How to find the average rate of change of a modeling function from a table

WEBVTT
Kind: captions
Language: en

00:00:00.039 --> 00:00:06.749
Let T of T, so capital T of lowercase T denote the temperature capital T in Windhoek, Namibia

00:00:07.540 --> 00:00:13.410
measured in degrees Celsius when it's T lowercase T hours after midnight on a given day.

00:00:13.780 --> 00:00:21.510
The table below lists a few selected values of the function. Alright, when it's 6 hours after midnight our temperature is 19 degrees Celsius

00:00:22.000 --> 00:00:29.279
Nine hours after midnight or 9 a.m. 25 degrees Celsius. 13 hours after midnight, which is the same thing as 1 p.m.

00:00:29.279 --> 00:00:35.398
Our temperature is 31 degrees Celsius. When did the temperature increase faster?

00:00:35.399 --> 00:00:40.889
So this is between 6 & 9 a.m.. So 6 & 9 a.m.. 9 a.m.. and 1 p.m.

00:00:40.930 --> 00:00:45.719
So that's between these two points, or did the temperature increase at the same rate over both intervals?

00:00:46.210 --> 00:00:51.509
Well if we want to figure out the rate of increase- so let's see rate of

00:00:52.719 --> 00:00:54.719
temperature change

00:00:54.730 --> 00:00:56.110
temperature

00:00:56.110 --> 00:00:57.940
change

00:00:57.940 --> 00:01:05.070
Well this is going to be equal to, our change in temperature capital T, over our change in time.

00:01:05.500 --> 00:01:11.699
This triangle, that's the Greek letter Delta, represents change in. So change in temperature over change in time.

00:01:12.010 --> 00:01:19.739
So let's think about what this rate of temperature change is, between these different intervals.

00:01:19.740 --> 00:01:21.740
So between 6:00 a.m. and 9:00 a.m.. So let me really see

00:01:21.880 --> 00:01:24.900
Change in temperature over change in time

00:01:25.390 --> 00:01:30.809
So what was our change in temperature? Our temperature went up, by 6 degrees Celsius.

00:01:32.020 --> 00:01:34.920
Let me just write it here

00:01:35.470 --> 00:01:41.789
6 degrees Celsius. And what is our change in time? Our time went up by 3 hours, plus 3 hours.

00:01:42.460 --> 00:01:47.579
So you're going to have a change of 6 degrees Celsius, positive change of 6 degrees Celsius

00:01:48.040 --> 00:01:51.720
over a positive change, we've Gone 3 hours into the future,

00:01:52.600 --> 00:01:59.969
Over 3 hours we increased our temperature by 6 degrees or you could say it's an average rate of change of

00:01:59.969 --> 00:02:01.570
6 divided by 3 is 2

00:02:01.570 --> 00:02:03.399
degrees Celsius, per hour.

00:02:03.399 --> 00:02:09.988
Notice, I'm just keeping the unit's the same degrees Celsius per hour, so that's the rate of change between 6 a.m.

00:02:09.989 --> 00:02:11.590
And 9:00 a.m.

00:02:11.590 --> 00:02:17.369
Now let's ask ourselves the same question between 9:00 a.m.. and 1:00 p.m.. So our change in

00:02:18.129 --> 00:02:20.129
temperature over change in time

00:02:20.620 --> 00:02:27.299
What is our change in temperature? Well our temperature goes up from 25 to 31 degrees Celsius.

00:02:27.819 --> 00:02:32.009
That's another 6 degrees Celsius.

00:02:32.739 --> 00:02:37.858
But it's no longer 3 hours to go from 9 hours after minute to 13 hours after midnight

00:02:37.859 --> 00:02:42.479
We're now doing it over 4 hours. So Plus 4 hours.

00:02:43.000 --> 00:02:46.529
So 6 degrees Celsius over 4 hours

00:02:47.379 --> 00:02:48.430
and

00:02:48.430 --> 00:02:53.040
We actually don't even have to calculate you see that you've had you've had the same change

00:02:53.040 --> 00:02:54.900
But you've had to do it over more hours

00:02:54.900 --> 00:02:55.840
So this is a lower

00:02:55.840 --> 00:03:02.519
Rate of change the temperature is increasing slower here. It took it four hours to increase 6 degrees Celsius well over here it took it

00:03:02.519 --> 00:03:05.219
only 3 hours. So immediately you might recognize

00:03:05.290 --> 00:03:12.150
that this is going to be faster. If you increase 6 degrees Celsius over 3 hours that's faster than increasing 6 degrees Celsius

00:03:12.159 --> 00:03:17.729
Over 4 hours. But just to make the comparison a little bit clearer

00:03:17.730 --> 00:03:22.169
Let's actually just do the math here. 6 divided by 4 well that's going to be 1.5

00:03:23.230 --> 00:03:26.720
1.5 degrees Celsius per hour.

00:03:26.720 --> 00:03:28.900
Here you can make the direct comparison.

00:03:29.000 --> 00:03:33.810
2 degrees Celsius per hour is faster than 1.5 degrees Celsius per hour.

00:03:33.810 --> 00:03:38.549
So that's why we liked this choice up here


How to match the features of a modeling function to their real-world meaning

WEBVTT
Kind: captions
Language: en

00:00:00.039 --> 00:00:05.639
Let T of [t], denote the temperature capital T in New York City measured in Celsius degrees

00:00:05.640 --> 00:00:12.179
Or degrees Celsius when it's t, lowercase t, hours after midnight on a given day. The function is graphed below

00:00:12.549 --> 00:00:19.829
The following table contains true statements. Match each statement with the feature on the graph that most closely corresponds to it

00:00:20.140 --> 00:00:25.560
So once again this is capital t as a function of lowercase t's temperature as a function of time

00:00:25.869 --> 00:00:30.539
So we see at time equals 0 the temperature is negative 3 degrees Celsius.

00:00:30.789 --> 00:00:33.058
And then as we go to

00:00:33.399 --> 00:00:37.589
8 hours later, the temperature is at Zero degrees Celsius

00:00:37.660 --> 00:00:43.220
and then it hits a, it hits at least a relative maximum point or at least from what we see it could even be a

00:00:43.220 --> 00:00:46.360
Global Maximum point but based on what we see, it's definitely a relative maximum point

00:00:47.020 --> 00:00:53.780
14 hours into this measurement at time equals 14 and then the temperature starts to go down again.

00:00:54.610 --> 00:01:00.599
So let's see, so in other words it is t hours after midnight, so this is this is at midnight

00:01:00.760 --> 00:01:07.680
This is going to be at 8:00 a.m.. This is going to be at noon and this is going to be at 2:00 p.m.

00:01:07.720 --> 00:01:11.099
And so on and so forth, but anyway. The feature, Y-intercept.

00:01:11.860 --> 00:01:14.639
So the y-intercept is right over here

00:01:14.640 --> 00:01:21.059
And we see when lowercase t when time is zero, zero hours after midnight the temperature in New York City is

00:01:21.159 --> 00:01:23.159
Negative 3 degrees Celsius

00:01:23.920 --> 00:01:27.809
So it was negative 3 degrees Celsius, at the beginning of the day

00:01:27.810 --> 00:01:32.100
Yeah, that's a true statement and especially if you consider the beginning of the day, the true beginning of the day is right at midnight

00:01:32.619 --> 00:01:33.729
so

00:01:33.729 --> 00:01:37.169
That's that one. The y-intercept tells us this true statement

00:01:37.509 --> 00:01:42.389
So positive or negative interval and these questions are a little bit tricky because you don't have to use

00:01:42.390 --> 00:01:44.399
It's the positive or the negative interval

00:01:44.399 --> 00:01:51.149
It doesn't have to be both, so if either the positive or negative interval helps you with one of these other two statements

00:01:51.640 --> 00:01:59.129
So let's see. It was getting warmer between 2:00 a.m.. and 2:00 p.m.. The temperature was above zero between 8 a.m.. and 8 p.m.

00:01:59.890 --> 00:02:05.040
So let's see. If we're talking about positive or negative intervals, so we're not talking about increasing or decreasing

00:02:05.040 --> 00:02:09.749
We're talking about positive or negative intervals, so we have a negative interval

00:02:09.750 --> 00:02:14.039
Sorry, we have a negative interval from time equals zero to time equals eight

00:02:14.040 --> 00:02:18.660
What do I mean by negative interval is the temperature is negative. It goes, it's below zero

00:02:18.940 --> 00:02:22.499
And then from, time, from eight hours from 8 a.m.

00:02:23.019 --> 00:02:28.679
To what is this this would be noon, this would be 8 p.m.. From 8 a.m.. to 8 p.m.

00:02:28.959 --> 00:02:34.589
Or the 20th hour if you're taking in military time, so to speak. We see that our temperature is

00:02:35.290 --> 00:02:37.950
positive and then it dips down to negative again.

00:02:38.380 --> 00:02:43.469
So a positive or negative interval tells us when our temperature was above or below zero.

00:02:43.810 --> 00:02:48.330
And we can see we have this positive interval where the temperature was above zero between 8 a.m.. and 8 p.m.

00:02:48.910 --> 00:02:53.039
So we're only using the positive interval this interval right over here where the function is positive?

00:02:53.590 --> 00:02:58.349
That means that the temperature was above zero degrees Celsius between 8 a.m.. and 8 p.m.

00:02:59.019 --> 00:02:59.790
So once again we don't-

00:02:59.790 --> 00:03:05.190
It's saying either the positive or the negative interval helped us make the statement. In this case

00:03:05.190 --> 00:03:11.609
it was only the positive interval helped us make this statement, and then finally we have increasing or decreasing interval.

00:03:11.610 --> 00:03:17.160
And it was getting warmer between 2 a.m.. and 2 p.m.. It was getting warmer between 2 a.m.

00:03:17.739 --> 00:03:19.540
Which is right over here.

00:03:19.540 --> 00:03:26.010
So we see that the function is increasing as t increases so does the temperature, all the way to 2 p.m.

00:03:26.010 --> 00:03:29.760
And that, right of there, is an increasing interval. So once again

00:03:29.760 --> 00:03:31.889
we're not using an increasing and a decreasing interval,

00:03:31.889 --> 00:03:35.699
we're just using the increasing interval, the decreasing interval isn't helping us to make the statement,

00:03:35.700 --> 00:03:42.149
but the increasing interval is helping us. It's letting us make the statement. It was getting warmer between 2:00 a.m.. And 2:00 p.m.

00:03:42.549 --> 00:03:46.499
From here to here or between 2:00 a.m.. and 2:00 p.m.

00:03:46.630 --> 00:03:50.939
We see that the function itself is increasing that is an increasing

00:03:51.760 --> 00:03:54.060
Increasing interval and we got it, right


Introduction to increasing, decreasing, positive or negative intervals

WEBVTT
Kind: captions
Language: en

00:00:00.150 --> 00:00:01.447
- [Voiceover] What I
hope to do in this video

00:00:01.447 --> 00:00:04.802
is look at this graph y is equal to f of x

00:00:04.802 --> 00:00:06.183
and think about the intervals

00:00:06.183 --> 00:00:08.923
where this graph is positive or negative

00:00:08.923 --> 00:00:10.561
and then think about the intervals

00:00:10.561 --> 00:00:14.369
when this graph is
increasing or decreasing.

00:00:14.369 --> 00:00:18.142
So first let's just think
about when is this function,

00:00:18.142 --> 00:00:20.725
when is this function positive?

00:00:21.590 --> 00:00:23.065
Well positive means that the value

00:00:23.065 --> 00:00:25.131
of the function is greater than zero.

00:00:25.131 --> 00:00:27.569
It means that the value
of the function this means

00:00:27.569 --> 00:00:30.483
that the function is
sitting above the x-axis.

00:00:30.483 --> 00:00:32.503
So it's sitting above the x-axis

00:00:32.503 --> 00:00:33.923
in this place right over here

00:00:33.923 --> 00:00:38.433
that I am highlighting in
yellow and it is also sitting

00:00:38.433 --> 00:00:40.683
above the x-axis over here.

00:00:42.693 --> 00:00:44.853
And if we wanted to, if we wanted

00:00:44.853 --> 00:00:48.150
to write those intervals mathematically.

00:00:48.150 --> 00:00:50.379
Well let's see, let's say that this point,

00:00:50.379 --> 00:00:54.570
let's say that this point
right over here is x equals a.

00:00:54.570 --> 00:00:58.100
Let's say that this right
over here is x equals b and

00:00:58.100 --> 00:01:01.989
this right over here is x equals c.

00:01:01.989 --> 00:01:05.890
Then it's positive, it's
positive as long as x is

00:01:05.890 --> 00:01:07.098
between a and b.

00:01:07.098 --> 00:01:10.372
At x equals a or at x equals b the value

00:01:10.372 --> 00:01:13.436
of our function is zero but it's positive

00:01:13.436 --> 00:01:16.519
when x is between a and b, a and b

00:01:18.591 --> 00:01:21.008
or if x is greater than c.

00:01:22.609 --> 00:01:25.430
X is, we could write it
there, c is less than x

00:01:25.430 --> 00:01:29.263
or we could write that
x is greater than c.

00:01:30.237 --> 00:01:34.439
These are the intervals when
our function is positive.

00:01:34.439 --> 00:01:38.000
Let me write this, f of x, f of x positive

00:01:38.000 --> 00:01:41.019
when x is in this
interval or this interval

00:01:41.019 --> 00:01:42.597
or that interval.

00:01:42.597 --> 00:01:45.070
So when is f of x negative?

00:01:45.070 --> 00:01:47.229
Let me do this in another color.

00:01:47.229 --> 00:01:50.062
F of x is going to be negative.

00:01:51.897 --> 00:01:56.715
Well, it's gonna be negative
if x is less than a.

00:01:56.715 --> 00:02:00.882
So this is if x is less than
a or if x is between b and c

00:02:02.137 --> 00:02:05.213
then we see that f of
x is below the x-axis.

00:02:05.213 --> 00:02:09.380
F of x is down here so this
is where it's negative.

00:02:10.473 --> 00:02:14.640
So here or, or x is between
b or c, x is between b and c.

00:02:17.950 --> 00:02:20.295
And I'm not saying less than or equal to

00:02:20.295 --> 00:02:24.462
because at b or c the value of
the function f of b is zero,

00:02:25.659 --> 00:02:26.950
f of c is zero.

00:02:26.950 --> 00:02:30.186
That's where we are actually
intersecting the x-axis.

00:02:30.186 --> 00:02:32.670
So that was reasonably straightforward.

00:02:32.670 --> 00:02:34.748
Now let's ask ourselves
a different question.

00:02:34.748 --> 00:02:38.022
When is the function
increasing or decreasing?

00:02:38.022 --> 00:02:41.355
So when is f of x, f of x increasing?

00:02:44.802 --> 00:02:47.380
Well increasing, one way to
think about it is every time

00:02:47.380 --> 00:02:49.887
that x is increasing then
y should be increasing

00:02:49.887 --> 00:02:53.510
or another way to think
about it, you have a,

00:02:53.510 --> 00:02:57.249
you have a positive rate of
change of y with respect to x.

00:02:57.249 --> 00:02:59.304
We could even think about it as imagine

00:02:59.304 --> 00:03:02.563
if you had a tangent line
at any of these points.

00:03:02.563 --> 00:03:04.687
If you had a tangent line at
any of these points the slope

00:03:04.687 --> 00:03:08.949
of that tangent line is
going to be positive.

00:03:08.949 --> 00:03:10.702
But the easiest way for
me to think about it is

00:03:10.702 --> 00:03:15.075
as you increase x you're
going to be increasing y.

00:03:15.075 --> 00:03:17.061
So where is the function increasing?

00:03:17.061 --> 00:03:19.522
Well I'm doing it in blue.

00:03:19.522 --> 00:03:22.469
So it's increasing right until we get

00:03:22.469 --> 00:03:26.552
to this point right over
here, right until we get

00:03:27.562 --> 00:03:30.929
to that point over there
then it starts decreasing

00:03:30.929 --> 00:03:33.228
until we get to this
point right over here and

00:03:33.228 --> 00:03:35.422
then it starts increasing again.

00:03:35.422 --> 00:03:38.589
It starts, it starts increasing again.

00:03:40.867 --> 00:03:43.886
So let me make some more labels here.

00:03:43.886 --> 00:03:47.808
So let's say that this,
this is x equals d and

00:03:47.808 --> 00:03:48.727
that this right over here,

00:03:48.727 --> 00:03:50.747
actually let me do that in green color,

00:03:50.747 --> 00:03:53.664
so let's say this is x equals d.

00:03:54.761 --> 00:03:58.325
Now it's not a, d, b but you
get the picture and let's say

00:03:58.325 --> 00:04:01.878
that this is x is equal to, x is equal to,

00:04:01.878 --> 00:04:06.638
let me redo it a little
bit, x is equal to e.

00:04:06.638 --> 00:04:08.205
X is equal to e.

00:04:08.205 --> 00:04:10.736
So when is this function increasing?

00:04:10.736 --> 00:04:14.903
Well it's increasing if x is
less than d, x is less than d

00:04:16.599 --> 00:04:18.097
and I'm not gonna say
less than or equal to

00:04:18.097 --> 00:04:21.127
'cause right at x equals
d it looks like just

00:04:21.127 --> 00:04:23.182
for that moment the
slope of the tangent line

00:04:23.182 --> 00:04:25.318
looks like it would be,
it would be constant.

00:04:25.318 --> 00:04:27.694
We're going from increasing to decreasing

00:04:27.694 --> 00:04:31.433
so right at d we're neither
increasing or decreasing.

00:04:31.433 --> 00:04:36.390
But then we're also increasing,
so if x is less than d

00:04:36.390 --> 00:04:40.557
or x is greater than e,
or x is greater than e.

00:04:42.950 --> 00:04:45.783
And where is f of x decreasing?

00:04:46.990 --> 00:04:50.450
So f of x, let me do this
in a different color.

00:04:50.450 --> 00:04:54.617
When is, let me pick a
mauve, so f of x decreasing,

00:04:59.053 --> 00:05:02.432
decreasing well it's going
to be right over here.

00:05:02.432 --> 00:05:06.054
It's gonna be right between d and e.

00:05:06.054 --> 00:05:08.330
Between x equals d and x equals e

00:05:08.330 --> 00:05:09.548
but not exactly at those points

00:05:09.548 --> 00:05:10.988
'cause at both of those points

00:05:10.988 --> 00:05:13.215
you're neither increasing nor decreasing

00:05:13.215 --> 00:05:15.641
but you see right over
here as x increases,

00:05:15.641 --> 00:05:18.380
as you increase your x
what's happening to your y?

00:05:18.380 --> 00:05:20.343
If you go from this point
and you increase your x

00:05:20.343 --> 00:05:21.663
what happened to your y?

00:05:21.663 --> 00:05:23.393
Your y has decreased.

00:05:23.393 --> 00:05:25.901
You increase your x, your y has decreased,

00:05:25.901 --> 00:05:27.781
you increase your x, y has decreased,

00:05:27.781 --> 00:05:30.580
increase x, y has decreased all the way

00:05:30.580 --> 00:05:32.553
until this point over here.

00:05:32.553 --> 00:05:36.720
So f of x is decreasing
for x between d and e.

00:05:38.997 --> 00:05:40.622
So hopefully that gives
you a sense of things.

00:05:40.622 --> 00:05:42.457
Notice, these aren't the same intervals.

00:05:42.457 --> 00:05:45.475
That we are, the intervals
where we're positive

00:05:45.475 --> 00:05:47.596
or negative don't perfectly coincide

00:05:47.596 --> 00:05:49.132
with when we are increasing or decreasing.

00:05:49.132 --> 00:05:52.301
So it's very important to
think about these separately

00:05:52.301 --> 00:05:54.867
even though they kinda sound the same.


How to find the domain and range of a piecewise function

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:04.990
So we have a piecewise linear function
right over here for different intervals

00:00:04.990 --> 00:00:05.589
of x.

00:00:05.589 --> 00:00:08.760
g of x is defined by a

00:00:08.760 --> 00:00:13.160
a line or the line changes depending
what interval of x we're actually in.

00:00:13.160 --> 00:00:17.090
And so let's think about its domain, and
then we'll think about its range.

00:00:17.090 --> 00:00:21.300
So the domain of this, this is a review.
The domain is the set of all inputs for which

00:00:21.330 --> 00:00:24.810
this function is defined, and our input
variable here

00:00:24.810 --> 00:00:29.010
is x. This is a set of all x values for which
this function is defined.

00:00:29.010 --> 00:00:34.600
And we see here. Anything, anything negative 6 or
lower, our function isn't defined. If it, if

00:00:34.640 --> 00:00:39.010
x is negative 6 or or lower than that. I don't --
it doesn't, it doesn't fall into one of

00:00:39.010 --> 00:00:40.270
these three intervals.

00:00:40.270 --> 00:00:44.010
So there is no definition for it.
It doesn't say hey do this in all other

00:00:44.010 --> 00:00:45.579
cases for x. It is just saying, look, if

00:00:45.579 --> 00:00:48.620
x falls into one of these three
conditions, apply this.

00:00:48.620 --> 00:00:52.309
And if x doesn't fall into one of those three
conditions, well this function g is just

00:00:52.309 --> 00:00:53.410
not defined.

00:00:53.410 --> 00:00:57.570
So, to fall into one of these three, you have
to be at least greater than negative 6.

00:00:57.570 --> 00:01:02.440
So this part right over here, the low end
of our domain is defined right over there,

00:01:02.440 --> 00:01:06.040
so we say, we could say, negative 6

00:01:06.040 --> 00:01:09.290
is less than x and I'm leaving --
so let's write it here.

00:01:09.290 --> 00:01:12.979
All real numbers -- actually
let me write this way

00:01:12.979 --> 00:01:16.119
x, I could write it more math-y. I could say

00:01:16.119 --> 00:01:19.460
x is a member of the real numbers

00:01:19.460 --> 00:01:23.090
such that, such that negative 6

00:01:23.090 --> 00:01:26.909
is less than x. Negative 6 is less than x

00:01:26.909 --> 00:01:30.439
and I also think about the upper bound.
So as x goes, I just wanna make sure

00:01:30.439 --> 00:01:32.540
that we fill in all the gaps between

00:01:32.540 --> 00:01:36.570
x being a greater than negative 6 and
x is less than or equal to 6.

00:01:36.570 --> 00:01:40.570
So let's see. As we go up to and including negative 3,
we're in this clause. As soon as we

00:01:40.570 --> 00:01:41.909
cross negative 3,

00:01:41.909 --> 00:01:46.200
we fall into this clause up to 4, but
as soon as we get 4, we're in this clause

00:01:46.280 --> 00:01:49.649
up to and including 6. So

00:01:49.649 --> 00:01:54.130
x at the high end is said to be less than
or equal to 6,

00:01:54.130 --> 00:01:57.469
less then or equal to

00:01:57.469 --> 00:02:00.740
6. Now another way to say this and

00:02:00.740 --> 00:02:04.759
kind of less math-y notation is x, x

00:02:04.759 --> 00:02:08.280
can be any

00:02:08.280 --> 00:02:13.210
real number, any the real number

00:02:13.210 --> 00:02:15.590
such that, such

00:02:15.590 --> 00:02:19.410
that negative 6 is less than x

00:02:19.410 --> 00:02:23.290
is less than or equal to 6. These two
statements are

00:02:23.290 --> 00:02:27.060
equivalent. So now let's think about
the range of this function.

00:02:27.060 --> 00:02:30.660
Let's think about the range, and the range
is, this is the set of all

00:02:30.660 --> 00:02:34.060
inputs , oh sorry, this is the set of all
outputs that

00:02:34.060 --> 00:02:37.740
this function can take on, or all the
values that this function can take on.

00:02:37.740 --> 00:02:41.390
And to do that, let's just think about
as x goes, but

00:02:41.390 --> 00:02:44.980
x varies or x can be any values in this
interval.

00:02:44.980 --> 00:02:48.450
What are the different values
that g of x could take on?

00:02:48.450 --> 00:02:53.520
Let's think about that. g of x is going
to be between what and what?

00:02:53.520 --> 00:02:56.840
g of x is going to be between what and
what?

00:02:56.840 --> 00:03:00.740
g of x is going to be between what and
what?

00:03:00.740 --> 00:03:05.320
And it might actually, this might be some
equal signs there but I'm gonna worry about

00:03:05.320 --> 00:03:06.130
that in a second.

00:03:06.130 --> 00:03:11.300
So when does this thing hit its low
point? o this thing hits, hits its low point

00:03:11.330 --> 00:03:16.250
when x is as small as possible. An x is
going to be as small as possible

00:03:16.250 --> 00:03:19.550
when x is approaching negative 6. So if

00:03:19.550 --> 00:03:23.180
x were equal to negative 6, it can't
equal negative 6 herer but if x is

00:03:23.180 --> 00:03:27.900
equal to negative 6, then this thing over here
would be equal to negative 6

00:03:27.900 --> 00:03:31.500
plus 7, would be, would be 1.

00:03:31.500 --> 00:03:35.420
So if x is greater than negative 6,  g of x

00:03:35.420 --> 00:03:40.800
is going to be greater than 1, or another way
to think about it is if negative 6 is less than x,

00:03:40.890 --> 00:03:43.989
then 1 is going to be less than g of x.

00:03:43.989 --> 00:03:47.630
And the reason I said that is if I put negative 6
into this, negative 6 plus 7

00:03:47.630 --> 00:03:52.850
is equal to 1. Now this gonna hit a
high end when it as large as possible.

00:03:52.850 --> 00:03:55.400
The largest value in this interval that
we can take on

00:03:55.400 --> 00:04:00.799
is x being equal to negative 3.
So when x is equal to negative 3,

00:04:00.799 --> 00:04:04.410
negative 3 plus 7 is equal to

00:04:04.410 --> 00:04:09.120
4, positive 4. And it can actually take on
that value because this is less than or

00:04:09.120 --> 00:04:11.930
equal to, so we can actually take on
x equals negative 3

00:04:11.930 --> 00:04:15.709
in which case g of x actually
will take on positive 4.

00:04:15.709 --> 00:04:19.150
So, let' do that for each of these.
Now here we have 1

00:04:19.150 --> 00:04:25.200
minus x, so this is going to take on
its smallest value when x is as large as possible.

00:04:25.280 --> 00:04:29.150
So the largest value x can approach for,
it can't quite take on for, but it's going

00:04:29.150 --> 00:04:30.200
to approach for.

00:04:30.200 --> 00:04:35.430
So if x, let's see, if we said x was 4,
although that's not this clause here,

00:04:35.430 --> 00:04:39.220
1 minus x, 1 minus 4 is negative 3.

00:04:39.220 --> 00:04:42.490
So as long as x is less than 4,

00:04:42.490 --> 00:04:46.160
then negative 3 is going to be less than

00:04:46.160 --> 00:04:49.740
g of x. I wanna make sure that makes sense
to you because it can be little bit confusing

00:04:49.740 --> 00:04:54.110
because this takes on its minimum value when x

00:04:54.110 --> 00:04:57.950
is approaching, or it's
approaching its minimum value

00:04:57.950 --> 00:05:01.700
when x is approaching its, when x is
approaching its maximum because

00:05:01.700 --> 00:05:02.630
we're subtracting it.

00:05:02.630 --> 00:05:06.190
So if you take the upper end, even though
this doesn't actually include

00:05:06.190 --> 00:05:09.290
4, but as we approach for, we could say, OK, 1 negative 4

00:05:09.290 --> 00:05:13.890
is negative 3 so that's, so g of x
is always going to be

00:05:13.890 --> 00:05:17.230
greater than that, as well it's going to be
going to be a less than.

00:05:17.230 --> 00:05:20.410
Well what happens as we approach x being

00:05:20.410 --> 00:05:23.570
equal to negative 3? So, 1 minus

00:05:23.570 --> 00:05:27.200
negative 3 is going to be positive

00:05:27.200 --> 00:05:30.780
4. So this is going to be positive
4 right over here.

00:05:30.780 --> 00:05:34.080
And these are both less than, not less
than or equal to

00:05:34.080 --> 00:05:38.810
because these are both less than right
over here. And now let's think about this

00:05:38.810 --> 00:05:39.840
right over here. So

00:05:39.840 --> 00:05:44.170
2x minus 11 is gonna hit its maximum value
when x is as large as possible.

00:05:44.170 --> 00:05:48.270
So its maximum value's going to be hit
when x is equal to 6

00:05:48.270 --> 00:05:52.240
So 2 times 6 is 12, minus 11.
Well that's going to be

00:05:52.240 --> 00:05:55.490
1. So its maximum value's going to be 1.
It's actually going to be able to hit

00:05:55.490 --> 00:05:57.770
because x can be equal to 6.

00:05:57.770 --> 00:06:02.520
Its minimum value is going to be when x
is equal to 4, and actually can be equal to

00:06:02.520 --> 00:06:03.350
4. We have this

00:06:03.350 --> 00:06:07.140
less than or equal sign right over there.
So 2 times 4

00:06:07.140 --> 00:06:15.000
is 8, minus 11 is negative 3. So, g of x in this case
can get as low as negative 3 when x is equal to 4.

00:06:15.090 --> 00:06:19.700
So now let's think about all of, all of the
values that g of x can take on.

00:06:19.700 --> 00:06:20.610
So we could say,

00:06:20.610 --> 00:06:24.720
we could write this a bunch of ways,
we could write g of x

00:06:24.720 --> 00:06:28.010
is going to be a member of the real
numbers

00:06:28.010 --> 00:06:33.050
such that -- let's see. What's the lowest
value g of x can take on? g of

00:06:33.050 --> 00:06:38.000
x can get as low as negative 3.
It can even be equal to negative 3.

00:06:38.000 --> 00:06:40.480
This one just has been greater the
negative 3, but here can be

00:06:40.480 --> 00:06:44.540
greater than or equal to negative 3.
So negative 3

00:06:44.540 --> 00:06:48.900
is less than or equal to g of x, and it
can get as high

00:06:48.900 --> 00:06:52.010
as, it can get as high as

00:06:52.010 --> 00:06:55.130
Let's see. It's defined all the way to
1 and then --

00:06:55.130 --> 00:06:58.260
or I shouldn't say it is defined all the way to 1.
It can take on values up to 1

00:06:58.260 --> 00:07:02.760
but it can also take on values beyond 1.
It can take on values all the way up to

00:07:02.760 --> 00:07:03.790
including 4

00:07:03.790 --> 00:07:07.210
over here. So it can take on values up to

00:07:07.210 --> 00:07:11.010
and including 4. So g of x is a member
real numbers such that

00:07:11.010 --> 00:07:14.300
negative 3 is less than or equal to g of x
is less than or

00:07:14.300 --> 00:07:18.660
equal to 4. So the set of all values
that g of x can take on

00:07:18.660 --> 00:07:21.750
between, including and including negative 3

00:07:21.750 --> 00:07:23.120
and positive 4.


WEBVTT
Kind: captions
Language: en

00:00:00.329 --> 00:00:04.759
I have a piecewise defined function here,
and my goal is to figure out its domain

00:00:04.759 --> 00:00:08.099
and its range. So first let's think about the
domain.

00:00:08.099 --> 00:00:11.990
And just a bit overview, the domain is
the set of all inputs

00:00:11.990 --> 00:00:15.440
for which our function is defined.
And over here, an input variable

00:00:15.440 --> 00:00:19.480
is x, so to think about, it's the set of all
the values that x can take on,

00:00:19.480 --> 00:00:23.359
and actually have this function be defined,
and actually figure out what f of x is.

00:00:23.359 --> 00:00:28.050
And when we look at this, we see, okay, if 0 is
less than x is less than or equal to 2. We're

00:00:28.050 --> 00:00:32.250
in this clause, it's x crosses 2 and it is greater than 2.
We follow this clause

00:00:32.250 --> 00:00:36.700
As we approach 6 but right when we get
to 6, we fall into this clause right over here,

00:00:36.730 --> 00:00:41.719
all the way up to and including 11.
But if we get larger than 11, the function is

00:00:41.719 --> 00:00:43.870
no longer defined. I don't know which of
these to use.

00:00:43.870 --> 00:00:47.390
And if we're 0 or less, the function is
longer defined as well.

00:00:47.390 --> 00:00:52.750
So in order for this to be defined, x has
to be greater than 0 or if we say 0 is less

00:00:52.750 --> 00:00:53.449
than x,

00:00:53.449 --> 00:00:58.700
and you see that part right over there.
And x has to be less than or equal to 11.

00:00:58.770 --> 00:01:02.340
and x has to be less then or equal to 11.

00:01:02.340 --> 00:01:05.890
It's defined for everything in between.
As we, as we see, once again,

00:01:05.890 --> 00:01:10.290
as we get to 2, we're here.
As we cross 2, between 2 and 6, we're

00:01:10.290 --> 00:01:13.180
here, and at 6, from 6 to 11 we're

00:01:13.180 --> 00:01:18.630
over here. So we're defined for all real
numbers in this interval. So our domain is --

00:01:18.630 --> 00:01:22.009
actually let me write this all, all

00:01:22.009 --> 00:01:25.340
real values, are all real

00:01:25.340 --> 00:01:30.030
all real values. maybe --

00:01:30.030 --> 00:01:34.290
Let me write that way.  All real value such
that, such

00:01:34.290 --> 00:01:38.380
that, 0 is less than x, is less than
or equal to 11.

00:01:38.380 --> 00:01:41.939
So now think about the range.
Let's think about

00:01:41.939 --> 00:01:45.040
the range of this piecewise
defined function.

00:01:45.040 --> 00:01:49.040
And that's a set of all values that this
function can actually take on.

00:01:49.040 --> 00:01:53.530
and this one is, is maybe deceptively
simple because there're only three values

00:01:53.530 --> 00:01:54.979
that this function can take on.

00:01:54.979 --> 00:01:58.479
You can take on, f of x can be equal to
1. It can be equal to 5,

00:01:58.479 --> 00:02:02.159
or it could be equal to negative 7.

00:02:02.159 --> 00:02:05.439
So the range here, we could say that f of x

00:02:05.439 --> 00:02:08.869
needs to be a member of,
this is just a fancy

00:02:08.869 --> 00:02:13.300
mathy symbol, just to say this is a member of the set

00:02:13.390 --> 00:02:17.730
1, 5, negative 7. f of x is going to take,

00:02:17.730 --> 00:02:21.930
is going to take on one of these three
values. Another way to say it

00:02:21.930 --> 00:02:25.379
is that f of x is going to be equal 1,

00:02:25.379 --> 00:02:28.400
5 or negative 7. This is maybe a little

00:02:28.400 --> 00:02:31.640
less -- a little -- a less

00:02:31.640 --> 00:02:35.040
a less mathy way, a less precise way

00:02:35.040 --> 00:02:38.720
of saying the same thing. But one way
or another, f of x can only take on

00:02:38.720 --> 00:02:40.879
one over these three values.


Introduction to decimals

WEBVTT
Kind: captions
Language: en

00:00:00.225 --> 00:00:02.165
So I have a number written here.

00:00:02.267 --> 00:00:04.277
It's a 2, a 3, and a 5.

00:00:04.314 --> 00:00:07.124
And we already have some
experience with numbers like this.

00:00:07.162 --> 00:00:09.212
We can think about
'what does it represent'.

00:00:09.214 --> 00:00:12.644
And to think about that we just have
to look at the actual place values.

00:00:12.662 --> 00:00:15.962
So this right-most place right over here.
This is the ones place.

00:00:15.998 --> 00:00:19.988
So this 5 represents five ones, or I guess
you could say that's just going to be 5.

00:00:20.864 --> 00:00:26.502
This 3, this is in the tens place. This is
the tens place, so we have three tens.

00:00:27.035 --> 00:00:29.205
So that's just going to be 30.

00:00:29.261 --> 00:00:32.934
And the 2 is in the hundreds place.

00:00:33.533 --> 00:00:37.467
So putting a 2 there means
that we have two hundreds.

00:00:38.363 --> 00:00:41.765
So this number we can view as
two hundred, thirty, five.

00:00:41.765 --> 00:00:46.470
Or you could view it as
two hundred plus thirty plus five.

00:00:46.470 --> 00:00:49.775
Now what I want to do in this video
is think about place values

00:00:49.775 --> 00:00:52.411
to the right of the ones place.
And you might say

00:00:52.411 --> 00:00:54.750
'wait, wait, I always thought
that the ones place

00:00:54.750 --> 00:00:56.673
was the place furthest to the right.'

00:00:56.673 --> 00:00:59.200
Well everything that we've done so far,
it has been.

00:00:59.200 --> 00:01:01.763
But to show that you can go
even further to the right

00:01:01.763 --> 00:01:06.321
I'm going to put a little dot. I'm going
to put a little dot right over here.

00:01:06.321 --> 00:01:08.144
We call that a 'decimal point'.

00:01:08.144 --> 00:01:11.681
And that dot means that anything
to the right of this is going to be

00:01:11.681 --> 00:01:15.562
place values that are smaller,
I guess you could say, than the

00:01:15.562 --> 00:01:18.378
ones place. So right to the left
you have the ones place

00:01:18.378 --> 00:01:20.338
and the tens place and
the hundreds place,

00:01:20.338 --> 00:01:23.206
and if you were to keep going
you'd go to the thousands place

00:01:23.206 --> 00:01:27.186
and the ten thousands place. But then if
you go to the right of the decimal point

00:01:27.186 --> 00:01:30.568
now you're going to divide by 10.
So what am I talking about?

00:01:30.568 --> 00:01:33.365
Well, right to the right of
the decimal point

00:01:33.365 --> 00:01:36.276
you are going to have-- find a new color--

00:01:36.276 --> 00:01:41.114
this is going to be the tenths place.

00:01:41.114 --> 00:01:42.778
Well what does that mean?

00:01:42.778 --> 00:01:44.372
Well whatever number I write here

00:01:44.372 --> 00:01:48.219
that tells us how many tenths
we're dealing with.

00:01:48.219 --> 00:01:54.980
So if I were to write the number 4
right over here, now my number is

00:01:54.980 --> 00:02:02.579
2 hundreds plus 3 tens plus 5 ones
plus 4 tenths.

00:02:02.579 --> 00:02:06.467
So you could view this a 4 times 1/10.

00:02:06.467 --> 00:02:13.409
Or you could write this as 4 tenths.
Not tens, 4 tenths.

00:02:13.409 --> 00:02:19.667
Or 4 tenths is the same thing as this
right over here.

00:02:19.667 --> 00:02:24.001
So this is a super important idea
in mathematics.

00:02:24.001 --> 00:02:28.777
I can now use our place values
to represent fractions.

00:02:28.777 --> 00:02:32.505
So this right over here, this 'point 4',
this is 4/10.

00:02:32.505 --> 00:02:34.585
So another way to write this number--

00:02:34.585 --> 00:02:39.555
I could write it this way, I could write
it as two hundred, thirty--

00:02:39.555 --> 00:02:48.834
let me do the thirty in blue-- two hundred
and thirty five and four tenths.

00:02:48.834 --> 00:02:51.900
So I could write it like this,
as a mixed number.

00:02:51.900 --> 00:02:56.210
So this up here would be a
decimal representation: 235.4

00:02:56.210 --> 00:02:57.864
And this right over here would be

00:02:57.864 --> 00:03:00.804
a mixed number representation:
235 and 4/10

00:03:00.804 --> 00:03:06.334
but they all represent 200
plus 30 plus 5 plus 4/10.

00:03:06.957 --> 00:03:09.023
Let's look at a few more examples of this.

00:03:09.023 --> 00:03:11.534
So let's say I were to write the number

00:03:14.674 --> 00:03:21.226
0.7--
and actually let me go one space

00:03:21.226 --> 00:03:25.863
even further to the right-- 0.76.

00:03:26.453 --> 00:03:29.265
So what would this be if I were
to write it as a fraction?

00:03:29.520 --> 00:03:31.550
So let's just think about the place value.

00:03:31.550 --> 00:03:32.833
We have our decimal point.

00:03:32.923 --> 00:03:35.404
To the left of the decimal point
is the ones place,

00:03:35.540 --> 00:03:38.696
but I have a zero there,
so this is 0 ones.

00:03:38.871 --> 00:03:43.110
Now I have 7 tenths, so this is
the tenths place.

00:03:44.408 --> 00:03:47.336
And then this is going to be this place
to the right of that.

00:03:47.342 --> 00:03:50.664
We're going to divide by 10 again.

00:03:50.664 --> 00:03:52.975
So this is going to be the
hundredths place.

00:03:52.975 --> 00:03:59.081
This space right over here is
going to be the hundredths place.

00:03:59.081 --> 00:04:03.732
So this number right over here is--
we can rewrite it as 0--

00:04:03.732 --> 00:04:12.072
let me write it this way-- we could
rewrite it as 0 ones plus 7 tenths--

00:04:12.752 --> 00:04:15.582
plus 7 tenths---

00:04:15.582 --> 00:04:19.721
not tens, tenths-- plus 6 hundredths--

00:04:20.510 --> 00:04:23.280
plus 6 hundredths--

00:04:24.471 --> 00:04:26.611
not hundreds, hundredths.

00:04:26.611 --> 00:04:34.261
Or we could write this as 0 plus 7/10--

00:04:34.825 --> 00:04:37.852
let me write that a little bit neater--
plus 7/10

00:04:37.869 --> 00:04:46.076
plus 6/100-- 6 over 100.

00:04:46.076 --> 00:04:49.760
So you could write this-- 7/10 plus
6/100 is exactly what this is.

00:04:49.760 --> 00:04:54.252
You could say this is 0 ones,
7 tenths, and 6 hundredths.

00:04:54.252 --> 00:04:57.397
Now another way we could
write this-- well look, if we wanted

00:04:57.397 --> 00:05:01.074
to write it as a fraction, or talk
about it as a fraction-- I could ignore

00:05:01.074 --> 00:05:04.245
the 0, that's not going to change the
value of the sum but I could

00:05:04.245 --> 00:05:09.965
add the 7/10 to the 6/100. So how could
I write 7/10 in terms of hundredths?

00:05:09.965 --> 00:05:14.863
Well 7 over 10 is the same thing as
70 over 100.

00:05:15.983 --> 00:05:19.803
7/10 is the same thing as 70 over 100.

00:05:19.834 --> 00:05:23.865
One way to thing about it is, if
I multiply the denominator by 10

00:05:23.865 --> 00:05:27.754
well, I can multiply the numerator by 10
as well and not change the value of it.

00:05:27.754 --> 00:05:32.813
7/10 is the same thing as 70/100.
And then you could add 6/100 to that--

00:05:33.473 --> 00:05:36.498
6/100 to that.
And what will that give you?

00:05:36.498 --> 00:05:39.409
Well that's going to give you 76/100--

00:05:40.069 --> 00:05:42.409
76/100.

00:05:42.409 --> 00:05:46.218
So this number up here--
a lot of people will call this--

00:05:46.218 --> 00:05:49.395
they might say
'zero point seven six'

00:05:49.443 --> 00:05:54.335
or they might call this
'seventy-six hundredths'.

00:05:54.355 --> 00:05:57.300
This is the hundredths place,
this is the tenths place.

00:05:57.313 --> 00:06:00.398
But each tenth is worth
ten hundredths.

00:06:00.412 --> 00:06:03.865
And you see that-- you could
either view this as 7 tenths.

00:06:03.878 --> 00:06:07.557
or you could view it
as 70 hundredths.

00:06:07.559 --> 00:06:10.007
So this is 76 hundredths.

00:06:10.007 --> 00:06:11.892
And you could keep going to the right.

00:06:11.892 --> 00:06:13.708
If you go to the right one more space

00:06:13.708 --> 00:06:15.802
you would get to the thousandths place

00:06:15.802 --> 00:06:17.481
and then the ten thousandths place.

00:06:17.481 --> 00:06:20.252
So you keep dividing by 10
each place you go to the right.

00:06:20.252 --> 00:06:23.912
And you multiply by 10
each place you go to the left.


Relating addition and subtraction

WEBVTT
Kind: captions
Language: en

00:00:00.381 --> 00:00:03.821
I have two green balls
and three purple balls

00:00:03.824 --> 00:00:07.274
so how many balls
do I have altogether?

00:00:07.274 --> 00:00:12.754
So two green balls
and three purple balls

00:00:12.754 --> 00:00:19.068
is going to be 5 balls altogether.

00:00:19.068 --> 00:00:24.328
We can count them;
one, two, three, four, five.

00:00:24.344 --> 00:00:27.864
Now what if we did something
a little different?

00:00:27.864 --> 00:00:34.994
Let's start with the five balls,
and let's take away two.

00:00:34.994 --> 00:00:41.984
Let's subtract two of these balls.

00:00:41.984 --> 00:00:47.643
Take away one and two,
and what are we left with?

00:00:47.643 --> 00:00:52.892
We are left with
one, two, three balls.

00:00:52.892 --> 00:00:56.883
We're left with three balls.

00:00:56.883 --> 00:01:00.883
You might see a pattern here;
two plus three is equal to five

00:01:00.883 --> 00:01:04.883
but if you start with five
and take away two,

00:01:04.883 --> 00:01:07.753
you're going to be left with three.

00:01:07.753 --> 00:01:10.073
Look how they're arranged.

00:01:10.073 --> 00:01:15.473
Two plus three is equal to five
and five minus two is three.

00:01:15.473 --> 00:01:18.351
It works the other way around.

00:01:18.351 --> 00:01:24.381
If I start with five
and I were to take away three,

00:01:24.392 --> 00:01:29.172
if I subtract one, two, three,
what am I left with?

00:01:29.188 --> 00:01:33.868
I'm left with one, two,
I'm left with two.

00:01:33.868 --> 00:01:38.720
So two plus three is five,
three plus two is five,

00:01:38.720 --> 00:01:42.720
which is another way of saying
that five minus two is three,

00:01:42.720 --> 00:01:45.060
or five minus three is two.

00:01:45.060 --> 00:01:46.550
Take a look at this,

00:01:46.550 --> 00:01:52.534
it's a really important
idea to understand in math.

00:01:52.534 --> 00:01:58.909
If you add two numbers
and you get another number,

00:01:58.913 --> 00:02:02.053
then you take the result
and subtract one of these

00:02:02.000 --> 00:02:05.010
you're going to get the other number.


Subtracting hundreds, tens, and ones

WEBVTT
Kind: captions
Language: en

00:00:00.465 --> 00:00:02.322
- [Voiceover] Let's think
about how many blocks

00:00:02.322 --> 00:00:03.583
we have here.

00:00:03.583 --> 00:00:07.406
What number these blocks represent.

00:00:07.406 --> 00:00:09.752
Well, I have four groups
of a hundred here,

00:00:09.752 --> 00:00:12.422
so I could say that I have four hundreds.

00:00:12.422 --> 00:00:14.480
Four hundreds.

00:00:15.650 --> 00:00:17.645
I have three groups of 10 here.

00:00:17.645 --> 00:00:20.216
So I could write plus three tens.

00:00:20.216 --> 00:00:22.708
Plus three tens.

00:00:22.708 --> 00:00:24.403
And then I have six ones here.

00:00:24.403 --> 00:00:27.026
So I could write plus six ones.

00:00:27.026 --> 00:00:30.185
Plus six ones.

00:00:30.185 --> 00:00:32.900
And, so I can write that as...

00:00:32.900 --> 00:00:34.271
I could write that as...

00:00:34.271 --> 00:00:35.943
I could write a six in the ones place

00:00:35.943 --> 00:00:37.940
for my six ones.

00:00:37.940 --> 00:00:40.958
Three in the tens place for my three tens.

00:00:40.958 --> 00:00:42.653
For my three tens.

00:00:42.653 --> 00:00:44.186
And then a four in the hundreds place

00:00:44.186 --> 00:00:46.669
for my four hundreds.

00:00:46.669 --> 00:00:48.328
So this is 436.

00:00:48.328 --> 00:00:49.991
Four hundred thirty-six.

00:00:49.991 --> 00:00:53.195
436.

00:00:53.195 --> 00:00:55.168
Now let's subtract from that.

00:00:55.168 --> 00:00:57.838
Let's subtract from that...

00:00:57.838 --> 00:00:59.385
Let's subtract from that

00:00:59.385 --> 00:01:01.568
the number...

00:01:01.568 --> 00:01:04.958
the number 300...

00:01:04.958 --> 00:01:08.274
312.

00:01:08.905 --> 00:01:12.202
312 and let's think about what we would do

00:01:12.202 --> 00:01:15.012
here for subtracting 300...

00:01:15.012 --> 00:01:19.470
If we are subtracting 312.

00:01:19.470 --> 00:01:21.211
Well, we had six ones before,

00:01:21.211 --> 00:01:22.697
we're taking away two ones.

00:01:22.697 --> 00:01:23.718
So let me write that down.

00:01:23.718 --> 00:01:27.944
We're gonna take away two ones.

00:01:27.944 --> 00:01:29.663
We're taking away one 10.

00:01:29.663 --> 00:01:32.496
So we're taking away one 10.

00:01:32.496 --> 00:01:35.244
So minus one 10.

00:01:35.244 --> 00:01:38.161
And we're taking away three hundreds.

00:01:38.161 --> 00:01:40.088
So we're taking away...

00:01:40.088 --> 00:01:43.620
So we're taking away three hundreds.

00:01:44.220 --> 00:01:45.591
Three hundreds.

00:01:45.591 --> 00:01:47.425
So what's that going to do to us?

00:01:47.425 --> 00:01:49.260
What's that going to do to us?

00:01:49.260 --> 00:01:51.605
Well, we had four hundreds,

00:01:51.605 --> 00:01:53.020
we're taking away three hundreds,

00:01:53.020 --> 00:01:53.996
so let's take them away.

00:01:53.996 --> 00:01:55.181
Let's take away...

00:01:55.181 --> 00:01:58.362
Let's take away the...

00:01:58.362 --> 00:02:02.508
Let's take way the one...

00:02:02.508 --> 00:02:04.296
two...

00:02:04.296 --> 00:02:05.968
and three hundreds

00:02:05.968 --> 00:02:08.151
and we're just left with one hundred.

00:02:08.151 --> 00:02:11.308
One hundred.

00:02:11.308 --> 00:02:12.214
We could do it over here.

00:02:12.214 --> 00:02:15.581
Four hundreds minus three hundreds

00:02:15.581 --> 00:02:17.972
is going to be one hundred.

00:02:17.972 --> 00:02:20.666
Now we had three tens,

00:02:20.666 --> 00:02:22.431
we're taking away one ten.

00:02:22.431 --> 00:02:24.660
So let's do that.

00:02:24.660 --> 00:02:26.796
So we're going to be left with two tens.

00:02:26.796 --> 00:02:29.698
So plus two tens.

00:02:29.698 --> 00:02:31.154
Three minus one is two.

00:02:31.231 --> 00:02:33.204
Three minus one is two.

00:02:33.204 --> 00:02:36.199
And then last,

00:02:36.199 --> 00:02:39.115
then we had six ones,

00:02:39.115 --> 00:02:41.367
we're taking away two ones,

00:02:41.367 --> 00:02:42.846
and we're going to be left with four ones.

00:02:42.846 --> 00:02:46.349
Six minus two is four.

00:02:47.288 --> 00:02:50.306
Six minus two is four.

00:02:50.306 --> 00:02:51.583
And we're done!

00:02:51.583 --> 00:02:55.430
436 minus 312 is 124.

00:02:55.430 --> 00:02:57.249
And you see how we did that.

00:02:57.249 --> 00:02:59.571
We took away three hundreds.

00:02:59.571 --> 00:03:01.150
We took away one ten.

00:03:01.150 --> 00:03:04.377
And we took away two ones.

00:03:04.377 --> 00:03:06.537
To get 124.

00:03:06.537 --> 00:03:10.537
Which is one hundred,
two tens, and four ones.


Counting with small numbers

WEBVTT
Kind: captions
Language: en

00:00:00.518 --> 00:00:03.350
- [Voiceover] Put seven
squirrels in the box.

00:00:03.350 --> 00:00:04.917
All right, so that's

00:00:04.917 --> 00:00:07.857
one, two,

00:00:07.857 --> 00:00:11.407
three, four

00:00:11.407 --> 00:00:14.890
five, six,

00:00:14.890 --> 00:00:16.214
seven squirrels.

00:00:16.214 --> 00:00:20.068
I've put seven squirrels in the box.

00:00:20.068 --> 00:00:22.111
Let's keep going, this is lots of fun.

00:00:22.111 --> 00:00:24.874
Put four squirrels in the box.

00:00:24.874 --> 00:00:26.964
That's even easier than putting seven.

00:00:26.964 --> 00:00:30.354
One, two, three, four.

00:00:30.354 --> 00:00:31.956
I just click on the squirrel

00:00:31.956 --> 00:00:34.069
and then drag it into the box.

00:00:34.069 --> 00:00:35.787
All right, hopefully we see more animals.

00:00:35.787 --> 00:00:37.389
Oh, here we go, horses.

00:00:37.389 --> 00:00:41.290
Put nine horses in the box.

00:00:41.290 --> 00:00:45.655
One, two, three, four,

00:00:45.655 --> 00:00:49.394
five, six, seven,

00:00:49.394 --> 00:00:53.318
eight, nine horses.

00:00:53.318 --> 00:00:54.543
It's a lot of fun.


Snow on Friday

WEBVTT
Kind: captions
Language: en

00:00:00.614 --> 00:00:03.214
- After Thursday's snow storm,

00:00:03.214 --> 00:00:08.212
the snow in Chillytown
was 55 centimeters deep.

00:00:08.212 --> 00:00:10.588
On Friday, the snow kept coming

00:00:10.589 --> 00:00:12.926
and several more centimeters fell.

00:00:12.926 --> 00:00:14.632
So, okay, they're getting a lot of snow.

00:00:14.632 --> 00:00:19.631
Now, the snow in Chillytown
is 84, 84 centimeters deep.

00:00:21.093 --> 00:00:24.242
How many centimeters
of snow fell on Friday?

00:00:24.242 --> 00:00:25.301
So let's look look at this diagram again,

00:00:25.301 --> 00:00:28.511
just so that we can imagine what went on.

00:00:28.511 --> 00:00:29.710
So, the first time...

00:00:29.710 --> 00:00:32.042
So the snow in Chillytown...

00:00:32.042 --> 00:00:33.802
There's a storm on Thursday,

00:00:33.802 --> 00:00:36.261
and then the show in Chillytown
was 55 centimeters deep

00:00:36.261 --> 00:00:38.008
and they show that to us right over here.

00:00:38.008 --> 00:00:41.311
So this is the 55 centimeters,

00:00:41.311 --> 00:00:43.426
is this point on the ruler.

00:00:43.426 --> 00:00:45.011
So this is how deep,

00:00:45.011 --> 00:00:48.607
this is how deep the
snow was after the storm,

00:00:48.607 --> 00:00:50.801
after the storm on Thursday.

00:00:50.801 --> 00:00:51.675
This is how deep.

00:00:51.675 --> 00:00:56.510
So this depth right over
here is 55 centimeters.

00:00:56.510 --> 00:01:00.004
But the snow did not stop, it kept coming.

00:01:00.005 --> 00:01:03.845
So now the snow in Chillytown
is 84 centimeters deep.

00:01:03.845 --> 00:01:06.445
So in the ruler, that's
this right over here,

00:01:06.445 --> 00:01:08.801
84 centimeters deep.

00:01:08.802 --> 00:01:10.915
So they wanna know how many centimeters

00:01:10.915 --> 00:01:13.516
of snow fell on Friday.

00:01:13.516 --> 00:01:15.161
So they want us to figure out

00:01:15.161 --> 00:01:16.888
what is this amount right over here?

00:01:16.888 --> 00:01:18.383
What is the amount that fell on Friday?

00:01:18.383 --> 00:01:20.027
They even put the question
mark right over here.

00:01:20.028 --> 00:01:22.202
Question mark centimeters.

00:01:22.202 --> 00:01:24.173
So how do we figure that out?

00:01:24.173 --> 00:01:25.697
Well, let's just right some...

00:01:25.697 --> 00:01:29.618
Let's write some things
that we know are true.

00:01:29.618 --> 00:01:32.890
So we know that 55 centimeters...

00:01:32.891 --> 00:01:37.361
55 centimeters, plus however
many centimeters this was,

00:01:37.361 --> 00:01:39.616
so I'll write question centimeters.

00:01:39.616 --> 00:01:42.298
So this is how many were
after Thursday's snow storm,

00:01:42.298 --> 00:01:44.251
plus however many fell on Friday.

00:01:44.251 --> 00:01:45.328
We don't know that, so I'm just gonna

00:01:45.328 --> 00:01:47.441
write a question mark over there.

00:01:47.441 --> 00:01:49.188
That's going to be equal to the total.

00:01:49.188 --> 00:01:51.910
That's equal to 84 centimeters.

00:01:51.910 --> 00:01:55.344
That's equal to 84 centimeters.

00:01:55.344 --> 00:01:58.940
So we just need to figure out
what's this question mark.

00:01:58.940 --> 00:02:01.602
Well, one way to figure out
what this question mark is

00:02:01.602 --> 00:02:06.112
if 55 plus question mark
is 84, then that means,

00:02:06.112 --> 00:02:11.112
that means that 84 minus 55,

00:02:11.252 --> 00:02:16.252
minus 55 is going to be
equal to the question mark,

00:02:16.474 --> 00:02:18.911
is going to be equal to the question mark.

00:02:18.912 --> 00:02:20.131
And one way to think about it is,

00:02:20.131 --> 00:02:21.698
if you had all this snow and you were to

00:02:21.698 --> 00:02:24.095
subtract out 55 centimeters of it,

00:02:24.095 --> 00:02:25.050
you're just going to be left

00:02:25.050 --> 00:02:27.508
with the question mark amount

00:02:27.508 --> 00:02:30.353
because the difference between 84 and 55

00:02:30.353 --> 00:02:32.120
is what fell on Friday.

00:02:32.120 --> 00:02:35.229
The difference between 84 and 55.

00:02:35.229 --> 00:02:37.017
So let's figure out what that is.

00:02:37.017 --> 00:02:38.825
So I can do that.

00:02:38.825 --> 00:02:41.933
I'm running out of space
here, I'll do it down here.

00:02:41.933 --> 00:02:46.301
So we have 84, 84

00:02:46.301 --> 00:02:51.301
minus 55, minus 55,

00:02:51.929 --> 00:02:56.033
and that's going to be
equal to our question mark.

00:02:56.033 --> 00:02:57.292
So let's look at our ones place.

00:02:57.293 --> 00:02:59.406
We have four ones minus five ones.

00:02:59.406 --> 00:03:01.072
Well, since we don't know how to do that,

00:03:01.072 --> 00:03:02.880
since four is less than five,

00:03:02.880 --> 00:03:04.079
so let's regroup.

00:03:04.079 --> 00:03:05.928
So let's take a 10 from the 10s place.

00:03:05.928 --> 00:03:07.208
Instead of having eight 10s there,

00:03:07.208 --> 00:03:09.221
let's have seven 10s there

00:03:09.221 --> 00:03:11.903
and then make that extra 10 we just took,

00:03:11.903 --> 00:03:13.382
turn that into 10 ones.

00:03:13.383 --> 00:03:16.247
So 10 plus four is 14.

00:03:16.247 --> 00:03:17.549
So now we can subtract.

00:03:17.549 --> 00:03:20.534
14 ones minus five ones.

00:03:20.534 --> 00:03:22.464
Let me move down a little bit.

00:03:22.464 --> 00:03:26.690
14 ones minus five ones is nine ones.

00:03:26.690 --> 00:03:30.591
Seven 10s minus five 10s is two 10s.

00:03:30.591 --> 00:03:34.736
So our mystery amount
of snow, 29 centimeters.

00:03:34.736 --> 00:03:38.820
So this right over here is 29 centimeters.

00:03:38.820 --> 00:03:40.465
And you can check that.

00:03:40.465 --> 00:03:45.423
55 plus 29 is indeed equal to 84.

00:03:45.423 --> 00:03:48.043
I encourage you to try to check

00:03:48.043 --> 00:03:50.156
that that does definitely work.

00:03:50.156 --> 00:03:52.474
So how many centimeters
of snow fell on Friday?

00:03:52.474 --> 00:03:56.434
29, 29 centimeters.

00:03:56.434 --> 00:03:58.101
The important thing for problems like this

00:03:58.101 --> 00:03:59.868
is just imagine the story.

00:03:59.868 --> 00:04:04.582
Imagine what's going on,
and then draw diagrams

00:04:04.582 --> 00:04:05.760
if they didn't draw it for you,

00:04:05.760 --> 00:04:08.786
and then think about writing
some math statements.

00:04:08.787 --> 00:04:11.205
Some people would call
them number sentences

00:04:11.205 --> 00:04:13.643
or equations that you
know are going to be true

00:04:13.643 --> 00:04:17.295
and then you can figure
out what you don't know.


Adding ten or one hundred

WEBVTT
Kind: captions
Language: en

00:00:00.534 --> 00:00:05.232
- So I have the number 235,
and we can think of that as

00:00:05.232 --> 00:00:07.357
2 hundreds, we have a
2 in the hundreds place

00:00:07.357 --> 00:00:10.696
so 2 hundreds, 3 tens, we
have a 3 in the tens place,

00:00:10.696 --> 00:00:14.623
so 3 tens, and 5 ones, we
have a 5 in the ones place.

00:00:14.623 --> 00:00:16.186
And over here I've actually drawn it out

00:00:16.186 --> 00:00:20.599
with that many boxes, this is 235 boxes.

00:00:20.705 --> 00:00:23.412
So you have 2 groups of a hundred boxes,

00:00:23.413 --> 00:00:28.051
so this is 2 hundreds,
then we have 3 tens, 1,2,3.

00:00:28.122 --> 00:00:30.160
And then we have 5 ones.

00:00:30.160 --> 00:00:32.585
Now let's try some
interesting things here.

00:00:32.830 --> 00:00:37.830
So what happens if I were to add 10 here.

00:00:38.689 --> 00:00:42.592
Well adding 10, we're just adding 1 ten.

00:00:42.592 --> 00:00:46.086
We're just adding 1 ten.

00:00:46.086 --> 00:00:48.494
So if we add 1 ten,
what's going to happen?

00:00:48.833 --> 00:00:51.193
Well, over here I need to add one more 10,

00:00:51.193 --> 00:00:52.815
I've got to add this 10.

00:00:53.031 --> 00:00:54.097
So what's it going to look like?

00:00:54.097 --> 00:00:56.469
Well that, the next 10 is
going to look like this.

00:00:56.469 --> 00:00:58.986
Let me see if I can draw it pretty fast.

00:00:59.499 --> 00:01:03.117
So we have one, two, three, four, five,

00:01:03.117 --> 00:01:06.529
six, seven, eight, nine, and ten.

00:01:06.529 --> 00:01:08.084
One, two, three, four, five,

00:01:08.084 --> 00:01:10.300
six, seven, eight, nine, ten.

00:01:10.569 --> 00:01:12.178
So I just added a ten.

00:01:12.178 --> 00:01:14.118
So what does that leave us with?

00:01:14.294 --> 00:01:18.755
Well if I had 3, if i had 3
tens, and I have 1 more ten now,

00:01:18.755 --> 00:01:23.146
now I'm going to have 4 tens,
now I'm going to have 4 tens.

00:01:23.146 --> 00:01:24.909
I'm gonna do that in that yellow color.

00:01:25.630 --> 00:01:29.838
So now I'm going to have 4 tens.

00:01:30.142 --> 00:01:32.741
I'm still going to have 5 ones, 5 ones,

00:01:32.741 --> 00:01:35.359
I'm still gonna have 5 ones,
I'm not adding any ones there.

00:01:35.858 --> 00:01:39.845
So I'm still going to have 5 ones, and

00:01:39.845 --> 00:01:40.882
I'm still going to have 2 hundreds,

00:01:40.882 --> 00:01:43.540
I'm not adding any hundreds there, so

00:01:43.540 --> 00:01:48.491
I'm still going to have
2 hundreds, 2 hundreds.

00:01:48.667 --> 00:01:50.079
So now it's going to be 2 hundreds,

00:01:50.079 --> 00:01:51.792
plus 4 tens, plus 5 ones.

00:01:51.793 --> 00:01:54.585
2 hundreds plus 4 tens plus 5 ones.

00:01:54.807 --> 00:01:56.432
Well what's that going to be?

00:01:56.432 --> 00:02:01.432
Well, 2 hundreds, plus 4 tens,

00:02:01.682 --> 00:02:06.103
plus 5 ones: 245.

00:02:06.291 --> 00:02:06.860
Let's do one more.

00:02:06.861 --> 00:02:09.489
Instead of adding a ten,
let's add a hundred.

00:02:09.792 --> 00:02:10.979
So let's do that, let me see

00:02:10.979 --> 00:02:14.827
if I can clear all this stuff out, so...

00:02:16.257 --> 00:02:20.416
So instead of adding 10 lets add 100 here,

00:02:20.708 --> 00:02:25.516
so I'm going to add, so plus, let's add

00:02:26.067 --> 00:02:29.579
1 hundred, 1,

00:02:30.540 --> 00:02:32.698
I'll actually do it in yellow and in blue.

00:02:32.699 --> 00:02:37.261
So I'm adding 1 hundred,
0 tens, and 0 ones.

00:02:37.261 --> 00:02:39.747
So essentially just adding 1 hundred.

00:02:40.167 --> 00:02:44.978
So this is going to be
equal to, so plus 1 hundred.

00:02:46.274 --> 00:02:51.274
So plus 1 hundred, so
what's that going to get me?

00:02:51.822 --> 00:02:54.483
Let me draw it out here,
let me copy and paste.

00:02:54.559 --> 00:02:58.038
I have 2 hundreds before but
now I'm going to add 1 hundred.

00:02:58.760 --> 00:03:03.025
So let me just copy that,
and let me paste it.

00:03:03.690 --> 00:03:06.470
So now, how many hundreds do I have?

00:03:07.573 --> 00:03:10.352
How many hundreds do I have now?

00:03:10.352 --> 00:03:13.832
Well now i have one, two, three hundreds.

00:03:14.473 --> 00:03:16.322
I have 3 hundreds.

00:03:16.322 --> 00:03:19.057
So over here, I had 2 hundreds,

00:03:19.057 --> 00:03:21.511
I'm adding one more
hundred right over here

00:03:21.511 --> 00:03:23.107
so now I have 3 hundreds.

00:03:23.469 --> 00:03:28.469
Over here I could say that
now I have 3, 3 hundreds.

00:03:29.053 --> 00:03:31.405
Now, are the number of
tens, have they changed?

00:03:31.405 --> 00:03:34.260
No, I had 3 tens before, and
I'm not adding tens this time

00:03:34.260 --> 00:03:36.810
so I'm still going to
have 3 tens, and I had

00:03:36.810 --> 00:03:39.652
5 ones before and I'm
not adding any more ones

00:03:39.652 --> 00:03:41.559
so I still have 5 ones.

00:03:41.615 --> 00:03:45.001
So I have 335, which is the same thing as

00:03:45.139 --> 00:03:50.139
3 hundreds plus 3 tens, plus 3 tens,

00:03:51.040 --> 00:03:56.040
plus 5 ones, plus 5 ones.


Picture graphs

WEBVTT
Kind: captions
Language: en

00:00:00.373 --> 00:00:04.296
- [Voiceover] What we have
here is a picture graph,

00:00:04.296 --> 00:00:08.162
and the reason why it's
called a picture graph is

00:00:08.162 --> 00:00:12.701
because it uses pictures
to give us information.

00:00:12.701 --> 00:00:16.597
In this case, it uses
the picture of a mouse

00:00:16.597 --> 00:00:19.237
and it tells us that one of
these little mouse pictures

00:00:19.237 --> 00:00:22.881
means one mouse, it equals one mouse.

00:00:22.881 --> 00:00:25.396
So what is this picture graph telling us?

00:00:25.396 --> 00:00:28.864
Well, the title says "Mice on Farm,"

00:00:28.864 --> 00:00:30.451
so this tells us how many mice are

00:00:30.451 --> 00:00:32.993
in the different places in a farm,

00:00:32.993 --> 00:00:34.549
and you can see here the different places

00:00:34.549 --> 00:00:36.175
where we might find mice.

00:00:36.175 --> 00:00:39.481
We could find mice in
the house, in the barn,

00:00:39.481 --> 00:00:42.865
in the pasture, or the corral.

00:00:42.880 --> 00:00:45.427
And then these, I guess
you could say these stacks

00:00:45.427 --> 00:00:47.226
of these mouse pictures tell us

00:00:47.226 --> 00:00:50.784
how many mice there are
in each of those places.

00:00:50.784 --> 00:00:52.810
So, for example, in the house, we see,

00:00:52.810 --> 00:00:54.946
well this, this they tell
us, that's one mouse,

00:00:54.946 --> 00:00:59.291
this is one plus one plus
one plus one, or four mice.

00:00:59.291 --> 00:01:03.190
Four mice in the house.

00:01:03.190 --> 00:01:06.033
That's four right over
there, write that over there.

00:01:07.125 --> 00:01:08.297
Four mice in the house.

00:01:08.297 --> 00:01:09.821
How many in the barn?

00:01:10.575 --> 00:01:15.360
Well, same idea, one, two,
let me use a different color,

00:01:15.360 --> 00:01:19.394
one, two, three, four, five, six,

00:01:19.394 --> 00:01:24.394
so there are six mice,
six mice, in the barn.

00:01:24.983 --> 00:01:26.315
So there's six mice in the barn,

00:01:26.315 --> 00:01:27.626
let me just write that down.

00:01:27.626 --> 00:01:31.729
So six mice in the barn,
what about the pasture?

00:01:31.729 --> 00:01:33.558
Well, we see that
there's three mice there,

00:01:33.558 --> 00:01:36.212
that's 'cause each of these
pictures is one mouse.

00:01:36.212 --> 00:01:38.074
The person who made the
picture graph could've made

00:01:38.074 --> 00:01:40.542
each of those pictures a
different number of mice,

00:01:40.542 --> 00:01:42.269
but here each one is one mouse,

00:01:42.269 --> 00:01:45.013
so there's three in the pasture.

00:01:45.013 --> 00:01:46.862
Three in the pasture.

00:01:46.862 --> 00:01:49.772
And then, finally, move over a little bit,

00:01:49.772 --> 00:01:53.902
and then finally there
are two in the corral.

00:01:53.902 --> 00:01:56.022
Two in the corral.

00:01:56.022 --> 00:01:56.730
So if someone asked you,

00:01:56.730 --> 00:01:59.792
"How many total mice
are there on this farm?"

00:01:59.792 --> 00:02:01.832
Well, you would add all of these up.

00:02:01.832 --> 00:02:05.857
You would say four plus six is 10,

00:02:05.857 --> 00:02:10.679
plus three is 13, plus 2 is 15.

00:02:10.693 --> 00:02:13.629
15 total mice on the farm.

00:02:13.629 --> 00:02:17.974
So once again, picture graph,
just a way to show data,

00:02:17.974 --> 00:02:19.251
to show information.

00:02:19.251 --> 00:02:24.251
This tells us how many mice
are in each part of the farm.

00:02:24.268 --> 00:02:25.183
If someone said, "How
many mice in the barn?"

00:02:25.183 --> 00:02:27.575
you say, okay, well, you
know, there's six mice.

00:02:27.575 --> 00:02:28.643
If someone says, "How many mice are

00:02:28.643 --> 00:02:32.016
"in the pasture and the corral combined?"

00:02:32.016 --> 00:02:33.563
Well, that's gonna be
the three in the pasture

00:02:33.563 --> 00:02:36.061
plus the two in the corral, or five.

00:02:36.061 --> 00:02:37.721
They said, "How many
mice in the whole farm?"

00:02:37.721 --> 00:02:40.723
and if these are all the
places where the mice could be,

00:02:40.723 --> 00:02:43.075
well, it's gonna be the four
plus six plus three plus two,

00:02:43.075 --> 00:02:45.948
or 15, and we're done.


Comparing small numbers on the number line

WEBVTT
Kind: captions
Language: en

00:00:00.282 --> 00:00:04.971
Voiceover:Which numbers
are greater than six?

00:00:04.971 --> 00:00:08.892
Select all that apply.

00:00:08.892 --> 00:00:11.962
We see six here on the number line,

00:00:11.962 --> 00:00:13.853
so the numbers that are greater than six

00:00:13.853 --> 00:00:16.374
are going to be the ones
that are to the right of six

00:00:16.374 --> 00:00:17.219
on the number line.

00:00:17.219 --> 00:00:19.399
We see that we're increasing beyond six

00:00:19.399 --> 00:00:20.750
as we go to the right.

00:00:20.750 --> 00:00:23.008
Six, seven, eight, nine, ten.

00:00:23.008 --> 00:00:24.297
Seven is greater than six,

00:00:24.297 --> 00:00:25.286
eight is greater than six,

00:00:25.286 --> 00:00:26.472
nine is greater than six,

00:00:26.472 --> 00:00:27.868
and 10 is greater than six,

00:00:27.868 --> 00:00:28.795
and we could keep going.

00:00:28.795 --> 00:00:29.677
11 is greater than six,

00:00:29.677 --> 00:00:31.271
and 12, and on and on and on.

00:00:31.271 --> 00:00:33.331
Which of these are greater than six?

00:00:33.331 --> 00:00:35.589
Well we see 10 is to the right,

00:00:35.589 --> 00:00:37.706
is on the right hand side of six,

00:00:37.706 --> 00:00:40.898
eight is also to the right of six

00:00:40.898 --> 00:00:43.130
but four is to the left of six.

00:00:43.130 --> 00:00:45.586
Four is less than six.

00:00:45.586 --> 00:00:48.413
These are the two numbers
that are greater than,

00:00:48.413 --> 00:00:51.714
the two choices that are greater than six.

00:00:51.714 --> 00:00:53.829
Which numbers are less than six?

00:00:53.829 --> 00:00:56.809
Well that's all of these
numbers right over here.

00:00:56.809 --> 00:00:58.435
The numbers to the left of six.

00:00:58.435 --> 00:01:02.207
Nine is definitely not less than six

00:01:02.207 --> 00:01:03.691
but four is.

00:01:03.691 --> 00:01:05.923
Notice four is to the left of six

00:01:05.923 --> 00:01:08.451
and three is even more to the left of six,

00:01:08.451 --> 00:01:12.303
so four and three are
definitely less than six.

00:01:12.303 --> 00:01:15.110
When every day life
when you're thinking of

00:01:15.110 --> 00:01:17.149
well if you have four things,
or you have three things

00:01:17.149 --> 00:01:19.550
you have less than someone
who has six things.

00:01:19.550 --> 00:01:23.349
In every day life if you have 10 bananas,

00:01:23.349 --> 00:01:25.778
you have a greater number of bananas

00:01:25.778 --> 00:01:28.181
than someone who has six bananas.


Skip counting by 100 example

WEBVTT
Kind: captions
Language: en

00:00:00.865 --> 00:00:03.152
- [Voiceover] When
counting by 100s from 43

00:00:03.152 --> 00:00:07.432
to 1,043, which numbers will we say?

00:00:07.432 --> 00:00:09.081
Select all that apply.

00:00:09.081 --> 00:00:10.359
So let's do that.

00:00:10.359 --> 00:00:14.851
Let's start with 43 , and then

00:00:14.851 --> 00:00:17.196
let's just keep adding 100 to 43.

00:00:17.196 --> 00:00:22.196
So if we add 100 to 43,
we're going to get 143.

00:00:25.441 --> 00:00:27.539
Now let's keep adding 100 to it.

00:00:27.539 --> 00:00:28.956
If we add another 100, we're now going to

00:00:28.956 --> 00:00:33.956
have two hundreds and 43, or 243.

00:00:34.458 --> 00:00:37.198
Now let's keep adding 100 to it.

00:00:37.198 --> 00:00:42.198
So if we add another 100
we're going to have 343.

00:00:43.595 --> 00:00:45.010
I think you see what's going on here.

00:00:45.010 --> 00:00:50.010
You add 100 again,
you're going to have 443.

00:00:50.892 --> 00:00:52.092
I'm coloring in the 100s place

00:00:52.092 --> 00:00:53.473
so that we can keep track of that.

00:00:53.473 --> 00:00:58.473
You add another 100,
you're going to have 543.

00:00:59.683 --> 00:01:04.683
Then add another 100,
you're going to have 643.

00:01:05.791 --> 00:01:10.791
Add another 100, you're going to have 743.

00:01:12.694 --> 00:01:16.478
Let's keep adding 100, keep adding 100.

00:01:16.478 --> 00:01:21.338
Add another 100, you're going to have 843.

00:01:21.338 --> 00:01:25.896
Add another 100, you're going to have 943.

00:01:25.896 --> 00:01:28.858
Or you can do that as
nine hundreds and 43.

00:01:28.858 --> 00:01:31.668
Or nine hundreds and
four tens and three ones.

00:01:31.668 --> 00:01:33.757
And let's add another 100.

00:01:33.757 --> 00:01:35.754
So if you add another
100, you could view this

00:01:35.754 --> 00:01:40.754
as 10 hundreds and 43.

00:01:42.035 --> 00:01:45.005
But 10 hundreds is the
same thing as 1,000.

00:01:45.005 --> 00:01:46.984
So we would read this as 1,043.

00:01:46.984 --> 00:01:49.223
And that's what they
wanted us to count up to

00:01:49.223 --> 00:01:54.223
1,043 is the same thing
as 10 hundreds and 43.

00:01:54.656 --> 00:01:56.955
So which of these numbers did we see

00:01:56.955 --> 00:02:01.608
when we counted from 43 to 1,043?

00:02:01.608 --> 00:02:03.599
Well we didn't see 44.

00:02:03.599 --> 00:02:05.539
We didn't see 123.

00:02:05.539 --> 00:02:09.858
But we did see 143, so we did see 143.

00:02:09.858 --> 00:02:14.745
We saw 543, we saw 543.

00:02:14.745 --> 00:02:17.858
And we saw 843.

00:02:17.858 --> 00:02:19.657
So we saw these numbers.

00:02:19.657 --> 00:02:20.817
Now you might have been able
to do this in your head.

00:02:20.817 --> 00:02:24.799
You might have said, "OK
43, then 143, then 243,

00:02:24.799 --> 00:02:29.799
343, 443, 543, 643, 743,
843, then 943 and then 1,043.

00:02:36.681 --> 00:02:39.441
Either way, these are the numbers

00:02:39.441 --> 00:02:44.441
that we said when we
counted from 43 to 1,043.

00:02:44.761 --> 00:02:47.924
These are three of the
numbers that we said.


Counting dollars

WEBVTT
Kind: captions
Language: en

00:00:01.007 --> 00:00:04.807
- [Voiceover] Zhang Tao
has two one dollar bills.

00:00:04.807 --> 00:00:06.351
So two one dollar bills.

00:00:06.351 --> 00:00:08.231
So let's just write that as one plus one,

00:00:08.231 --> 00:00:09.798
because each one dollar bill

00:00:09.798 --> 00:00:11.691
represents one dollar.

00:00:11.691 --> 00:00:13.711
So, one dollar plus one dollar.

00:00:13.711 --> 00:00:15.394
That's the two one dollar bills.

00:00:15.394 --> 00:00:17.634
One five dollar bill.

00:00:17.634 --> 00:00:19.178
One five dollar bill.

00:00:19.178 --> 00:00:21.570
So that represents five dollars.

00:00:21.570 --> 00:00:24.217
And three 10 dollar bills.

00:00:24.217 --> 00:00:26.515
Three 10 dollar bills.

00:00:26.515 --> 00:00:31.333
So that's plus ten plus ten plus ten.

00:00:31.333 --> 00:00:33.817
How much money does he have in all?

00:00:33.817 --> 00:00:34.468
Well, let's see.

00:00:34.468 --> 00:00:36.755
One plus, let's see if
we can calculate it.

00:00:36.755 --> 00:00:38.415
One plus one is two.

00:00:38.415 --> 00:00:40.083
Two plus five is seven.

00:00:40.083 --> 00:00:42.756
So, these are going to be seven dollars.

00:00:42.756 --> 00:00:45.350
And then 10 plus 10 plus
10, that's three 10's.

00:00:45.350 --> 00:00:48.073
So that's going to be 30 dollars.

00:00:48.073 --> 00:00:50.929
Now what's seven ones plus three 10's?

00:00:50.929 --> 00:00:52.949
Or what's seven plus 30?

00:00:52.949 --> 00:00:56.350
Well, that's going to be 37 dollars.

00:00:56.350 --> 00:01:00.646
So Zhang Tao has 37 dollars in all.

00:01:00.646 --> 00:01:02.107
Let's do another one of these.

00:01:04.895 --> 00:01:07.205
Diya has six one dollar bills.

00:01:07.205 --> 00:01:08.888
And actually, let's draw it out.

00:01:08.888 --> 00:01:10.715
Diya has six one dollar bills.

00:01:10.715 --> 00:01:14.527
So that's one, two, three,

00:01:15.112 --> 00:01:19.464
four, five and six.

00:01:19.464 --> 00:01:21.112
And these are one dollar bills.

00:01:21.112 --> 00:01:23.910
So, one dollar bills.

00:01:23.910 --> 00:01:27.590
This is my rough drawing
of one dollar bills.

00:01:27.590 --> 00:01:29.773
Have to draw a lot of
one dollar bills here.

00:01:29.773 --> 00:01:33.302
So, she has six one dollar bills.

00:01:33.302 --> 00:01:35.932
She has three five dollar bills.

00:01:35.932 --> 00:01:37.586
Can we do that in a different color?

00:01:38.339 --> 00:01:41.489
So, three five dollar bills.

00:01:41.489 --> 00:01:44.667
So one, so that's a five dollar bill.

00:01:45.650 --> 00:01:48.390
Two, that's a five dollar bill.

00:01:49.098 --> 00:01:51.917
And three five dollar bills.

00:01:53.517 --> 00:01:54.487
That little circle in ....

00:01:54.487 --> 00:01:56.385
That's supposed to be
the picture of someone.

00:01:56.385 --> 00:01:57.477
Let me make it clear

00:01:58.215 --> 00:02:01.428
these are pictures of folks.

00:02:01.428 --> 00:02:03.989
Of famous historical figures.

00:02:03.989 --> 00:02:08.029
So, these are pictures of
famous historical figures here.

00:02:08.029 --> 00:02:11.581
And then finally she
has one 10 dollar bill.

00:02:11.581 --> 00:02:13.589
One 10 dollar bill.

00:02:13.589 --> 00:02:15.122
So one 10 dollar bill.

00:02:15.122 --> 00:02:17.072
So it's worth 10 dollars.

00:02:17.072 --> 00:02:20.601
A picture of a famous historical
figure right over there.

00:02:20.601 --> 00:02:23.120
So, how much money does she have in all?

00:02:23.120 --> 00:02:24.513
Well, the six one dollar bills

00:02:24.513 --> 00:02:26.940
that's going to be six dollars.

00:02:26.940 --> 00:02:28.519
The three five dollar bills

00:02:28.519 --> 00:02:31.730
that's going to be worth
five, 10, 15 dollars.

00:02:32.237 --> 00:02:34.826
So plus 15 dollars.

00:02:34.826 --> 00:02:36.370
And then the one 10 dollar bill.

00:02:36.370 --> 00:02:38.348
That's going to be worth 10 dollars.

00:02:38.348 --> 00:02:39.389
10 dollars.

00:02:39.912 --> 00:02:42.353
So, what's six plus 15 plus 10?

00:02:43.046 --> 00:02:44.629
Well, let's see, six plus fifteen

00:02:44.629 --> 00:02:46.904
is going to be 21.

00:02:46.904 --> 00:02:51.049
21 plus 10 is going to
be equal to 31 dollars.

00:02:51.049 --> 00:02:52.964
Is equal to 31 dollars.

00:02:52.964 --> 00:02:54.586
And you could have also added it this way,

00:02:54.586 --> 00:02:56.426
You could have said 15

00:02:56.426 --> 00:02:59.790
plus 10 plus 10

00:02:59.790 --> 00:03:02.878
plus six plus six.

00:03:03.570 --> 00:03:04.809
Add them together

00:03:04.809 --> 00:03:07.025
and we would have gotten five

00:03:07.025 --> 00:03:09.502
plus zero plus six is 11.

00:03:09.502 --> 00:03:11.457
That's one one and one 10.

00:03:11.457 --> 00:03:13.244
And then you have three 10's here.

00:03:13.244 --> 00:03:15.763
So, 31 dollars.


Losing tennis balls

WEBVTT
Kind: captions
Language: en

00:00:00.871 --> 00:00:04.928
- Mrs. Grange's second grade
class has 45 tennis balls.

00:00:04.996 --> 00:00:06.489
Whenever I do a word problem like this

00:00:06.489 --> 00:00:07.657
I just like to write down

00:00:07.657 --> 00:00:09.100
the information that they're giving me.

00:00:09.100 --> 00:00:13.169
So we're gonna start with 45 tennis balls.

00:00:13.740 --> 00:00:17.253
They then tell us that 19 tennis balls,

00:00:17.253 --> 00:00:20.077
so she starts with 45 tennis
balls to play with at recess.

00:00:20.077 --> 00:00:22.414
They then tell us that 19 tennis balls

00:00:22.414 --> 00:00:25.179
get thrown over the fence

00:00:25.179 --> 00:00:28.831
and 6 tennis balls just disappear.

00:00:28.933 --> 00:00:32.604
How many tennis balls does
Mrs. Grange's class have left?

00:00:32.604 --> 00:00:34.150
So what I'd like to do, when I do

00:00:34.151 --> 00:00:35.235
these type of word
problems, let's say, okay,

00:00:35.235 --> 00:00:37.091
first of all where am I starting from?

00:00:37.091 --> 00:00:39.664
Well, I'm starting from 45 tennis balls

00:00:39.664 --> 00:00:40.519
and then whats happening?

00:00:40.519 --> 00:00:43.528
Am I gaining tennis balls
or am I losing tennis balls?

00:00:44.017 --> 00:00:45.428
Well I started with 45 or

00:00:45.428 --> 00:00:47.943
Mrs. Grange's class started with 45

00:00:47.943 --> 00:00:50.504
then 19 tennis balls get
thrown over the fence

00:00:50.505 --> 00:00:52.331
so they're losing the balls.

00:00:52.331 --> 00:00:54.149
And then another 6 tennis balls disappear,

00:00:54.149 --> 00:00:56.244
so both of these are losing tennis balls.

00:00:56.536 --> 00:00:57.844
And since I'm losing something,

00:00:57.844 --> 00:00:59.302
since I'm losing the tennis balls,

00:00:59.302 --> 00:01:01.511
or some of the tennis
balls that I started with

00:01:01.511 --> 00:01:03.063
I should subtract here.

00:01:03.063 --> 00:01:05.925
Another clue that subtraction
might be interesting

00:01:06.186 --> 00:01:07.763
is because it says how many tennis balls

00:01:07.763 --> 00:01:10.961
does Mrs. Grange's class have
left, which implies that,

00:01:10.961 --> 00:01:12.843
well, she doesn't have
all the tennis balls

00:01:12.843 --> 00:01:14.232
that she started with so we're probably

00:01:14.232 --> 00:01:15.303
going to subtract some balls.

00:01:15.303 --> 00:01:16.541
But it comes straight
out of the word problem.

00:01:16.541 --> 00:01:20.384
19 tennis balls get thrown
over the fence, she loses them,

00:01:20.384 --> 00:01:22.758
so we should subtract that
from the original amount.

00:01:22.957 --> 00:01:24.773
And then another 6 just
disappear, so we should then

00:01:24.774 --> 00:01:28.178
subtract that on top of
the 19, so let's do that.

00:01:28.795 --> 00:01:30.535
So let's first subtract the 19.

00:01:31.119 --> 00:01:33.322
So we start with 45 tennis balls,

00:01:33.645 --> 00:01:37.851
and then 19 balls get
thrown over the fence, so

00:01:37.852 --> 00:01:40.611
19 get thrown over the fence
so we're subtracting them.

00:01:40.611 --> 00:01:44.239
They lose those balls, so
what is this going to be?

00:01:44.345 --> 00:01:45.761
Well if we look in the ones place

00:01:45.761 --> 00:01:47.645
we have 5 ones minus 9 ones.

00:01:47.645 --> 00:01:51.280
Well, since 5 is less than 9
we want to do some regrouping.

00:01:51.537 --> 00:01:53.150
So let's go to the tens place.

00:01:53.150 --> 00:01:55.444
Instead of 4 tens, lets
try that as 3 tens,

00:01:55.444 --> 00:01:57.797
and then take that 10 , turn it into ones

00:01:57.797 --> 00:01:59.355
so it's gonna be 10 ones.

00:01:59.355 --> 00:02:02.652
10 ones plus 5 ones is 15.

00:02:03.159 --> 00:02:05.125
Now we're ready to subtract.

00:02:05.255 --> 00:02:10.255
15 minus 9, well that's gonna
be 6 because 9 plus 6 is 15.

00:02:10.649 --> 00:02:15.649
And then 3 tens minus 1 ten
is going to be 2 tens, so 26.

00:02:16.724 --> 00:02:18.572
But are we done? No.

00:02:18.573 --> 00:02:20.851
This is just the number
of balls we have left

00:02:20.851 --> 00:02:23.346
after the 19 get thrown over the fence

00:02:23.346 --> 00:02:27.096
but before we count the
6 that just disappeared.

00:02:27.096 --> 00:02:28.969
So we also have to take away 6

00:02:28.969 --> 00:02:30.516
for the 6 balls that disappeared.

00:02:30.516 --> 00:02:31.846
So let's do that.

00:02:31.847 --> 00:02:34.663
So after 19, so we started with 45,

00:02:34.663 --> 00:02:39.179
19 get thrown over the fence,
we have 26 at that point.

00:02:39.855 --> 00:02:42.893
Then, 6 balls just disappear
so we have, once again

00:02:42.893 --> 00:02:46.353
they disappear, they're not,
you know, magically appearing.

00:02:46.353 --> 00:02:48.784
Since they're disappearing we subtract.

00:02:49.401 --> 00:02:52.845
So we, they're disappearing,
so we want to subtract those.

00:02:52.845 --> 00:02:55.034
So how many do we have left?

00:02:55.034 --> 00:02:59.231
Well, 6 ones minus 6 ones is 0 ones.

00:02:59.639 --> 00:03:02.478
2 tens minus, well nothing here,

00:03:02.478 --> 00:03:04.039
that's just going to be 2 tens.

00:03:04.506 --> 00:03:06.241
So how many do we have left?

00:03:06.719 --> 00:03:09.755
Well, or how many does Mrs. Grange's

00:03:09.755 --> 00:03:11.702
second grade class have left?

00:03:11.702 --> 00:03:13.365
Well you see it right here.

00:03:13.365 --> 00:03:18.365
They have 20 tennis balls,
20 tennis balls left.

00:03:19.265 --> 00:03:21.915
Started with 45, 19 get
thrown over the fence

00:03:21.915 --> 00:03:23.947
so they lose them,
that's why we subtracted.

00:03:23.947 --> 00:03:26.845
That gets us to 26,
then 6 of them disappear

00:03:26.845 --> 00:03:31.448
so we take the 6 away
from the 26 to get to 20.


Counting by tens examples

WEBVTT
Kind: captions
Language: en

00:00:00.488 --> 00:00:03.205
- Count by 10s starting with 40.

00:00:03.205 --> 00:00:04.923
Fill in the blanks.

00:00:04.923 --> 00:00:09.817
So, let's count by 10s, starting with 40.

00:00:10.357 --> 00:00:11.982
So, 40 is four 10s.

00:00:11.982 --> 00:00:13.514
So, if we add 10 to that,

00:00:13.514 --> 00:00:16.417
we're gonna get five 10s, or 50.

00:00:16.417 --> 00:00:17.879
Now, if we add 10 to that,

00:00:17.879 --> 00:00:21.060
we're going to get six 10s, or 60.

00:00:21.060 --> 00:00:22.500
Now, if we add 10 to that,

00:00:22.500 --> 00:00:24.381
we're going to get seven 10s, or 70,

00:00:24.381 --> 00:00:25.495
but we don't have to type that in,

00:00:25.495 --> 00:00:27.794
because they've already
put that there for us.

00:00:27.794 --> 00:00:30.348
Let's keep adding 10s.

00:00:30.348 --> 00:00:33.645
So, if we add 10 to
70, or 10 to seven 10s,

00:00:33.645 --> 00:00:36.292
we're going to have eight 10s, or 80.

00:00:36.292 --> 00:00:37.592
Add 10 to that,

00:00:37.592 --> 00:00:40.239
you're going to have nine 10s, or 90.

00:00:40.239 --> 00:00:41.493
And then we're done.

00:00:41.493 --> 00:00:43.020
Let's do one more of these.

00:00:43.629 --> 00:00:47.693
Count by 10s starting with 656.

00:00:47.693 --> 00:00:49.063
Fill in the blanks.

00:00:49.063 --> 00:00:52.569
Well, each time we want to add one 10.

00:00:52.569 --> 00:00:55.750
So, this 656, it's six hundreds,

00:00:55.750 --> 00:00:58.490
five tens, and six ones.

00:00:58.490 --> 00:01:00.417
Well, now we're going to add 10 to it.

00:01:00.417 --> 00:01:02.460
So instead of having five 10s,

00:01:02.460 --> 00:01:05.246
we want to have six 10s,

00:01:05.246 --> 00:01:08.474
so it's gonna be 666,

00:01:08.474 --> 00:01:12.160
six hundreds, six tens, and six ones.

00:01:12.631 --> 00:01:14.186
Let's add 10 again.

00:01:14.186 --> 00:01:15.417
Well, now, in the last one,

00:01:15.417 --> 00:01:18.366
we had six 10s, now we
want to have seven 10s.

00:01:18.366 --> 00:01:22.702
So, it's gonna be 676.

00:01:23.427 --> 00:01:25.448
Now, let's add another 10.

00:01:25.448 --> 00:01:27.073
Well, now we're gonna have six hundreds,

00:01:27.073 --> 00:01:31.545
eight tens, and six ones, or 686.

00:01:31.635 --> 00:01:33.714
Now, let's add 10 again.

00:01:33.714 --> 00:01:36.639
So, now, we're gonna have six hundreds,

00:01:36.639 --> 00:01:39.170
instead of eight 10s,
we're gonna have nine 10s,

00:01:39.170 --> 00:01:40.587
and six ones.

00:01:40.587 --> 00:01:43.489
So, 696.

00:01:43.489 --> 00:01:46.136
Now, the next one is
going to be interesting.

00:01:46.136 --> 00:01:48.829
We want to add another 10 to this.

00:01:48.829 --> 00:01:51.801
But how do we have, if we
add another 10 to this,

00:01:51.801 --> 00:01:54.588
what do we have in the tens place?

00:01:54.588 --> 00:01:56.677
Well, one way to think about this is,

00:01:56.677 --> 00:01:59.115
you can view six hundreds and nine tens

00:01:59.115 --> 00:02:01.275
as 69 tens.

00:02:01.275 --> 00:02:04.572
So if you have 69 tens,
and you add one more ten,

00:02:04.572 --> 00:02:06.731
you're going to have 70 tens,

00:02:06.731 --> 00:02:10.980
so you're going to have
70 tens, and six ones.

00:02:10.980 --> 00:02:14.138
So, 696, you add 10 to it,

00:02:14.138 --> 00:02:16.715
you get 706.

00:02:16.715 --> 00:02:20.547
Once again, you can view
this as 65 tens and six ones,

00:02:20.547 --> 00:02:21.940
So, 65 tens,

00:02:21.940 --> 00:02:23.310
66 tens,

00:02:23.310 --> 00:02:25.724
67 tens,

00:02:25.724 --> 00:02:27.837
68 tens,

00:02:27.837 --> 00:02:29.579
69 tens,

00:02:29.579 --> 00:02:31.436
and 70 tens.

00:02:31.436 --> 00:02:33.753
706.


Fewer word problems

WEBVTT
Kind: captions
Language: en

00:00:00.553 --> 00:00:03.722
In a basketball game, the Giraffes scored

00:00:03.722 --> 00:00:07.232
13 fewer points than the Penguins.

00:00:07.270 --> 00:00:12.270
The Penguins scored, the
Penguins scored 87 points.

00:00:13.588 --> 00:00:16.531
How many points did the Giraffes score?

00:00:16.900 --> 00:00:19.542
Now before we even try to do this,

00:00:19.542 --> 00:00:22.447
let's just think that
based on what they told us,

00:00:22.447 --> 00:00:24.234
will the Giraffes score more points

00:00:24.234 --> 00:00:27.623
than the Penguins, or less
points than the Penguins?

00:00:28.399 --> 00:00:29.293
Well, they tell us.

00:00:29.293 --> 00:00:32.906
The Giraffes scored 13 fewer points.

00:00:33.194 --> 00:00:37.681
13 fewer, fewer points than the Penguins.

00:00:37.806 --> 00:00:41.215
So the Giraffes scored less
points than the Penguins.

00:00:41.424 --> 00:00:43.557
So whatever number is here, the Giraffes,

00:00:43.557 --> 00:00:46.050
this is gonna be less than 87.

00:00:46.197 --> 00:00:48.773
Because the Penguins scored 87 points.

00:00:49.102 --> 00:00:50.479
And how much less is it?

00:00:50.627 --> 00:00:52.183
Well, it's gonna be 13 less.

00:00:52.183 --> 00:00:53.944
It's gonna be 13 fewer.

00:00:54.193 --> 00:00:56.672
So this right over here,
this right over here

00:00:56.672 --> 00:00:58.313
is whatever the Penguins scored.

00:00:58.581 --> 00:01:01.868
Whatever the penguins scored,
which they tell us is 87,

00:01:02.077 --> 00:01:04.286
cause they tell us the
Penguins scored 87 points,

00:01:04.331 --> 00:01:06.424
it's gonna be 13 fewer than that,

00:01:06.424 --> 00:01:08.248
or 13 less than that.

00:01:08.476 --> 00:01:12.128
So it's going to be 87 minus 13 points.

00:01:12.255 --> 00:01:13.794
So let's figure out what that is.

00:01:14.653 --> 00:01:17.737
So the Penguins scored 87 points,

00:01:18.614 --> 00:01:23.450
and the Giraffes scored 13 fewer.

00:01:23.450 --> 00:01:26.863
So if we subtract 13 from the 87,

00:01:26.863 --> 00:01:28.753
that's going to give
us the number of points

00:01:28.753 --> 00:01:30.434
that the Giraffes scored.

00:01:30.621 --> 00:01:31.998
So what is that going to be?

00:01:32.511 --> 00:01:34.136
Let's see, let's look at the ones place.

00:01:34.136 --> 00:01:38.642
Seven ones minus three ones is four ones.

00:01:39.074 --> 00:01:40.374
Now go to the tens place.

00:01:40.374 --> 00:01:44.168
Eight tens minus one ten is seven tens.

00:01:44.904 --> 00:01:49.064
So 13 fewer than 87 is 74 points.

00:01:49.314 --> 00:01:51.751
And the Giraffes scored 13
fewer than the Penguins,

00:01:51.751 --> 00:01:53.087
this is the Penguins,

00:01:53.315 --> 00:01:55.769
this is what the Penguins
scored right over here.

00:01:56.750 --> 00:01:58.801
And the Giraffes scored
13 fewer than that,

00:01:58.801 --> 00:02:03.801
so that is going to be 74 points.

00:02:05.038 --> 00:02:05.954
And it makes sense.

00:02:05.954 --> 00:02:09.808
74 is 13 fewer than 87.


Subtraction word problems within 10

WEBVTT
Kind: captions
Language: en

00:00:00.584 --> 00:00:03.487
- [Voiceover] There were
10 sharks at the reef.

00:00:03.487 --> 00:00:06.817
And we see, there's one,
two, three, four, five,

00:00:06.817 --> 00:00:11.317
six, seven, eight, nine, 10
sharks, just like they told us.

00:00:11.317 --> 00:00:14.917
Two sharks swam away, I
was gonna say ran away,

00:00:14.917 --> 00:00:16.751
(laughs), they swam away.

00:00:16.751 --> 00:00:18.883
How many sharks are left?

00:00:18.883 --> 00:00:20.383
So, let's think about it.

00:00:20.383 --> 00:00:23.050
There were 10 sharks at the reef

00:00:23.050 --> 00:00:26.983
and two of them, two of them, swam away,

00:00:26.983 --> 00:00:29.383
so how many sharks are left?

00:00:29.383 --> 00:00:31.150
How many sharks are left?

00:00:31.150 --> 00:00:34.450
Well, what's 10 minus
two going to be equal to?

00:00:34.450 --> 00:00:36.817
Well, we could look up
here if we had 10 pictures

00:00:36.817 --> 00:00:39.617
of sharks and we were to take away,

00:00:39.617 --> 00:00:42.750
we were to take away one and then two,

00:00:42.750 --> 00:00:44.750
how many are going to be left?

00:00:44.750 --> 00:00:47.883
Well, we see we have one, two, three,

00:00:47.883 --> 00:00:52.382
four, five, six, seven, and eight.

00:00:52.382 --> 00:00:56.417
So, 10 minus two, 10
taking away two sharks,

00:00:56.417 --> 00:00:58.418
the two sharks that swam away,

00:00:58.418 --> 00:01:01.464
is going to be equal to eight.


Spots on a die

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.496
- [Voiceover] If you up all of the spots

00:00:02.496 --> 00:00:05.863
on these dominoes, there are 63 spots.

00:00:05.863 --> 00:00:08.208
So if you counted all of these dots,

00:00:08.208 --> 00:00:10.193
they're telling us that would be 63.

00:00:10.193 --> 00:00:11.597
So we don't have to do that.

00:00:11.598 --> 00:00:15.231
And then they tell us,
that is 42 more spots

00:00:15.231 --> 00:00:18.794
than the total number of
spots on a six sided die.

00:00:18.819 --> 00:00:21.430
So they're saying if you
took a six sided die,

00:00:21.431 --> 00:00:25.318
so a six sided die, might
look something like this.

00:00:25.354 --> 00:00:27.409
It's a cube, it might have one dot

00:00:27.409 --> 00:00:29.974
and then it might have
three and then over here

00:00:29.974 --> 00:00:33.271
it might have maybe two
and so if you count up

00:00:33.271 --> 00:00:35.871
all of these dots, the ones we see

00:00:35.871 --> 00:00:38.019
and then the ones on
the other side as well,

00:00:38.019 --> 00:00:40.536
that's the number of
spots on a six sided die.

00:00:40.887 --> 00:00:43.725
Now what they're saying
is that the 63 spots,

00:00:43.725 --> 00:00:44.855
they said if you add up all the spots

00:00:44.855 --> 00:00:47.409
on these dominoes there is 63.

00:00:47.409 --> 00:00:49.998
And they're saying that is 42 more spots

00:00:49.998 --> 00:00:53.757
than the total number of
spots on a six sided die.

00:00:55.457 --> 00:00:57.431
Let's now figure out how many spots

00:00:57.431 --> 00:00:59.891
are on a six sided die.

00:01:00.554 --> 00:01:03.108
The first thing you should
always ask yourselves

00:01:03.108 --> 00:01:04.931
is what they're asking for.

00:01:04.931 --> 00:01:08.437
Is this going to be more or less

00:01:08.437 --> 00:01:12.335
than the number of spots that
you have on these dominoes?

00:01:12.523 --> 00:01:14.798
You might say, "Hey, I'm reading 42 more

00:01:14.798 --> 00:01:18.697
"I see this word more, there
must be 42 more spots."

00:01:18.838 --> 00:01:21.717
But you have to be very,
very, very, very careful.

00:01:21.717 --> 00:01:22.994
Whenever you do a word problem,

00:01:22.994 --> 00:01:24.515
don't just look for words and say

00:01:24.515 --> 00:01:27.168
"Oh it's more I'll add or if
it's fewer I'll subtract."

00:01:27.168 --> 00:01:29.594
Because that's not going
to be the case always.

00:01:29.594 --> 00:01:31.532
You really need to think about

00:01:31.533 --> 00:01:34.190
what is the problem asking for?

00:01:34.273 --> 00:01:38.313
Because they're saying that
this 63 dots right over here

00:01:38.313 --> 00:01:41.795
that is 42 more than the
total number of spots

00:01:41.795 --> 00:01:43.676
on a six sided die.

00:01:43.676 --> 00:01:45.951
Another way of thinking about it is,

00:01:45.951 --> 00:01:48.308
if you took the number of
spots on a six sided die

00:01:48.308 --> 00:01:50.467
that's what they're asking for.

00:01:50.467 --> 00:01:52.592
So if we say that's question mark.

00:01:52.592 --> 00:01:56.167
If you took the number of
spots on a six sided die

00:01:56.167 --> 00:02:01.167
and you had 42 more than that,

00:02:01.496 --> 00:02:03.086
then that is going to be equal to

00:02:03.086 --> 00:02:04.955
the number of spots on these dominoes

00:02:04.955 --> 00:02:07.972
and they tell us that is 63.

00:02:08.136 --> 00:02:12.733
The number of spots on a
six sided die plus 42 is 63

00:02:12.733 --> 00:02:17.609
or 63 spots is 42 more
spots than the total

00:02:17.609 --> 00:02:19.652
number of spots on a six sided die.

00:02:19.652 --> 00:02:21.730
Always think about what they're saying.

00:02:21.730 --> 00:02:23.471
So now we can try to figure out

00:02:23.472 --> 00:02:25.828
what this question mark is.

00:02:25.828 --> 00:02:29.206
Well if question mark
plus 42 is equal to 63,

00:02:29.206 --> 00:02:32.430
that's the same thing as
saying that question mark

00:02:32.431 --> 00:02:36.608
is equal to 63,

00:02:37.411 --> 00:02:40.391
63 minus 42.

00:02:43.840 --> 00:02:45.110
So what's that going to be?

00:02:45.110 --> 00:02:46.933
We can calculate it.

00:02:46.933 --> 00:02:50.393
So 63, six tens three ones.

00:02:50.393 --> 00:02:54.141
42, four tens and two ones.

00:02:54.141 --> 00:02:56.498
We want to subtract that, and when we do

00:02:56.498 --> 00:02:58.762
we're going to figure
out the number of spots

00:02:58.762 --> 00:03:00.919
on the six sided die.

00:03:01.246 --> 00:03:03.220
We'll go to ones place, three ones

00:03:03.220 --> 00:03:06.064
minus two ones is one one.

00:03:06.064 --> 00:03:10.046
Six tens minus four
tens, those are two tens.

00:03:10.046 --> 00:03:13.319
So two tens and one one that's 21 spots.

00:03:13.319 --> 00:03:18.319
21 spots on a six sided die.