Now that you know that fractions are special numbers written like division problems,
we’re going to learn about some different types of fractions, and where they’re located on the number line.
Because fractions are division problems, their value depends on the top and bottom numbers and the relationship between them.
There are a few basic rules about that relationship that will help us estimate the value of a fraction
and know about where it should be on the number line.
The first rule is: If the top number of a fraction is zero,
then the value of the fraction is always zero, no matter what the bottom number is.
For example, zero over two and zero over twenty-thousand are both just zero.
I like to call these fractions, “zero fractions” … ya know… cuz they equal zero.
Oh.. and by the way… you can never have zero as the bottom number of a fraction
because you can’t divide something into zero parts, so don’t even try it.
The next rule is this: If the bottom number is bigger than the top number
then the value of the fraction will be greater than zero but less than one.
That means it’ll be somewhere in this section of the number line.
Any fractions that have values in this range are called ‘Proper Fractions’
because we can use these values to represent smaller parts of things.
Our third rule is this: If the top number and the bottom number are the same,
then the value of the fraction is always just one.
So, whether you have one over one, five over five or one-hundred over one-hundred, the value is always just one.
I’m going to call this kind of fraction a “whole fraction” because its value represents one-whole.
Oh… and in case you’re wondering, this rule doesn’t apply to zero over zero,
because like I told you, having a zero on the bottom of a fraction is a big no-no.
Okay, our last rule is this:
If the top number is greater than the bottom number, then the value of the fraction will be bigger than one.
That means it’ll be somewhere in this section of the number line, which goes on forever.
These are called ‘Improper Fractions’, because even though they’re written like regular fractions,
since their value is greater than one, they aren’t really used to represent smaller parts of things.
Alright, these rules show that we have four main types of fractions:
We have ‘zero fractions’, ‘proper fractions’, ‘whole fractions’, and ‘improper fractions’.
Knowing that these main types are in order from smallest to largest on the number line
allows you to do some very simple comparisons between the four types of fractions.
That’s because we know that a zero fraction is always less than a proper fraction,
and a proper fraction is always less than a whole fraction,
and a whole fraction is always less than an improper fraction.
Let’s do a few comparisons to get the hang of it…
Here we have 1/5 and 0/8:
Since 1/5 is a proper fraction and 0/8 is a zero fraction,
1/5 is greater than 0/8
Now let’s do 3/8 and 2/2:
3/8 is a proper fraction and 2/2 is a whole fraction,
so that means that 3/8 is less than 2/2
Now what about 9/9 and 32/32 ?
Ah… now that’s easy.
Since they are both whole fractions, and whole fractions are always equal to one,
these fractions are equal.
And finally, what about 1/2 and 5/4 ?
Now we know that 1/2 is a proper fraction,
but 5/4 is an improper fraction because its top number is bigger that its bottom number.
So that means that 1/2 is less than 5/4.
Now that we know that there are four basic types of fractions, and we’ve learned where they fit on the number line,
let’s learn more about how the relationship between the top and bottom numbers effects the value of a fraction.
Let’s go on a journey down our number line.
Now we’re gonna start with this ‘zero fraction’ (zero over twenty)
and its value puts us here at zero on the number line.
To get moving, all we have to do is start changing the value of our fraction by increasing the top number.
We’re going to leave the bottom number the same the whole time though.
Alright! Let’s go!
We haven’t gotten very far from zero yet,
and you might have noticed that the top number is still very small compared to the bottom number.
But, as the top number gets bigger, the value of our fraction is increasing.
That tells us that if a fraction’s top number is a lot smaller than its bottom number,
then its value is going to be close to zero. …in this part of the number line.
Look at this… we’re almost to ten on top,
and since ten is half of twenty, we’re almost to one-half on the number line.
It’s pretty easy to figure out what half of something, or double of something is,
and we can use that to help us compare fractions.
…like we know that 9 over 20 is going to be really close to 1/2 on our number line.
Alright, so we’ve passed one-half now, and we’re making our way to the number ‘1’.
Notice that our top number keeps increasing, and it’s getting closer and closer to twenty.
In fact, when it reaches 20, we’ll have arrived at '1' because twenty over twenty is a whole fraction.
Knowing this can also help you estimate a fraction’s value.
Whenever you see a fraction with a top and bottom number that are almost the same… like 19 over 20,
you know that the value is close to ‘1’.
There, we’ve passed ‘1’ now, but we’re still going and our top number is now bigger than our bottom number,
which means we have an improper fraction.
You can see that the bigger the number gets, the bigger the value of the fraction,
and we could keep on going forever and ever, but that might take all day! [laughter]
Okay, so our journey showed us some pretty useful regions of the number line:
…the region near zero, where the top number is much smaller than the bottom number.
…the region near one-half, where the top number is about one-half of the bottom number.
…the region near ‘1’, where the top number and bottom number are about the same.
…and the region past ‘1’ where the top number is bigger than the bottom number, and it keeps on going forever.
Knowing about these regions can sometimes help you quickly estimate the value of some fractions.
For example, you can estimate that 1 over 16 is going to be pretty small. …close to zero on the number line.
And you can estimate that 29 over 31 is going to be almost ‘1’
because there is not much difference between the top and bottom numbers.
And if you have the fraction 14 over 30,
you can estimate that it’ll be about one-half since 14 is close to 15 and 15 is half of 30.
Alright, that wraps up this section,
and I hope it’s helped you understand the different types of fractions and where they are on the number line.
You’ll understand even better if you do the exercises for this section.
Good luck and I’ll see you next time.
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