WEBVTT

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>> Arlene Mitchell: A ratio
is a cognitive process.

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It's not a writing process.

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Just because you give them
a problem and you ask them

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to write the ratio does not mean
that they understand a ratio.

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When you are asking them to work
with a ratio, you are asking

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for them to pull
apart attributes

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that are being described and
how those characteristics

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of that attribute
relate to one another.

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When we look at this
idea of ratios,

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a lot of times it connects
to the idea of fractions,

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and it links to it through
a lot of different ways.

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They share the same notation;
fractions are ratios,

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but not all ratios
are fractions.

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As you look at this,
it's the building block

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because we are trying to get
kids to think proportionally.

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You have to start
with ratios to be able

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to build up to proportions.

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You need to give kids a lot of
experiences with ratios in a lot

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of different contexts,
which is saying,

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make yourself a little
bit more familiar

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with the different
types of ratios.

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One of them is that
part-to-whole.

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The part-to-whole makes it look

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like it is fraction;
it is a fraction.

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And as you are doing
the fraction piece here,

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you are comparing two
measures of the same type.

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So some of the examples there,
the nonfiction books to all

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of the books in the library.

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When you set that up as
a fraction, as a ratio,

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you have the part
that is nonfiction,

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but the nonfiction is
also included in all

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of the books in the library.

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Another one would be
part-to-part, and this truly,

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then, is a ratio when you
look at it as part-to-part.

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You are expressing one
part to another part,

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but they are of the
same measure.

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So to follow along with
the same type of examples,

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it would be nonfiction
books to fiction books.

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They are still books, but
you are classifying it

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as either nonfiction or
fiction to do the comparison.

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Another one that comes into
play is ratios as rate.

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When you have a ratio
that is a rate,

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you now are putting
together two measurements,

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and those measurements
can be different.

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So it can be two balloons
for three dollars.

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The balloons and the dollars
are the two different units.

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Miles per gallon, inches per
feet-all of those become a rate,

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and a rate is a comparison

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of two different
types of measurements.

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So here is the one I would
like for you guys to work on,

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and I do have this one
written up for you.

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So the idea for this one
is, come up with a unit rate

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for the two camps that are
dividing up the pizzas.

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This comes back then to that
sharing, and you are looking

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at it from the idea of
which one of them is going

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to give you the better deal.

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If you start off in
looking at the Bear Camp,

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what's one of the unit rates
that we might end up with?

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And unit rate means that we
have a one, and then it means

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that label becomes critical.

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>> Teacher 1: One camper
gets two-thirds of a pizza.

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>> Mitchell: And
how did you do that?

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>> Teacher 1: Because three
two-thirds equals two pizzas.

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So one is equal to
two-thirds of a pizza.

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>> Mitchell: All right, so
what's it going to be over here?

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One camper to how many pizzas?

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>> Teacher 1: Three-fifths
of a pizza.

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>> Mitchell: All right, if
you were to judge right now,

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who's got the better deal
when it comes to the pizza?

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Is it the Bear Camp or
is it the Raccoon Camp?

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>> Teacher 1: The Bears.

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>> Mitchell: Why the Bears?

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>> Teacher 1: Because
they get 66 percent of the pizza,

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and the Raccoons only
get 60 percent of a pizza.

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>> Mitchell: Now, that's not
the only way to think about it

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because as soon as-this
is the way you're thinking

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about it-you know you've got
that kid in the back of the room

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who goes, "But I want to
think about it as one pizza."

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If it's one pizza,
how many campers will

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that feed for the Bear Camp?

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>> Teacher 1: One and a half.

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>> Mitchell: And I'll write
it as three-halves campers.

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On this one over here, one
pizza will feed how many?

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>> Teacher 2: Three-fifths?

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>> Other teachers in
background: Five-thirds.

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>> Mitchell: Which one is it?

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>> Other teachers: Five-thirds.

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>> Mitchell: Okay, this is the
one that's going to become one,

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so I have to divide
both sides by the three.

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So five-thirds of the campers.

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Which one is the better deal?

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>> Teachers: The Raccoons.

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>> Mitchell: Raccoons?

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This one is for one
and a half campers.

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One pizza will go for every
one and a half campers.

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Over here, one pizza
will go for every one

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and two-thirds campers.

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This is where you really
have to stop and think

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about what's the relationship

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and where do I answer
the question from.

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It's not just going
for the bigger number,

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but it's the relationship that's
existing within the problem.

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This is the first
problem on the sheet.

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It sets it up as saying, "A
person who weighs 160 pounds

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on Earth will weigh 416
pounds on the planet Jupiter.

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How much will a person
weigh on Jupiter

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who weighs 120 pounds on earth?"

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In setting it up, all I need
to know is the relationship

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that exists from the problem.

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In this case, the Earth
weight of 160 and it's given

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that Jupiter, then,
I would weigh 416.

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Now, all I am going to do is use
that table, and I am just going

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to start to think of
it multiplicatively.

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I am also going to keep
in mind where I need

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to end up, 120 for what?

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For Earth.

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One way I could think about this
is this: What if I just went

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and took 160 and the 416
and divided it by two?

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Is my relationship the same?

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What you do to one,
do it to both.

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You've still got that
same relationship.

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All right, I'm at 80.

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If I go down to 40, which
is, again, dividing by two.

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The 80 divided by two.

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The 208 divided by two.

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Where do I want to be?

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>> Teacher 3: 120.

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>> Mitchell: 120.

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Anything jumping out at you?

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>> Mitchell: Add the what?

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>> Teacher 1: 80 and 40 is 120.

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Add the 208 and the 104,
then you have 312...

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>> Mitchell: I elected
to multiply, but...

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there's yours.

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>> Teacher 4: Or
the 160 minus 40.

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>> Mitchell: Oh, nice.

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160 get to here to the one 40.

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Subtract these two, and that
would also give you your 120.

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>> Teacher 4: 16 minus 104 312.

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>> Mitchell: Nice.

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The idea of this is that
there isn't just one way

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to complete the chart.

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But the multiplicative
comparison is based off

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of that unit rate, and it allows
you to use that unit rate to do

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that comparison, but
the questions are going

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to look different than what
you have had from before.

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All right, so that's one that
we will build on for tomorrow

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as we start to look at this
thing called proportion.