Zero pairs worked example

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:01.800
- [Instructor] We're
told, "This is the key

00:00:01.800 --> 00:00:03.720
for the integer chips."

00:00:03.720 --> 00:00:07.020
So this yellow circle with
a plus is equal to one.

00:00:07.020 --> 00:00:09.930
This, I guess, pinkish
circle, peach circle,

00:00:09.930 --> 00:00:13.417
with a minus, that is
equal to negative one.

00:00:13.417 --> 00:00:15.570
"Consider the following image."

00:00:15.570 --> 00:00:16.740
And so, there we have a bunch

00:00:16.740 --> 00:00:18.900
of the positive yellow circles,

00:00:18.900 --> 00:00:21.480
and then we have even more of the pink,

00:00:21.480 --> 00:00:24.127
or peach-looking negative circles.

00:00:24.127 --> 00:00:27.000
"Complete the description of the image.

00:00:27.000 --> 00:00:32.000
There are blank zero pairs
and blank units left over."

00:00:32.190 --> 00:00:33.630
If you're actually doing this exercise,

00:00:33.630 --> 00:00:35.730
this is a screenshot from the exercise,

00:00:35.730 --> 00:00:37.110
you would fill in something here,

00:00:37.110 --> 00:00:39.270
and there's a dropdown over here.

00:00:39.270 --> 00:00:40.807
So the first question you might ask is,

00:00:40.807 --> 00:00:44.040
"What is a zero pair?"

00:00:44.040 --> 00:00:48.270
So a zero pair is when you
take two opposite numbers,

00:00:48.270 --> 00:00:51.570
and they essentially, when you add them,

00:00:51.570 --> 00:00:53.520
they cancel out to get to zero.

00:00:53.520 --> 00:00:57.840
An example of that would
be one plus negative one.

00:00:57.840 --> 00:01:01.410
These two numbers are opposites,
so they are a zero pair,

00:01:01.410 --> 00:01:04.320
because when you add them
together, you get zero.

00:01:04.320 --> 00:01:05.880
Why does that make sense?

00:01:05.880 --> 00:01:09.330
Well, imagine if positive
values were walking forward,

00:01:09.330 --> 00:01:11.820
and negative values
were walking backwards.

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So you could view this as one step forward

00:01:14.130 --> 00:01:15.810
plus one step backward.

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That's just going to get you
back to where you were before.

00:01:18.450 --> 00:01:20.280
You could have other zero pairs.

00:01:20.280 --> 00:01:25.280
You could have things like
positive two plus negative two.

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That's also a zero pair.

00:01:27.240 --> 00:01:30.060
You walk two steps forward
and then two steps backward.

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That's just going to get you
back to where you were before.

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You will not have moved after all,

00:01:34.080 --> 00:01:36.840
or you'd be back to where you were before.

00:01:36.840 --> 00:01:38.639
So let's think about how many zero pairs

00:01:38.639 --> 00:01:40.500
there are over here.

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Well, we know that each of
the ones forms a zero pair

00:01:43.890 --> 00:01:45.240
with each of the negative ones.

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So that's one zero pair, two, three,

00:01:49.500 --> 00:01:52.350
four, five zero pairs.

00:01:52.350 --> 00:01:53.790
So I'll just write a five right over here.

00:01:53.790 --> 00:01:55.470
You would type that in
if you were doing this.

00:01:55.470 --> 00:01:58.500
And then how many units are left over?

00:01:58.500 --> 00:02:00.600
Well, you could see right over here,

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we see that we have two of the
negative one units left over.


Common percentages

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:01.450
- [Instructor] What I would like you to do

00:00:01.450 --> 00:00:04.010
is pause this video and
see if you can calculate

00:00:04.010 --> 00:00:09.010
each of these percentages, and
ideally do it in your head.

00:00:10.110 --> 00:00:12.540
All right, now let's do it together.

00:00:12.540 --> 00:00:14.630
Now I said, how are you
going to do it in your head?

00:00:14.630 --> 00:00:16.570
You might be tempted to write down these

00:00:16.570 --> 00:00:19.140
as multiplication problems
and have to write it down.

00:00:19.140 --> 00:00:21.140
And that might be a reasonable thing,

00:00:21.140 --> 00:00:23.490
but these are particular percentages

00:00:23.490 --> 00:00:25.640
that you might see a lot of in life

00:00:25.640 --> 00:00:28.450
and so it's useful to think
about them in your head.

00:00:28.450 --> 00:00:30.774
For example, 1%.

00:00:30.774 --> 00:00:35.600
1% is the same thing as 1 over 100.

00:00:35.600 --> 00:00:38.370
So 1% of 900 is the same thing

00:00:38.370 --> 00:00:41.230
as 1/100 of 900.

00:00:41.230 --> 00:00:44.250
And so this question boils down to really

00:00:44.250 --> 00:00:48.990
what is 900 divided by 100.

00:00:48.990 --> 00:00:52.510
And that of course is equal to 9.

00:00:52.510 --> 00:00:56.680
Let's do another example.
What's 10% of 630?

00:00:56.680 --> 00:00:59.100
Well 10%

00:00:59.100 --> 00:01:02.560
is equal to 10 over 100,

00:01:02.560 --> 00:01:05.290
which is the same thing as 1 over 10.

00:01:05.290 --> 00:01:07.560
So if I were to say 10% of 630,

00:01:07.560 --> 00:01:11.220
that's the same thing
as saying 1/10 of 630.

00:01:11.220 --> 00:01:15.320
So this all boils down
to 630 divided by 10,

00:01:15.320 --> 00:01:19.090
which you would recognize as 63.

00:01:19.090 --> 00:01:24.090
All right, let's do this
next one, 20% of 45.

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You might recognize already,
and if you haven't already,

00:01:26.430 --> 00:01:28.720
it's good to recognize that 20%

00:01:28.720 --> 00:01:31.160
is the same thing as 20 over 100,

00:01:31.160 --> 00:01:33.850
or that it's the same thing as 1 over 5.

00:01:33.850 --> 00:01:36.740
It's good to just know
that hey, 20% is 1/5.

00:01:36.740 --> 00:01:39.150
So if I'm saying 1/5 of 45,

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that's the same thing as 45 divided by 5,

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which is of course, equal to 9.

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Let's keep going. This is too much fun.

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25%, you might recognize
that's the same thing as 1/4,

00:01:53.141 --> 00:01:56.320
25% is 25 over 100.

00:01:56.320 --> 00:01:59.050
If you divide the numerator
and the denominator by 25,

00:01:59.050 --> 00:02:00.900
you're going to get 1 over 4.

00:02:00.900 --> 00:02:04.120
So this is equivalent to
saying what's 1/4 of 28.

00:02:04.120 --> 00:02:09.110
Well, 28 divided by 4 is, of course, 7.

00:02:09.110 --> 00:02:13.350
Let's keep going. 50% of 128.

00:02:13.350 --> 00:02:16.223
You might recognize 50%
is the same thing as 1/2.

00:02:17.070 --> 00:02:20.923
It's 50 over 100, which is equal to 1/2.

00:02:21.840 --> 00:02:24.690
And so we're really just
saying what's half of 128,

00:02:24.690 --> 00:02:27.020
or what's 128 divided by 2.

00:02:27.020 --> 00:02:29.533
And that of course would be 64.

00:02:30.600 --> 00:02:35.600
And then last but not least, 400% of 8.

00:02:35.610 --> 00:02:40.500
Well, 400%, that's the
same thing as 400 over 100,

00:02:40.500 --> 00:02:42.330
or it's equal to 4.

00:02:42.330 --> 00:02:45.120
So that's really saying what's 4 times 8.

00:02:45.120 --> 00:02:50.120
So 4 times 8 is, of course, equal to 32.

00:02:50.350 --> 00:02:51.353
And we are done.


Plotting inequalities on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:02.784
- [Instructor] We're told
that Pierre has 48 minutes

00:00:02.784 --> 00:00:06.600
until he needs to get
ready for his dance lesson.

00:00:06.600 --> 00:00:10.410
Graph how many minutes he can
spend playing with his pet,

00:00:10.410 --> 00:00:12.570
before getting ready.

00:00:12.570 --> 00:00:15.720
If you are so inspired, I
encourage you to be so inspired,

00:00:15.720 --> 00:00:17.993
pause the video and see if
you can work through that.

00:00:17.993 --> 00:00:20.160
All right, now let's think
through this together.

00:00:20.160 --> 00:00:23.100
So first of all, before I
even graph it, I could say,

00:00:23.100 --> 00:00:27.180
let's say that M is how many minutes

00:00:27.180 --> 00:00:31.290
he is playing with his
pet, before getting ready.

00:00:31.290 --> 00:00:32.700
So let's just call it,

00:00:32.700 --> 00:00:35.279
that's what M is going to be equal to.

00:00:35.279 --> 00:00:37.230
It's how many minutes he spends.

00:00:37.230 --> 00:00:41.370
And so how would M
relate to the 48 minutes?

00:00:41.370 --> 00:00:42.660
Well, he has 48 minutes,

00:00:42.660 --> 00:00:46.080
so he could spend anywhere from zero,

00:00:46.080 --> 00:00:49.110
up to and including 48 minutes.

00:00:49.110 --> 00:00:52.140
So you might say M is less than 48.

00:00:52.140 --> 00:00:53.160
But it's not just less than,

00:00:53.160 --> 00:00:55.620
it could even be exactly 48 minutes.

00:00:55.620 --> 00:00:57.297
As long as immediately after that,

00:00:57.297 --> 00:00:59.400
he gets ready for his dance lesson.

00:00:59.400 --> 00:01:02.130
So M is less than or equal to 48.

00:01:02.130 --> 00:01:04.260
So if I were to put that on a number line,

00:01:04.260 --> 00:01:06.633
let me construct a number line like this.

00:01:09.018 --> 00:01:13.560
And let me put this at
zero right over here.

00:01:13.560 --> 00:01:18.560
And let's say this is 10, 20,
30, 40, 50, 60, keep going.

00:01:28.170 --> 00:01:29.760
This could be 70.

00:01:29.760 --> 00:01:32.100
This is going to be
essentially the values of M

00:01:32.100 --> 00:01:33.060
that are going to be okay,

00:01:33.060 --> 00:01:35.372
as long as he's prepared
for his dance lesson.

00:01:35.372 --> 00:01:38.340
M is less than or equal to 48.

00:01:38.340 --> 00:01:42.960
So what we can do is, we can
go to 48, which is, let's see,

00:01:42.960 --> 00:01:43.793
this would be 45,

00:01:43.793 --> 00:01:45.810
so 40 eight's gonna be right around there.

00:01:45.810 --> 00:01:46.950
So that's 48.

00:01:46.950 --> 00:01:48.840
And since it can be equal to 48,

00:01:48.840 --> 00:01:50.100
we're gonna fill in the circle.

00:01:50.100 --> 00:01:52.980
If it just said purely less than 48,

00:01:52.980 --> 00:01:54.540
but not less than or equal,

00:01:54.540 --> 00:01:56.100
then we would put an open circle here.

00:01:56.100 --> 00:01:58.830
But because it's less than or equal to,

00:01:58.830 --> 00:02:01.050
we're going to include
48 right over there.

00:02:01.050 --> 00:02:03.660
I'll write the number 48 to
make it clear what that is,

00:02:03.660 --> 00:02:05.640
and less than or equal to.

00:02:05.640 --> 00:02:08.520
So it's gonna be all of
the values less than that.

00:02:08.520 --> 00:02:11.343
And so it would look something like this.

00:02:13.680 --> 00:02:15.960
I'm doing that light blue color.

00:02:15.960 --> 00:02:20.195
It would look something
like then, like that.

00:02:20.195 --> 00:02:23.070
And if we wanted to be clear
that we're including zero,

00:02:23.070 --> 00:02:25.860
we could actually put a dot here as well.

00:02:25.860 --> 00:02:26.790
It wouldn't make sense

00:02:26.790 --> 00:02:30.240
to go and include negative values as well.

00:02:30.240 --> 00:02:33.573
But let's do another
example, a different example.

00:02:33.573 --> 00:02:37.470
So here, we are told
that the Harris family

00:02:37.470 --> 00:02:39.420
needs to heat their leftover gumbo,

00:02:39.420 --> 00:02:42.300
to a minimum of 74 degrees Celsius,

00:02:42.300 --> 00:02:44.250
to be sure it is safe to eat.

00:02:44.250 --> 00:02:47.220
Graph temperatures to which
they could heat their food,

00:02:47.220 --> 00:02:49.470
so that it is safe to eat.

00:02:49.470 --> 00:02:50.820
So once again, pause this video,

00:02:50.820 --> 00:02:51.990
and see if you could think about that,

00:02:51.990 --> 00:02:53.613
before we do this together.

00:02:55.080 --> 00:02:58.680
All right, well let's imagine,

00:02:58.680 --> 00:03:02.160
let's see, maybe we'll
say T for temperature.

00:03:02.160 --> 00:03:03.510
So T for temperature.

00:03:03.510 --> 00:03:07.320
And let's say T is the
temperature that they heat to,

00:03:07.320 --> 00:03:12.320
temperature that they heat their gumbo to,

00:03:12.737 --> 00:03:14.850
that they heat to.

00:03:14.850 --> 00:03:16.890
And now let's do a number line.

00:03:16.890 --> 00:03:18.510
We see, well, before
I even do number line,

00:03:18.510 --> 00:03:21.300
let me express it as some
kind of an inequality.

00:03:21.300 --> 00:03:25.230
So they need a minimum
of 74 degrees Celsius.

00:03:25.230 --> 00:03:29.580
So that means it has to
be at least 74, or higher.

00:03:29.580 --> 00:03:34.580
So that means T is not just
greater than 74 degrees Celsius.

00:03:37.230 --> 00:03:40.200
It can also be exactly 74 degrees Celsius,

00:03:40.200 --> 00:03:43.474
'cause it says as long
as it's a minimum of 74.

00:03:43.474 --> 00:03:44.550
So if it's exactly 74,

00:03:44.550 --> 00:03:47.070
that is apparently
going to be safe to eat.

00:03:47.070 --> 00:03:48.420
And anything higher than that,

00:03:48.420 --> 00:03:49.860
is also going to be safe to eat.

00:03:49.860 --> 00:03:51.540
At some point you can get
to such a high temperature,

00:03:51.540 --> 00:03:54.210
that you essentially
turn your food into ash.

00:03:54.210 --> 00:03:56.670
It might not be a delicious gumbo anymore,

00:03:56.670 --> 00:03:58.930
but it would probably be safe to eat,

00:03:58.930 --> 00:03:59.790
if you're just eating gumbo powder

00:03:59.790 --> 00:04:02.040
of some gumbo ash I guess.

00:04:02.040 --> 00:04:03.450
Actually, I'm not sure
if that's safe to eat.

00:04:03.450 --> 00:04:04.920
But let's just assume it is.

00:04:04.920 --> 00:04:06.060
I don't recommend doing that.

00:04:06.060 --> 00:04:08.190
But let's put this on the number line.

00:04:08.190 --> 00:04:12.810
So let me, and actually here, actually,

00:04:12.810 --> 00:04:16.200
let me do it in white so that
I can, in that reddish color,

00:04:16.200 --> 00:04:19.637
I can actually put the
values that we care about.

00:04:19.637 --> 00:04:22.380
And we can have negative temperature,

00:04:22.380 --> 00:04:23.880
if we're talking about degrees Celsius.

00:04:23.880 --> 00:04:28.770
So let's say that this is zero degrees,

00:04:28.770 --> 00:04:33.770
this is 10, 20, 30, 40, 50, 60, 70.

00:04:39.390 --> 00:04:40.590
Let me label that one.

00:04:40.590 --> 00:04:42.300
That's 70 degrees.

00:04:42.300 --> 00:04:44.550
This would be 80 degrees right over here.

00:04:44.550 --> 00:04:48.030
If we wanted to, this would
be negative 10, negative 20.

00:04:48.030 --> 00:04:49.110
And we have to be greater than,

00:04:49.110 --> 00:04:51.510
or equal to 74 degrees Celsius.

00:04:51.510 --> 00:04:55.260
If we just had T is greater
than 74 degrees Celsius,

00:04:55.260 --> 00:04:57.990
we would go to 74, which
would be right around there.

00:04:57.990 --> 00:04:59.490
We put an open circle,

00:04:59.490 --> 00:05:01.620
and then we would go greater than that.

00:05:01.620 --> 00:05:06.420
So that's if T was strictly
greater than 74 degrees Celsius.

00:05:06.420 --> 00:05:08.580
But it's greater than or equal to.

00:05:08.580 --> 00:05:10.080
And so because of that,

00:05:10.080 --> 00:05:12.300
we are going to fill in
this dot right over there.

00:05:12.300 --> 00:05:17.300
And to be clear, that is at
74 degrees and we're done.


Signs of sums on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:02.460
- [Instructor] Let's give
ourselves some intuition

00:00:02.460 --> 00:00:05.160
and then some practice
adding negative numbers.

00:00:05.160 --> 00:00:08.503
So let's start with -11 plus -3.

00:00:10.290 --> 00:00:13.410
So first we can visualize
what -11 looks like

00:00:13.410 --> 00:00:14.730
on a number line like this.

00:00:14.730 --> 00:00:17.610
I intentionally have not
marked off any of the numbers.

00:00:17.610 --> 00:00:19.290
We can just build an intuition

00:00:19.290 --> 00:00:22.200
for what's going to happen here.

00:00:22.200 --> 00:00:23.850
So if we're talking about -11,

00:00:23.850 --> 00:00:27.870
you could think about that
as 11 to the left of zero,

00:00:27.870 --> 00:00:30.240
or if you start at zero right over here,

00:00:30.240 --> 00:00:33.540
we are going to go 11 to the left,

00:00:33.540 --> 00:00:35.040
which I'm just estimating.

00:00:35.040 --> 00:00:36.090
I have nothing marked here,

00:00:36.090 --> 00:00:40.350
but let's just say that
gets us right about there.

00:00:40.350 --> 00:00:45.350
Now to that, I am going to add -3.

00:00:45.840 --> 00:00:50.250
So if I start at -11 and I add -3,

00:00:50.250 --> 00:00:52.260
I'm gonna go three more to the left.

00:00:52.260 --> 00:00:56.130
I'm gonna go three more even
more in the negative direction.

00:00:56.130 --> 00:00:59.310
So it's going to go something like that,

00:00:59.310 --> 00:01:02.190
and I'm going to end
up even more negative,

00:01:02.190 --> 00:01:06.390
even further from zero,
further to the left from zero

00:01:06.390 --> 00:01:08.160
than I was at -11.

00:01:08.160 --> 00:01:11.160
You might already be
guessing what that is,

00:01:11.160 --> 00:01:14.340
but we'll think about
that more in a little bit.

00:01:14.340 --> 00:01:16.290
Let's do another example.

00:01:16.290 --> 00:01:21.290
Let's say, -11 plus 3,
what would that look like?

00:01:21.300 --> 00:01:25.830
Well, we have the same -11,
going on right over here,

00:01:25.830 --> 00:01:27.900
which we already drew,

00:01:27.900 --> 00:01:30.324
but now let me do plus
3 in a different color,

00:01:30.324 --> 00:01:32.700
let me do it in green so
we don't get confused.

00:01:32.700 --> 00:01:34.860
So negative 11 plus 3.

00:01:34.860 --> 00:01:37.620
So if we start at -11 right over here,

00:01:37.620 --> 00:01:39.390
but now we're adding 3.

00:01:39.390 --> 00:01:42.870
So now I'm gonna go to the right.

00:01:42.870 --> 00:01:46.830
So notice, here, I'm still ending up left

00:01:46.830 --> 00:01:49.620
of zero on the number line,
so I'm still going to end up

00:01:49.620 --> 00:01:52.890
in a negative place,
but it is less negative.

00:01:52.890 --> 00:01:57.890
It's left, it's less to the
left that negative 11 was.

00:01:57.900 --> 00:01:59.160
But that's interesting,

00:01:59.160 --> 00:02:01.050
I'm still gonna get a negative value.

00:02:01.050 --> 00:02:06.050
Now, let's imagine if I
were to have +11 plus -3,

00:02:07.380 --> 00:02:09.150
think about what would happen there.

00:02:09.150 --> 00:02:12.210
So +11, lemme do this in this red color,

00:02:12.210 --> 00:02:14.380
it might look something like this,

00:02:14.380 --> 00:02:19.380
+11 would look something like that.

00:02:19.410 --> 00:02:23.010
I'm starting at zero
and I go 11 to the right

00:02:23.010 --> 00:02:25.643
and now I'm going to add -3 to that,

00:02:27.024 --> 00:02:28.740
I'm going to add negative three.

00:02:28.740 --> 00:02:32.016
So where I left off here,
I'm now going to go three.

00:02:32.016 --> 00:02:34.212
Am I going to the right or to the left?

00:02:34.212 --> 00:02:35.217
What's a -3?

00:02:35.217 --> 00:02:38.250
I'm going to go to the left over here.

00:02:38.250 --> 00:02:41.970
So it might get us someplace
right around there.

00:02:41.970 --> 00:02:46.170
So I went 11 to the right
and then I go 3 to the left.

00:02:46.170 --> 00:02:47.550
So I'm still going to end up to the right.

00:02:47.550 --> 00:02:50.010
I'm still going to have a positive value,

00:02:50.010 --> 00:02:54.000
but it's going to be less
positive than that 11 over there.

00:02:54.000 --> 00:02:55.500
Now, let's give one more scenario

00:02:55.500 --> 00:02:57.240
and this is one that you're
probably familiar with

00:02:57.240 --> 00:02:58.440
for many years.

00:02:58.440 --> 00:03:02.250
If I had +11 plus 3.

00:03:02.250 --> 00:03:06.210
Well, I have that +11 in
red already over there.

00:03:06.210 --> 00:03:09.570
And then if I were to add 3.

00:03:09.570 --> 00:03:12.030
it would get me even more positive

00:03:12.030 --> 00:03:13.920
and I think you know what that probably is

00:03:13.920 --> 00:03:16.110
if you go even more positive.

00:03:16.110 --> 00:03:18.360
So there's a couple of
interesting patterns.

00:03:18.360 --> 00:03:21.420
If we are adding numbers
of the same signs,

00:03:21.420 --> 00:03:22.890
if we're adding two negatives,

00:03:22.890 --> 00:03:24.300
we still end up with a negative.

00:03:24.300 --> 00:03:27.750
If we add two positives, we
still end up with a positive.

00:03:27.750 --> 00:03:30.930
But when we're adding
numbers of different signs,

00:03:30.930 --> 00:03:33.450
you actually end up taking on the sign

00:03:33.450 --> 00:03:37.530
of whichever one is further from zero.

00:03:37.530 --> 00:03:41.793
So for example, when I took -11 plus 3,

00:03:46.260 --> 00:03:48.840
I still end up with an answer
that is to the left of zero,

00:03:48.840 --> 00:03:50.370
something that is negative

00:03:50.370 --> 00:03:54.420
because 11 is further to the left of zero

00:03:54.420 --> 00:03:57.570
than 3 is moving in the right direction.

00:03:57.570 --> 00:03:59.850
And then when we did 11 plus -3,

00:03:59.850 --> 00:04:02.010
it was the other way around.

00:04:02.010 --> 00:04:03.990
Hopefully we've built up some intuition

00:04:03.990 --> 00:04:05.070
and now what I'm gonna do

00:04:05.070 --> 00:04:07.050
is tackle these exact same problems,

00:04:07.050 --> 00:04:09.030
but we're gonna do it with a number line

00:04:09.030 --> 00:04:11.130
that actually has the number marked off

00:04:11.130 --> 00:04:13.950
and we can actually compute
what these are going to be.

00:04:13.950 --> 00:04:16.350
So let me delete all of this

00:04:16.350 --> 00:04:21.350
and then let me give ourselves,
let's mark these things off,

00:04:21.390 --> 00:04:23.790
and then let's do the
same ones over again.

00:04:23.790 --> 00:04:28.790
So if I were to say -11 plus
-3, what would that get us?

00:04:31.230 --> 00:04:33.300
I'll try to remember
the colors I just used.

00:04:33.300 --> 00:04:36.810
- 11, I can start at zero here

00:04:36.810 --> 00:04:39.720
and I'm going to go to the left, 11,

00:04:39.720 --> 00:04:42.870
and I get to, whoops, I
went a little bit too far.

00:04:42.870 --> 00:04:46.920
I'm gonna go all the way
-5, -10, right over here,

00:04:46.920 --> 00:04:50.190
- 11, gets me right over there.

00:04:50.190 --> 00:04:55.190
And then to that I am going to add -3.

00:04:56.100 --> 00:04:58.800
So to that I'm going to, different color,

00:04:58.800 --> 00:05:00.390
let me do this carefully.

00:05:00.390 --> 00:05:02.850
To that, I am going to add -3.

00:05:02.850 --> 00:05:06.240
So I start there and I'm gonna
go three more to the left.

00:05:06.240 --> 00:05:08.100
So 1, 2, 3.

00:05:08.100 --> 00:05:11.460
So I am going to end up right over there.

00:05:11.460 --> 00:05:12.780
And where do I end up?

00:05:12.780 --> 00:05:15.120
That is at -14.

00:05:15.120 --> 00:05:17.553
So this is equal to negative,

00:05:17.553 --> 00:05:22.293
lemme do this in a neutral
color, it's equal to -14.

00:05:23.370 --> 00:05:25.890
Now, let's, actually, let's
do this in different orders

00:05:25.890 --> 00:05:27.930
just to see what's going on here.

00:05:27.930 --> 00:05:30.120
Let's do the, essentially,
something very similar,

00:05:30.120 --> 00:05:31.560
but let's go to the
right of the number line.

00:05:31.560 --> 00:05:35.220
Let's do 11 plus 3.

00:05:35.220 --> 00:05:38.193
So 11 plus 3, we could start with +11.

00:05:39.690 --> 00:05:44.690
So let me do that in red,
so +11 would look like that,

00:05:46.800 --> 00:05:48.600
that's that right over there.

00:05:48.600 --> 00:05:51.840
And then I am going to add 3.

00:05:51.840 --> 00:05:56.383
So I'm going to add 1,
2, 3, and I get to +14.

00:05:58.680 --> 00:06:02.040
So notice, here I ended up
14 to the right of zero,

00:06:02.040 --> 00:06:05.370
here I ended up 14 to the left of zero.

00:06:05.370 --> 00:06:06.630
But now let's look at the scenarios

00:06:06.630 --> 00:06:09.000
where we had mixed signs.

00:06:09.000 --> 00:06:14.000
So let's say we have -11 plus 3.

00:06:14.010 --> 00:06:17.220
What do we think that
is going to be equal to?

00:06:17.220 --> 00:06:20.310
So we already have drawn -11.

00:06:20.310 --> 00:06:22.560
That is negative 11, right over there.

00:06:22.560 --> 00:06:25.470
And now we want to think about plus 3,

00:06:25.470 --> 00:06:27.570
so we're gonna start right over here,

00:06:27.570 --> 00:06:30.480
and I'll do it a little bit
higher so we can see ourselves.

00:06:30.480 --> 00:06:31.680
And then plus three,

00:06:31.680 --> 00:06:33.420
we're gonna go three to
the right from that point.

00:06:33.420 --> 00:06:36.150
So 1, 2, 3.

00:06:36.150 --> 00:06:37.920
Where do we end up?

00:06:37.920 --> 00:06:41.970
Well, this right over here is -8,

00:06:41.970 --> 00:06:46.650
so we are now at -8.

00:06:46.650 --> 00:06:49.620
Now let's do it, let's
do the other way around.

00:06:49.620 --> 00:06:54.620
If we had +11 plus -3 what
is that going to be equal to?

00:06:56.520 --> 00:06:59.850
Well, we've already drawn +11 here,

00:06:59.850 --> 00:07:02.603
but now we're adding -3.

00:07:02.603 --> 00:07:05.460
- 3 is in this, like,
salmon color looking thing.

00:07:05.460 --> 00:07:07.290
They're very close, but different.

00:07:07.290 --> 00:07:12.240
So now we're going to add
-3 to +11, so we start here,

00:07:12.240 --> 00:07:14.970
but we're gonna go three to
the left 'cause it's negative.

00:07:14.970 --> 00:07:17.100
So 1, 2, 3.

00:07:17.100 --> 00:07:18.450
And where do we end up?

00:07:18.450 --> 00:07:21.483
We end up at +8.


Subtracting with integer chips

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:02.340
- [Instructor] Let's say
that we wanna figure out

00:00:02.340 --> 00:00:05.400
what negative eight minus negative two is.

00:00:05.400 --> 00:00:07.050
Now there's a lot of
ways to approach this,

00:00:07.050 --> 00:00:09.150
but what we're going to
focus on in this video

00:00:09.150 --> 00:00:10.890
is to really build the intuition,

00:00:10.890 --> 00:00:11.910
and we're going to do that

00:00:11.910 --> 00:00:13.740
with something called number chips

00:00:13.740 --> 00:00:15.570
which you might have seen before.

00:00:15.570 --> 00:00:16.560
So the first thing is,

00:00:16.560 --> 00:00:20.370
I'm going to represent negative
eight with number chips.

00:00:20.370 --> 00:00:21.203
Well number chips,

00:00:21.203 --> 00:00:24.630
I can either have a negative
number chip like that

00:00:24.630 --> 00:00:28.410
or I could have a positive
number chip like that,

00:00:28.410 --> 00:00:30.780
and if you have one
negative and one positive,

00:00:30.780 --> 00:00:32.130
if you were to combine these two,

00:00:32.130 --> 00:00:35.490
they could essentially
cancel out to just zero

00:00:35.490 --> 00:00:37.830
because they're opposites of each other,

00:00:37.830 --> 00:00:39.990
and we've looked at that in other videos,

00:00:39.990 --> 00:00:42.600
but let's go to actually
representing this up here.

00:00:42.600 --> 00:00:44.670
So negative eight could be represented

00:00:44.670 --> 00:00:47.940
as eight of these negative number chips.

00:00:47.940 --> 00:00:49.110
So let me do that over here.

00:00:49.110 --> 00:00:54.110
So that's one, two, three, four,

00:00:54.480 --> 00:00:59.310
five, six, seven, and eight.

00:00:59.310 --> 00:01:02.910
Now if we're subtracting
a negative number,

00:01:02.910 --> 00:01:04.650
this negative number right over here,

00:01:04.650 --> 00:01:08.730
you could view this as
two negative number chips,

00:01:08.730 --> 00:01:10.290
but we're going to be taking those away

00:01:10.290 --> 00:01:12.720
because we have a
subtraction right over here.

00:01:12.720 --> 00:01:14.910
So if we start with negative eight,

00:01:14.910 --> 00:01:17.040
if we wanna subtract negative two,

00:01:17.040 --> 00:01:20.910
we can just take away two of
these negative number chips,

00:01:20.910 --> 00:01:22.560
so let's just do that.

00:01:22.560 --> 00:01:26.010
And then how many negative
number chips are we left with?

00:01:26.010 --> 00:01:29.190
We have one, two, three, four, five, six.

00:01:29.190 --> 00:01:30.300
So this is going to be,

00:01:30.300 --> 00:01:31.980
is that going to be six or negative six?

00:01:31.980 --> 00:01:33.750
Well these are negative number chips,

00:01:33.750 --> 00:01:37.650
so this is going to be
equal to negative six.

00:01:37.650 --> 00:01:39.600
Now let's up it a little bit

00:01:39.600 --> 00:01:40.500
and let's start with something

00:01:40.500 --> 00:01:41.760
that might seem a little bit intuitive,

00:01:41.760 --> 00:01:43.830
and once again, there's
many ways to approach this.

00:01:43.830 --> 00:01:47.010
Maybe the number chip technique
will resonate with you

00:01:47.010 --> 00:01:49.200
for building a little bit of an intuition.

00:01:49.200 --> 00:01:52.290
Let's say we wanna figure
out four minus seven.

00:01:52.290 --> 00:01:53.990
What is that going to be equal to?

00:01:54.840 --> 00:01:56.850
Well four is a positive number,

00:01:56.850 --> 00:01:59.310
so I think I'm gonna use
some positive number chips

00:01:59.310 --> 00:02:00.750
to represent four.

00:02:00.750 --> 00:02:05.373
I could represent that as one, two,

00:02:06.240 --> 00:02:11.240
three, and four positive number chips.

00:02:11.910 --> 00:02:15.090
Now, if I'm taking away seven,
I could say hey, I'm gonna,

00:02:15.090 --> 00:02:17.130
and I'm taking away a positive seven here,

00:02:17.130 --> 00:02:18.090
so that would be equivalent

00:02:18.090 --> 00:02:21.090
to taking away seven
positive number chips,

00:02:21.090 --> 00:02:23.580
but I don't have seven
positive number chips here,

00:02:23.580 --> 00:02:25.800
so what could I do?

00:02:25.800 --> 00:02:27.660
Well, I talked before,

00:02:27.660 --> 00:02:31.200
if I have one positive and
one negative, they cancel out,

00:02:31.200 --> 00:02:33.060
so I can add these pairs

00:02:33.060 --> 00:02:35.010
of positive and negative number chips

00:02:35.010 --> 00:02:38.640
until I get to seven so that
I can then take them away.

00:02:38.640 --> 00:02:39.720
So let me do that.

00:02:39.720 --> 00:02:44.720
So I'm gonna add a positive and
I'm going to add a negative.

00:02:46.500 --> 00:02:48.810
Let me make sure I'm
doing that same color.

00:02:48.810 --> 00:02:50.970
So this, once again, this is still four,

00:02:50.970 --> 00:02:53.880
because these two right
over here, these cancel out.

00:02:53.880 --> 00:02:56.790
These are just zero, so you're
just left with positive four.

00:02:56.790 --> 00:02:59.490
So I can do that, frankly,
as many times as I want,

00:02:59.490 --> 00:03:00.960
and I still have four.

00:03:00.960 --> 00:03:02.940
So let me do that again.

00:03:02.940 --> 00:03:06.180
This is still equal to
four, because once again,

00:03:06.180 --> 00:03:09.360
each of these pairs
cancel out to just a zero.

00:03:09.360 --> 00:03:10.830
And let me do that again.

00:03:10.830 --> 00:03:11.663
And once again,

00:03:11.663 --> 00:03:14.460
my goal is to have seven
positive number chips

00:03:14.460 --> 00:03:16.890
'cause I need to take
away a positive seven,

00:03:16.890 --> 00:03:19.680
so let me put the
negative right over here.

00:03:19.680 --> 00:03:22.770
So this is all, right over
here, equivalent to four.

00:03:22.770 --> 00:03:24.510
It's equivalent to what we had before,

00:03:24.510 --> 00:03:25.980
but by adding these pairs,

00:03:25.980 --> 00:03:30.840
I now have seven positive number chips.

00:03:30.840 --> 00:03:34.590
Now, if I'm taking away a positive seven,

00:03:34.590 --> 00:03:38.220
I can take away seven, let
me do this in another color,

00:03:38.220 --> 00:03:40.980
I can take away seven
positive number chips,

00:03:40.980 --> 00:03:42.660
and I have seven now, so
I'm gonna take away one,

00:03:42.660 --> 00:03:45.990
two, three, four, five, six, seven.

00:03:45.990 --> 00:03:47.310
Now, what am I left with?

00:03:47.310 --> 00:03:49.710
Well all the positive
number chips are gone,

00:03:49.710 --> 00:03:52.740
and all I have left are
three negative number chips,

00:03:52.740 --> 00:03:55.500
so this is going to be negative three.

00:03:55.500 --> 00:03:57.030
Three negative number chips

00:03:57.030 --> 00:03:59.370
is the same thing as negative three.

00:03:59.370 --> 00:04:01.020
Let's do another example.

00:04:01.020 --> 00:04:06.020
Let's say that I wanna do
positive five minus negative one.

00:04:06.570 --> 00:04:07.830
Why don't you pause this video

00:04:07.830 --> 00:04:10.503
and see if you can do
this with number chips.

00:04:11.640 --> 00:04:12.840
All right?

00:04:12.840 --> 00:04:15.720
Well let's first represent
that positive five

00:04:15.720 --> 00:04:18.030
with five positive number chips,

00:04:18.030 --> 00:04:23.030
so that's one, two, three,
four, and then five.

00:04:25.920 --> 00:04:28.080
Now, we're not taking
away a positive number.

00:04:28.080 --> 00:04:29.640
If we were taking away a positive number,

00:04:29.640 --> 00:04:32.280
we could just take away some
of these positive number chips.

00:04:32.280 --> 00:04:33.780
We're taking away a negative number,

00:04:33.780 --> 00:04:35.850
so we need to have negative number chips

00:04:35.850 --> 00:04:37.530
in order to take 'em away.

00:04:37.530 --> 00:04:40.080
So, and we're only taking
away one negative one,

00:04:40.080 --> 00:04:42.780
or we're taking away one negative
number chip, I should say,

00:04:42.780 --> 00:04:44.010
so let's add one of those pairs.

00:04:44.010 --> 00:04:45.360
Remember, we can keep adding pairs

00:04:45.360 --> 00:04:47.700
'cause it doesn't change the
value because they cancel out.

00:04:47.700 --> 00:04:48.540
So let's add a pair.

00:04:48.540 --> 00:04:51.990
This is one positive number chip,

00:04:51.990 --> 00:04:56.490
and then let's put a negative
number chip right over there.

00:04:56.490 --> 00:04:58.620
This quantity that's
represented with number chips

00:04:58.620 --> 00:05:01.110
is still five because
these last two cancel out,

00:05:01.110 --> 00:05:04.080
and you're just left with
one, two, three, four, five.

00:05:04.080 --> 00:05:06.180
The whole reason why
I added this pair here

00:05:06.180 --> 00:05:08.520
is now I can take away a negative one.

00:05:08.520 --> 00:05:10.080
That negative one is represented

00:05:10.080 --> 00:05:12.840
by this negative number
chip right over here,

00:05:12.840 --> 00:05:14.520
so if I take that away,

00:05:14.520 --> 00:05:16.470
this taking away negative one,

00:05:16.470 --> 00:05:19.050
that's taking away a negative number chip.

00:05:19.050 --> 00:05:20.820
And now what am I left with?

00:05:20.820 --> 00:05:23.040
Well I'm now left with one, two,

00:05:23.040 --> 00:05:26.160
three, four, five, six
positive number chips,

00:05:26.160 --> 00:05:28.953
or that's just a positive six.