Students engage with the content standards by using the eight
mathematical practices described in the Standards for Mathematical
Practice. The practice standards describe the ways in which
mathematically proficient students approach the discipline of
mathematics.
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning
of others.
4 Model with mathematics.
5 Use appropriate tools strategically.
6 Attend to precision.
7 Look for and make use of structure.
8 Look for and express regularity in repeated reasoning.
Considering the math practices can help you structure problems and
activities in a way that makes the most of your instructional time.
1. Begin with a contextual problem. (MP1, MP2)
a. Begin with problems in which the numbers are fairly simple.
b. As students become more comfortable with these kinds of
problems and the visual model, increase the difficulty of the
numbers.
2. The context of the problem suggests a visual model. (MP4, MP5)
3. Represent the problem with the visual model and solve. (MP1, MP4,
MP5, MP6)
4. Record the problem and its solution with an equation. (MP4, MP5)
5. Provide opportunities for students to look for patterns, make
generalizations, and summarize their learning. (MP3, MP7, MP8)
Wisconsin Mathematics Council, Annual Conference, 2015, Green Lake, WI
Martha Ruttle (The Math Learning Center)
Understanding Fraction Multiplication & Division with Visual Models
1
Objective: Modeling fractions and mixed numbers as quotients of two
whole numbers
5.NF.3. Interpret a fraction as division of the numerator by the denominator (a/b
= a ÷ b). Solve word problems involving division of whole numbers leading to
answers in the form of fractions or mixed numbers, e.g., by using visual fraction
models or equations to represent the problem.
Context: Sharing Sour Straws candy
Visual model: length
Example Problems
Problem 1a Eloise, Alex, Trace, and Dylan have a package of Sour Straws,
but there is just 1 straw left. If they share it equally, how much does each
kid get? (1 whole shared 4 ways  unit fraction)
Problem 1b Eloise, Alex, Trace, and Dylan have a package of Sour Straws,
and there are 3 straws left in it. If they share the straws equally, how much
does each kid get? (3 wholes shared 4 ways  fraction)
Problem 1c Eloise, Alex, Trace, and Dylan have a package of Sour Straws,
and there are 6 straws in it. If they share the straws equally, how much
does each kid get? (6 wholes shared 4 ways  mixed number)
Your contexts, models, and problems
Wisconsin Mathematics Council, Annual Conference, 2015, Green Lake, WI
Martha Ruttle (The Math Learning Center)
Understanding Fraction Multiplication & Division with Visual Models
2
Connecting visual models, equations, and language.
Problem 1a Eloise, Alex, Trace, and Dylan have a package of Sour Straws,
but there is just 1 straw left. If they share it equally, how much does each
kid get?
Each kid gets ¼ of a straw.
1÷ 4 = ¼
“1 straw divided into 4 equal shares produces shares that are each equal
to ¼ .”
Problem 1b Eloise, Alex, Trace, and Dylan have a package of Sour Straws,
and there are 3 straws left in it. If they share the straws equally, how much
does each kid get?
Each kid gets ¾ of a straw.
3÷ 4 = ¾
“3 straws divided into 4 equal shares produces shares that are each equal
to 3/4 .”
Problem 1c Eloise, Alex, Trace, and Dylan have a package of Sour Straws,
and there are 6 straws in it. If they share the straws equally, how much
does each kid get?
Each kid gets 1 ½ (6/4) straws.
6 ÷ 4 = 6/4 = 1 ½
“6 straws divided into 4 equal shares produces shares that are each equal
to 1 ½ .”
Wisconsin Mathematics Council, Annual Conference, 2015, Green Lake, WI
Martha Ruttle (The Math Learning Center)
Understanding Fraction Multiplication & Division with Visual Models
3
Objective: Modeling multiplication of whole numbers or fractions by
fractions
5.NF.4. Apply and extend previous understandings of multiplication to multiply a
fraction or whole number by a fraction.
a. Interpret the product (a/b) x q as a parts of a partition of q into b equal
parts; equivalently, as the result of a sequence of operations a x q ÷ b.
b. Find the area of a rectangle with fractional side lengths by tiling it with unit
squares of the appropriate unit fraction side lengths, and show that the area is
the same as would be found by multiplying the side lengths. Multiply fractional
side lengths to find areas of rectangles, and represent fraction products as
rectangular areas.
Context: Finding the area of rectangular objects in the environment
Visual model: rectangular array
Example Problems
Problem 2a Locate a region that has an area of 1 square inch. Label its
dimensions and area.
Problem 2b Locate a region that has an area of ½ (¼, 1/8) square inch.
Label its dimensions and area.
Problem 2c What are the dimensions and areas of these selected
regions?
Problem 2d What are the dimensions and area of these objects?
Rectangular Objects Scavenger Hunt
• Find an object with an area of exactly 1 sq. inch.
• Find an object with dimensions that are both fractions of an inch.
• Find an object with one dimension that is a fraction and the other
that is a mixed number.
• Find an object with dimensions that are both mixed numbers.
Your contexts, models, and problems
Wisconsin Mathematics Council, Annual Conference, 2015, Green Lake, WI
Martha Ruttle (The Math Learning Center)
Understanding Fraction Multiplication & Division with Visual Models
4
Objective: Modeling division with unit fractions and whole numbers
5.NF.7. Apply and extend previous understandings of division to divide
unit fractions by whole numbers and whole numbers by unit fractions.
a. Interpret division of a unit fraction by a non-zero whole number, and
compute such quotients.
b. Interpret division of a whole number by a unit fraction, and compute such
quotients.
c. Solve real world problems involving division of unit fractions by non-zero
whole numbers and division of whole numbers by unit fractions, e.g., by using
visual fraction models and equations to represent the problem.
Note Students able to multiply fractions in general can develop strategies to
divide fractions in general, by reasoning about the relationship between
multiplication and division. But division of a fraction by a fraction is not a
requirement at this grade.
Context: Running fractions of a mile
Visual model: number line
Example Problems
Problem 3a Frank, Amaya, Jim, and Cecilia are running a short relay race
that is ½ mile long. If they divide the legs of the relay evenly, how far does
each person have to run? (Dividing a unit fraction by a whole number 
fraction)
Problem 3b Amaya wanted to get better at running, so she decided to
run ½ mile every day on her own. How many days will it take her to run 6
miles? (Dividing a whole number by 1/2  whole number)
Problem 3c Cecilia liked Amaya’s idea, so she decided to run every day
too. She ran 1/3 mile every day. How long will it take her to run 6 miles?
(Dividing a whole number by a different unit fraction  whole number)
Your contexts, models, and problems
Wisconsin Mathematics Council, Annual Conference, 2015, Green Lake, WI
Martha Ruttle (The Math Learning Center)
Understanding Fraction Multiplication & Division with Visual Models
5
Connecting visual models, equations, and language.
Problem 3a Frank, Amaya, Jim, and Cecilia are running a short relay race
that is ½ mile long. If they divide the legs of the relay evenly, how far does
each person have to run?
F
A
J
0
C
1/2
1
Each person ran 1/8 mile.
1/2 ÷ 4 = 1/8
"When you divide 1/2 into 4 equal parts, each part has a value of 1/8 of
the whole."
Problem 3b Amaya wanted to get better at running, so she decided to
run ½ mile every day on her own. How many days will it take her to run 6
miles?
It would take 12 days for her to run 6 miles.
6 ÷ 1/2 = 12
"When you divide 6 miles into lengths of 1/2 mile each, there are 12 of
those lengths."
Problem 3c Cecilia liked Amaya’s idea, so she decided to run every day
too. She ran 1/3 mile every day. How long will it take her to run 6 miles?
1/3
0
1
2
3
4
5
6
It would take 18 days for her to run 6 miles.
6 ÷ 1/3 = 18
"When you divide 6 miles into lengths of 1/3 mile each, there are 18 of
those lengths."
Wisconsin Mathematics Council, Annual Conference, 2015, Green Lake, WI
Martha Ruttle (The Math Learning Center)
Understanding Fraction Multiplication & Division with Visual Models
6
Students use context to build understanding of fractions and operations
with fractions.
Students use models, including physical objects, to facilitate problem
solving and as tools for communicating about their mathematical
thinking.
We can help students to formalize their thinking by connecting it explicitly
to symbolic notation and careful language.
For the free geoboard app, go to:
http://catalog.mathlearningcenter.org/apps
Wisconsin Mathematics Council, Annual Conference, 2015, Green Lake, WI
Martha Ruttle (The Math Learning Center)
Understanding Fraction Multiplication & Division with Visual Models
7
February
|
Calendar Grid Activity 1
NAME
| DATE
Fraction Multiplication Grid Area Key
1
4
2
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4
1
5
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11
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Number Corner Grade 5 Student Book
45
© The Math Learning Center | mathlearningcenter.org
February
|
Calendar Grid class set, plus more as needed (optional)
Three-by-Three-Inch Grids
1
4
2
4
3
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1
5
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2
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1
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3
Number Corner Grade 5 Teacher Masters
T2
1
4
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© The Math Learning Center | mathlearningcenter.org
Fraction Grid
1
4
1
4
1
2
3
4
1
1
14
1
12
3
14
2
1
24
1
22
3
24
3
1
34
1
32
3
34
4
1
44
1
42
3
44
5
1
54
1
52
3
54
6
1
64
1
62
3
64
7
1
74
1
72
3
74
8
1
2
3
4
1
1
1
3
14 12 14
2
1
1
3
24 22 24
3
1
1
3
34 32 34
4
1
1
44 42
3
44
1
1
3
5 54 52 54
6