Math Facilitator Meeting
January 17, 2013
Multiplication and Division
of Fractions and Decimals
Session 1
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How do we help students develop conceptual
understanding of operations with decimals
and fractions?
How does our work with multiplication and
division of whole numbers relate to decimals
and fractions?
What is “flexibility” with fractions and
decimals?
Why is flexibility in working with decimals
and fractions important for solving problems?
Look at the fraction standards from grades
1-5
 What standards are new at each grade level?
 With a partner, make a list of the concepts
that should be mastered before learning to
reason with multiplying and dividing decimals
and fractions
 What other standards are important in
building that relationship?
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864,352.79
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What place value understanding do students
need when describing this number?
Write the number in expanded form
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Multiplication does not always make things
bigger

Multiplication is not “just” repeated
addition
The meaning of “times” 3 x 4 = 4 x 3. Are
they the same? (think about groups)
 Is 3/4 of a group of 3 the same as 3
groups of 3/4?
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Translating multiplication expressions 5 x 6
could be 5 groups of 6 or 5 taken 6 times.
We need pictorial representations when it
comes to fractions!!- the idea of 1/2 taken
1/4 times makes no sense. 1/2 a group of
1/4 makes more sense.
If students can connect multiplication
equations to real things, it will help them
make sense of problems
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Students shouldn’t be focused on just the
numbers, but make sense of the magnitude
of the fractions. Example: 3 1/2 x 3 1/2
The answer can’t be more than 4 x 4 or less
than 3 x 3.
There is a real connection between
multiplication and division of fractions (they
are not just opposites)
Example: 10 x 1/2 is the same thing as 10 ÷ 2
and 10 ÷ 1/2 is the same thing as 10 x 2
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Understand a fraction 1/b as the quantity
formed by 1 part when a whole is
partitioned into b equal parts;
Understand a fraction a /b as the quantity
formed by a parts of size 1/b
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A unit fraction is a proper fraction with
a numerator of 1 and a whole number
denominator
1
5
to
is the unit fraction that corresponds
2
5
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or to
3
5
or to
17
5
As there are 3 one-inches in 3 inches, there
are 3 one-eighths in
3
8
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Unit fractions are formed by partitioning a whole
into equal parts and naming fractional parts with
unit fractions 1/3 +1/3 = 2/3
1/5 + 1/5 + 1/5 = ?
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Unit fractions are the basic building blocks of
fractions, in the same sense that the number 1
is the basic building block of whole numbers
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We can obtain any fraction by combining a sufficient
number of unit fractions
1
b
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If the yellow hexagon represents one whole,
how might you partition the whole into
equal parts? Name the fractional parts with
unit fractions
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Name the unit fractions that equal one whole
Hexagon
1/3
1/2
1/3
1/6
1/6
1/6
1/6
1/6
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Two yellow hexagons = 1 whole
How might you partition the whole into equal parts?
Name the unit fraction for one triangle; one hexagon;
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What is the value of the red trapezoid, the green
triangle and the yellow hexagon?
Show and explain your answer
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What part is red?
15
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If the blue rhombus is ¼, build the whole.
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If the red trapezoid is 3/8, build the whole.
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Find the missing values.
1¾
n
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1¾
n
n
x
1½
Figures that are the same size and shape must have the same value.
Adapted from Wheatley and Abshire, Developing Mathematical Fluency, Mathematical Learning, 2002
17
 How
many different ways
can you model 5/4?
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Understand a fraction a/b as a multiple of 1/b
5
1
is the product of x
4
4
5 ( )
5
4
=5x
1
4
Understand
a multiple of a/b as a multiple of 1/b,
and use this understanding to multiply a fraction
by a whole number
3 sets of
2
5
is the same as 6 sets of
1
5
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How much is shaded?
◦ How could you name the amount as a fraction?
◦ As a whole number and a fraction?
(¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ ) = 15/4
(4 x 1/4) + (4 x 1/4) + (4 x 1/4) + (3 x 1/4) = 15/4
4/4
1
+
+
4/4
1
+
+
4/4
1
+ ¾
+ ¾
= 15/4
= 3¾
21
Young Mathematicians at Work:
How is multiplication and division connected to
fractions?
What is meant by “there are two wholes when
dividing fractions?”
The cafeteria made lunches for the fourth graders
going on a field trip. They were in four different
groups so the number of sandwiches differed. The
sandwiches were all the same size.
Group
Group
Group
Group
One had 4 students sharing 3 subs
Two had 5 students sharing 4 subs
Three had 8 students sharing 7 subs
Four had 5 students sharing 3 subs
Did each student get a “fair share?”
If not, which group ate the least? Most? How
do you know?
Next trip we want to guarantee that each
student will receive 2/3 of a sub
Using large paper, create a chart for the
cafeteria to help them know how many subs
to make for up to 15 students
 What patterns do you notice?
 What strategy could cafeteria workers use for
any number of students?
 If you knew there were 8 subs made, how
could you figure out how many students
could each get 2/3 sub?
Model this situation using numbers and symbols.
5 x 1/3
•Write a story problem that matches this
expression
•Solve the problem using two different
strategies
 Write
a story problem that
matches this expression
 Solve
the problem using two
different strategies
Nicholas is helping to paint a wall at a park near his
house as part of a community service project. He had
painted half of the wall yellow when the park director
walked by and said, This wall is supposed to be
painted red.”
Nicholas immediately started painting over the yellow
portion of the wall. By the end of the day, he had
repainted 5/6 of the yellow portion red. What
fraction of the entire wall is painted red at the end of
the day?
www.Illustrativemathematics.org
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Read the commentary that goes
with this task.
How does the pictorial
representation help make sense
of the problem?
How can you prove the following:
5 ÷ 2/3 = 5 x 3/2
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Keep the following guidelines in mind when developing
computational strategies for fractions:
◦ Begin with simple contextual problems
◦ Connect the meaning of fraction computation with
whole-number computation
◦ Let estimation and informal methods play a big role
in the development of strategies
◦ Explore each of the operations with models
(Van de Walle, Karp, & Bay-Williams, 2010, p.310)
We must go beyond
how we were taught
and teach how we
wish we had been
taught.
Miriam Leiva, NCTM Addenda Series, Grade 4, p. iv
Multiplying and Dividing fractions is so easy
when you just use the procedure.
Multiplication: multiply numerator x
numerator and denominator x
denominator.
Division: Just invert the second fraction and
multiply.
So why don’t we just teach it that way?