WEBVTT

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[Music]

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I am Yukari Okamoto, professor
of education at the University

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of California, Santa Barbara.

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I was a member of the
fractions Practice Guide panel.

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One of the problems
students have

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when they are first
introduced to fractions is

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that they think a fraction is
made up of two whole numbers.

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For example, when asked to
add two fractions like, say,

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two-thirds and one-fifth,

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some students add the numerators
together, then the denominators,

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and then come up
with three-eighths.

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Another example of a common
misconception is the students

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think three-eighths is
bigger than three-fifths

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because eight is
bigger than five.

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So why do students have
this kind of misconception?

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Part of the problem is that
there is little emphasis

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on the development of
rational number understanding

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in early grades.

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Fractions are numbers
that can be used

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to measure quantities
just like whole numbers.

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Fractions expand the number
system beyond whole numbers.

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And fractions also provide
more precise units of measure.

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It seems like a simple
idea to adults,

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but it's not for
young students.

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So when teaching fractions, we
often see a pie or area model

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to convey fraction concepts.

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The idea is that a
fraction represents a part

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or parts of a whole.

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In this example, we see a pizza
cut into three equal pieces,

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with two of them
shaded or faded away.

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So if you ate two pieces, then
you ate two-thirds of the pizza.

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This explanation may make sense
to some students but not to all.

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So what happens is that
the students count each

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of the three pieces as if
they are whole numbers.

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So they don't necessarily
realize

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that each part is
one-third of a unit,

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and together they
make the unit of one.

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Here's a more problematic
example.

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Here's a picture
of six cookies

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of different shapes and flavors.

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One is a chocolate cookie, and
the rest are raisin cookies.

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The task is to write a
fraction for the relation

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of the chocolate
cookie to all cookies.

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To me, it's problematic that
all the students need to do

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to succeed is to count
the number of cookies

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and put the resulting whole
numbers in the blanks above

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and below the division marker.

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I am personally concerned
that this type

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of instruction may lead to
students' misconception

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that fractions are made
up of whole numbers.

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Fractions themselves are
numbers with magnitudes

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that extend the number system.

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It's not easy to see
this in this example,

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and this example also
doesn't give you a good sense

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of the unit of measure.

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Is it the single cookie or
the entire set of cookies?

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It's not clear.

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In this example, we want to
show students what it means

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to add two fractions,
one-third and one-third,

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using the part-whole approach.

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We can first shade one-third
and shade another third

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and then come up with
the answer, two-thirds.

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This is a good way to
use a part-whole approach.

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But as you can see below,

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students could interpret
this situation incorrectly.

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They might add these two,

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and they might add the
numerators together

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and the denominators together
and come up with two-sixths.

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How can we help students
develop the idea

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that fractions are numbers?

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We, the Fractions
Panel, strongly recommend

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that teachers use number lines

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as a representational tool to help
students understand important

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fraction concepts.

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I think measurement activities
are particularly useful

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for this purpose.

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Here, I have some
strips of paper.

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Using something like
this, or fraction strips,

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we can find out how many
strips of paper we need

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to measure different things.

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Here is a pencil.

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Let's measure this
using these paper strips.

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And this pencil is
longer than one strip

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but shorter than two strips.

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How can we express
this extra amount?

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The class can discuss various
methods for doing this.

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Folding this strip like
this works in this case.

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Then the teacher can
talk about the notion

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of one and a half strips.

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When doing this sort of
activity, it's important

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that teachers use strips

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of different length representing
different units of measure.

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Ask the students to measure an
object using different strips.

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The length of the object could
be expressed differently, say,

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one-and-a-half blue
strips in this case,

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or if you are using red
strips of different length,

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it could be three
red strips long.

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This type of activity should
help students become aware

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of the importance of
measurement units.

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The number line is also useful

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in helping students understand
what it means to add fractions.

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The addition of fractions like
1/3 plus 1/3 could be introduced

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like this, using
the number line.

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Here, the number line is
partitioned into three parts.

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So the first one-third
is marked here,

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and then you add
another one-third to come

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up with the answer, two-thirds.

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Here, parallel number
lines are partitioned

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into different fractional parts.

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Something like this can also
help students learn equivalence

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of fractions.

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Using a straight edge like
a pencil, students will find

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out one-half is the
same as two-fourths

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and also same as six-twelfths.

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Or you line it up like
this, and students will find

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out one-quarter is the
same as three-twelfths.

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Again, you can line
up like this and see

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that three-fourths is the
same as nine-twelfths.

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By partitioning a
number line repeatedly,

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students realize that
there are an infinite number

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of numbers between any two
adjacent whole numbers.

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This is a very important
notion called fraction density.

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Number lines are also useful

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in helping students translate
among various notations

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of fractions such as
decimals and percents.

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Number lines convey
important properties

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of fractions that other
methods,

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such as the part-whole
approach, do not.

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[Music]