Multiplication
with Fractions
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Monday July 27, 2015
Isn’t this everything
I need to know?
Unfortunately students also extend their
whole number misunderstandings to fractions.
“The rush to tell students how to perform procedures
prevents them from establishing a solid foundation of
operation sense for fractions.
It is time to shift the emphasis and redefine the goal of
fraction instruction in elementary school from learning
computational rules to developing fraction operation
sense.”
– Huinker (2014, 2002)
Operation Sense for Fractions
1. Understand the meanings of operations.
2. Recognize and describe real-world situations for
specific operations.
3. Understand the meaning of symbols and formal
mathematical language.
4. Translate easily among representations.
5. Understand relationships among operations.
6. Compose and decompose numbers and use
properties of operations.
7. Understand the effects of an operation on a pair
of numbers.
(Huinker, 2014, 2002)
Making Sense of
Multiplication with Fractions
Learning Targets
We are learning to…
• Apply and extend our understanding of multiplication
with whole numbers to multiplication with fraction
through use of context (word problem situations) and
use of the underlying mathematical structure.
The Fudge Shop
Sam loves fudge!! While at the Dells,
he bought 3 packages of fudge at the Fudge Shop.
Each package contained 4 pounds of fudge.
How many pounds of fudge did Sam buy?
Morgan also loves fudge!! However,
she didn’t have as much money as Sam.
She bought 3 packages of fudge at the Fudge Shop.
Each package contained 4/5 of a pound of fudge.
How many pounds of fudge did Morgan buy?
Sam loves fudge!! While at the Dells,
he bought 3 packages of fudge at the Fudge Shop.
Each package contained 4 pounds of fudge.
How many pounds of fudge did Sam buy?
Equation:
3
Contextual
Meaning:
Packages of
fudge Sam
bought.
x
4
Pounds of
fudge in each
package.
=
12
Pounds of
fudge in all
the packages
combined.
Morgan also loves fudge!! She bought 3 packages
of fudge at the Fudge Shop. Each package
contained 4/5 of a pound of fudge. How many
pounds of fudge did Morgan buy?
Equation:
Contextual
Meaning:
3
x
=
Packages of
Pounds of
fudge Morgan fudge in each
bought.
package.
Pounds of
fudge in all
the packages
combined.
The Underlying Mathematical Structure
of Multiplication
Cluster: Represent and solve problems involving
multiplication and division.
Standard 3.OA.1. Interpret products of whole
numbers, e.g., interpret 5 × 7 as the total number
of objects in 5 groups of 7 objects each.
Standard for Mathematical Practice #7
“Look for and make use of structure”
Mathematically proficient students look
closely to discern a pattern or structure....
They also can step back for an overview and
shift perspective.
Mathematics has far more consistent
structure than our language, but too often
it is taught in ways that don’t make that
structure easily apparent for students.
Sam loves fudge!! While at the Dells,
he bought 3 packages of fudge at the Fudge Shop.
Each package contained 4 pounds of fudge.
How many pounds of fudge did Sam buy?
Equation:
3
x
4
=
12
Contextual
Meaning:
Packages of
fudge Sam
bought.
Pounds of
fudge in each
package.
Pounds of
fudge in all
the packages
combined.
Structural
Meaning:
Number of
Groups
Size of
Each Group
Total
Amount
Morgan also loves fudge!! She bought 3 packages
of fudge at the Fudge Shop. Each package
contained 4/5 of a pound of fudge. How many
pounds of fudge did Morgan buy?
Equation:
Contextual
Meaning:
Structural
Meaning:
3
x
=
Packages of
Pounds of
fudge Morgan fudge in each
bought.
package.
Number of
Groups
Size of
Each Group
Pounds of
fudge in all
the packages
combined.
Total
Amount
However.... we are living in the world of fractions...
Structural
Meaning:
Number of
Groups
Complete
Groups
Partial
Group
Size of
Each Group
Total
Amount
Professional Reading and
Reflection (PRR)
PRR: A Unified Approach to Multiplying
Fractions (Jack Ott, 1990)
Read the article (only 3 pages) and study the figures.
Ott confirms the “relative difficulty of explaining to
students the meaning of addition and multiplication of
fractions” (p. 47)
Choose one sentence from the article that impacted your
thinking on language difficulty when understanding or
explaining the multiplication of fractions. Re-write the
sentence in your notebook and explain why you selected it.
CCSSM Standards
Standard 3.NF.1
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.
Standard 4.NF.3
Understand a fraction a/b
with a > 1 as a sum of
fractions 1/b.
“6 parts of size one fifth”
Standard 3.NF.1
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.
“6 parts of size one fifth”
Standard 4.NF.3
Understand a fraction a/b
with a > 1 as a sum of
fractions 1/b.
Standard 4.NF.4a
Understand a fraction a/b as
a multiple of 1/b.
= 6x
6
x
1
=
5
I am making 3 batches of bean soup.
Each batch calls for 2/5 cup of dried
beans. How many cups of beans will
I need for the three batches?
1. State what each quantity means in the story.
2. Using your “1/5” measuring cup, demonstrate how to
measure out the correct amount of beans for one batch.
3. Write at least two expressions to represent the amount
of beans needed for one batch of soup.
4. Demonstrate the scoops for 2 batches? 3 batches?
5. Write at least three expressions to represent the amount
of beans needed for 3 batches of soup.
Bean Soup: Keeping Track of Quantity
• How many groups of 2/5 cup did you measure out?
How do you know?
• How many total scoops of 1/5 cup did you measure
out? How do you know?
4.NF.4b. Understand a multiple of a/b as a
multiple of 1/b, and use this understanding to
multiply a fraction by a whole number.
For example, use a visual fraction model to express
3 × (2/5) as 6 × (1/5), recognizing this product as 6/5.
(In general, n × (a/b) = (n × a)/b.)
4.NF.4b. Understand a multiple of a/b as a multiple of 1/b, and
use this understanding to multiply a fraction by a whole number.
For example, use a visual fraction model to express
3 × (2/5) as 6 × (1/5), recognizing this product as 6/5.
(In general, n × (a/b) = (n × a)/b.)
3x
=
Three batches;
2-fifths of a cup
in each group.
3x2
5
= (3 x 2) x
=
Three batches with two Six scoops of
scoops in each batch;
size one-fifth
size of a scoop is oneof a cup.
fifth of a cup.
A Glimpse into a Classroom: Ribbons
Extend Understanding of Repeated Addition to
Multiplication of a Fraction by a Whole Number
https://www.teachingchannel.org/videos/multiplying-fractions-by-whole-numberslesson
Possible Solutions to 4 × ⅔
• How did Ms. Spies create opportunities for
students to share and learn from each other?
• What did students learn by critiquing the
teacher’s solutions?
• How did Ms. Spies help her students further
their understanding of multiplication as
repeated addition?
Moving towards Those
“Partial” Groups
Another Classroom: Fraction Number Talk
https://www.teachingchannel.org/videos/fraction-multiplication-intro-sbac
½ of 20
1/4 of 20
1/5 of 20
3/5 of 20
1/8 of 20
1/7 of 21
3/7 of 21
6/7 of 21
Mental Math
Making Lasagna
In order to encourage her family to eat
more vegetables Melissa decides
to include spinach in her lasagna.
The recipe calls for ¾ pound of spinach
for each batch of lasagna.
Referent Whole:
What quantity is represented by one paper strip?
Size of Each Group:
What quantity comprises one “complete” group?
In order to encourage her family to eat more
vegetables Melissa decides to include spinach
in her lasagna. The recipe calls for ¾ pound of
spinach for each batch of lasagna.
How many pounds of spinach does Melissa
need to make 2 batches of lasagna?
Context: Restate the problem in your own words.
Physical  Visual Model: Use new fraction strips!
Partition as needed, then cut or tear them apart
to just show the amount of spinach in each batch.
Make a sketch to record your reasoning.
If time allows, also solve using a number line.
In order to encourage her family to eat more
vegetables Melissa decides to include spinach
in her lasagna. The recipe calls for ¾ pound of
spinach for each batch of lasagna.
How many pounds of spinach does Melissa
need to make 2 batches of lasagna?
Equation:
Write the equal that models the problem situation.
Label each number with phrases to describe:
(1) Contextual meaning of the numbers.
(2) Structural meaning of the numbers.
A fraction strip represents
1 whole pound of spinach
1 batch of lasagna
uses ¾ of a pound of spinach
1 batch of lasagna
uses ¾ of a pound of spinach
Total of 6 parts of size ¼ of a pound
or 6/4 pounds of spinach.
Janine just called she will be joining them for dinner.
Melissa decides to make 2 ⅓ batches of her lasagna.
Given that the recipe calls for ¾ pound of
spinach for each batch, now how many
pounds of spinach does Melissa need to buy?
1. Context: Restate in your own words.
2. Define the Referent Whole.
3. Describe the Size of Each Group.
4. Use the fraction strips!! (Make a drawing to
record; try a number line if time allows.)
5. Write a “descriptive” equation.
A fraction strip represents
1 whole pound of spinach
1 batch of lasagna uses
¾ of a pound of spinach
1 batch of lasagna uses
¾ of a pound of spinach
⅓ batch of lasagna uses
¼ of a pound of spinach
____
Total of 7 parts of size ¼ of a pound or 7/4 pounds of spinach.
Melissa rethinks her decision. She finally settles on
making 2 ⅔ batches of lasagna. The recipe calls for
¾ pound of spinach for each batch, now how
many pounds of spinach does she need?
1. Context: Restate in your own words.
2. Define the Referent Whole.
3. Describe the Size of Each Group.
4. Use the fraction strips!! (Make a drawing to
record; try a number line if time allows.)
5. Write a “descriptive” equation.
A fraction strip represents
1 whole pound of spinach
1 batch of lasagna uses
¾ of a pound of spinach
1 batch of lasagna uses
¾ of a pound of spinach
2/3 batch of lasagna uses
2/4 of a pound of spinach
____
Total of 8 parts of size ¼ of a pound,
which is 8/4 or 2 pounds of spinach.
Summing Up
“Symbols can become tools for thinking when students
use them as records of actions and things they already
know. Without this understanding, students
manipulate symbols without meaning rather than
thinking of symbols as quantities and actions to be
performed or records of actions already performed.
Huinker (2002) p. 73
Learning Targets
We are learning to…
• Apply and extend our understanding of multiplication
with whole numbers to multiplication with fraction
through use of context (word problem situations) and
use of the underlying mathematical structure.
Day 6 Reflections (Log)
Revisiting our learning intentions:
• A key connection in extending understanding of
multiplication with whole numbers to multiplication
with fractions.
• The importance of connecting representations to
support students’ understanding of multiplication with
fractions.
How about your students?
Analyzing Student Reasoning
Felicia is running on the track at school on
a Saturday. The distance around the track
is ¾ of a mile. Felicia ran 2 1/3 laps around
the track. How many miles did she run?
• Solve using a visual model.
• Solve using an equation.
“I’m introducing multiplication with
fractions today. I hope at least half
the class will understand the lesson.”
Disclaimer
Core Mathematics Partnership Project
University of Wisconsin-Milwaukee, 2013-2016
This material was developed for the Core Mathematics Partnership project through the University of
Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material
may be used by schools to support learning of teachers and staff provided appropriate attribution and
acknowledgement of its source. Other use of this work without prior written permission is prohibited—
including reproduction, modification, distribution, or re-publication and use by non-profit organizations
and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and
Science Partnerships.