Frequency tables and dot plots

WEBVTT
Kind: captions
Language: en

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- [Voiceover] What I have
here is a list of ages

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of the students in a class.

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And what I wanna explore in this video

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is different ways of
representing this data,

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and then see if we can answer
questions about the data.

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The first way we can think about
it is as a frequency table.

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Frequency table.

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Frequency table.

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What we're gonna do is,
we're gonna look at each ...

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for each age, for each possible age

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that we've measured here,

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see how many students in
the class are of that age.

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So we could say, the age is one column,

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and then the number,

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the number of students of that age ...

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Or we could even say, the frequency.

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Frequency.

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When people say, "How
frequent do you do something?"

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They're saying, "How often does it happen?

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"How often do you do that thing?"

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Frequency.

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Or we could also say ...

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Actually, I'll just write "number."

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I'm always a fan of the simpler.

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Number at age, which
we could also consider

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the frequency at that age.

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Frequency of students.

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All right. So what's the
lowest age that we have here?

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Well, the lowest age is five.

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So I'll start with five.

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And how many students in
the class are age five?

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How frequent is the number five?

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Let's see, there is one, two.

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Let me keep scanning.

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Looks like there's only two fives.

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So I could write a two here.

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There are two fives.

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And now let's go to six.

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How many sixes are there?

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Let's see, there is one six.

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There's only one
six-year-old in the class.

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All right. Seven-year-olds.

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See, there's one, two,

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three, four seven-year-olds.

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Four seven-year-olds.

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Now, what about eight-year-olds?

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Eight-year-olds, I'm gonna use a color

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that I have not used yet.

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Eight-year-olds, we
have no eight-year-olds.

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Zero eight-year-olds.

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And then we have nine-year-olds.

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Let's see.

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Nine-year-olds, we have one, two, three,

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four nine-year-olds.

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Four nine-year-olds.

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10-year-olds? What do we have?

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We have one 10-year-old, right over there.

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And then 11-year-olds.

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11-year-olds, there are no 11-year-olds.

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And then, let me scroll up a little bit.

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And then finally, 12-year-olds.

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12-year-olds, there are
one, two 12-year-olds.

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So what we have just constructed
is a frequency table.

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It's a frequency table.

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You can see, you can see for each age,

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how many students are at that age?

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So it's giving you the same
information as we have up here.

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You could take this table

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and construct what we have up here.

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You would just write down two fives,

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one six, four sevens, no eights,

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four nines, one 10, no 11s, and two 12s,

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and then you would just
have this list of numbers.

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Now, a way to visually
look at a frequency table

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is a dot plot.

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So let me draw a dot plot right over here.

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A dot plot.

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And a dot plot, we essentially just take

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the same information, and even
think about it the same way.

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But we just show it visually.

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In a dot plot, what we would have ...

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In a dot plot, what we would have ...

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Actually, let me just not
draw an even arrow there.

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We have the different age groups,

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so five, six, seven,

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eight, nine, 10,

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11, and 12,

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and we have a dot to represent,

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or we use a dot for each
student at that age.

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So there's two five-year-olds,
so I'll do two dots.

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One, and two.

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There's one six-year-old,

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so that's gonna be one dot,

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right over here.

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There's four seven-year-olds,

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so one, two,

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three, four dots.

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There's no eight-year-olds.

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There's four nine-year-olds,

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so one, two,

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three, and four.

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There's one 10-year-old.

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So let's put a dot, one dot,

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right over there for that one 10-year-old.

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There's no 11-year-olds.

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I'm not gonna put any dots there.

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And then there's two 12-year-olds.

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So one 12-year-old,

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and another 12-year-old.

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So there you go, we
have a frequency table,

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dot plot, list of numbers.

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These are all showing the same data,

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just in different ways.

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Once you have it represented
in any of these ways,

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we can start to ask questions about it.

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So we could say, "What is
the most frequent age?"

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Well, the most frequent age,

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when you look at it visually,

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or the easiest thing might be

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just to look at the dot plot
because you see visually,

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the most frequent age are
the two highest stacks.

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There's actually seven and nine

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are tied for the most frequent age.

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You'd have also seen it here,

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where seven and nine are tied at four.

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And if you just had this data,

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you would actually, you'd
have to count all of them

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to kind of come up with this again

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and say, "Okay, there's
four sevens, four nines.

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"That's the largest number."

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So this is, if you're looking for,

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what's the most frequent age?

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When you just visually inspect here,

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probably pops out at you the fastest.

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But there's other questions
we can ask ourselves.

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We can ask ourselves,

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"What is the range?

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"What is the range of
ages in the classroom?"

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And this is once again
where maybe the dot plot

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jumps out at you the most,

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because the range is
just the maximum age ...

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or, the maximum data point
minus the minimum data point.

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So what's the maximum age here?

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Well, the maximum age here,

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we see it from the dot plot, is 12.

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And the minimum age
here, you see, is five.

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So there's a range of seven.

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The difference between the
maximum and the minimum is seven.

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But you could have also
done that over here.

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You could say, "The
maximum age here is 12.

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"Minimum age here is five.

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"And so let's subtract ..."

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You find the difference between
12 and five, which is seven.

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Here, you'd have done ...

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You still could have done it.

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You'd say, "Okay, what's the lowest?

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"Let's see, five. Are
there any fours here?

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"Nope, there's no fours.

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"So five's the minimum age.

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"And what's the largest?

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"Is it seven? No.

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"Is it nine? Nine, maybe 10.

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"Oh, 12. 12.

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"Are there any 13s? No.

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"12 is the maximum."

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So you say, "12 minus five
is seven" to get the range.

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But then we could ask
ourselves other questions.

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We could say, "How many ...

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"How many older ...

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"older than nine?" is a question

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we could ask ourselves.

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And then, if we were to
look at the dot plot,

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we'd say, "Okay, this is nine."

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And we'd care about how
many are older than nine.

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So that would be this one, two and three.

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Or you could look over here.

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How many are older than nine?

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Well, it's the one person who's 10

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and then the two who are 12.

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So there are three.

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And over here, if you said,
"How many are older than nine?"

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Well, then you'd just have
to go through the list

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and say, "Okay, no,
no, no, no, no, no, no,

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"okay, here, one, two, three."

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And then not that person right over there.

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So hopefully you ...

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This is just an appreciation

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for yet another two
ways of looking at data,

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frequency tables and dot plots.


Values to make absolute value inequality true

WEBVTT
Kind: captions
Language: en

00:00:00.267 --> 00:00:02.523
- [Voiceover] We have three
inequalities here that involve

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absolute values and then
below them we have potential

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values for x,

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and what I want to do in this
video is see which of these

00:00:09.654 --> 00:00:13.270
potential values for x actually
make the inequality true.

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So let's start with
this first one in green,

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here on the left.

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It says the absolute
value of x is less than

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the absolute value of negative seven.

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So let's think about
which of these x values

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would make this true.

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And before I even try out the x values,

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let's see if we can
simplify this inequality.

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Now the thing that might jump
out at you and encourage you

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to try these on your own
before I work through them,

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what might jump out at you is that we know

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what the absolute value
of negative seven is.

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The absolute value of
negative seven is how far

00:00:40.068 --> 00:00:43.116
from zero is negative seven?

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Well, it's seven to the left.

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The distance from negative
seven to zero is seven,

00:00:47.382 --> 00:00:49.109
or another way to think about it is

00:00:49.109 --> 00:00:53.051
the absolute value of any
number is always going to be

00:00:53.051 --> 00:00:55.225
the non-negative version of it.

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So this right over here,
absolute value of negative seven

00:00:58.415 --> 00:01:00.548
is just going to be seven.

00:01:00.548 --> 00:01:02.844
So an equivalent inequality
would be that the

00:01:02.844 --> 00:01:06.724
absolute value of x needs
to be less than seven.

00:01:06.724 --> 00:01:09.873
So let's see if its true
for x equals negative eight.

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So if x equals negative eight,

00:01:11.864 --> 00:01:14.953
then wherever we see the x we
put a negative eight there.

00:01:14.953 --> 00:01:17.309
So let's see, is the absolute
value of negative eight

00:01:17.309 --> 00:01:20.560
less than seven?

00:01:20.560 --> 00:01:22.003
Is that true?

00:01:22.003 --> 00:01:23.750
Well, the absolute value of negative eight

00:01:23.750 --> 00:01:26.086
is just going to be eight,

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so is eight less than seven?

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No, it's greater than seven.

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So that, x equals negative eight does not

00:01:32.507 --> 00:01:34.579
satisfy the inequality.

00:01:34.579 --> 00:01:37.078
Now x equals negative two?

00:01:37.078 --> 00:01:39.089
Well, wherever we see an x,
let's put a negative two.

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So the absolute value
of negative two needs

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to be less than seven.

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What's the absolute value of negative two?

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Well it's going to be positive two.

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Is positive two less than seven?

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Sure, two is less than seven.

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So this, this is x equals negative two,

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satisfies our inequality.

00:02:00.079 --> 00:02:03.147
The absolute value of negative
two is going to be less than

00:02:03.147 --> 00:02:05.220
the absolute value of negative seven.

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Then, finally, x equals six.

00:02:07.658 --> 00:02:10.340
So the absolute value is
the absolute value of six.

00:02:10.340 --> 00:02:13.123
Once again, everywhere I see
the x I just put the six there.

00:02:13.123 --> 00:02:14.484
X equals six.

00:02:14.484 --> 00:02:15.906
We're going to say the
absolute value of six,

00:02:15.906 --> 00:02:18.731
is that less than seven?

00:02:18.731 --> 00:02:21.839
Well the absolute value of
six is once again, just six,

00:02:21.839 --> 00:02:24.419
Six is six to the right of zero.

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Is six less than seven?

00:02:26.248 --> 00:02:28.787
Yeah, sure, six is less than seven.

00:02:28.787 --> 00:02:32.201
So x equals six and x equals negative two

00:02:32.201 --> 00:02:35.100
both satisfy the inequality.

00:02:35.100 --> 00:02:38.595
Now let's do this one here
in this magenta color.

00:02:38.595 --> 00:02:40.586
And once again, encourage
you to pause the video

00:02:40.586 --> 00:02:42.719
and try to work through
it out on your own.

00:02:42.719 --> 00:02:44.954
Let's try x equals negative four.

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So if x equals negative four,

00:02:47.392 --> 00:02:49.769
we're going to say the
absolute value of negative four

00:02:49.769 --> 00:02:52.105
is greater than five.

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Absolute value of negative four,

00:02:53.670 --> 00:02:55.559
well that's just going to be four.

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Is four greater than five?

00:02:58.871 --> 00:03:00.801
No, four is less than five.

00:03:00.801 --> 00:03:02.975
So that doesn't work.

00:03:02.975 --> 00:03:04.499
Now, x equals three.

00:03:04.499 --> 00:03:07.120
Everywhere we see the x,
replace it with a three.

00:03:07.120 --> 00:03:08.644
The absolute value of three,

00:03:08.644 --> 00:03:10.391
is that greater than five?

00:03:10.391 --> 00:03:12.626
Absolute value of three is just three,

00:03:12.626 --> 00:03:14.353
so is three greater than five?

00:03:14.353 --> 00:03:16.791
No, three is less than five.

00:03:16.791 --> 00:03:18.599
I think you're seeing, hopefully you're

00:03:18.599 --> 00:03:19.899
getting the hang of it.

00:03:19.899 --> 00:03:22.043
So finally, if x equals negative nine.

00:03:22.043 --> 00:03:25.029
The absolute value of
negative nine, that would

00:03:25.029 --> 00:03:27.244
need to be greater than five.

00:03:27.244 --> 00:03:29.215
Absolute value of negative nine,

00:03:29.215 --> 00:03:30.942
well that's just going
to be positive nine.

00:03:30.942 --> 00:03:32.404
So it's just going to be nine,

00:03:32.404 --> 00:03:34.802
and is that greater than five?

00:03:34.802 --> 00:03:36.915
Well, sure, nine is greater than five.

00:03:36.915 --> 00:03:40.531
So x equals negative nine
satisfies the inequality.

00:03:40.531 --> 00:03:42.725
All right, now let's do these ones

00:03:42.725 --> 00:03:44.981
in this light purple color.

00:03:44.981 --> 00:03:48.333
The absolute value of x
needs to be greater than

00:03:48.333 --> 00:03:49.877
negative 16.

00:03:49.877 --> 00:03:52.600
So there's something very
interesting about this one.

00:03:52.600 --> 00:03:54.469
We don't even have to
look at the choices here.

00:03:54.469 --> 00:03:58.918
Can you think of any x for
which this would not be true?

00:03:58.918 --> 00:04:01.194
Well just think about it a little bit.

00:04:01.194 --> 00:04:02.799
The absolute value of a number,

00:04:02.799 --> 00:04:05.471
is it ever going to be negative?

00:04:05.471 --> 00:04:08.376
No, the absolute value
of a number is going

00:04:08.376 --> 00:04:10.997
to be zero or positive.

00:04:10.997 --> 00:04:12.541
It's going to be non-negative,

00:04:12.541 --> 00:04:16.360
so this right over here
is zero or positive,

00:04:16.360 --> 00:04:18.514
or we could call that non-negative.

00:04:18.514 --> 00:04:21.338
Zero or positive.

00:04:21.338 --> 00:04:23.878
So if this thing over
here is zero or positive,

00:04:23.878 --> 00:04:26.174
something that is zero or greater,

00:04:26.174 --> 00:04:28.815
something that is zero
or positive is always

00:04:28.815 --> 00:04:30.989
going to be greater
than a negative number.

00:04:30.989 --> 00:04:34.869
So this is actually true for all x's.

00:04:34.869 --> 00:04:36.109
We don't even have to try them out.

00:04:36.109 --> 00:04:37.876
We could try them out,
actually we will try them out,

00:04:37.876 --> 00:04:38.912
just so you see that,

00:04:38.912 --> 00:04:40.436
but it's going to be true for all x's,

00:04:40.436 --> 00:04:41.818
because when you take the absolute value,

00:04:41.818 --> 00:04:44.560
if you take, if x is zero
it's going to be zero,

00:04:44.560 --> 00:04:47.445
but for any other value,
any non-zero value of x,

00:04:47.445 --> 00:04:50.351
the absolute value of it
is going to be positive.

00:04:50.351 --> 00:04:52.382
And let's just see that,
we could put any x there,

00:04:52.382 --> 00:04:54.536
and this statement's going to be true.

00:04:54.536 --> 00:04:57.015
Absolute value of x equals negative 15.

00:04:57.015 --> 00:04:58.762
Well, the absolute value of negative 15,

00:04:58.762 --> 00:05:01.342
is that greater than negative 16?

00:05:01.342 --> 00:05:02.744
Well, the absolute value of negative 15

00:05:02.744 --> 00:05:05.832
is positive 15, and of course positive 15

00:05:05.832 --> 00:05:07.783
is going to be greater than negative 16.

00:05:07.783 --> 00:05:10.343
A positive is always going to
be greater than a negative.

00:05:10.343 --> 00:05:12.090
So this is true.

00:05:12.090 --> 00:05:14.518
If x equals three,
absolute value of three,

00:05:14.518 --> 00:05:16.326
is that greater than negative 16?

00:05:16.326 --> 00:05:18.399
Oh, whoops let me write it that way.

00:05:18.399 --> 00:05:22.584
Absolute value of three, is
that greater than negative 16?

00:05:22.584 --> 00:05:24.291
Sure, absolute value of three is three.

00:05:24.291 --> 00:05:26.853
And three is positive, so it's going to be

00:05:26.853 --> 00:05:27.877
greater than a negative number.

00:05:27.877 --> 00:05:29.075
So that works.

00:05:29.075 --> 00:05:31.109
And as I said, any x would work there.

00:05:31.109 --> 00:05:33.060
And finally if x equals nine.

00:05:33.060 --> 00:05:36.432
Well, absolute value of
nine, if x equals nine,

00:05:36.432 --> 00:05:39.317
is that greater than negative 16?

00:05:39.317 --> 00:05:41.085
Well sure, that's just nine,

00:05:41.085 --> 00:05:43.503
and that is greater than negative 16.

00:05:43.503 --> 00:05:45.799
Even if x was zero, then
you would have zero is

00:05:45.799 --> 00:05:49.923
greater than negative 16,
which is absolutely true.

00:05:49.923 --> 00:05:53.923
So any x here works.


Comparing absolute values on number line

WEBVTT
Kind: captions
Language: en

00:00:00.525 --> 00:00:03.233
- [Voiceover] What I have
here are three numbers

00:00:03.233 --> 00:00:04.800
plotted on the number line.

00:00:04.800 --> 00:00:08.767
We have the number a, the
number c, the number b.

00:00:08.767 --> 00:00:11.100
And then we have three -- (laughs)

00:00:11.100 --> 00:00:13.133
we have four inequalities, actually.

00:00:13.133 --> 00:00:15.900
Four inequalities that
involve absolute value.

00:00:15.900 --> 00:00:17.199
And what I want to do is figure out

00:00:17.199 --> 00:00:19.900
which is these inequalities are true,

00:00:19.900 --> 00:00:24.233
given where a, c and b
are on the number line.

00:00:24.233 --> 00:00:26.017
And I encourage you to pause the video

00:00:26.017 --> 00:00:29.240
and try to think through it on your own.

00:00:29.240 --> 00:00:30.667
All right, let's look at this first one.

00:00:30.667 --> 00:00:33.166
It says that, "a is less than b."

00:00:33.166 --> 00:00:35.766
So if we look at a and we look at b,

00:00:35.766 --> 00:00:39.667
a is clearly to the left
of b on the number line.

00:00:39.667 --> 00:00:42.933
So we know that this is true.

00:00:42.933 --> 00:00:44.333
Even more we know that a is negative,

00:00:44.333 --> 00:00:45.400
it's to the left of zero,

00:00:45.400 --> 00:00:46.801
while b is positive.

00:00:46.801 --> 00:00:48.433
Which is, if one thing is negative and

00:00:48.433 --> 00:00:49.333
the other thing is positive,

00:00:49.333 --> 00:00:50.866
the negative thing is
definitely going to be

00:00:50.866 --> 00:00:53.200
less than the positive thing.

00:00:53.200 --> 00:00:54.900
But even easier than that,

00:00:54.900 --> 00:00:57.401
a is to the left of b on the number line.

00:00:57.401 --> 00:00:59.366
If you're to the left of
something else on the number line

00:00:59.366 --> 00:01:01.200
you're less than that other thing.

00:01:01.200 --> 00:01:02.133
Because the number line,

00:01:02.133 --> 00:01:03.634
at least the way we've constructed it,

00:01:03.634 --> 00:01:07.113
it increases from left to right.

00:01:07.113 --> 00:01:08.233
All right, the next statement,

00:01:08.233 --> 00:01:10.466
"The absolute value of a is

00:01:10.466 --> 00:01:13.735
greater than the absolute value of b."

00:01:13.735 --> 00:01:14.433
Well, let's just think about

00:01:14.433 --> 00:01:16.496
where these are on the number line.

00:01:16.496 --> 00:01:19.600
So we've already said
a is three hash marks

00:01:19.600 --> 00:01:21.667
to the left of zero.

00:01:21.667 --> 00:01:23.694
That is a.

00:01:23.694 --> 00:01:26.202
So what is going to be
the absolute value of a?

00:01:26.202 --> 00:01:27.833
Well, the absolute value of a is the

00:01:27.833 --> 00:01:29.800
distance that a is from zero.

00:01:29.800 --> 00:01:31.900
So the distance that a is from zero is

00:01:31.900 --> 00:01:35.330
one, two, three hash marks.

00:01:35.330 --> 00:01:37.984
So the absolute value
of a is just going to be

00:01:37.984 --> 00:01:41.000
that same distance on the positive side.

00:01:41.000 --> 00:01:43.467
So the point that we marked as c

00:01:43.467 --> 00:01:47.333
is also the absolute value of a.

00:01:47.333 --> 00:01:49.866
So that is also the absolute value of a.

00:01:49.866 --> 00:01:51.366
The absolute value of a -- sorry,

00:01:51.366 --> 00:01:52.934
a is three to the left of zero.

00:01:52.934 --> 00:01:56.167
Absolute value of a is going
to be three to the right.

00:01:56.167 --> 00:02:00.800
It's just a measure of, how
many hash marks is it from zero?

00:02:00.800 --> 00:02:02.466
Well, it's three hash marks from zero

00:02:02.466 --> 00:02:04.766
so we put it right over here.

00:02:04.766 --> 00:02:07.133
So is the absolute value of a greater than

00:02:07.133 --> 00:02:08.800
the absolute value of b?

00:02:08.800 --> 00:02:10.840
Or what's the absolute value of b?

00:02:10.840 --> 00:02:15.840
Well, b is one, two, three,
four, five, six, seven,

00:02:16.066 --> 00:02:19.867
eight hash marks to the right of zero.

00:02:19.867 --> 00:02:21.933
And so the absolute value of b

00:02:21.933 --> 00:02:24.033
is going to be on the eighth hash mark.

00:02:24.033 --> 00:02:25.767
Because it's eight hash
marks to the right.

00:02:25.767 --> 00:02:28.433
So this is also the absolute value of b.

00:02:28.433 --> 00:02:29.533
And this is consistent with

00:02:29.533 --> 00:02:32.033
what we've learned about absolute value.

00:02:32.033 --> 00:02:33.732
Absolute value of a positive number

00:02:33.732 --> 00:02:35.400
is just going to be that number again.

00:02:35.400 --> 00:02:37.433
Absolute value of a negative number

00:02:37.433 --> 00:02:40.466
is going to be the
opposite of that number.

00:02:40.466 --> 00:02:43.333
And absolute value of zero
is just going to be zero.

00:02:43.333 --> 00:02:44.600
So is the absolute value of a

00:02:44.600 --> 00:02:46.596
greater than the absolute value of b?

00:02:46.596 --> 00:02:47.367
Well, no.

00:02:47.367 --> 00:02:49.900
Absolute value of a is to the left

00:02:49.900 --> 00:02:52.400
of the absolute value
of b on our number line.

00:02:52.400 --> 00:02:55.362
It is less than the absolute value of b.

00:02:55.362 --> 00:02:58.300
So this is not true.

00:02:58.300 --> 00:02:59.733
All right, next statement.

00:02:59.733 --> 00:03:03.463
"Absolute value of a is less
than the absolute value of c."

00:03:03.463 --> 00:03:05.918
Well, we already know that
the absolute value of a

00:03:05.918 --> 00:03:08.684
is the same value as c.

00:03:08.684 --> 00:03:10.525
So what's the absolute value of c?

00:03:10.525 --> 00:03:11.866
Well, the absolute value
of a positive number

00:03:11.866 --> 00:03:13.234
is just going to be that number.

00:03:13.234 --> 00:03:15.000
So this point right over here is also

00:03:15.000 --> 00:03:17.566
the absolute value of c.

00:03:17.566 --> 00:03:19.266
So we see that the absolute value of a

00:03:19.266 --> 00:03:22.166
is equal to the absolute value of c.

00:03:22.166 --> 00:03:23.533
It's not less than.

00:03:23.533 --> 00:03:26.900
So we are going to mark that off.

00:03:26.900 --> 00:03:29.133
We could have written,
"Absolute value of a

00:03:29.133 --> 00:03:30.433
is equal to absolute value of c."

00:03:30.433 --> 00:03:32.133
That would have been true.

00:03:32.133 --> 00:03:34.433
All right, last one.

00:03:34.433 --> 00:03:36.761
a is less than c.

00:03:36.761 --> 00:03:41.100
Well, once again a is to the
left of c on the number line.

00:03:41.100 --> 00:03:43.934
So that is true, because our
number line is increasing

00:03:43.934 --> 00:03:45.733
as we go from left to right.

00:03:45.733 --> 00:03:47.267
If one number is to the
left of another number,

00:03:47.267 --> 00:03:49.066
it is less than the other number.

00:03:49.066 --> 00:03:52.467
So a is indeed less than c.

00:03:52.467 --> 00:03:54.262
And we are done.


Sorting absolute values on number line

WEBVTT
Kind: captions
Language: en

00:00:01.230 --> 00:00:03.620
What we have here is four different values

00:00:03.666 --> 00:00:05.603
some of them expressed with negative numbers

00:00:05.603 --> 00:00:09.153
or absolute value, and what I want to do is sort

00:00:09.243 --> 00:00:11.786
them from least to greatest, where the least is on

00:00:11.786 --> 00:00:14.053
the left and the greatest is on the right. So sort

00:00:14.053 --> 00:00:15.513
from least to greatest.

00:00:15.893 --> 00:00:18.183
And like always, I encourage you to pause the

00:00:18.183 --> 00:00:20.363
video and try to work through it on your own.

00:00:20.983 --> 00:00:23.303
So assuming you have, you had a try at it,

00:00:23.493 --> 00:00:26.013
so the easiest way I can think about it, well let's

00:00:26.013 --> 00:00:27.913
just, let's just plot them all on a number line.

00:00:27.913 --> 00:00:29.993
And you might want to draw it out or eventually

00:00:29.993 --> 00:00:31.443
you kind of have a number line in your head,

00:00:31.443 --> 00:00:32.413
and you can think about it that way.

00:00:32.601 --> 00:00:35.061
So let's first think about the absolute value of

00:00:35.072 --> 00:00:39.072
value of -27. So what is the absolute value of -27?

00:00:39.095 --> 00:00:42.785
Well it's just going to be +27. This right over here

00:00:42.785 --> 00:00:47.545
is the same thing as +27, because -27 is 27 to the

00:00:47.546 --> 00:00:50.526
left of 0. 27 to the left of 0.

00:00:51.392 --> 00:00:53.822
So let's plot the absolute value of -27.

00:00:53.822 --> 00:00:56.072
Well it's really 27, so it's going to be right

00:00:56.072 --> 00:00:59.032
over here. So that is the absolute value of

00:00:59.032 --> 00:01:06.812
-27. It is +27. Alright, then we have -17.

00:01:06.962 --> 00:01:10.752
Well, -17, let's see each hash mark here is

00:01:10.764 --> 00:01:17.774
3, so this is -9 and -12, -15. So -17

00:01:17.789 --> 00:01:19.529
is gonna be right around there.

00:01:19.796 --> 00:01:21.786
So that is -17.

00:01:23.263 --> 00:01:24.993
And then we have -28.

00:01:25.619 --> 00:01:29.029
So we have -27 and each hash mark is 3, so -28

00:01:29.029 --> 00:01:33.029
is going to be right over there. -28.

00:01:33.468 --> 00:01:35.928
And then finally we have -22.4.

00:01:36.779 --> 00:01:42.119
So let's see, we have 18, 21, 22 would be here, so

00:01:42.119 --> 00:01:45.089
22.4 is going to be roughly right over there.

00:01:45.112 --> 00:01:47.832
So that is 22.4.

00:01:48.339 --> 00:01:50.609
So if we want to sort it from least to greatest,

00:01:51.110 --> 00:01:52.710
well, we just have to start at the left end

00:01:52.714 --> 00:01:54.024
of the number line.

00:01:54.030 --> 00:01:56.490
The smallest of them, or the least of them,

00:01:56.500 --> 00:02:03.070
is -28. Then we go to -17. -17.

00:02:03.421 --> 00:02:07.521
Then we go to 22.4. Then we go to 22.4.

00:02:08.582 --> 00:02:12.332
And then we go to the absolute value of -27,

00:02:13.400 --> 00:02:15.010
and we are done.


Interpreting absolute value

WEBVTT
Kind: captions
Language: en

00:00:00.367 --> 00:00:03.102
- [Voiceover] So we have three
folks: Jenna,Sarah and Bill

00:00:03.102 --> 00:00:06.233
and they're keeping track of their weight.

00:00:06.233 --> 00:00:10.166
And in particular they're
keeping track of their weight

00:00:10.166 --> 00:00:11.620
over a month.

00:00:11.620 --> 00:00:13.418
And right over here we
have their weight change

00:00:13.418 --> 00:00:14.423
over the month.

00:00:14.423 --> 00:00:16.206
So if you take their ending weight

00:00:16.206 --> 00:00:18.666
and from that you subtract
their starting weight.

00:00:18.666 --> 00:00:21.874
So, let me just write that,
that's your end weight

00:00:21.874 --> 00:00:23.930
minus your starting weight.

00:00:23.930 --> 00:00:27.584
That's going to be your change
in weight over that month.

00:00:27.584 --> 00:00:30.300
And so we have the change in weights here.

00:00:30.300 --> 00:00:32.911
So, Jenna's change in weight

00:00:32.911 --> 00:00:35.382
is negative three pounds

00:00:35.382 --> 00:00:38.133
which means that she lost weight.

00:00:38.133 --> 00:00:41.300
Sarah's change in weight
is positive 2 1/2 pounds

00:00:41.300 --> 00:00:43.766
which means she gained 2 1/2 pounds.

00:00:43.766 --> 00:00:48.634
And then Bill's change in
weight is positive four pounds

00:00:48.634 --> 00:00:51.166
which means he gained four
pounds over the month.

00:00:51.166 --> 00:00:53.700
So, given that, let's
answer some questions.

00:00:53.700 --> 00:00:55.567
Who lost the most weight?

00:00:55.567 --> 00:00:57.702
Who had the largest change in weight?

00:00:57.702 --> 00:00:59.425
And who had the smallest change in weight?

00:00:59.425 --> 00:01:00.876
And I encourage you to pause the video

00:01:00.876 --> 00:01:03.363
and try to work through it on your own.

00:01:03.363 --> 00:01:07.100
Well let's just plot all of these

00:01:07.100 --> 00:01:09.907
on a number line here to
help us think through it,

00:01:09.907 --> 00:01:13.502
although some of you might
already have an answer here.

00:01:13.502 --> 00:01:17.333
So Jenna's weight change
is negative three pounds.

00:01:17.333 --> 00:01:19.502
So that's Jenna right over there.

00:01:19.502 --> 00:01:21.666
I'll say J for Jenna.

00:01:21.666 --> 00:01:25.234
Sarah's weight change,
and she had a weight gain

00:01:25.234 --> 00:01:28.800
is 2 1/2 pounds so
that's right over there.

00:01:28.800 --> 00:01:31.702
Positive 2 1/2 pounds for Sarah.

00:01:31.702 --> 00:01:34.300
And then finally Bill gained four pounds

00:01:34.300 --> 00:01:37.468
so that is Bill right over there.

00:01:37.468 --> 00:01:40.401
Now let's answer the
question, who lost the most?

00:01:40.401 --> 00:01:43.206
So when we talk about lost,
where we care about directions,

00:01:43.206 --> 00:01:45.399
we said hey, who moved most

00:01:45.399 --> 00:01:48.299
to the left of zero?

00:01:48.299 --> 00:01:51.094
Well Sarah and Bill didn't move
to the left of zero at all.

00:01:51.094 --> 00:01:52.333
They gained weight.

00:01:52.333 --> 00:01:53.800
That's why they're to the right of zero.

00:01:53.800 --> 00:01:55.876
Jenna's the only person who lost anything.

00:01:55.876 --> 00:01:59.468
So the person who lost the most is Jenna.

00:02:00.068 --> 00:02:02.470
Now who had the largest change?

00:02:02.470 --> 00:02:03.771
And if we think about the largest change,

00:02:03.771 --> 00:02:05.000
one way to think about it is,

00:02:05.000 --> 00:02:07.469
who is the furthest away from zero?

00:02:07.469 --> 00:02:08.866
So one way to think about it is,

00:02:08.866 --> 00:02:12.169
which of these is the
largest absolute value?

00:02:12.169 --> 00:02:14.833
Well the absolute value
here, the absolute value

00:02:14.833 --> 00:02:18.333
of negative three is,
and we can just call this

00:02:18.333 --> 00:02:20.830
the absolute change,

00:02:23.416 --> 00:02:25.601
is just going to be the absolute value.

00:02:25.601 --> 00:02:29.034
The absolute value of negative
three is positive three.

00:02:29.034 --> 00:02:32.915
Absolute value of 2.5

00:02:32.915 --> 00:02:35.478
is 2.5.

00:02:35.478 --> 00:02:38.701
Let me make this a little bit clearer.

00:02:38.701 --> 00:02:42.733
Absolutue value of four
is just equal to four.

00:02:42.733 --> 00:02:45.001
So who had the largest absolute change?

00:02:45.001 --> 00:02:47.158
Well Bill had the largest absolute change

00:02:47.158 --> 00:02:48.650
and then second place was actually Jenna.

00:02:48.650 --> 00:02:50.749
She lost weight, but she
had a pretty big change.

00:02:50.749 --> 00:02:52.289
She had a three pound change

00:02:52.289 --> 00:02:54.217
although it was a three pound loss.

00:02:54.217 --> 00:02:55.834
Bill's was a four pound gain.

00:02:55.834 --> 00:02:59.617
But the largest change is definitely Bill.

00:03:00.833 --> 00:03:03.267
Now who had the least change?

00:03:03.267 --> 00:03:03.973
Well another way to think about it,

00:03:03.973 --> 00:03:05.870
this is the least absolute change.

00:03:05.870 --> 00:03:09.467
Who changed the least from zero?

00:03:09.467 --> 00:03:10.284
Well if you look at here,

00:03:10.284 --> 00:03:14.100
Sarah had the smallest absolute change.

00:03:14.100 --> 00:03:17.966
Jenna lost weight so the
weight change of Jenna,

00:03:17.966 --> 00:03:21.369
she has negative three
vs. Sarah's positive 2.5.

00:03:21.369 --> 00:03:24.386
But if we talk about who had
the least change from zero.

00:03:24.386 --> 00:03:25.586
If your weight change was zero, that means

00:03:25.586 --> 00:03:26.976
you didn't gain or lose weight

00:03:26.976 --> 00:03:29.139
from the beginning of the
month to the end of the month.

00:03:29.139 --> 00:03:30.038
So who had the least change?

00:03:30.038 --> 00:03:32.564
Well that's the least absolute change?

00:03:32.564 --> 00:03:33.971
Well, we see that right over here.

00:03:33.971 --> 00:03:38.534
2.5 is less than three or four.

00:03:38.534 --> 00:03:43.246
So the least change is Sarah.

00:03:43.246 --> 00:03:45.047
And we're done.


Ordering rational numbers

WEBVTT
Kind: captions
Language: en

00:00:00.291 --> 00:00:01.173
- [Voiceover] What I'd like to do

00:00:01.173 --> 00:00:03.972
in this video is order these six numbers

00:00:03.972 --> 00:00:05.259
from least to greatest.

00:00:05.259 --> 00:00:07.233
So the least of them being
on the left hand side,

00:00:07.233 --> 00:00:09.011
and the greatest on the right.

00:00:09.011 --> 00:00:11.821
And I encourage you to pause this video,

00:00:11.821 --> 00:00:14.096
and see if you can do it on your own,

00:00:14.096 --> 00:00:16.836
before we work through it together.

00:00:16.836 --> 00:00:18.252
So assuming you've had a go at it,

00:00:18.252 --> 00:00:19.831
so let's do it together,

00:00:19.831 --> 00:00:21.932
and to help us there,
let's plot these numbers

00:00:21.932 --> 00:00:24.220
on a number line, and I have a number line

00:00:24.220 --> 00:00:25.892
up here, so there you go,

00:00:25.892 --> 00:00:28.677
there is a handy number line.

00:00:28.677 --> 00:00:30.511
And let's just take them one by one.

00:00:30.511 --> 00:00:31.730
So the first number here,

00:00:31.730 --> 00:00:33.495
we have 7/3.

00:00:33.495 --> 00:00:34.901
So let's see if we can express that

00:00:34.901 --> 00:00:36.594
in a different way, if we can write that

00:00:36.594 --> 00:00:39.160
as a mixed number.

00:00:39.160 --> 00:00:42.376
So 7/3, how many wholes are here?

00:00:42.376 --> 00:00:44.418
And the whole is going to be 3/3.

00:00:44.418 --> 00:00:48.111
So this is going to be 3/3,

00:00:48.111 --> 00:00:50.107
plus another 3/3,

00:00:50.107 --> 00:00:52.186
is going to get us to 6/3.

00:00:52.186 --> 00:00:54.589
And so you're going to
have one more third left.

00:00:54.589 --> 00:00:55.386
So this is 7/3.

00:00:55.386 --> 00:00:57.818
Three plus three, plus one is seven.

00:00:57.818 --> 00:01:00.011
So this is 3/3 is one whole,

00:01:00.011 --> 00:01:02.554
three thirds is one whole, so this is two

00:01:02.554 --> 00:01:04.505
and 1/3.

00:01:04.505 --> 00:01:07.418
So seven thirds, same thing as one, two,

00:01:07.418 --> 00:01:11.207
and you see, between consecutive integers,

00:01:11.207 --> 00:01:15.653
we have three spaces,

00:01:15.653 --> 00:01:18.194
so we are essentially marking off thirds,

00:01:18.194 --> 00:01:19.671
so two and 1/3,

00:01:19.671 --> 00:01:20.678
is going to be 1/3 of the way

00:01:20.678 --> 00:01:21.769
between two and three,

00:01:21.769 --> 00:01:23.371
so it's going to be right over there,

00:01:23.371 --> 00:01:27.330
so that is 7/3.

00:01:27.330 --> 00:01:29.617
Then we have negative 5/2,

00:01:29.617 --> 00:01:30.778
so same logic.

00:01:30.778 --> 00:01:33.830
Negative, let me do that
in that green color.

00:01:33.830 --> 00:01:35.362
So, I can do it over here.

00:01:35.362 --> 00:01:38.588
Negative five over two,

00:01:38.588 --> 00:01:40.308
well that's the same thing as the negative

00:01:40.308 --> 00:01:42.526
of 5/2, and 5/2 is going

00:01:42.526 --> 00:01:46.391
to be 2/2 plus another 2/2,

00:01:46.391 --> 00:01:49.907
plus 1/2 so this is two

00:01:49.907 --> 00:01:51.312
and 1/2, this is one, this is one,

00:01:51.312 --> 00:01:52.581
and that's 1/2.

00:01:52.581 --> 00:01:55.786
So this is going to be one, plus one,

00:01:55.786 --> 00:01:57.459
plus 1/2, two and 1/2,

00:01:57.459 --> 00:01:58.552
we have our negative out there,

00:01:58.552 --> 00:02:01.733
so it's negative two and 1/2.

00:02:01.733 --> 00:02:04.656
Let's see, negative one, negative two,

00:02:04.656 --> 00:02:06.227
and the negative two and 1/2 is going

00:02:06.227 --> 00:02:07.628
to be halfway between negative two,

00:02:07.628 --> 00:02:09.231
and negative three, so it's going

00:02:09.231 --> 00:02:11.808
to be right over there.

00:02:11.808 --> 00:02:16.139
So that is negative 5/2.

00:02:16.139 --> 00:02:18.854
Then we have zero, not too difficult.

00:02:18.854 --> 00:02:21.467
It's actually labeled on
our number line for us.

00:02:21.467 --> 00:02:23.347
Then we have negative two.

00:02:23.347 --> 00:02:25.310
Negative two, once again,

00:02:25.310 --> 00:02:26.831
on our number line for us.

00:02:26.831 --> 00:02:29.431
Two, two steps, two whole numbers

00:02:29.431 --> 00:02:31.171
to the left of zero.

00:02:31.171 --> 00:02:34.876
So negative two is going
to put us right over there.

00:02:34.876 --> 00:02:37.151
Then we have negative 12/4

00:02:37.151 --> 00:02:38.695
so it might jump out at you immediately,

00:02:38.695 --> 00:02:40.471
12 divided by four is three,

00:02:40.471 --> 00:02:43.792
so this is going to be the
same thing as negative three.

00:02:43.792 --> 00:02:47.876
So this is negative 12/4

00:02:47.876 --> 00:02:48.821
or if you want to use the type

00:02:48.821 --> 00:02:51.203
of logic that we used in
these first two numbers,

00:02:51.203 --> 00:02:54.715
you could say negative 12/4

00:02:54.715 --> 00:02:56.397
is the same thing as the negative

00:02:56.397 --> 00:03:00.181
of 12/4 which is 4/4

00:03:00.181 --> 00:03:01.104
plus another 4/4

00:03:01.104 --> 00:03:02.579
so that gets us to 8/4

00:03:02.579 --> 00:03:04.042
plus another 4/4

00:03:04.042 --> 00:03:05.248
that's 12/4.

00:03:05.248 --> 00:03:08.754
This is one, two, and three,

00:03:08.754 --> 00:03:10.972
or negative three.

00:03:10.972 --> 00:03:14.200
So negative three is the
same thing as negative 12/4.

00:03:14.200 --> 00:03:16.162
And then finally, one more number,

00:03:16.162 --> 00:03:18.786
negative 3.25.

00:03:18.786 --> 00:03:21.038
So you could view this,
this is the same thing,

00:03:21.038 --> 00:03:24.997
negative 3.25 is the same thing as

00:03:24.997 --> 00:03:29.558
negative three and 25/100

00:03:29.558 --> 00:03:31.974
and 25/100 is the same thing as 1/4.

00:03:31.974 --> 00:03:34.292
25 is 1/4 of 100.

00:03:34.292 --> 00:03:36.695
So this is the same
thing as negative three

00:03:36.695 --> 00:03:38.599
and 1/4.

00:03:38.599 --> 00:03:40.886
So let's see, let's go negative one,

00:03:40.886 --> 00:03:42.697
negative two, negative three,

00:03:42.697 --> 00:03:44.485
and then we have to go a fourth,

00:03:44.485 --> 00:03:45.396
and we're not going to be able

00:03:45.396 --> 00:03:47.286
to do it super precisely,

00:03:47.286 --> 00:03:49.282
but it's going to be less than 1/3,

00:03:49.282 --> 00:03:51.673
so it's going to be right over there.

00:03:51.673 --> 00:03:56.339
So that right over
there, is negative 3.25.

00:03:56.339 --> 00:03:58.026
So we wanted the numbers ordered

00:03:58.026 --> 00:03:59.405
from least to greatest,
well, we've done that.

00:03:59.405 --> 00:04:01.785
The least is negative 3.25,

00:04:01.785 --> 00:04:04.827
Then negative 12/4, then negative 5/2,

00:04:04.827 --> 00:04:07.231
then negative two, then zero,

00:04:07.231 --> 00:04:12.231
then 7/3, and we are done.


Comparing negatives with variables

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.066
- [Voiceover] So we
have a number line here

00:00:02.066 --> 00:00:05.100
with zero at the center and
then on that number line

00:00:05.100 --> 00:00:07.068
we've marked off some numbers.

00:00:07.068 --> 00:00:08.500
So, to the left of the number line

00:00:08.500 --> 00:00:10.167
we have the number a,

00:00:10.167 --> 00:00:11.867
and then we have the number b here,

00:00:11.867 --> 00:00:13.166
a little bit closer to zero,

00:00:13.166 --> 00:00:15.133
and then on the right side of zero

00:00:15.133 --> 00:00:17.300
we have the number c.

00:00:17.300 --> 00:00:18.766
And then after that, we have a bunch

00:00:18.766 --> 00:00:22.166
of statements dealing with inequalities.

00:00:22.166 --> 00:00:24.433
Now, what I want you to
do is pause the video

00:00:24.433 --> 00:00:27.000
and think about which of
these statements are true,

00:00:27.000 --> 00:00:28.900
which of these statements are false,

00:00:28.900 --> 00:00:30.633
and maybe which of these
statements you don't have

00:00:30.633 --> 00:00:33.733
enough information to figure out.

00:00:33.733 --> 00:00:35.466
So, I'm assuming you've had a go at it.

00:00:35.466 --> 00:00:37.467
You've tried to figure out
which of these are true,

00:00:37.467 --> 00:00:39.000
which of these are
false, and which of these

00:00:39.000 --> 00:00:40.466
you can't figure out.

00:00:40.466 --> 00:00:42.333
So, let's do them together.

00:00:42.333 --> 00:00:44.133
So, this first statement says

00:00:44.133 --> 00:00:47.652
negative b is less than negative c.

00:00:47.652 --> 00:00:51.300
So, we do know for a fact
that b is less than c.

00:00:51.300 --> 00:00:56.299
We know that b is less than c.

00:00:56.299 --> 00:00:57.563
How do we know that?

00:00:57.563 --> 00:01:00.133
Well, b is to the left
of c on the number line.

00:01:00.133 --> 00:01:01.066
It's that straight forward.

00:01:01.066 --> 00:01:04.000
So, we know this. This is definitely true.

00:01:04.000 --> 00:01:05.272
But what about negative b?

00:01:05.272 --> 00:01:08.740
Is negative b less than negative c?

00:01:08.740 --> 00:01:10.633
So, let's think about where negative b is

00:01:10.633 --> 00:01:12.200
on this number line.

00:01:12.200 --> 00:01:15.533
So, negative b I will do in yellow.

00:01:15.533 --> 00:01:19.033
So, negative, negative b
means the opposite of b.

00:01:19.033 --> 00:01:22.433
So if b is one hash mark
to the left of zero,

00:01:22.433 --> 00:01:24.566
negative b is going to be one hash mark

00:01:24.566 --> 00:01:26.734
to the right of zero.

00:01:26.734 --> 00:01:30.166
So, that right over there,
is going to be negative b.

00:01:30.166 --> 00:01:32.200
And now, where is negative c?

00:01:32.200 --> 00:01:34.266
Once again, negative
c, this literally means

00:01:34.266 --> 00:01:35.933
the opposite of c.

00:01:35.933 --> 00:01:39.633
C is one, two, three, four, five

00:01:39.633 --> 00:01:43.100
hash marks to the right of zero.

00:01:43.100 --> 00:01:44.566
And so negative c is going to be

00:01:44.566 --> 00:01:47.533
one, two, three, four, five

00:01:47.533 --> 00:01:50.300
hash marks to the left of zero.

00:01:50.300 --> 00:01:52.133
And actually, let me do
this in a different color.

00:01:52.133 --> 00:01:54.766
So, negative c I will do in purple.

00:01:54.766 --> 00:01:59.185
This right over here is negative c.

00:01:59.185 --> 00:02:03.900
So, let's compare, is negative
b less than negative c?

00:02:03.900 --> 00:02:06.200
No, negative b is to the right of

00:02:06.200 --> 00:02:07.642
negative c on the number line.

00:02:07.642 --> 00:02:10.733
Negative b is greater than negative c.

00:02:10.733 --> 00:02:12.667
So, this is not true.

00:02:12.667 --> 00:02:15.900
Negative b is to the right of negative c.

00:02:15.900 --> 00:02:19.081
Negative b is greater than negative c.

00:02:19.081 --> 00:02:21.102
And if this is a little
confusing, just think about it.

00:02:21.102 --> 00:02:23.271
Since b is a negative number,

00:02:23.271 --> 00:02:27.100
negative b is going to
be a positive number.

00:02:27.100 --> 00:02:29.396
And since c is a positive number,

00:02:29.396 --> 00:02:31.502
negative c is going to
be a negative number.

00:02:33.434 --> 00:02:35.233
So, it makes complete sense
that a positive number

00:02:35.233 --> 00:02:36.830
is going to be greater
than a negative number.

00:02:36.830 --> 00:02:40.033
And you see it here,
negative b is to the right of

00:02:40.033 --> 00:02:42.366
negative c on the number line.

00:02:42.366 --> 00:02:44.033
So, we can rule this one out.

00:02:44.033 --> 00:02:48.009
So the next question, is
negative b greater than zero?

00:02:48.009 --> 00:02:49.600
Well, we already plotted negative b,

00:02:49.600 --> 00:02:51.133
it's going to be one to the right,

00:02:51.133 --> 00:02:52.533
or one hash mark to the right,

00:02:52.533 --> 00:02:55.633
we don't know how much each
of these hash marks represent,

00:02:55.633 --> 00:02:58.000
but it's going to be to the right of zero.

00:02:58.000 --> 00:02:59.567
So, it is greater than zero.

00:02:59.567 --> 00:03:03.515
This is true. That is true.

00:03:03.515 --> 00:03:07.533
All right, now is a greater than b?

00:03:07.533 --> 00:03:08.558
Well, let's look at it.

00:03:08.558 --> 00:03:11.233
A is to the left of b on the number line.

00:03:11.233 --> 00:03:14.200
A is more negative than b.

00:03:14.200 --> 00:03:16.956
So, a is less than b, not greater than b.

00:03:16.956 --> 00:03:19.466
So, this is not going to be the case.

00:03:19.466 --> 00:03:21.098
In order for something to be greater than

00:03:21.098 --> 00:03:23.400
something else, it would have
to be to the right of it.

00:03:23.400 --> 00:03:24.733
For a to the greater than b,

00:03:24.733 --> 00:03:26.433
it would have to be to the right of b.

00:03:26.433 --> 00:03:28.000
But we see, a is to the left of b.

00:03:28.000 --> 00:03:30.100
A is less than b.

00:03:30.100 --> 00:03:32.133
All right, one more to think about.

00:03:32.133 --> 00:03:35.671
Negative a is greater than c.

00:03:35.671 --> 00:03:39.133
So, we know that a isn't greater than c,

00:03:39.133 --> 00:03:40.933
a is to the left of c.

00:03:40.933 --> 00:03:42.966
A is a negative number,
it's to the left of zero,

00:03:42.966 --> 00:03:46.200
c is a positive number,
it's to the right of zero.

00:03:46.200 --> 00:03:48.000
But what about negative a?

00:03:48.000 --> 00:03:50.100
Well, let's draw that.

00:03:50.100 --> 00:03:52.366
Let me do this in a color
that I haven't used yet.

00:03:52.366 --> 00:03:54.567
Negative a, where would that be?

00:03:54.567 --> 00:03:59.567
Well a is one, two, three, four, five, six

00:03:59.747 --> 00:04:03.366
hash marks to the left of zero.

00:04:03.366 --> 00:04:04.800
And so negative a is going to be

00:04:04.800 --> 00:04:06.700
six hash marks to the right of zero.

00:04:06.700 --> 00:04:07.800
So, let's count that.

00:04:07.800 --> 00:04:12.266
One, two, three, four, five, six.

00:04:12.266 --> 00:04:15.800
So, negative a is going
to be right over there.

00:04:15.800 --> 00:04:20.033
And notice, negative a
is to the right of c,

00:04:20.033 --> 00:04:22.600
so negative a is greater than c.

00:04:22.600 --> 00:04:23.867
This is true.

00:04:23.867 --> 00:04:25.000
And if you get confused, if you say,

00:04:25.000 --> 00:04:26.767
wait, this looks like a negative,

00:04:26.767 --> 00:04:28.534
how can it be larger than a positive?

00:04:28.534 --> 00:04:32.233
Remember, negative a itself
is a negative number.

00:04:32.233 --> 00:04:35.500
And a itself is six
hash marks to the left.

00:04:35.500 --> 00:04:37.178
So if you take the opposite of that,

00:04:37.178 --> 00:04:38.533
you're going to get a positive number.

00:04:38.533 --> 00:04:40.833
You're going to get six
hash marks to the right.

00:04:40.833 --> 00:04:42.700
And c, which was already
a positive number,

00:04:42.700 --> 00:04:44.840
is only five hash marks to the right.

00:04:44.840 --> 00:04:48.771
And so, negative a, this is
going to be a positive number,

00:04:48.771 --> 00:04:51.399
and it's going to be greater than c.

00:04:51.399 --> 00:04:53.357
It's to the right of c.


Deflate gate

WEBVTT
Kind: captions
Language: en

00:00:00.543 --> 00:00:04.525
Ever since the AFC championship game,
and into the run-up to the Super Bowl,

00:00:04.525 --> 00:00:07.408
there's been a lot of talk
about deflated footballs.

00:00:07.408 --> 00:00:11.968
And in particular, to what degree
can the pressure of a football go down--

00:00:11.968 --> 00:00:15.532
to what degree can it get deflated--
based on temperature differentials

00:00:15.532 --> 00:00:19.294
from where it was inflated
to where it is actually played.

00:00:19.294 --> 00:00:22.957
Especially if it's inflated in a
warmer environment, say indoors,

00:00:22.957 --> 00:00:26.265
and then it's played outdoors,
in a colder environment.

00:00:26.265 --> 00:00:27.943
And so to think about this,

00:00:27.943 --> 00:00:31.787
we'll take out one of my
favorite laws or equations,

00:00:31.787 --> 00:00:34.202
the ideal gas law.

00:00:34.202 --> 00:00:38.214
And it tells us that
pressure times volume

00:00:38.214 --> 00:00:44.114
is equal to the number of molecules
of our ideal gas we're dealing with--

00:00:44.114 --> 00:00:49.029
it's measured in terms of moles, which is
the measure of the number of molecules--

00:00:49.029 --> 00:00:53.273
times our ideal gas constant,
times our temperature.

00:00:53.273 --> 00:00:56.954
Or if we just want to rearrange it--
and I do want to rearrange it,

00:00:56.954 --> 00:00:59.231
because I just want to
solve for P here--

00:00:59.231 --> 00:01:02.823
we can divide both sides by V,
and we would be left with

00:01:02.823 --> 00:01:04.825
P is equal to nRT over V.

00:01:07.825 --> 00:01:11.189
Now let's think about
the exact case of what happened

00:01:11.189 --> 00:01:13.237
with the New England football--

00:01:13.237 --> 00:01:17.034
what the argument is for why
they might have deflated naturally--

00:01:17.034 --> 00:01:19.515
why there might not be foul play.

00:01:19.515 --> 00:01:25.082
The argument is maybe they were
inflated indoors where the temperature

00:01:25.082 --> 00:01:30.963
was room temperature--
so roughly 75 degrees Fahrenheit,

00:01:30.963 --> 00:01:37.265
which is roughly 24 degrees Celsius,
and in order to apply the ideal gas law,

00:01:37.265 --> 00:01:42.739
we'd want to convert it to Kelvin,
so this is approximately 297 Kelvin.

00:01:42.739 --> 00:01:45.409
That's the indoor temperature.

00:01:45.409 --> 00:01:48.517
So if we're infated here, indoors,

00:01:53.567 --> 00:01:57.520
and then we play outdoors,

00:01:57.520 --> 00:02:02.565
where the temperature is
roughly 50 degrees Fahrenheit--

00:02:02.565 --> 00:02:07.123
and that, from what I've read,
was actually the temperature

00:02:07.123 --> 00:02:10.337
on the field during
the New England game.

00:02:10.337 --> 00:02:14.923
50 degrees Fahrenheit, which is
approximately 10 degrees Celsius,

00:02:14.923 --> 00:02:17.698
which is approximately 283 Kelvin.

00:02:19.988 --> 00:02:23.549
So the argument being made is,
look, you have this temperature drop,

00:02:23.549 --> 00:02:28.294
could that temperature drop, if this
temperature drops by a certain percent,

00:02:28.294 --> 00:02:32.781
could that account for the pressure drop
that was actually measured?

00:02:32.781 --> 00:02:36.515
And the pressure drop that was
actually measured, gauge pressure--

00:02:36.515 --> 00:02:39.592
and this is an important point,
because it was glossed over

00:02:39.592 --> 00:02:41.800
in some of the initial analyses--

00:02:41.800 --> 00:02:44.689
the gauge pressure indoors,
where the ball was inflated,

00:02:44.689 --> 00:02:49.530
was 12 point 5 pounds per square inch,

00:02:49.530 --> 00:02:54.029
and the gauge pressure outdoors,
where it was felt the balls were

00:02:54.029 --> 00:02:57.072
deflated somehow, or where they
had less pressure,

00:02:57.072 --> 00:02:59.469
was 10 point 6 pounds per square inch.

00:03:02.152 --> 00:03:05.783
So, to see if these numbers are
consistent with no foul play,

00:03:05.783 --> 00:03:08.577
let's simplify our
ideal gas law a little bit--

00:03:08.577 --> 00:03:11.653
make it a little bit particular for this
circumstance.

00:03:11.653 --> 00:03:14.874
So in this circumstance,
we're going to assume no foul play.

00:03:14.874 --> 00:03:18.156
If we assume no foul play,
that means no air is let into or out of.

00:03:18.156 --> 00:03:20.563
So the number of molecules
aren't going to change.

00:03:20.563 --> 00:03:22.966
The ideal gas constant
obviously shouldn't change,

00:03:22.966 --> 00:03:26.087
it's a universal gas constant,
so this should be constant.

00:03:26.087 --> 00:03:27.332
And then our volume--

00:03:27.332 --> 00:03:31.373
let's just assume the football is made
out of leather, it's relatively rigid--

00:03:31.373 --> 00:03:33.766
its volume might have
changed a little bit,

00:03:33.766 --> 00:03:37.219
but for the sake of our analysis,
we'll say that it pretty much

00:03:37.219 --> 00:03:38.250
held up its shape.

00:03:38.250 --> 00:03:42.308
So its volume stayed constant
as we went from indoors to outdoors.

00:03:42.308 --> 00:03:46.951
So to simplify this equation, let's just
call all of this stuff in green K.

00:03:46.951 --> 00:03:51.376
If you take your n, multiply it times R,
divide it by V, that's just equal to

00:03:51.376 --> 00:03:53.165
some constant called K.

00:03:57.189 --> 00:04:00.930
So we can simplify all of this to
P, pressure, in this case,

00:04:00.930 --> 00:04:06.635
should be some constant times temperature.

00:04:06.635 --> 00:04:12.572
Because once again, we're assuming
volume doesn't change,

00:04:12.572 --> 00:04:15.525
we're assuming no air is let in or out.

00:04:15.525 --> 00:04:17.466
So when you look at it this way,

00:04:17.466 --> 00:04:22.220
any percent change on temperature should
have the same percent change in pressure.

00:04:22.220 --> 00:04:25.955
So let's see if the percent change
in temperature is consistent with

00:04:25.955 --> 00:04:29.853
the percent change in pressure
that was actually observed.

00:04:29.853 --> 00:04:34.509
And so when we look at the temperature,
we should be doing it in Kelvin,

00:04:34.509 --> 00:04:40.768
so we should look at this change here,
going from 297 Kelvin to 283 Kelvin.

00:04:40.768 --> 00:04:43.182
So what is the drop here?

00:04:43.182 --> 00:04:44.616
Let's see.

00:04:44.616 --> 00:04:48.432
It's a drop of 14 Kelvin,
let me get my calculator out.

00:04:51.822 --> 00:04:54.363
It's a drop of 14 Kelvin.

00:04:54.363 --> 00:04:59.292
And we started at 297, so divided by 297,

00:04:59.292 --> 00:05:03.654
let's just say if we round
to the nearest percentage,

00:05:03.654 --> 00:05:09.258
gets us zero point zero four seven,
that's rougly a five percent drop.

00:05:09.258 --> 00:05:12.864
So this is a five percent drop.

00:05:16.886 --> 00:05:19.622
And I'll write approximately
5 percent drop.

00:05:19.622 --> 00:05:22.732
Now let's see what
the drop in pressure is.

00:05:22.732 --> 00:05:26.317
So this is going from 12 point 5 psi
to 10 point 6 psi.

00:05:26.317 --> 00:05:28.955
That's 1.9 psi drop.

00:05:28.955 --> 00:05:32.584
So that's 1.9 psi drop
divided by our start,

00:05:32.584 --> 00:05:35.361
which is 12 point 5,

00:05:35.361 --> 00:05:40.746
which looks like a 15 point 2
percent drop, or roughly

00:05:40.746 --> 00:05:42.982
a 15 percent drop.

00:05:42.982 --> 00:05:47.064
So things are starting to look shady.

00:05:47.064 --> 00:05:50.009
This looks like a 15 percent drop.

00:05:50.009 --> 00:05:53.483
And this is actually what some of the
initial analyses did,

00:05:53.483 --> 00:05:56.362
they said, look, something
clearly shady happened,

00:05:56.362 --> 00:05:59.456
because temperature by itself
should only account for

00:05:59.456 --> 00:06:03.425
a 5 percent drop in pressure, but
there was a 15 percent drop in pressure,

00:06:03.425 --> 00:06:06.168
so maybe some air was let out somehow.

00:06:06.168 --> 00:06:09.260
But there was actually a mistake
in that initial analysis.

00:06:09.260 --> 00:06:15.089
Gauge pressure, the 12 point 5 psi,
that's actually not the absolute pressure.

00:06:15.089 --> 00:06:18.141
What gauge pressure is a measure of

00:06:18.141 --> 00:06:23.489
is how much more pressure
you have inside than outside.

00:06:23.489 --> 00:06:27.551
Outside the ball, you do have pressure,
standard atmospheric pressure.

00:06:27.551 --> 00:06:30.751
And it might change depending on
the weather, et cetera.

00:06:30.751 --> 00:06:32.044
But there is pressure.

00:06:32.044 --> 00:06:34.958
The pressure is caused by
the weight of the atmosphere.

00:06:34.958 --> 00:06:37.172
So, in order to figure absolute pressure,

00:06:37.172 --> 00:06:39.734
and to figure out the correct
percentage change,

00:06:39.734 --> 00:06:43.190
you need to add how much
more pressure there is inside the ball

00:06:43.190 --> 00:06:45.038
to what the outside pressure is.

00:06:45.038 --> 00:06:49.083
And I don't know the exact readings
for that day in New England,

00:06:49.083 --> 00:06:53.939
but standard atmospheric pressure
is 14 point 7 psi.

00:06:57.818 --> 00:07:01.870
So plus 14 point 7 psi gets us to,

00:07:01.870 --> 00:07:04.076
27 point 2 psi.

00:07:07.056 --> 00:07:09.425
And if we add the 14 point 7 over here,

00:07:09.425 --> 00:07:11.922
again, to get the true absolute pressure--

00:07:11.922 --> 00:07:15.067
once again, the gauge reading is just
how much more pressure

00:07:15.067 --> 00:07:16.935
inside the football than outside.

00:07:16.935 --> 00:07:20.600
So if you want the true pressure,
the absolute pressure, you add these.

00:07:20.600 --> 00:07:22.810
And this is going to be 25 point 2 psi.

00:07:25.857 --> 00:07:29.207
Now we can calculate
the actual percentage drop,

00:07:29.207 --> 00:07:31.617
and compare it to this 5 percent drop.

00:07:31.617 --> 00:07:35.577
Minor mistake, this is 25 point 3 psi.

00:07:39.849 --> 00:07:41.853
So we still have the same drop,

00:07:41.853 --> 00:07:43.976
we have a 1 point 9 psi drop,

00:07:43.976 --> 00:07:46.667
to go from 27 point 2
to 25 point 3.

00:07:46.667 --> 00:07:48.573
But it's over a larger base.

00:07:48.573 --> 00:07:51.017
So it should be a lower percentage.

00:07:51.017 --> 00:07:53.669
Let's calculate what that is.

00:07:53.669 --> 00:07:58.506
We have 1 point 9 psi drop,
divided by 27 point 2 psi.

00:07:58.506 --> 00:08:02.427
Is equal to a little under
a seven percent drop.

00:08:02.427 --> 00:08:05.540
So approximately a seven percent drop.

00:08:09.320 --> 00:08:11.783
Now these two things are not exact.

00:08:11.783 --> 00:08:13.807
But they are a lot closer.

00:08:13.807 --> 00:08:15.784
And this degree, you're like, okay,

00:08:15.784 --> 00:08:18.027
there might have been
other factors at play.

00:08:18.027 --> 00:08:20.952
Maybe some of the measurements
weren't done exactly right.

00:08:20.952 --> 00:08:22.694
Maybe actually the ambient pressure

00:08:22.694 --> 00:08:24.776
was somehow different inside and outside.

00:08:24.776 --> 00:08:27.997
There's also some possibility,
especially when you're outside,

00:08:27.997 --> 00:08:30.708
that some water,
some precipitation is on the ball.

00:08:30.708 --> 00:08:33.352
As that precipitation evaporates,

00:08:33.352 --> 00:08:35.582
it might cool down the ball further,

00:08:35.582 --> 00:08:38.170
which would drop the pressure even more.

00:08:38.170 --> 00:08:39.735
So this isn't definitive,

00:08:39.735 --> 00:08:43.768
but at least the numbers here,
when we apply the ideal gas law properly,

00:08:43.768 --> 00:08:47.985
they pretty much account for
most of the pressure drop.

00:08:47.985 --> 00:08:50.967
So at least here,
I wouldn't get too caught up

00:08:50.967 --> 00:08:53.397
in all of the conspiracy theories.


Negative symbol as opposite

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:02.775
- [Voiceover] In another video
we've already talked about

00:00:02.775 --> 00:00:04.344
what an opposite of a number is,

00:00:04.344 --> 00:00:06.012
but we'll review it a little bit.

00:00:06.012 --> 00:00:07.671
If we start with a positive number,

00:00:07.671 --> 00:00:10.212
say, the number four.

00:00:10.212 --> 00:00:14.733
If you have a number

00:00:14.733 --> 00:00:15.907
and then we want to think about

00:00:15.907 --> 00:00:18.244
what the opposite of that number is.

00:00:18.244 --> 00:00:19.800
Opposite.

00:00:19.800 --> 00:00:21.611
This is just the symbol for number

00:00:21.611 --> 00:00:22.642
or shorthand for number.

00:00:22.642 --> 00:00:24.043
So if the number is four,

00:00:24.043 --> 00:00:25.511
what is its opposite?

00:00:25.511 --> 00:00:28.273
Well, four is one, two, three, four

00:00:28.273 --> 00:00:29.480
to the right of zero.

00:00:29.480 --> 00:00:31.610
Positive four is four
to the right of zero.

00:00:31.610 --> 00:00:33.033
So its opposite is going to be

00:00:33.033 --> 00:00:36.610
one, two, three, four to the left.

00:00:36.610 --> 00:00:39.640
So it's going to be negative four.

00:00:39.640 --> 00:00:40.533
Or another way to think about it,

00:00:40.533 --> 00:00:41.809
if you have a positive number,

00:00:41.809 --> 00:00:44.810
its opposite is going to be
the negative of that number.

00:00:44.810 --> 00:00:46.474
So another way of thinking about it is,

00:00:46.474 --> 00:00:49.910
is that this negative
literally means opposite.

00:00:49.910 --> 00:00:51.341
One way to think about
it, this is the number

00:00:51.341 --> 00:00:53.567
that is the opposite of four.

00:00:53.567 --> 00:00:55.144
So let me write this down.

00:00:55.144 --> 00:00:58.805
So if I write negative four.

00:00:58.805 --> 00:01:01.074
Negative, let me write this,

00:01:01.074 --> 00:01:04.405
negative four.

00:01:04.405 --> 00:01:05.776
That literally means...

00:01:07.537 --> 00:01:11.440
So you can take this negative symbol

00:01:11.440 --> 00:01:13.107
as meaning opposite.

00:01:13.107 --> 00:01:14.809
Opposite.

00:01:14.809 --> 00:01:19.809
Opposite of four. Opposite of four.

00:01:19.909 --> 00:01:23.145
If you were to say negative,

00:01:23.145 --> 00:01:24.805
instead of saying a specific number

00:01:24.805 --> 00:01:27.334
let's just say a letter that
could represent a number.

00:01:27.334 --> 00:01:29.710
So if I said negative A,

00:01:29.710 --> 00:01:33.465
that means the opposite of the number A.

00:01:33.465 --> 00:01:38.333
That means the opposite...

00:01:39.581 --> 00:01:42.044
Opposite of...

00:01:43.133 --> 00:01:45.115
Let me do that in blue color.

00:01:45.115 --> 00:01:48.612
Opposite of A. Of A.

00:01:48.612 --> 00:01:50.215
And if this confuses you
a little bit just look,

00:01:50.215 --> 00:01:52.870
A could be any number right over here.

00:01:52.870 --> 00:01:53.903
So let me draw a number line

00:01:53.903 --> 00:01:55.746
just so this can be a little bit,

00:01:55.746 --> 00:01:58.209
make a little bit more sense hopefully.

00:01:58.209 --> 00:02:00.345
So this is a number line.

00:02:00.345 --> 00:02:02.342
Let me put some tick marks here.

00:02:02.342 --> 00:02:05.043
I don't know much each
of these tick marks jump

00:02:05.043 --> 00:02:09.703
but let's say that A is some
number that is right over here.

00:02:11.075 --> 00:02:14.045
Well negative A is going
to be the opposite of A.

00:02:14.045 --> 00:02:16.667
So if A is three tick marks to the right

00:02:16.667 --> 00:02:19.638
negative A is going to be
three tick marks to the left.

00:02:19.638 --> 00:02:22.313
One, two, three.

00:02:22.313 --> 00:02:24.100
And so the opposite of A

00:02:24.100 --> 00:02:26.539
is going to be this value right over here.

00:02:26.539 --> 00:02:29.503
And we can write that, we can
write opposite of A over here.

00:02:29.503 --> 00:02:31.610
We could literally write opposite,

00:02:31.610 --> 00:02:35.533
opposite of A

00:02:35.533 --> 00:02:37.606
is that number right over there.

00:02:37.606 --> 00:02:41.670
Or as a shorthand, we can just write--

00:02:41.670 --> 00:02:44.942
We can just write negative.

00:02:44.942 --> 00:02:49.381
We can just write this is
negative A right over here.

00:02:49.381 --> 00:02:53.276
Negative A.

00:02:53.276 --> 00:02:54.081
So with that in mind,

00:02:54.081 --> 00:02:56.500
if we literally view this negative symbol

00:02:56.500 --> 00:02:59.308
as meaning the opposite
of whatever this is,

00:02:59.308 --> 00:03:01.142
let's try something interesting.

00:03:01.142 --> 00:03:04.010
What would be the negative--

00:03:04.010 --> 00:03:06.411
Let me do this in another color.

00:03:06.411 --> 00:03:10.404
What would be the negative of

00:03:10.404 --> 00:03:13.144
negative three?

00:03:13.144 --> 00:03:16.143
And I encourage you to pause
the video and think about it.

00:03:16.143 --> 00:03:19.340
Well we just said, this
negative means the opposite.

00:03:19.340 --> 00:03:21.343
So you can think about this as meaning

00:03:21.343 --> 00:03:24.138
this means the opposite...

00:03:27.738 --> 00:03:31.140
Opposite of negative--

00:03:31.140 --> 00:03:33.338
The opposite of negative three.

00:03:33.338 --> 00:03:35.070
So what is the opposite of negative three?

00:03:35.070 --> 00:03:38.348
Well negative three is
three to the left of zero.

00:03:38.348 --> 00:03:39.745
One, two, three.

00:03:39.745 --> 00:03:41.809
So it's opposite is going to
be three to the right of zero.

00:03:41.809 --> 00:03:43.172
One, two, three.

00:03:43.172 --> 00:03:46.382
So it's going to be positive three.

00:03:46.382 --> 00:03:50.779
So this is equal to positive three.

00:03:50.779 --> 00:03:53.175
Or we could just write positive three

00:03:53.175 --> 00:03:56.547
like that.

00:03:56.547 --> 00:03:58.574
So hopefully this gives
you a better appreciation

00:03:58.574 --> 00:04:01.409
for what opposite means

00:04:01.409 --> 00:04:03.644
and also how it relates to the

00:04:03.644 --> 00:04:06.442
actual negative symbol.

00:04:06.442 --> 00:04:08.329
We could keep going.

00:04:08.329 --> 00:04:10.012
We could do something like

00:04:10.012 --> 00:04:13.503
what is the negative

00:04:13.508 --> 00:04:15.206
of the negative

00:04:15.206 --> 00:04:18.303
of negative--

00:04:18.303 --> 00:04:20.732
Let me do a different
number. Of negative two.

00:04:22.040 --> 00:04:25.280
Well, this part right over here

00:04:25.280 --> 00:04:28.571
the negative, the
opposite of negative two,

00:04:28.571 --> 00:04:31.081
which is really the opposite
of the opposite of two,

00:04:31.081 --> 00:04:32.270
well that's just going to be two.

00:04:32.270 --> 00:04:35.979
Every time you say opposite
you flip over the number line.

00:04:35.979 --> 00:04:38.942
So this flips you over the
number line two to the left

00:04:38.942 --> 00:04:40.772
and this flips you back two to the right

00:04:40.772 --> 00:04:42.537
so all of this is going to be two

00:04:42.537 --> 00:04:44.143
but then you're going to
take the opposite of two

00:04:44.143 --> 00:04:48.808
so that's just going to be negative two.

00:04:48.808 --> 00:04:50.538
If you threw another
negative in front of this,

00:04:50.538 --> 00:04:51.738
it would be the opposite of all of this.

00:04:51.738 --> 00:04:52.908
So it would be the
opposite of negative two

00:04:52.908 --> 00:04:54.109
and then all of a sudden
it would become positive.

00:04:54.109 --> 00:04:56.545
So every time you put that
negative in front there

00:04:56.545 --> 00:05:00.634
you're flipping on the other
side of the number line.


Number opposites practice

WEBVTT
Kind: captions
Language: en

00:00:00.233 --> 00:00:01.966
- [Voiceover] So let's
do a couple of examples

00:00:01.966 --> 00:00:04.833
of the number opposites
exercise on Khan Academy.

00:00:04.833 --> 00:00:07.293
Just to make sure that
we've really digested

00:00:07.293 --> 00:00:09.866
what it means to be the
opposite of a number.

00:00:09.866 --> 00:00:11.667
So they have a number line here,

00:00:11.667 --> 00:00:14.566
and then they've marked some
of the numbers with letters.

00:00:14.566 --> 00:00:16.500
And this is A, B, C, D, E.

00:00:16.500 --> 00:00:19.233
And then they say, "What
can we say about point B?"

00:00:19.233 --> 00:00:24.233
And point B is one hash mark
to the left of zero, all right.

00:00:24.489 --> 00:00:25.833
"Select all that applies."

00:00:25.833 --> 00:00:28.600
So the first thing, they say
is, "B is equal to negative D."

00:00:28.600 --> 00:00:31.766
So that means that B needs
to be the opposite of D.

00:00:31.766 --> 00:00:32.566
Let's see if that's true.

00:00:32.566 --> 00:00:33.600
D is here.

00:00:33.600 --> 00:00:36.692
D is one hash mark to the right of zero.

00:00:36.692 --> 00:00:40.266
The opposite of D, or negative D,

00:00:40.266 --> 00:00:43.233
should be one hash mark
to the left of zero.

00:00:43.233 --> 00:00:44.717
Which is B.

00:00:44.717 --> 00:00:46.617
B is the opposite of D.

00:00:46.617 --> 00:00:50.855
B is negative D, so this is true.

00:00:50.855 --> 00:00:53.711
The next statement is,
"B is the opposite of E."

00:00:53.711 --> 00:00:56.112
Let's see, E is one, two, three hash marks

00:00:56.112 --> 00:00:57.481
to the right of zero.

00:00:57.481 --> 00:01:00.666
So the opposite of E should be one, two,

00:01:00.666 --> 00:01:03.095
three hash marks to the left of zero.

00:01:03.095 --> 00:01:05.812
That is not B; that is A.

00:01:05.812 --> 00:01:07.666
B is not the opposite of E.

00:01:07.666 --> 00:01:09.357
A is the opposite of E,

00:01:09.357 --> 00:01:10.310
but that isn't what they write here

00:01:10.310 --> 00:01:11.633
so we're not going to check that.

00:01:11.633 --> 00:01:13.866
And then they say, "B
is equal to negative B."

00:01:13.866 --> 00:01:16.719
That's saying that B is the
same thing as the opposite of B.

00:01:16.719 --> 00:01:18.266
That is not true.

00:01:18.266 --> 00:01:20.333
The opposite of B is going to go on the

00:01:20.333 --> 00:01:21.533
other side of the number line.

00:01:21.533 --> 00:01:23.466
That's D, which is not B.

00:01:23.466 --> 00:01:25.200
So this is also not true.

00:01:25.200 --> 00:01:26.267
This would have been true if they said,

00:01:26.267 --> 00:01:28.866
"C is equal to negative C."

00:01:28.866 --> 00:01:32.900
Because C is at zero,
it's zero away from zero.

00:01:32.900 --> 00:01:34.267
C is zero.

00:01:34.267 --> 00:01:36.033
The opposite of zero is going to be

00:01:36.033 --> 00:01:38.300
zero away on the other
side of the number line.

00:01:38.300 --> 00:01:39.748
Well, that's still zero away.

00:01:39.748 --> 00:01:42.200
So zero is negative zero.

00:01:42.200 --> 00:01:45.335
Or we could have said C
is equal to negative C.

00:01:45.335 --> 00:01:49.366
But the only number for
which that is true is zero.

00:01:49.366 --> 00:01:50.776
It's not going to be true for B,

00:01:50.776 --> 00:01:53.792
because B is clearly not zero.

00:01:53.792 --> 00:01:56.733
So let's do a couple of here.

00:01:56.733 --> 00:01:58.133
"What can we say about negative C?"

00:01:58.133 --> 00:02:00.100
All right, so now C is zero.

00:02:00.100 --> 00:02:01.319
"Select all that apply."

00:02:01.319 --> 00:02:03.540
"Negative C is zero."

00:02:03.540 --> 00:02:07.017
Yeah, the negative of zero
is going to still be zero.

00:02:07.017 --> 00:02:09.166
The negative of zero,
the opposite of zero,

00:02:09.166 --> 00:02:11.392
is still going to be zero.

00:02:11.392 --> 00:02:13.800
The opposite of zero, that's
not going to be this number.

00:02:13.800 --> 00:02:15.600
This is some non-zero number.

00:02:15.600 --> 00:02:18.666
So that's that. All right.

00:02:18.666 --> 00:02:20.100
"What can we say about point T?"

00:02:20.100 --> 00:02:21.766
T once again is at zero.

00:02:21.766 --> 00:02:23.433
So we say, "T is zero."

00:02:23.433 --> 00:02:26.133
"Zero is the opposite of zero."

00:02:26.133 --> 00:02:28.302
It is equal to negative zero.

00:02:28.302 --> 00:02:30.866
Both of them -- one is zero, you could say

00:02:30.866 --> 00:02:32.533
zero to the left, one
is zero to the right.

00:02:32.533 --> 00:02:34.907
Well, that still means
that they're still on zero.

00:02:34.907 --> 00:02:36.817
And they're right, they are zero.

00:02:36.817 --> 00:02:40.233
"T is equal to the opposite
of the opposite of T."

00:02:40.233 --> 00:02:43.592
Well, this would actually
be true for any number.

00:02:43.592 --> 00:02:45.333
For any number, the
opposite of the opposite

00:02:45.333 --> 00:02:46.379
is going to be that number.

00:02:46.379 --> 00:02:48.529
We could have put the Q here, the R here.

00:02:48.529 --> 00:02:51.039
Really for any number the
opposite of the opposite

00:02:51.039 --> 00:02:52.533
is going to be that number again.

00:02:52.533 --> 00:02:56.200
And so it's definitely going
to be true for zero as well.

00:02:56.200 --> 00:02:59.534
So that's definitely going to be true.

00:02:59.534 --> 00:03:01.400
So hopefully that kind
of gives you a little bit

00:03:01.400 --> 00:03:03.933
more practice with number opposites and

00:03:03.933 --> 00:03:06.529
helps you digest things a little bit more.


Opposite of a number

WEBVTT
Kind: captions
Language: en

00:00:00.215 --> 00:00:01.935
- [Voiceover] What I want
to do in this video is

00:00:01.935 --> 00:00:06.893
think about what it means to
have an opposite of a number.

00:00:06.893 --> 00:00:10.063
Let me draw a number line here.

00:00:10.063 --> 00:00:12.660
Let's draw a number line.

00:00:12.662 --> 00:00:15.239
And let's put some numbers
on this number line.

00:00:15.239 --> 00:00:17.422
We can start at zero,

00:00:17.422 --> 00:00:19.778
and if we go to the right
we have positive numbers.

00:00:19.778 --> 00:00:24.778
One, two, three, four, five.

00:00:25.641 --> 00:00:28.138
As we go to the left we
get more and more negative.

00:00:28.138 --> 00:00:33.094
So negative one, negative
two, negative three,

00:00:33.094 --> 00:00:37.097
negative four, and I can
keep going on and on and on.

00:00:37.354 --> 00:00:39.560
Let's pick one of these numbers.

00:00:39.560 --> 00:00:42.694
Let's say that we pick the number three.

00:00:43.048 --> 00:00:47.141
What is going to be the
opposite of the number three?

00:00:47.141 --> 00:00:50.822
Well the opposite of the
number is a number that's

00:00:50.822 --> 00:00:54.964
the same distance from
zero but on the other side.

00:00:55.105 --> 00:00:57.995
So three is three to the right of zero.

00:00:57.995 --> 00:01:00.015
One, two, three.

00:01:00.015 --> 00:01:02.535
So its opposite is going to
be three to the left of zero.

00:01:02.535 --> 00:01:05.682
One, two, three.

00:01:05.682 --> 00:01:10.081
So the opposite of
three is negative three.

00:01:10.081 --> 00:01:11.880
Let me make a little table here.

00:01:11.880 --> 00:01:15.082
If we have the number, the number.

00:01:15.734 --> 00:01:18.520
And then we have its opposite.

00:01:18.520 --> 00:01:21.655
We have its opposite.

00:01:21.655 --> 00:01:25.448
So we just figured out that
if you have the number three,

00:01:25.448 --> 00:01:30.093
its opposite is going
to be negative three.

00:01:30.571 --> 00:01:33.484
Now what if your number is negative?

00:01:33.484 --> 00:01:37.267
What if your number, let's
say the number negative four.

00:01:37.409 --> 00:01:38.744
What's the opposite of that?

00:01:38.744 --> 00:01:40.287
And I encourage you to pause the video

00:01:40.287 --> 00:01:42.342
and try to think about it on your own.

00:01:42.342 --> 00:01:45.489
Well, you say, okay, negative
four is right over here.

00:01:45.489 --> 00:01:46.928
That's negative four.

00:01:46.928 --> 00:01:48.623
It is four to the left of zero.

00:01:48.623 --> 00:01:52.465
One, two, three, four to the left of zero.

00:01:52.465 --> 00:01:55.763
So its opposite is going to
be four to the right of it.

00:01:55.763 --> 00:01:59.142
So one, two, three, four.

00:01:59.142 --> 00:02:01.429
It's going to be positive.

00:02:01.429 --> 00:02:03.565
It's going to be positive four.

00:02:03.565 --> 00:02:06.362
So you're probably starting
to see a pattern here.

00:02:06.362 --> 00:02:10.250
The opposite of a number is going to be

00:02:10.250 --> 00:02:13.048
the opposite sign of that number.

00:02:13.786 --> 00:02:15.894
If you have a positive three here,

00:02:15.894 --> 00:02:17.960
its opposite is going
to be negative three.

00:02:17.960 --> 00:02:19.620
If you start with negative four,

00:02:19.620 --> 00:02:22.580
its opposite is going to be positive four.

00:02:22.580 --> 00:02:25.274
One way to think about
it, it's going to have the

00:02:25.274 --> 00:02:29.233
same absolute value but
have a different sign.

00:02:29.233 --> 00:02:30.951
Or another way to think about it is,

00:02:30.951 --> 00:02:33.412
however if this is three
to the right of zero,

00:02:33.412 --> 00:02:36.034
its opposite is going to be
three to the left of zero.

00:02:36.034 --> 00:02:38.775
Or if the number is four
to the left of zero,

00:02:38.775 --> 00:02:42.791
its opposite is going to be
four to the right of zero.

00:02:42.791 --> 00:02:44.973
So we'll do one last one.

00:02:44.973 --> 00:02:47.816
What is the opposite of...

00:02:48.051 --> 00:02:51.647
What's going to be the opposite of one?

00:02:52.194 --> 00:02:55.167
Well one is one to the right of zero,

00:02:55.167 --> 00:02:57.848
so its opposite is going to
be one to the left of zero,

00:02:57.848 --> 00:02:59.229
or negative one.

00:02:59.229 --> 00:03:01.912
Or another way to think
about it, one is positive

00:03:01.912 --> 00:03:04.431
so its opposite is where you're
going to change the sign.

00:03:04.431 --> 00:03:06.277
Instead of being positive
it's going to be negative.

00:03:06.277 --> 00:03:08.332
It's going to be negative one.

00:03:08.332 --> 00:03:10.709
Either way that you can think about it.


Idea behind hypothesis testing

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.850
- [Voiceover] Let's say that
you have a cholesterol test,

00:00:02.850 --> 00:00:05.090
and you know, you somehow magically know

00:00:05.090 --> 00:00:08.050
that the probability that it is accurate,

00:00:08.050 --> 00:00:13.050
that it gives the correct
results is 99, 99%.

00:00:14.820 --> 00:00:17.720
You have a 99 out of
100 chance that any time

00:00:17.720 --> 00:00:21.320
you apply this test, that
it is going to be accurate.

00:00:21.320 --> 00:00:23.650
Now let's say that you, and
you just magically know that,

00:00:23.650 --> 00:00:25.350
we're just assuming that.

00:00:25.350 --> 00:00:28.390
Now let's just say that you
get 100 folks into this room

00:00:28.390 --> 00:00:31.320
and you apply this test
to all 100 of them.

00:00:31.320 --> 00:00:36.320
So apply, apply test 100 times.

00:00:38.820 --> 00:00:41.450
So what are some of the
possible outcomes here?

00:00:41.450 --> 00:00:45.990
Is it for sure that 99,
exactly 99 out of the 100

00:00:45.990 --> 00:00:47.950
are going to be accurate
and that 1 out of the 100

00:00:47.950 --> 00:00:49.420
is gonna be inaccurate?

00:00:49.420 --> 00:00:51.850
Well that's definitely
a likely possibility,

00:00:51.850 --> 00:00:53.550
but it's also possible
you get a little lucky

00:00:53.550 --> 00:00:56.450
and all 100 are accurate,
or you get a little unlucky

00:00:56.450 --> 00:01:00.190
and that 98 are accurate
and that two are inaccurate.

00:01:00.190 --> 00:01:02.580
And actually, I calculated the
probabilities ahead of time,

00:01:02.580 --> 00:01:03.650
and the goal of this video isn't to go

00:01:03.650 --> 00:01:06.090
into the probability
and combinatorics of it,

00:01:06.090 --> 00:01:07.050
but if you're curious about it,

00:01:07.050 --> 00:01:08.490
there's a lot of good
videos on probability

00:01:08.490 --> 00:01:10.290
and combinatorics on Khan Academy,

00:01:10.290 --> 00:01:12.120
but I calculated ahead of time,

00:01:12.120 --> 00:01:14.450
and the probability, if you have something

00:01:14.450 --> 00:01:16.650
that has a 99% chance of being accurate,

00:01:16.650 --> 00:01:19.520
and you apply it 100 times,

00:01:19.520 --> 00:01:22.100
the probability that it is accurate,

00:01:22.530 --> 00:01:26.950
that it is accurate 100
out of the 100 times,

00:01:26.950 --> 00:01:29.320
is approximately equal to,

00:01:29.320 --> 00:01:32.720
approximately equal to 36.6%.

00:01:32.720 --> 00:01:36.020
I rounded to the nearest
tenth of a percent.

00:01:36.020 --> 00:01:37.220
So it's a little better
than a third chance

00:01:37.220 --> 00:01:39.990
that you'll actually
get, all of the people

00:01:39.990 --> 00:01:41.150
are going to get an accurate result,

00:01:41.150 --> 00:01:42.390
even though for any one of them

00:01:42.390 --> 00:01:44.990
there's a 99% chance that it is accurate.

00:01:44.990 --> 00:01:47.050
Now we could keep going,

00:01:47.050 --> 00:01:48.750
the probability that it is accurate,

00:01:48.750 --> 00:01:50.050
I'm just gonna put these quotes here

00:01:50.050 --> 00:01:51.950
so I don't have to rewrite
accurate over and over again,

00:01:51.950 --> 00:01:56.690
the probability that it is
accurate 99 out of 100 times,

00:01:56.690 --> 00:02:01.690
I calculated it ahead of time,
it is approximately 37.0%.

00:02:02.150 --> 00:02:03.520
So this is what you would expect,

00:02:03.520 --> 00:02:05.750
getting 100 out of 100
doesn't seem that unlikely

00:02:05.750 --> 00:02:07.290
if each of the times you apply it

00:02:07.290 --> 00:02:09.590
has a 99% chance of being accurate,

00:02:09.590 --> 00:02:12.790
but it makes sense that
you would expect 99 out 100

00:02:12.790 --> 00:02:15.510
to be even more likely,
slightly more likely.

00:02:15.510 --> 00:02:17.720
And we can of course keep going,

00:02:17.720 --> 00:02:20.020
the probability that it is accurate

00:02:20.020 --> 00:02:25.020
98 out of 100 times is
approximately 18.5%.

00:02:28.650 --> 00:02:30.350
And I'm just gonna do a few more.

00:02:30.350 --> 00:02:33.090
The probability that it is accurate

00:02:33.090 --> 00:02:35.620
97 out of 100 times, and once again

00:02:35.620 --> 00:02:37.810
I calculated all of these ahead of time,

00:02:37.810 --> 00:02:42.150
is 6%, so it's definitely
in the realm of possibility,

00:02:42.150 --> 00:02:44.190
but it's, the probability is much lower

00:02:44.190 --> 00:02:48.950
than having 99 out of 100 or
100 out of 100 being accurate,

00:02:48.950 --> 00:02:51.790
and then the probability, let
me put the double quotes here,

00:02:51.790 --> 00:02:52.920
the probability that it is accurate

00:02:52.920 --> 00:02:57.920
96 out of 100 times is approximately 1.5%,

00:02:59.710 --> 00:03:02.080
and then the probability,
and I'll just do one more,

00:03:02.080 --> 00:03:03.450
I could keep going, the probability,

00:03:03.450 --> 00:03:04.420
you know, there's some probability

00:03:04.420 --> 00:03:06.550
that even though each
test has a 99% chance

00:03:06.550 --> 00:03:09.020
you just get super unlucky
and that, you know,

00:03:09.020 --> 00:03:10.650
very few of them are
accurate, well I'll just,

00:03:10.650 --> 00:03:11.990
and you see, you see what's happening

00:03:11.990 --> 00:03:14.760
to the probabilities as we have

00:03:14.760 --> 00:03:16.950
fewer and fewer of them being accurate,

00:03:16.950 --> 00:03:19.650
it becomes less and less probable.

00:03:19.650 --> 00:03:24.520
So the probability that 95 out of the 100

00:03:24.520 --> 00:03:29.520
are accurate is, is approximately 0.3%.

00:03:32.280 --> 00:03:33.520
So this was just kind of a,

00:03:33.520 --> 00:03:34.790
I guess you could say
a thought experiment.

00:03:34.790 --> 00:03:36.620
If we had a test that we know for sure

00:03:36.620 --> 00:03:37.750
that every time you administer it,

00:03:37.750 --> 00:03:40.790
the probability that
it is accurate is 99%,

00:03:40.790 --> 00:03:41.850
then these are the probabilities

00:03:41.850 --> 00:03:43.520
that if you administered it 100 times,

00:03:43.520 --> 00:03:45.810
that you get 100 out 100 accurate,

00:03:45.810 --> 00:03:48.380
the probability that you
get 99 out of 100 accurate,

00:03:48.380 --> 00:03:50.210
and so on and so forth.

00:03:50.210 --> 00:03:51.450
So let's just keep that in mind,

00:03:51.450 --> 00:03:53.950
and then think a little bit
about hypothesis testing,

00:03:53.950 --> 00:03:57.050
and how we can use this framework.

00:03:57.050 --> 00:03:58.950
So let's put all that in
the back of our minds,

00:03:58.950 --> 00:04:03.410
and let's say that you
have devised a new test,

00:04:03.410 --> 00:04:07.820
you have a new test, and you
don't know how accurate it is.

00:04:07.820 --> 00:04:09.120
You have a new cholesterol test,

00:04:09.120 --> 00:04:11.510
you don't know how accurate it is,

00:04:11.510 --> 00:04:13.920
you know that in order
for it to be approved

00:04:13.920 --> 00:04:17.330
by whatever governing body
it has to be accurate 99,

00:04:17.330 --> 00:04:21.090
the probability of it being
accurate has to be 99%.

00:04:21.090 --> 00:04:26.090
So needs to have probability of accurate,

00:04:29.550 --> 00:04:32.220
accurate equal to 99%.

00:04:32.220 --> 00:04:33.150
You don't know if this is true,

00:04:33.150 --> 00:04:35.520
you just know that that's
what it needs to be.

00:04:35.520 --> 00:04:36.490
And so you have your test,

00:04:36.490 --> 00:04:38.020
and let's you set up a hypothesis,

00:04:38.020 --> 00:04:39.450
and your hypothesis
could be a lot of things,

00:04:39.450 --> 00:04:42.520
and once you get deeper into
statistics, there's, you know,

00:04:42.520 --> 00:04:44.690
null hypothesis and alternate hypotheses,

00:04:44.690 --> 00:04:46.650
but let's just start with
just a simple hypothesis,

00:04:46.650 --> 00:04:51.650
you're hopeful, your hypothesis
is that the probability

00:04:52.590 --> 00:04:57.590
that your new test is accurate is,

00:05:00.020 --> 00:05:02.290
this is your hypothesis,
because you want that

00:05:02.290 --> 00:05:04.520
to be your hypothesis cause
if you feel good about it,

00:05:04.520 --> 00:05:06.350
then you're like, okay,
yeah, maybe I'll get approved

00:05:06.350 --> 00:05:08.150
by the appropriate governing body.

00:05:08.150 --> 00:05:10.390
So you say, "Hey, my hypothesis is that

00:05:10.390 --> 00:05:13.650
"my new test is accurate
99, the probability

00:05:13.650 --> 00:05:16.580
"of it being accurate is 99%."

00:05:16.580 --> 00:05:19.850
So then you go off and
you apply it 100 times.

00:05:19.850 --> 00:05:22.590
So you apply your new test,

00:05:22.590 --> 00:05:25.120
you don't know the actual
probability of it being accurate,

00:05:25.120 --> 00:05:30.120
you apply the test 100 times.

00:05:30.890 --> 00:05:33.360
And let's say out of those 100 times

00:05:33.360 --> 00:05:36.220
you get that they are accurate,

00:05:36.220 --> 00:05:38.950
you get that it is
accurate, and you're able

00:05:38.950 --> 00:05:40.650
to use some other test that you, you know,

00:05:40.650 --> 00:05:43.320
some for-sure test, some
super accurate test,

00:05:43.320 --> 00:05:45.520
to verify your own test results,

00:05:45.520 --> 00:05:50.050
and you see that it is accurate
95 out of the 100 times.

00:05:50.050 --> 00:05:51.920
So the question you have is, well,

00:05:51.920 --> 00:05:54.020
does the hypothesis make sense to you?

00:05:54.020 --> 00:05:56.520
Will you accept this hypothesis?

00:05:56.520 --> 00:05:59.750
Well what you say is, well,
"If my hypothesis was true,

00:05:59.750 --> 00:06:03.520
"if my test were accurate 99,

00:06:03.520 --> 00:06:07.990
"if the probability of my
test being accurate is 99%,

00:06:07.990 --> 00:06:11.590
"what's the probability of
me getting this outcome?"

00:06:11.590 --> 00:06:12.650
Well, we figured that out.

00:06:12.650 --> 00:06:16.520
If it really was accurate 99% of the time,

00:06:16.520 --> 00:06:21.520
then the probability of getting
this outcome is only 0.3%.

00:06:22.590 --> 00:06:27.590
So if you assume true,
if you assume hypothesis,

00:06:29.220 --> 00:06:32.010
hypothesis, I'll just write "hyp,"

00:06:32.190 --> 00:06:34.590
if you assume the hypothesis is true,

00:06:34.590 --> 00:06:37.190
the probability of the outcome you got,

00:06:37.190 --> 00:06:42.190
probability of observed
outcome is approximately 0.3%.

00:06:48.750 --> 00:06:50.420
And so you say, "Look, you know,

00:06:50.420 --> 00:06:52.650
"maybe, it's definitely possible

00:06:52.650 --> 00:06:55.750
"that I just got very,
very, very, very unlikely,

00:06:55.750 --> 00:07:00.050
"but based on this, I probably
should reject my hypothesis,

00:07:00.050 --> 00:07:04.250
"because the probability
of me getting this outcome,

00:07:04.250 --> 00:07:05.210
"if the hypothesis was true,

00:07:05.210 --> 00:07:07.120
"is very, very, very, very low."

00:07:07.120 --> 00:07:08.850
And as we go deeper into statistics,

00:07:08.850 --> 00:07:11.720
you'll see that there are
thresholds that people often set,

00:07:11.720 --> 00:07:14.250
for, you know, if the probability
of something happening

00:07:14.250 --> 00:07:16.920
or not happening is above
or below some threshold,

00:07:16.920 --> 00:07:19.550
then we might reject a certain hypothesis.

00:07:19.550 --> 00:07:21.750
But in this world, you
could see that, look,

00:07:21.750 --> 00:07:24.190
if my test really was
accurate 99% of the time,

00:07:24.190 --> 00:07:26.750
for me to get, when I
apply it to 100 people,

00:07:26.750 --> 00:07:28.820
it's only accurate 95 out of 100,

00:07:28.820 --> 00:07:31.390
if my hypothesis is true,
that would have only,

00:07:31.390 --> 00:07:34.090
there's only a 0.3% chance

00:07:34.090 --> 00:07:36.090
that I would have seen this observation.

00:07:36.090 --> 00:07:38.650
So based on that, it might be
completely reasonable to say,

00:07:38.650 --> 00:07:41.820
"You know what, I might
reject my hypothesis,

00:07:41.820 --> 00:07:44.020
"look for a new test, I don't feel good

00:07:44.020 --> 00:07:47.370
"about this new cholesterol
test that I constructed."