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Follow along in the book, Lesson 2-2 on page 91.
The target is: "I can compare and order rational numbers."
Remember rational numbers are any numbers that can be written as fractions.
And for comparing purposes, it says here "sometimes you can use estimation to compare rational numbers.
Another method is to rename each fraction using the least common denominator and then compare the numerators."
The denominator is the bottom of a fraction, denominator on the bottom, the numerator on top.
Now I like to do it a little bit differently than the book. They say the least common denominator,
well really, all we really need is a common denominator.
Once the bottoms are the same, you look at the top numbers and figure out which one's bigger, which one's smaller.
So let's take a look at our first example.
They say, uh, which--what replaces the little circle there, with less than, greater than, or equal to to make that statement a true statement.
And really what you're trying to figure out is: five-eighths, is that less than, greater than, or equal to three-fourths.
Well, here's the simple version on how to do this.
This works with all fractions being compared.
You could do cross multiplication from the bottom to the top. So watch what I do here.
8 times 3 is 24, 4 times 5 is 20.
Right now, I know this 24 right here is larger than 20.
Therefore, this fraction's going to be larger.
So how would we write that? We would say five-eighths is less than three-fourths, because three-fourths is the larger.
Remember the larger one gets eaten by the inequality sign.
Now what you really did here, is you created a fraction where the bottoms--
notice I'm taking 4 times 5 here to get 20, I'm taking 8 times 3 to get 24,
and then here's what you do to make the fraction-- the bottom, the denominator part.
You do 8 times 4, and that's 32. So you really made these two fractions right here, let me circle it in green.
You made 24/32nds and 20/32nds.
And you know that 24 is greater than 20.
Now that's common denominator, not least common denominator, but it works so why not use it? It's simple.
You can try it down here. Stop the video, come on back and see how you did.
Alright, I'll use the same method here that I used before.
Now, this one you can use the common denominator. I know that the common denominator is going to be 12,
so I'll show it what they do in the book.
What you would do is change this 4 to a 12 by multiplying by 3.
We have to multiply the top by 3 as well.
So you have nine-twelfths. So numerator is 9, denominator is 12.
You don't change the other fraction, but notice you now have 12s on the bottom.
Well which one is greater, 9/12ths or 7/12ths? Well, 9/12ths is greater.
There you go.
Let's look at B here.
This one, I won't find a common denominator, I'm just going to use--or a, least common denominator--
I'm going to use a common denominator. Just do 6 times 7, that's 42. 8 times 5, that's 40.
Those two numbers, I know that this one's going to be larger, so there you go.
Five-sixths is going to be less than this seven-eighths. Seven-eighths is the larger.
Let's look at c here. Now they give you mixed numbers.
Well, I can see that the whole number part in front is 1 on both of them, so I could just compare the 4/9ths and
2/5ths and see which one of those is bigger, and then I'll know that that particular mixed number is bigger.
So if we look at four-ninths and two-fifths, let's see here. I'll just do my cross multiplication from down to up,
5 times 4 is 20, I'll just put that number there. 9 times 2 is 18. This number is going to be larger.
So 1 and 4/9ths is going to be the larger part. One and four-ninths is going to be greater than one and two-fifths
Another version of that would be taking 9 times 1 plus the 4 and making that 13/9ths.
And then taking 5 times 1 plus 2 and that's 7/5ths, then doing the same thing.
Larger numbers, you'll get the same result.
Let's take a look at more comparing. It says, "You can also compare and order rational numbers by expressing them as decimals."
A lot of kids like to take the fractions and turn them into decimals. Not always the easiest, but you can do it that way.
Take a look at the example they have here. It says "Replace this circle with less than, greater than, or equal
to make 8/9ths, and you have to determine what goes in there to 0.8, which is eight-tenths, a true sentence."
Well, I put the whole answer up here from the book so you can kinda see what they've done.
They've taken this 8/9ths, they've turned it into a decimal. Well 8 divided by 9 is a repeating 8.
So, you have to go out so far--and I should say, and 0.8 is actually 0.8000.
So if you look at these two, you've got 8 repeating forever, and then you have 0.8 with zeros after it.
Well, take a look: in the hundreths place, which is that second spot in red, you can see that this 8 is larger
than this 0.
Therefore, this is going to be larger than this one.
And that's what that says right there. Now,
It's easier to look at it on a vertical scale, so if you put 0.8888 and again, you know, this keeps going on and on,
and then you have 0.8 and you put your zeroes in, you can see right here, which one of these is bigger?
You can almost drop off the decimal and go, "Well 888 is larger than 800."
And that's another method to do that.
Go ahead and stop the video; try these three and see how you do.
Alright, one-third. That's 0.3333 and on.
Zero point three is 0.3 and then there's nothing there.
As you can see, the 333 is larger than the 300. This 3 and this 0 are being compared,
so this 3 wins out right here over this 0. So this one's larger.
And that represents the 1/3. So one-third is going to be greater than a 0.3 or three-tenths.
Uh, let's see here, E: 0.22, again that's 0.22 and there's a zero after that.
And then 11/50ths.
Now this one, I have to try to calculate what 11/50ths is.
I would say, that's too hard. I would rather to do this version.
Can you say this point two-two? I'm saying that slang wise. Can you say it mathematically?
It's twenty-two-hundreths.
There it is. This other one is eleven-fiftieths.
Now, I can do my cross multiplication, or I can find a common denominator between 50 and 100,
And that's going to be 100 which means I have to change this by multiplying it by 2, top by 2.
That'll be 22 on top and I'd have 100 on the bottom.
And then take a look over here, this is the same one. So I don't change this one.
Now I have common denominators on the bottoms, I can look at the tops. Hey, they're the same.
So what goes in between here? An equal sign. Those are the same thing.
Alright, let's take a look at this last one.
Two and five-twelfths. Alright, well comparison-wise, I would have to change this five--if I want to do it in decimal I have to change the 5/12ths to a decimal.
Alright, so if I do 5/12ths to a decimal, I'd have to take 5 divided by 12.
How many 12s go into 5? None. Then I put a dot here, then I have my 12--how many 12s go into 50?
That'd be 4, that's 48, subtract you got 2. Bring down a zero again. Twelve goes into 20 one time
and that's going to be, uh, 12. Now you got yourself 8, bring another zero. Oh, wait, let me see here.
So I noticed this: I've got 0.41, so I've got 2 point, and then I put this part in, 41
and then we're comparing it to the 2.42. Now this does go on, there's other numbers here. But I know that this 1 is less than this 2,
so therefore, this one's going to be larger. I stopped after here, I didn't have to go any further because I noticed
that that hundreths spot, the 2 would win out.
Alright, let's take a look at the third example here. They talk about ordering rational numbers.
It says, "The average life expectancies of males for several countries are shown in the table. Order the countries from least to greatest male life expectancy."
Now if you look over on the right there, uh, they're all generally in the 70s. But some of them are written
in decimal form, and some of them are written in, uh, mixed number form.
So what are we going to do? Well the book has you change these to decimals.
Now you can compare the decimals. It wants you to look at the decimals, hopefully you can figure out which one's the smallest, which one's the greatest.
Or the least, which one's the greatest, and then organize those.
So if you look, which one's the least here?
Well, that's going to be this right here. 74.25
Doesn't look so good for us here in the United States.
And then what's the largest? Well, that's going to be 76.90
Of course, then you put these in order, and you'll notice that it says--it goes from United States, France, United Kingdom, Spain and then Australia.
Easy to compare decimals, just remember to line up your decimals when you compare.
How 'bout you give it a shot? Turn off the video here and then come on back and see how you do.
Sophia has five wrenches measuring 3/8 inch, 1/4 inch, 5/16 of an inch--I don't know why that didn't come in, there's an actual fraction bar there.
1/2 inch and 3/4 of an inch. What is the order of the measures from least to greatest?"
Well, let's change all these to decimals. Alright, so we've got 3/8ths.That's going to be, hmm.
Use my calculator in my head, that's going to be 0.375
1/4 is going to be  0.25. Let's see here, 5/16ths--that might be a tough one.
I know the 1/2. That's going to be 0.5
And then 3/4, that's going to be 0.75.
Alright, so what's this 5/16ths?
How do we do that one? Let's do that one. Five-sixteenths.
16 goes into 5 none--no times. Put our decimal point there. 16 goes into 50, let's see here, that's 3 times.
That's 48. And then we've got ourselves another zero that we bring down--oops, that's 2.
Bring down our zero, that's 20. That's one time it goes in there, that's 16. Minus out, that's 4. Here's another zero,
bring it on down, how many 16s go into 40? That's going to be 2. And then we got, uh that's 32. Subtract it out,
that's 8. There's zero, bring that down. How many 16s go into 80? Well magically that's 5 times exactly.
So 5 times 16 is another 80, subtract that out and that's when you get zero.
There it is. So it's 0.3125.
All these should have zeros in them. I'm getting kinda lazy, I'm not putting them in there.
I should.
Now let's line them up. So we've got 0.375, we got uh, 0.25, we've got the 0.3125, we've got the 0. 5, and we've
got the 0.75.
Let's see here, which one comes first?
So I would say that this from least to greatest, we're going to have--if we look down this row right here,
all of them have zeros so there's nothing to compare here. So we go to the next one.
So this won't be the smallest.
So here's, here's number--we'll call it the first, which is the least.
And then we go on to the second. Let's go here, we've got a 3 here and a 3 here, but this one has a 1 after it,
and this one has a 7 after it. So this one would be the 2nd one right here.
And then we'd go with the 3rd one right here, because that still has the 3.
Then we'd go to the 5 right here, and this would be our fourth.
And then the last one would be our fifth.
Which is 0.75.
Alright, last part.
It says, "Just as positive and negative integers can be represented on a number line, so can positive and negative rational numbers."
You can see they have number lines there with the fractions and then the decimals.
So, it says, "You can use a number line to help you compare and order negative rational numbers."
Take a look at the example. Again, looking at number four. Basically, is -2.4 less than, greater than, or equal to -2.45?
Well, graph the decimals on a number line.
You've gotta remember that this negative 2.4 is actually
a negative 2.40.
So there's where they located those. There's your -2.40,
and there's your -2.45.
Well, if you read the student tip over here, it says:
"Any number to the left is always less than a number to the right."
Since -2.4 is the right--- is to the right of -2.45,
this is how we'd state it: -2.4 is greater than -2.45.
Alright, you give it a shot here.
Oh excuse me, no, I'm going to do this example for you. Negative 7/8ths and comparing it to negative 6/8ths.
Is that going to be less than, greater than, or equal to?
Here's the key when you have negative fractions: always put the negative on the top of your fraction.
You'll see I actually put that in there, that's something you'll probably write in your notes.
So the first thing I would do is rewrite these fractions. Negative seven-eighths, what is that compared to a negative six-eighths?
So I changed the problem to this. Say, ok, what goes in that little circle? Less than, greater than, or equal to?
The negatives are now on the top, the numerator.
Well, since the denominators are the same, you notice their eighths, all you have to do is compare the numerators.
And you know that -7 is less than -6. So, -7/8 is less than -6/8.
So again, if the denominators are the same, it's real easy. Just compare those top numbers.
Let's take a look at these two examples, give your sh--self a shot at these and then come on back.
Alright, so -3.15 and -3.17. Well I know on a number line, if I am looking at negative 3.15 I know that negative 3.16
is here, and then negative 3.17 would be roughly there.
So I know this one is to the right, this one's to the left. Which means this one's bigger. This one gets eaten.
There ya go.
How 'bout when it's fractions? Again, let's put those negatives on top.
But don't get fooled. Notice the bottoms aren't the same, so you have to change the bottoms to the same.
Well let me multiply that by 2.
You've got negative 7/10 and now you have negative 8/10. Hmm. I know that -7 is bigger than -8,
because it's to the right of -8 on a number line.
So this one would have a greater than sign in it.
Don't forget, you can rewatch this video, read the examples in the book, or of course watch any of the personal tutor videos on the online text book.
And don't forget, this is always a wonderful Friday Shoes Production.
