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[Music] Welcome to the overview on
The Conceptual Basis for Fractions.

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Students are never too
young to practice working

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with the fundamental ideas
underlying fractions.

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These students in Mrs. Malik's
first-grade class won't be

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formally working with
fractions for a few years,

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but they're dealing with ideas
about fractions all the time.

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Young children come to school
with an informal understanding

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of sharing and proportionality.

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Teachers in the primary
grades can build

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on that informal understanding
to develop fraction concepts.

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Here's how Mrs. Malik is
challenging her first-grade

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students to use what they already 
know to develop fraction concepts.

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One activity involves dividing
groups of objects equally.

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She says to her students,
"Four friends have 12 pretzels.

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They want to share,
but each of them needs

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to get the exact
number of pretzels.

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How many pretzels
should each person get?"

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To figure out the answer,
some children draw pictures;

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others count out
pretzels one by one.

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Regardless of how they
work out the answer,

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they are dividing 12 objects
into four equal shares.

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Mrs. Malik changes the problem
to require different partitioning:

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"Two more of their friends come over.
They want some of the pretzels too.

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How many pretzels does
each person get now?"

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Once Mrs. Malik's students
show that they are comfortable

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with dividing sets equally,
she introduces the new

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challenge of unit fractions.

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She poses a problem that
requires the partitioning

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of a single object: "What
if there is only one apple

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and three children
want to share it?

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How much should each get?"

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The problems get
gradually more complex:

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"What about three
children and four apples?"

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While working through problems
like these, Mrs. Malik begins

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to introduce fraction
names, such as one-third.

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Similar problems can help young
students compare fractions.

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Students can be asked
to try different ways

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to partition objects so that
each receives equal shares.

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Here, one student
breaks both candy bars

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in half while another
breaks each into four pieces

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and groups two pieces
together for each child.

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Children can check to see that 
each approach results

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in equivalent shares.

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There are many variations
on this kind of problem,

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and they can grow
increasingly sophisticated

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as students develop
their understanding

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about the equivalence
of fractions.

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"If four children in one
group share six apples,

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how many apples do
they each get?

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If six children want the
same amount of apples

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as the first group, how many
apples will they need to share?"

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These exercises encourage
students to explore the concept

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of equal partitioning by
using different approaches.

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During such activities,
teachers can use fraction names

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and help children
compare fractional parts.

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"Is one-third of an
apple larger or smaller

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than one-fourth of
the same apple?

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In addition to working
with students on sharing

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and dividing, researchers and
mathematics educators suggest

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that teachers build on children's 
informal understandings

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of proportional relationships.

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This will lay the foundation for
learning more advanced concepts

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of ratio and proportion in
the upper elementary grades.

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Early proportional thinking
can develop as teachers point

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out relationships
between objects that occur

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in children's stories,
such as relationships based

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on size and pattern.

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Or, help children
prepare recipe mixtures

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that involve relationships
among quantities.

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Or, show examples of how
objects vary together--

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in size or length, for example.

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Learning is effective
when it builds

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on children's existing
knowledge, making connections

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to what they already understand
in real-world contexts.

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Given the difficulty that
many older children experience

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with rational numbers, it
is especially important

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to start early to build the
mathematical thinking associated

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with fractions.

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To learn more about The
Conceptual Basis for Fractions,

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please explore the
additional resources

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on the Doing What Works website.