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Dear Mitch,

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?

Yours truly,

Heather Stochas

p.s. I don't have any brothers, but I have a sister who should help me but doesn't.


Dear Heather,

There are a few important things to know about denominators, and I'm not so sure I can do it all as fast as you'd probably like, but I'll try to keep it as short as I can because something tells me if you see a very long answer you might not read it. And it is summer, right?

Still, I have an idea: Instead of fast, I think you'll be just as happy reading this if I break it up into parts so if you prefer you can read one or two little parts each week of the summer, and by the time you go back to school, you'll have the basic ideas in your head.

So here goes:

1. You don't have to worry about who was the person responsible for introducing the world to fractions, as long as you appreciate how lucky we are that she (or he) did!

2. You are right that the word 'denominator' refers to the part of a fraction that is 'under' the horizontal or sloped line of a fraction, and "numerator" is the name of the part above the line.

3. You need to know what it means when a number is "above" or "below" that line. But before that, we need to go over what a fraction is and why it exists at all. That way, just as a person would have an easier time understanding what tires are and what a dashboard is if he first understands what a car is and why people have them before trying to learn about tires, you will know the big ideas that give all the little ideas their purpose.

4. So a fraction is a part of something or part of a group. Since you have one sister and no brothers, I'll bet you know exactly what "half" means. So if you get half a cookie and she gets half, you each get an equal-sized piece of what was the whole cookie.

But, if you and your sister were playing a game against two other kids, and you and your sister were one team, then you (Heather) are half of that team and your sister is the other half of that team, with each of you being a part of the team, and together making a whole team. So, you can each be a part of something, even though you are each a whole person, okay?

Now, back to the cookie. The bottom part, called the denominator, tells you how many equal sized parts the whole cookie was divided into.

The top part tells you how many of those you're dealing with.

Between 2 people, and we call each of the evenly divided pieces a "half" (1/2).

If your mom wanted a piece too, she might have tried to break it up into 3 pieces (each about the same size) and you would get 1/3 of the cookie or 1 out of those 3 pieces. What kind of piece would each be? The kind we call "a third" (1/3), which means it's the fair amount when there are 3 people sharing it. You need three thirds to rebuild a whole cookie, or you need 2 halves, or 4 fourths. Okay, now we're ready to move up to the next level.

You need to know why I bothered to tell you that the 'bottom' tells you what "kind" of piece you have. The reason you need to know is that it can be VERY TRICKY and confusing if you are adding or subtracting different kinds of parts. So, if you took your piece (the half cookie) and put it in your pocket for later, and you met a friend who has two sisters and she reaches into her pocket and pulls out her snack, she would probably have a third of a cookie (and let's say it's the very same kind of cookie that your mom split up). So that third is one of that cookie's three parts. Now, if someone asked you how much of a
cookie you two girls (you with a half and your friend with a third) would have when you put them would you answer? Well, you might not have an answer right away because it is hard to figure out.

When your teacher says "common denominator", the word 'common' means "the same". Okay, so, you may have noticed that when a question in math asks you to add 1/5 + 1/5, some kids make the mistake of saying 2/10, right? But, if you think of a fifth (1/5) as a kind of something, like a strange creature, and make up a new name for it to give your brain a new start, like a "fithsh" instead of a "fifth", and you catch one of these creatures, and then you catch two more of these creatures, these "fithshes" and you put all three of them in the same bag, what do you have in there? ANSWER: You have three "fithshes"-- not 2-tenths or 2-turtles or 2-frogs. So, since you caught creatures that were "common" to each other, they were the same kind of creature, and so you could just add them up to find out how many you had altogether.

Nothing had to be changed into a different kind of creature to add them together.

There is another creature in the world, and it is called a "therd". (It is pronounced just like the word "third", but therds look more like birds, except without feathers).

Now, if you catch one therd and then catch another therd, together you have two therds. And guess what? The same is true for thirds. One third (1/3) plus one third (1/3) equals two thirds (2/3) – NOT 2/6.

And if you put the two therds in a bag and one gets taken away by a flying minus, then you have two therds minus one therd, and that leaves one therd. Of course, the same is true for thirds (1/3's).

There is another way the line in the middle of a fraction can be read. And that is this way: "divided by".

So: One giant cookie divided by 3 boys, can be written mathematically as 1/3, with the giant cookie represented (shown) on the top, and it is being divided up by the three boys (shown by the 3 on the bottom), and if you close your eyes for a third of a second and then reopen them, you will discover the answer to an important question, "How much of the cookie would each of the boys get?"


Here's another example of that: TWO (2) giant cookies divided up amongst three (3) boys. Mathematically, it looks like this: 2/3, with the number on the top being the number of things being divided up, and the number on the bottom being the number of people (or whatever) that the things on the top have to be divided up by.

So, 2 cookies divided fairly by 3 boys. How much will each boy get?

ANSWER: 2/3 of a cookie. Like magic!

So that's how easy the work is to get right when you have fractions with common (the same) denominators. (Remember, the same denominators make the fractions into the same kind of creatures: therds, fithshes, and even something called "hevs" – pronounced like "halves", (Though they are never found near schools, or anyway else, actually).) So, as your teacher probably told you, when the fractions have DIFFERENT denominators, the best thing to do is to find a way to change things so that you end up with fractions that do have the same denominators, and the new denominator is usually called the "Lowest Common Denominator". Basically, this means that the new denominator is the smallest number that will work easily with the different numbers you were given.
One way to find lowest common denominators uses something called "prime factorization". It sounds kind of hard and weird, I know, but I promise you that after a little practice, it's actually kind of fun.

And I am going to explain how to do it.

But, not today.

Today, I'm stopping right here!



Remember you asked if I could teach everything you would need to know about denominators "really fast"? And remember I said I wasn't sure about "fast" but I knew I would be able to divide up the explanation into parts so you could read a little at a time over the summer? Well, that should be enough to think about for a few days. Then I will post the rest of what you need to know about all this.

But guess what? Before I say goodbye, there's something I want you to know:

You do NOT need to find common denominators. You really don't, I promise -- even though your teacher might be disappointed to hear that. There really is a way around it, a simple trick that frees you from ever having to find the common denominator again ... ever, for the rest of your life. All you have to do is five very easy steps and you can go from question to answer to recess. And I will show you how to do it. BUT, as much as I love teaching magical new (and old) ways to solve math problems, I also enjoy what Hollywood directors call "cliffhangers". This means finding ways to leave the audience in
suspense for at least a little while. (It almost always makes things more interesting than they'd be without that suspense.) SO... stay tuned, because within one week of today, along with the method I recommend to find the lowest common denominator for two fractions, I will post the secret to the method that lets you avoid the whole process. And I will do that right here on this website. I will also explain why you need to understand the method of prime factorization, and how to use it, even though you may never need to actually do so.

Until then, practice using fractions the standard ways that fractions are most often used. After all, no matter which method you use, if you do things correctly and carefully you will always get the correct answer in math. And you don't have to take my word on that or anyone else's, because in math the good news is you can check for yourself!

I hope that helps.

Until next time,

Have fun learning!