished with the math problem of adding 1/18 + 5/12 just by finding the lowest common denominator. No: You must then convert each of those fractions into new fractions that have a denominator of 36.
How do you make those new fractions? Well, I will explain, but in order to do a thorough job of that, it would require another lesson, and since you did not ask it, I will save it for another time.
Next: I mentioned that if you cannot remember how to find the lowest common denominator, you do not have to find one. You just have to find a common denominator and not worry about reducing everything until the end. To do that, you just multiply the two denominators together! That will give you a common denominator, but of course there's no guarantee that it is the lowest (smallest) one or even close.
And, finally, I also mentioned that I would show you how to add and subtract fractions that have different denominators WITHOUT doing anything as complicated as anything we did here, and without even having to think much about denominators at all!
And I will, because it is one of those helpful tips that are worth knowing, AFTER you understand how to work with numbers in the standard ways that help you have an easier time grasping more and more math as you get older. So, since you originally asked me (last week) if I could do all this really fast, and I said probably not but would instead break down into small pieces the explanation of some of the most important math you'll learn in elementary school and middle school, I think this is a good place to take a break.
So, until next time, when we do get to that famously magical trick that helps you work with fractions without even really thinking, I HOPE THIS HELPS.
Until next time,