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MARK HARTMAN: Can we grab
the straws, the long straws?

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00:00:24,600 --> 00:00:28,710
I want your group to
measure the length of one

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of these chalkboards-- well, not
chalkboards, but whiteboards.

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Your challenge to start
with is measure the width.

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00:00:39,420 --> 00:00:43,560
And again, this is linear
width, like we're actually going

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00:00:43,560 --> 00:00:44,890
to measure it with a ruler.

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I want you to measure how
wide each of these whiteboards

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00:00:47,670 --> 00:00:50,890
is using this unit.

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00:00:50,890 --> 00:00:52,620
We are going to give
you premade units.

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00:00:52,620 --> 00:00:57,690
They may look like straws, but
they're actually premade units.

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00:00:57,690 --> 00:00:59,460
So what I want you
to do, so I want you

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00:00:59,460 --> 00:01:02,520
to measure the width of each--

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00:01:02,520 --> 00:01:06,910
well, just pick one of these
from one side to the other,

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00:01:06,910 --> 00:01:09,330
and I want you to
measure by counting out.

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00:01:09,330 --> 00:01:12,480
And you can just put the straws
right next to each other.

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00:01:12,480 --> 00:01:17,060
I want you measure how wide
each of those boards are.

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00:01:20,310 --> 00:01:27,740
Measuring linear
width of a whiteboard.

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00:01:31,410 --> 00:01:35,340
All right, so what did you guys
have as your width measurement

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00:01:35,340 --> 00:01:37,005
using long straws?

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00:01:37,005 --> 00:01:38,169
AUDIENCE: 12 and 1/2.

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00:01:38,169 --> 00:01:39,210
MARK HARTMAN: 12 and 1/2.

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00:01:39,210 --> 00:01:50,340
So width equals
12.5 long straws.

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00:01:50,340 --> 00:01:53,339
What was your width
measurement in short straws?

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00:01:53,339 --> 00:01:53,880
AUDIENCE: 37.

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00:01:53,880 --> 00:01:54,660
MARK HARTMAN: 37.

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00:01:54,660 --> 00:02:06,240
So width equals also
37 short straws.

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00:02:06,240 --> 00:02:08,759
These are two observations.

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00:02:08,759 --> 00:02:13,590
You made a measurement, which is
measurements are observations.

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00:02:13,590 --> 00:02:16,164
You could say that object is
red-- that's an observation.

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00:02:16,164 --> 00:02:17,580
But you could also
say that object

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00:02:17,580 --> 00:02:21,100
is 12 and 1/2 centimeters wide.

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00:02:21,100 --> 00:02:22,230
These are two observations.

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00:02:25,589 --> 00:02:26,880
Somebody throw this out for me.

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00:02:30,520 --> 00:02:32,610
Thank you.

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00:02:32,610 --> 00:02:34,220
Don't like those.

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00:02:40,800 --> 00:02:43,770
I'm going to give you a model.

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00:02:43,770 --> 00:02:46,950
Think of converting
between units

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00:02:46,950 --> 00:02:51,820
as using a model of how these
two units relate to each other.

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00:02:51,820 --> 00:02:58,950
I'm going to tell you
that one long straw is

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00:02:58,950 --> 00:03:04,125
equal to three short straws.

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00:03:06,990 --> 00:03:09,870
And again, this is my
number and this is a unit.

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00:03:09,870 --> 00:03:13,860
The unit is long straw, just
like inches or centimeters.

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00:03:13,860 --> 00:03:18,180
It just happens to be
called a long straw.

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00:03:18,180 --> 00:03:20,070
And I'm telling
you that for a fact

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00:03:20,070 --> 00:03:23,250
we know that that's equal in
length to three short straws.

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00:03:23,250 --> 00:03:26,580
So this is my model.

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00:03:26,580 --> 00:03:28,980
It tells me how
those two quantities

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00:03:28,980 --> 00:03:32,400
relate to each other.

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00:03:32,400 --> 00:03:35,970
I can use this model to make
some predictions, right?

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00:03:35,970 --> 00:03:41,020
We actually just observed
this relationship,

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00:03:41,020 --> 00:03:44,490
but we didn't have to go through
and measure the 37 short straws

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00:03:44,490 --> 00:03:47,470
if we had known this
in the first place.

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00:03:47,470 --> 00:03:48,640
Let's check it out.

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00:03:48,640 --> 00:03:51,950
Here's how I'm going to want
you to do every factor label

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00:03:51,950 --> 00:03:54,450
conversion, unit conversions--
we're going to do it in a way

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00:03:54,450 --> 00:03:56,220
called factor label conversion.

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00:03:56,220 --> 00:03:57,550
I'll tell you why.

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00:03:57,550 --> 00:04:04,230
So if I said I wanted to
start with my width equal to--

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00:04:04,230 --> 00:04:06,690
and you're going to do
this all on one line,

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00:04:06,690 --> 00:04:09,210
and then you're going to
simplify on each line below.

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00:04:09,210 --> 00:04:18,570
If I start with
12.5 short straws,

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00:04:18,570 --> 00:04:22,110
now I'm going to multiply by
a number that's equal to 1,

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00:04:22,110 --> 00:04:24,814
because I don't want to change
the actual quantity that I'm

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00:04:24,814 --> 00:04:26,730
working with, I just
want to change the units.

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00:04:26,730 --> 00:04:28,940
AUDIENCE: So you
multiply by 3, right?

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00:04:28,940 --> 00:04:30,360
Multiply 12.5 by 3?

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00:04:30,360 --> 00:04:32,280
MARK HARTMAN: You could
multiply 12.5 by 3,

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00:04:32,280 --> 00:04:35,400
and that gets you
to about 37, right?

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00:04:35,400 --> 00:04:37,450
But that's not how
we're going to do it.

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00:04:37,450 --> 00:04:39,420
We want you to not
just multiply numbers.

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00:04:39,420 --> 00:04:41,400
Numbers don't mean anything.

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00:04:41,400 --> 00:04:43,420
I'm going to say
that once, and I want

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00:04:43,420 --> 00:04:44,940
you to listen to it forever.

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00:04:44,940 --> 00:04:47,830
Because anytime you're
talking about a quantity,

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00:04:47,830 --> 00:04:50,761
the number and the units
are equally important.

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00:04:50,761 --> 00:04:52,260
So any time you
write down a number,

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00:04:52,260 --> 00:04:54,630
if you don't write down
units, I will yell at you,

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00:04:54,630 --> 00:04:57,300
these guys will yell at
you, we will call you out.

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00:04:57,300 --> 00:05:00,030
Because in science, there's
no such thing as just

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00:05:00,030 --> 00:05:00,900
a plain number.

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00:05:00,900 --> 00:05:03,540
If there is, we'll tell you.

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00:05:03,540 --> 00:05:10,770
So I'm going to multiply by a
fraction that is equal to 1.

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00:05:10,770 --> 00:05:12,440
This is a quantity.

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00:05:12,440 --> 00:05:15,655
I want to multiply by a
fraction that is equal to 1,

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00:05:15,655 --> 00:05:19,290
and I want to obtain a unit.

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00:05:19,290 --> 00:05:22,500
I want to obtain the same
quantity in units of--

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00:05:22,500 --> 00:05:25,300
oops, that's not
12.5 short straws--

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00:05:25,300 --> 00:05:29,070
12.5 long straws.

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00:05:29,070 --> 00:05:32,440
And I want to end up
with short straws.

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00:05:36,250 --> 00:05:39,580
So think about it in this way--

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00:05:39,580 --> 00:05:42,580
if it helps you, you
can make each one

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00:05:42,580 --> 00:05:43,900
of these factors a fraction.

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00:05:43,900 --> 00:05:49,335
You can say 12.5 long
straws divided by 1 times--

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00:05:49,335 --> 00:05:52,360
let me move my times
down to the middle.

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00:05:52,360 --> 00:05:55,390
I want to be able to
cancel out long straws

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00:05:55,390 --> 00:05:58,090
on the top and the bottom.

104
00:05:58,090 --> 00:06:02,170
And I want my answer
to be in short straws.

105
00:06:02,170 --> 00:06:05,320
If I know that this
number, this quantity,

106
00:06:05,320 --> 00:06:10,360
is equal to that quantity,
I can make two fractions

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00:06:10,360 --> 00:06:11,710
that are equal to 1.

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00:06:11,710 --> 00:06:15,820
I can say-- again,
this is our model--

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00:06:15,820 --> 00:06:26,170
I could say one long straw
over three short straws,

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00:06:26,170 --> 00:06:29,530
that fraction is
equal to the number 1.

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00:06:29,530 --> 00:06:32,390
Because this quantity is
the same as that quantity.

112
00:06:32,390 --> 00:06:36,190
The numbers are different,
but the number and the unit

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00:06:36,190 --> 00:06:39,100
together, one long straw, that's
a physical measurement, that's

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00:06:39,100 --> 00:06:41,650
the same thing as
three short straws.

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00:06:41,650 --> 00:06:50,470
I can also say that one is also
equal to three short straws

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00:06:50,470 --> 00:06:52,240
over one long straw.

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00:06:55,330 --> 00:06:59,890
That's like saying this is
still the same quantity.

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00:06:59,890 --> 00:07:02,230
Three short straws,
bing bang boom.

119
00:07:02,230 --> 00:07:04,261
One long straw, bang.

120
00:07:04,261 --> 00:07:05,260
Still the same quantity.

121
00:07:05,260 --> 00:07:09,520
So the same quantity divided
by the same quantity, these

122
00:07:09,520 --> 00:07:11,410
are all equal to 1.

123
00:07:11,410 --> 00:07:14,340
So I'm not changing what
I'm working with over here,

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00:07:14,340 --> 00:07:16,810
I'm just changing the units.

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00:07:16,810 --> 00:07:17,740
So I can choose.

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00:07:17,740 --> 00:07:21,970
Do I want to multiply that
number by a fraction that

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00:07:21,970 --> 00:07:22,600
represents 1?

128
00:07:22,600 --> 00:07:26,830
Do I multiply by this one or
do I multiply by that one?

129
00:07:26,830 --> 00:07:30,580
If I want short straws
to come out on top,

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00:07:30,580 --> 00:07:36,070
we're going to want
to use this fraction.

131
00:07:36,070 --> 00:07:40,590
Again, I'm going
to throw that out.

132
00:07:40,590 --> 00:07:44,100
So I want to say
three short straws--

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00:07:46,820 --> 00:07:49,200
I want you guys to
write this down--

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00:07:49,200 --> 00:07:52,470
over one long straw.

135
00:07:56,440 --> 00:07:59,520
So I follow my work over.

136
00:07:59,520 --> 00:08:01,897
I've got long
straws divided by 1.

137
00:08:01,897 --> 00:08:03,730
That's just to make it
look like a fraction.

138
00:08:08,620 --> 00:08:10,214
Let's write the whole thing out.

139
00:08:10,214 --> 00:08:11,630
If you multiply
two fractions, you

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00:08:11,630 --> 00:08:13,790
write the top times the top.

141
00:08:13,790 --> 00:08:27,110
So 12.5 long straws
times three short straws,

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00:08:27,110 --> 00:08:33,030
divided by 1, times
long one long straw.

143
00:08:36,470 --> 00:08:38,809
All I did was I
multiplied the tops

144
00:08:38,809 --> 00:08:40,340
together, put them together.

145
00:08:40,340 --> 00:08:42,809
Multiply the bottoms
together, put them together.

146
00:08:42,809 --> 00:08:44,990
Make sense?

147
00:08:44,990 --> 00:08:47,000
Now I'm going to
simplify down below.

148
00:08:47,000 --> 00:08:48,890
And when I simplify,
I always just want

149
00:08:48,890 --> 00:08:52,170
to keep my equal signs
in the same place.

150
00:08:52,170 --> 00:08:55,190
And I just want to
simplify this expression.

151
00:08:55,190 --> 00:08:58,910
When you're dealing with
units, units work the same way

152
00:08:58,910 --> 00:09:00,830
that numbers do.

153
00:09:00,830 --> 00:09:05,060
You can cancel if you have
12.5 long straws on top and so

154
00:09:05,060 --> 00:09:07,030
many long straws on the bottom.

155
00:09:07,030 --> 00:09:10,160
I can cancel out
long straws, and I'm

156
00:09:10,160 --> 00:09:13,601
left with-- what
is 12.5 times 3?

157
00:09:13,601 --> 00:09:15,002
AUDIENCE: 37.

158
00:09:15,002 --> 00:09:22,840
MARK HARTMAN: 37
short straws over--

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00:09:22,840 --> 00:09:26,890
1 times 1 is 1.

160
00:09:26,890 --> 00:09:30,760
So that gives me
37 short straws.

161
00:09:30,760 --> 00:09:36,404
That makes sense
with our observation.

162
00:09:36,404 --> 00:09:38,320
That if we measured, if
we actually physically

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00:09:38,320 --> 00:09:42,270
measure them, this makes sense.