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## 7th grade (Illustrative Mathematics)

### Course: 7th grade (Illustrative Mathematics) > Unit 5

Lesson 1: Lesson 1: Interpreting negative numbers# Comparing rational numbers

Comparing Rational Numbers.

## Want to join the conversation?

- Is "zero" is a Rational number?

Is "zero" Is Negative or Positive?

(90 votes)- Zero is rational since it has no decimal places. It is neither positive nor negative. Positive is defined as being more than 0 and negative as less than 0, and clearly zero is neither more or less than itself.(128 votes)

- Question: How does Sal put his screen like that and change the colors and stuff and how does he write SO good with a mouse. Like I try and it's just awful.(16 votes)
- Me three!

But he does it on a tablet, that's why it looks so neat. And he probably uses a touch screen pen or something.(14 votes)

- What IS a rational number? Is it a negative number?(5 votes)
- A rational number is a ratio such as 8 in 8/1.Howeversome of these numbers can be negative number such as -8 in -8/1.(11 votes)

- We're told to look at the rational numbers below, order them from least to greatest. They really didn't have to tell us this first sentence, I would have known to look at the rational numbers to order them from least to greatest. Well anyway, they tell us 1/2, negative five, three point three, zero, 21 over 12, negative five point five, and two and 1/8ths. So the easiest way to visualize this might just be to make a number line that's long enough that it actually can contain all of these numbers, and then we can think about how we can compare them. So let me just draw a huge number line over here. So, take up almost all the entire screen. I'll stick with, we have some negative numbers here, we go as low as negative five point five, and we have some positive numbers here, looks like we go as high as three point three. This thing is still a little less than two, so we go about as high as three point three, so I can put, I can safely I think put zero right here in the middle, I can go a little bit to the right since we have our negative numbers go more negative. So zero, and let's make this negative one, negative two, negative three, negative four, negative five, and well that should be enough. Negative five, and then in the positive direction we have, one, two, three, in the positive direction. And let's see if we can plot these. So, to start off, to start off let's look at 1/2. Where does 1/2 sit, so it sits, let me actually make the scale a little bit better. So this is one, two and three, and four. Alright, so let's start with 1/2. 1/2 is directly in-between zero and one, it is half of a whole. This right here would be one whole. This would be one whole, let me label that. This over here is one. So 1/2 is directly between zero and one. So 1/2 is gonna sit right over here. So that is, let me write that a little bit bigger, you probably have trouble reading that, Alright, one over two, which is also zero point five. So this is also zero point five, anyway that's where it sits. Then we have negative five. Negative five, well this is negative one, negative two, negative three, negative four, negative five. Negative five sits right over there. And then we have three point three. Positive three point three, I'll do that in blue. Positive three point three. So this is one, two, three, and then we want to do another point three. So point three is about a third of the way, a little less than a third of the way, it would be three point three three three forever, if it was a third of the way. So a third of the way, that looks like about right over here. This is three, this right over here would be three point three, let me label. What I'm gonna do is I'm gonna label the numbers on the number line up here so it's one, two, three, four, this is zero, negative one, negative two, negative three, negative four, negative five, and so on and so forth. And then we get to zero, which is one of the numbers that we've already written down. Zero is obviously right over there on the number line, so I'll just write this zero in orange to make it clear, it's this zero. Then we have 21 over 12, which is an improper fraction, and to think about where we should place that on the number line, to think about where to place it on the number line, let me do this in this blue color. To think about where to place this on the number line let's change it into a mixed number, makes it a little bit easier to visualize, at least for my brain. So 12 goes into 21, well it goes into it one time. One times 12 is 12. If you subtract you get a remainder of, well we could actually regroup here, or borrow, if you don't want to do this in your head, you would get nine, but let's do this. So if we borrow one from the two, the two becomes a one, this becomes 11, or we're really regrouping a 10. Anyway, 11 minus two is nine, one minus one is zero. So we have a remainder of nine. So this thing, written as a mixed number, 21 over 12 written as a mixed number is one and 9/12ths. You get one 12/12ths in there and then you get 9/12ths left over. So one and 9/12ths we can also write that, actually we could've simplified this right from the get go, cause both 21 and 12 are divisible by three, but now we can just divide nine, we can simplify 9/12ths, divide both the numerator and the denominator by three, we then get one and three over four, one and 3/4ths. And just to make it clear, I could have simplified this right from the get go, 21 divided by three, is equal to seven, and 12 divided by three, is equal to four. So this is the same thing as 7/4ths, and if you were to divide four into seven, four goes into seven one time, subtract, one times four is four, subtract to get a remainder of three, one and 3/4ths. So going back to where do we plot this? Well it is, it's one, and then we have 3/4ths, we're going to go three fourths of the way. This is half way, this is one fourths, two fourths, three fourths, would be right over there. So this is our 21 over 12, which is the same thing as 7/4ths, which is the same thing as one and 3/4ths. And then we have negative five point five. Negative five point five, I'll do that in magenta again, running out of colors. Negative five point five, well this is negative five, so negative five point five is going to be between negative five and negative six. So let me add negative six to our number line, right here just to make it clear. So let me go a little bit further, let's say that this is negative six. Negative six, and our number line will keep going to smaller values. Let me scroll to the left a little bit. Negative six, so if we go to negative five point five, it's smack dab in-between negative five and negative six. So this is negative five point five, right over there. And then finally we have two and 1/8ths. I'll do that in orange again, or I'll do it in blue. Two and 1/8ths, so it's two and then 1/8th. And so if we want to find the exact place we could divide this into eighths, this would be 4/8ths, this would be 2/8ths, and that would be 6/8ths, and then 1/8th would sit right over here. So that right over there is two and 1/8th. So we've actually plotted, as best as we could, the exact locations. You didn't have to plot the exact locations if you were just trying to order them, but it doesn't hurt to see exactly where they sit when we order them. So now we've essentially ordered them cause we stuck them all on this number line. The order is negative five point five is the smallest, then negative five, then a zero, and then positive 1/2, then 21 over 12, then two and 1/8th, and then three point three. And we're done.(4 votes)
- what do you mean by the consept of the equation(5 votes)
- is there a way without a number line(3 votes)
- Yes, a number line makes it a lot less confusing(5 votes)

- We're no strangers to love

You know the rules and so do I (do I)

A full commitment's what I'm thinking of

You wouldn't get this from any other guy

I just wanna tell you how I'm feeling

Gotta make you understand

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

We've known each other for so long

Your heart's been aching, but you're too shy to say it (say it)

Inside, we both know what's been going on (going on)

We know the game and we're gonna play it

And if you ask me how I'm feeling

Don't tell me you're too blind to see

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

We've known each other for so long

Your heart's been aching, but you're too shy to say it (to say it)

Inside, we both know what's been going on (going on)

We know the game and we're gonna play it

I just wanna tell you how I'm feeling

Gotta make you understand

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you(3 votes) - why dont they give me more help with the math.(4 votes)
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⠀⠀⠀⠀⠀⠀⣸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡛⠛⠉⠀⠀⠀

⠀⠀⠀⢀⣴⣿⠋⠁⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠈⠙⠻⣶⣄⠀⠀⠀

⠀⢀⣴⣿⣯⣴⡆⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢰⣶⣬⣻⣷⡀⠀

⠀⣾⠏⢸⣿⠿⠃⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠈⠻⢿⡿⠹⣿⡀

⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣤⣤⣀⣀⠀⢀⣀⣠⣤⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇

⠀⠻⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⣿⠟⠛⠻⣿⣯⣥⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠃

⠀⠀⠀⠉⠛⠛⠿⠿⠶⠶⠶⠶⠶⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠶⠶⠶⠶⠶⠾⠿⠛⠛⠋

this is the all mighty kirby(4 votes) - We're no strangers to love

You know the rules and so do I (do I)

A full commitment's what I'm thinking of

You wouldn't get this from any other guy

I just wanna tell you how I'm feeling

Gotta make you understand

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

We've known each other for so long

Your heart's been aching, but you're too shy to say it (say it)

Inside, we both know what's been going on (going on)

We know the game and we're gonna play it

And if you ask me how I'm feeling

Don't tell me you're too blind to see

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

We've known each other for so long

Your heart's been aching, but you're too shy to say it (to say it)

Inside, we both know what's been going on (going on)

We know the game and we're gonna play it

I just wanna tell you how I'm feeling

Gotta make you understand

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you

Never gonna give you up

Never gonna let you down

Never gonna run around and desert you

Never gonna make you cry

Never gonna say goodbye

Never gonna tell a lie and hurt you(4 votes)

## Video transcript

- We're told to look at
the rational numbers below, order them from least to greatest. They really didn't have to
tell us this first sentence, I would have known to look
at the rational numbers to order them from least to greatest. Well anyway, they tell
us 1/2, negative five, three point three, zero, 21 over 12, negative five point
five, and two and 1/8ths. So the easiest way to
visualize this might just be to make a number line that's long enough that it actually can contain
all of these numbers, and then we can think about
how we can compare them. So let me just draw a huge
number line over here. So, take up almost all the entire screen. I'll stick with, we have
some negative numbers here, we go as low as negative five point five, and we have some positive numbers here, looks like we go as high
as three point three. This thing is still a
little less than two, so we go about as high
as three point three, so I can put, I can safely I think put zero right here in the middle, I can go a little bit to
the right since we have our negative numbers go more negative. So zero, and let's make this negative one, negative two,
negative three, negative four, negative five, and well
that should be enough. Negative five, and then in the
positive direction we have, one, two, three, in
the positive direction. And let's see if we can plot these. So, to start off, to start
off let's look at 1/2. Where does 1/2 sit, so it sits, let me actually make the
scale a little bit better. So this is one, two and three, and four. Alright, so let's start with 1/2. 1/2 is directly in-between zero and one, it is half of a whole. This right here would be one whole. This would be one whole,
let me label that. This over here is one. So 1/2 is directly between zero and one. So 1/2 is gonna sit right over here. So that is, let me write
that a little bit bigger, you probably have trouble reading that, Alright, one over two, which is also zero point five. So this is also zero point five, anyway that's where it sits. Then we have negative five. Negative five, well this is
negative one, negative two, negative three, negative
four, negative five. Negative five sits right over there. And then we have three point three. Positive three point three,
I'll do that in blue. Positive three point three. So this is one, two, three, and then we want to do
another point three. So point three is about
a third of the way, a little less than a third of the way, it would be three point
three three three forever, if it was a third of the way. So a third of the way, that looks like about right over here. This is three, this
right over here would be three point three, let me label. What I'm gonna do is I'm
gonna label the numbers on the number line up here
so it's one, two, three, four, this is zero,
negative one, negative two, negative three, negative
four, negative five, and so on and so forth. And then we get to zero,
which is one of the numbers that we've already written down. Zero is obviously right over
there on the number line, so I'll just write this zero
in orange to make it clear, it's this zero. Then we have 21 over 12,
which is an improper fraction, and to think about where
we should place that on the number line, to think about where to
place it on the number line, let me do this in this blue color. To think about where to
place this on the number line let's change it into a mixed
number, makes it a little bit easier to visualize,
at least for my brain. So 12 goes into 21, well
it goes into it one time. One times 12 is 12. If you subtract you get a remainder of, well we could actually
regroup here, or borrow, if you don't want to do this in your head, you would get nine, but let's do this. So if we borrow one from the
two, the two becomes a one, this becomes 11, or we're
really regrouping a 10. Anyway, 11 minus two is
nine, one minus one is zero. So we have a remainder of nine. So this thing, written as
a mixed number, 21 over 12 written as a mixed number
is one and 9/12ths. You get one 12/12ths in there and then you get 9/12ths left over. So one and 9/12ths we can also write that, actually we could've simplified
this right from the get go, cause both 21 and 12
are divisible by three, but now we can just divide
nine, we can simplify 9/12ths, divide both the numerator and the denominator by three, we then get one and three
over four, one and 3/4ths. And just to make it clear,
I could have simplified this right from the get
go, 21 divided by three, is equal to seven, and
12 divided by three, is equal to four. So this is the same thing as 7/4ths, and if you were to divide four into seven, four goes into seven one time, subtract, one times four is four, subtract to get a remainder
of three, one and 3/4ths. So going back to where do we plot this? Well it is, it's one,
and then we have 3/4ths, we're going to go three
fourths of the way. This is half way, this is
one fourths, two fourths, three fourths, would be right over there. So this is our 21 over 12,
which is the same thing as 7/4ths, which is the same
thing as one and 3/4ths. And then we have negative five point five. Negative five point five,
I'll do that in magenta again, running out of colors. Negative five point five,
well this is negative five, so negative five point
five is going to be between negative five and negative six. So let me add negative
six to our number line, right here just to make it clear. So let me go a little bit further, let's say that this is negative six. Negative six, and our
number line will keep going to smaller values. Let me scroll to the left a little bit. Negative six, so if we go
to negative five point five, it's smack dab in-between
negative five and negative six. So this is negative five point five, right over there. And then finally we have two and 1/8ths. I'll do that in orange again, or I'll do it in blue. Two and 1/8ths, so it's
two and then 1/8th. And so if we want to find the exact place we could divide this into eighths, this would be 4/8ths,
this would be 2/8ths, and that would be 6/8ths, and then 1/8th would sit right over here. So that right over there is two and 1/8th. So we've actually plotted,
as best as we could, the exact locations. You didn't have to plot
the exact locations if you were just trying to order them, but it doesn't hurt to
see exactly where they sit when we order them. So now we've essentially
ordered them cause we stuck them all on this number line. The order is negative five
point five is the smallest, then negative five, then a zero, and then positive 1/2, then 21
over 12, then two and 1/8th, and then three point three. And we're done.