WEBVTT

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[Music] Welcome to the
overview on Making Sense

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of Computational Procedures.

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There is a growing consensus about
the benchmarks that elementary

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and middle school students must
achieve in order to be proficient

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in mathematics and ready for algebra.

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Several of those benchmarks
involve computation with fractions.

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In the intermediate grades
students should master addition

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and subtraction of fractions
and decimals.

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In the middle grades, they should
become proficient in multiplying

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and dividing fractions and decimals.

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Research shows that when students
understand why procedures work,

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they are more likely to
become fluent in computation.

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Teaching fractions by connecting
conceptual understanding

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with procedural fluency will
strengthen student learning.

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Conceptual understanding of fraction
operations can be developed by:

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Using visual representations
to demonstrate concepts,

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Presenting problems in
real-world situations,

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Giving students opportunities
to use estimations

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and make reasonable predictions,

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and Directly addressing
common misconceptions.

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Teachers can use different types of
visual representations and manipulatives

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to help students gain insight into
why computational procedures work.

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For example, area models like fraction
circles can help students see the need

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for common denominators
when adding fractions.

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Here, we see a visual representation
that shows addition of fractions

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with different denominators by
identifying a common denominator

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and translating the problem
into equivalent fractions.

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Pictorial representations can
help students grasp the concept

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of multiplying fractions.

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In this example, the challenge is to
decide how much of a cake can be frosted

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with one-fourth of a
cup of icing if one cup

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of icing covers only
two-thirds of the cake.

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A pictorial representation allows
students to partition the cake

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from one whole, to two-thirds,
to one-fourth of two- thirds.

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In short, to find a fraction
of a fraction.

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Simple ribbons or strips
of paper can be useful

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in demonstrating division
with fractions.

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In this problem, students cut
same-length ribbons into fourths

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and halves to find what is
one-half divided by one-fourth.

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Two-fourths fit into one-half

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of the ribbon...so one-half
divided by one-fourth is two.

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Number lines are the representational
tool of choice for many educators

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because they offer flexibility
with all types of fractions,

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including improper fractions
and negative fractions.

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The overview on Recognizing Fractions

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as Numbers includes more
information about number lines.

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Problems such as frosting
a cake or measuring and

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cutting ribbons help students
see how computation is used

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in real-life situations
and allows them to build

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on their own intuitive
understanding of fractions.

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Students can strengthen their reasoning
about fractions when they use estimation

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to predict or judge the reasonableness
of an answer to a problem.

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Teachers can provide opportunities
and discuss strategies for students

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to make estimates and compare those
estimates to their completed solutions.

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Estimations help students
improve their reasoning

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and make more accurate predictions.

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Often students hold misconceptions
about fractions that get in the way

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of understanding or learning
computational procedures.

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The most common misconceptions
involve treating fractions

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as if they were whole numbers
or misapplying a procedure

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from a different fraction operation.

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A common mistake students make when
adding or subtracting fractions is

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to add or subtract the numerators
and denominators separately.

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Meaningful contexts can help students
recognize their misunderstanding.

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If a student has a piece of wood
that is three- fourths of a foot long

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and cuts off a piece that is half a
foot long, but ends up with one foot

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of board, he can clearly
see that he's made a mistake

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and needs to rethink his process.

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Adding and subtracting fractions
requires a common unit fraction.

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Not recognizing that different
denominators indicate different-sized

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unit fractions leads to
many common misconceptions,

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such as adding only numerators
in an equation.

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Use of fraction strips or number lines
can help students see how different

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denominators indicate different
fractional parts of the whole.

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Another mistake can occur when students
take what they've learned about addition

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of fractions and apply
it to multiplication.

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Pictorial representations can
help students see the meaning

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of two- thirds of one-third.

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Students quite often have
difficulty with the "invert

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and multiply" procedure
for dividing fractions.

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The problem arises when students
are taught this procedure

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without an understanding
of why the procedure works.

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Students are less likely 
to make errors if

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the steps involved
are clearly explained.

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They need to see that
multiplying a number

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by its reciprocal yields
a product of one.

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Then they need to understand
that dividing a number

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by one doesn't change the number.

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Once this is clear, teachers
can show that "invert

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and multiply" involves
multiplying both fractions

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by the reciprocal of the divisor.

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Teachers should feel
comfortable taking the time needed

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to build students' conceptual
understanding

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as they also teach the procedures for
performing operations with fractions.

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Representations, estimations,
and real-world contexts will help

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to strengthen students'
grasp of fractional concepts.

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Most important, teachers should
expect students to make mistakes

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in computations with fractions.

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These mistakes can offer a window into
students' thinking and an opportunity

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to use mistakes in learning.

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To learn more about Making Sense
of Computational Procedures,

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please explore the additional resources
on the Doing What Works website.