WEBVTT

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[Music]

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I'm Jim Lewis.

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I'm a professor of mathematics

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at the University
of Nebraska-Lincoln.

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Over the past year, year
and a half, I have worked

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with the fractions panel

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to create a fractions practice
guide for Doing What Works.

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Most of us buy orange juice
by buying frozen concentrate.

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Almost always the instructions
say use one can of concentrate

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and three cans of water.

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What if we wanted to
make enough orange juice

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that we use four
cans of concentrate?

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Or let's suppose we started
with 15 cans of water,

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and how much concentrate
should we put in to go with it?

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Or what if you want to make
12 cans of orange juice?

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How much concentrate
should you use?

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How much water should you use?

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Let's work through a problem
that involves a recipe,

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so that's something
that involves a ratio.

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Perhaps we want three
cups of flour and one

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and one-quarter cups of water.

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We might start analyzing
the situation

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by building a ratio table.

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So we log in three
cups of flour and one

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and a quarter cups of water.

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If we make it again,
that's six cups of flour two

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and a half cups of water, then
nine cups of flour and three

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and three-quarter cups of water.

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If we by chance were asked about
five cups of water matching

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up with 12 cups of
flour, that would show

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up in our ratio table.

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What if we wanted to
use ten cups of flour?

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Then our ratio table
is not going to all

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that automatically
give us an answer.

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We could adapt our ratio table
doing what we call "going by way

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of one" to put in one here.

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That forces us to think.

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We have divided three
by three to get one.

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Let's divide one and a
quarter, or five-fourths,

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by three to get five-twelfths.

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Then if we say this is a
multiple of ten to get to here,

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we want to multiply by ten to
decide what goes right there:

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5/12 times 10 = 50/12
= 4 and 1/6.

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Using our ratio table, we
are working with proportions.

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We could do the same
thing with a number line.

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We could put flour here, water,
zero is our starting point.

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And we might locate
three- three cups of flour,

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one and one-quarter
cups of water.

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Now, the interval
from zero to three,

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we think of breaking it apart:
Here is two; here is one.

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And so, to stay in proportion
we have got to take this amount

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of water-whatever number we
put right here-take one-third

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of it right here.

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If this is five-fourths,
we want five-twelfths here.

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So out here in our number
line somewhere we find ten,

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and we are interested in
what number goes with it.

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We see a multiple of ten here,
we need a multiple of ten here.

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We got fifty-twelfths,
five times ten being 50,

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so that we know that
our answer is going

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to be four and one sixth.

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Now both of these situations
are essentially the same

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as the ratio: three is to one
and one-quarter as ten is to X.

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But it would really
help if we put in a unit

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so that we know we set
up the correct ratio:

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flour to water, flour to water.

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The students need to see
one problem worked in more

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than one way so that they see
that both techniques work.

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They are going to gain a sense

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of which problem works-which
approach works best

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for them most often,
but it's never going

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to be a completely clean
strategy, "In this situation,

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you should always
do the following."

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And that's part of the
nature of mathematics,

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that it involves the reasoning
and judgment, and so the person

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with more experience is
going to do a better job

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of picking the better
tool out of their toolbox.

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[Music]