Transcript Fractions - RandolphK

Fractions:
Getting the Whole
Picture
Fraction Hot Topic Workshop
November 1, 2012
Complete “Fractions of Words” sheet
Fraction Understandings
 What
misconceptions do
students have about fractions?
 Why are fractions so difficult for
students to understand?
 What are the important fraction
concepts that students need to
understand?
Fractions
Researchers have concluded
that fractions is a complex topic and
causes more trouble for elementary
and middle school students than any
other area of mathematics.
Why do you think this is the case?
What makes fractions difficult for
students?
Reasons for Difficulties
in Learning Fractions

Material is being taught:
 too abstractly
 too procedurally
 outside meaningful contexts
 through rote memorization of procedures
 more attention on algorithms and less
attention on number sense and reasoning
 without connections
 with limited models
EOG Weight Distribution
Fractions
can make
SENSE!
What is a
FRACTION?
Visualizing Fractions
 In
your mind, picture three quarters.
 What
image did you create?
 Create
a different picture of three
quarters.
 What
image did you create?
Visualizing Fractions

What image did you create?
three quarters
3
4
Fraction Understandings
 What
early experiences do students
have with fractions?
There’s a quarter moon tonight.
 You can have half of my cookie.
 It’s a quarter past one.
 The recipe calls for two-thirds cup of sugar.
 The dishwasher is less than half full.
 I earned half a dollar.

Fractions in the Real World
 Find
examples of fractions
in the real world
 Illustrate or take pictures of
examples
 Fraction Scavenger Hunt
Fractions Decimals Percents
Meanings of Fractions

Values



Operators



Fractions are rational numbers that can be
counted and ordered
They can represent parts of a region or a set
or a point on the number line
One can find a fractional part of a value
A fraction can represent a division problem
Ratios

A comparison of two quantities
12
Fractions are NUMBERS
 Name
an amount (quantity), a part of
a specified whole
 Name a point on a number line
 An infinite amount of fractions exists
between any two whole numbers
 Can be counted
Partitioning

Partitioning is KEY to understanding and
generalizing concepts related to fractions
Partitioning in Grades 1 & 2
First Grade Geometry
1.G.3 Partition circles and rectangles into
two and four equal shares, describe the
shares using the words halves, fourths,
and quarters, and use the phrases half of,
fourth of, and quarter of. Describe the
whole as two of, or four of the shares.
Understand for these examples that
decomposing into more equal shares
creates smaller shares.
Partitioning in Grades 1 & 2
Second Grade Geometry
2.G.2 Partition a rectangle into rows
and columns of same-size squares
and count to find the total number of them.
2.G.3 Partition circles and rectangles into two,
three, or four equal shares, describe the
shares using the words halves, thirds, half
of, a third of, etc, and describe the whole as
two halves, three thirds, four fourths.
Recognize that equal shares of identical
wholes need not have the same shape.
Second Grade

Ms. Nim gave her students a picture of a
rectangle. Then she asked them to shade
in one half of the rectangle. Which one
shows one half?
Second Grade

Which pictures show
one half of the shape
shaded?
Partitioning in Grade 3
Third Grade Geometry
3.G.2 Partition shapes into parts with equal
areas. Express the area of each part as a
unit fraction of the whole. For example,
partition a shape into 4 parts with equal
area, and describe the area of each part as
¼ of the area of the shape.
Relationship of a Fraction to its Whole

Fractions are defined in relation to a whole


Need to understand what the fraction is “of”
The whole can be

One object

A collection of multiple objects

A quantity
Models & Representations
• Area/Region Models
• Linear/Measurement Models
• Set Models
Model
added in
4th grade
• Symbols (with meaning)
3
4
Models
introduced
in 3rd grade
7
8
1
2
A Fraction Represents…

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
From Models to Symbols

Top Number (numerator)


Bottom Number (denominator)




The counting number. It tells how many shares or
parts of a certain size are being counted.
Numerator -- Latin word meaning number
Tells fractional part being counted
If a 4, counting fourths. If a 6, counting sixths…
The number of equal parts into which the whole is
partitioned; parts or shares of the whole
Denominator -- Latin word meaning namer
How many of what
type of parts
how many
what
Unit Fraction

The amount formed by 1 of the parts when
a whole is divided into b equal parts; 1/b of
a whole
1
Two Fifths

⅕
⅕
What does the
denominator
represent?


⅕
⅕
⅕

1 whole object is split
into 5 equal parts
Each part is ⅕ of 1
whole object
What does the
numerator represent?

2 parts of 1 whole object,
where the size of each part
is ⅕
Unit Fractions

A unit fraction is a proper fraction with a
numerator of 1 and a whole number
denominator
2
1

is the unit fraction that corresponds to
5
5
3
or to
or to 17
5
5

As there are 3 one-inches in 3 inches,
3
there are 3 one-eighths in
8
Unit Fractions

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

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 = ?

We can obtain any fraction by
combining a sufficient number of unit
fractions
1
b
Unit Fractions

The numerator 3 of ¾ shows that 3 is the
number you get by combining 3 of the 1/4 ’s
together when the whole is divided into 4 equal
parts

A fraction such as 5/3 shows combining 5 parts
together when the whole is divided into 3 equal
parts – best shown on a number line
Unit Fractions

Decompose the following fractions in as
many ways as you can
3
4
5
8
2
1
3
Unit Fraction Counting
 Fractional
Parts Counting
Display pie-piece – tell what fraction this
represents of the whole and count as a
class
 Example: each pie-piece is one-third

What is another way we can say eight-thirds?
Problem Solving
 Some
girls were sharing some
bananas so that each person got the
same amount. Each girl got ¼ of a
banana. How many bananas and
how many girls could there have
been?
How would students solve this problem?
 What models/representations would they
use?

Fractions

Third Grade


Fourth Grade


Halves, thirds, fourths, sixths, eighths
Halves, thirds, fourths, fifths, sixths,
eighths, tenths, twelfths, hundredths
Fifth Grade

No limits specified
Naming Fractions
Students need many opportunities to model
and name non-unit fractions
 This process begins with regional
representations and is extended to include
number lines
 The advantage of regional representations
is their familiar contexts (pizza slices,
sections of apples, cake etc.) for
students.

Area/Region Models
 Region
is cut into smaller parts
 Examples: pattern blocks, grid
paper, geoboards, circles, rectangles,
triangles
REGIONS as Models for Fractions
Identify the Thirds

Which of these regions is divided into
thirds? Why or why not?
Identify the Fourths
Area Models
 Geoboard
Models
Divide the geoboard into half
 Make halves on the geoboard and record all
of your ideas on dot paper

How do you know the two parts are halves?
How do you know the two parts are equal?

Repeat with fourths and eighths
How will the size of each piece change?

Extension:
Combine fractions to create a design
Fractional Parts of a Whole (Region)
How many ways can you show fourths on a
geoboard?
Fractional Parts of a Whole (Region)
Does region show fourths of the square?
 How do you know?

Fractions and Geoboards
Videos:
Annenberg Learner - Learner Express
http://www.learner.org/vod/vod_windo
w.html?pid=905
http://www.learner.org/series/modules
/express/pages/ccmathmod_07.html
Fractional Parts of a Whole

If the yellow hexagon represents one whole,
how might you partition the whole into
equal parts? Name the fractional parts with
unit fractions.
Fractional Parts of a Whole

Name the unit fractions that equal one
whole hexagon
Fractional Parts of a Whole

Two yellow hexagons = 1 whole
•
How might you partition the whole into
equal parts? Name the unit fraction for one
triangle; one hexagon; one trapezoid and
one rhombus
Fractional Parts of a Whole

One blue rhombus = 1 whole
•
What is the value of the red trapezoid, the
green triangle and the yellow hexagon?
Show and explain your answer
•
Caution

Be sure to identify the
whole or whole unit.
The unit is not
consistent and may
change.
Want half of a candy bar?
Identifying Fractional Parts

What part is red? Blue? Green? Yellow?
49
Area Models
 Pattern

Blocks
Extensions:
 Create
a pattern block design. Assign one
block the value of 1 and find the value of
your entire picture.
 Change the whole. How does this change the
value of your design?
 Challenge students to create a design with a
predetermined value. (Ex. – If the hexagon is
1, create a design with a value of 24 1/3.)
Create the whole knowing a part…

If the blue rhombus is ¼, build the whole.

If the red trapezoid is 3/8, build the whole.
If you know a fractional part,
can you make the whole?
Make the whole line if this is one third.
Make the whole shape if this is three fourths.
c
c
c
c
c
c
Fraction Area
Name each piece as a fraction.
What are the mathematical understandings
involved in solving this problem?

Mrs. Frances drew a picture on the board.

When she asked her students what fraction
it represents.



Emily said that the picture represents two-sixths.
Raj said that the picture represents two-thirds.
Alejandra said that the picture represents 2.
www.illustrativemathematics.org
Illustrative Mathematics
http://illustrativemathematics.org/
Illustrative Mathematics
http://illustrativemathematics.org/
Illustrative Mathematics
http://illustrativemathematics.org/
Caution

Be careful about using
limited models with
students. It is important
to have students model
fractions with
manipulatives but also
draw a fraction
representation.
Linear Models
 Linear
region is cut into smaller
parts; lengths are compared
 Examples: fraction tiles, paper
strips, cuisenaire rods, number lines,
rulers
0
1
2
3
4
Number Lines
Fractions on a Number Line
Develop understanding of fractions as numbers.
3.NF.2 Understand a fraction as a number on the
number line; represent fractions on a number line
diagram.
a. Represent a fraction 1/b on a number line diagram
by defining the interval from 0 to 1 as the whole and
partitioning it into b equal parts. Recognize that each
part has size 1/b and that the endpoint of the part
based at 0 locates the number 1/b on the number line.
Fractions on a Number Line
Develop understanding of fractions as numbers.
3.NF.2 Understand a fraction as a number on the
number line; represent fractions on a number line
diagram.
b. Represent a fraction a/b on a number line diagram
by marking off a lengths 1/b from 0. Recognize that
the resulting interval has size a/b and that its
endpoint locates the number a/b on the number line.
Fractions on a Number Line
On a line segmented into fourths, show that
three 1/4’s equals 3/4
Understanding Number Lines

Number lines represent the order of
numbers and their magnitude

Numbers to the right of any given number
are greater in value; numbers to the left of
any given number are less in value

Once two numbers are marked on the
number line, the location of all other
numbers is fixed
Shaughnessy (2011)
Number Line – A Linear Model


Differs from other models

An identified length represents the unit

Can represent iteration of the unit

Can simultaneously illustrate subdivisions of
all iterated units
There is no visual separation between
consecutive units

Model is continuous
Unit Fractions on a Number Line

Fractions allow for more precise
measurement of quantities, including
fractional parts greater than 1 whole.
Fractions on a Number Line

Number lines show relative magnitude of
fractions.

Where would you place 1/3?
0 a
b
c
d
e
1
What fraction would be at point a?
 How far apart are a and b?

69
Placing Fractions on a Number Line

In turn, place your fraction on the number
line
0
1
2

Explain your reasoning for placement

How can a Number Line be a useful tool for
making sense of fractions?
Fractions on a Number Line
 Parallel
number lines support students
in identifying equivalent fractions
Equivalence – Third Grade
Develop understanding of fractions as numbers.
3.NF.3 Explain equivalence of fractions in special cases,
and compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if
they are the same size, or the same point on a
number line.
b. Recognize and generate simple equivalent fractions,
e.g., ½ = 2/4, 4/6 = 2/3). Explain why the fractions
are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize
fractions that are equivalent to whole numbers.
Examples: Express 3 in the form 3 = 3/1; recognize
that 6/1 = 6; locate 4/4 and 1 at the same point of a
number line diagram.
Equivalence – Fourth Grade
Extend understanding of fraction
equivalence and ordering.
4.NF.1 Explain why a fraction a/b is equivalent
to a fraction (n x a)/(n x b) by using visual
fraction models, with attention to how the
number and size of the parts differ even
though the two fractions themselves are the
same size. Use this principle to recognize and
generate equivalent fractions.
Equivalent Fractions
The Concept:
Two fractions are equivalent if they are
representations for the same amount or
quantity – if they are the same number.
The Method:
Students should EXPLORE equivalent fractions
rather than depending on procedures,
techniques or algorithms.
Intuitive methods are always best at first.
Van de Walle and Lovin, Teaching Student-Centered Mathematics, Grades 3-5
Equivalent Fractions

Every fraction equal to infinitely many other fractions.
Explanation 1
 Divide the 4 equal parts into 5 small, equal
parts (20); 3 parts subdivided into 5 equal
parts becomes 15 parts out of 20.
Equivalent Fractions
Equivalent Fractions

Compare the numbers and the size of the
parts.
Equivalent Fractions

Every fraction equal to infinitely many other fractions.
Explanation 2

Multiply A/B by 1 in the form of N/N
5
=1
5
A N
·
B N
3 5

4 5
15
20
Equivalent Fractions

What algebraic property does this represent?
Equivalent Fractions
NCTM Illuminations
http://illuminations.nctm.org/
Simplest Form
What fractions are in simplest form?
 How do you know?

Simplest Form

There is no whole
number other than
1 that divides both
the numerator and
the denominator
evenly
Comparing – Third Grade
Develop understanding of fractions as numbers.
3.NF.3 Explain equivalence of fractions in special
cases, and compare fractions by reasoning about
their size.
d. Compare two fractions with the same numerator
or the same denominator by reasoning about their
size. Recognize that comparisons are valid only
when the two fractions refer to the same whole.
Record the results of comparisons with the symbols
>, =, or <, and justify the conclusions, e.g., by
using a visual fraction model.
Comparing – Fourth Grade
Extend understanding of fraction
equivalence and ordering.
4.NF.1 Compare two fractions with different
numerators and different denominators, e.g.,
by creating common denominators or
numerators, or by comparing to a benchmark
fraction such as ½. Recognize that
comparisons are valid only when the two
fractions refer to the same whole. Record the
results of comparisons with symbols >, =, or
<, and justify the conclusions, e.g., by using a
visual fraction model.
Comparing Fractions

Using the fraction cards, sort the fractions using
the benchmarks 0, 1/2, and 1
1
2
0
0
½
1
1
Common Student Errors
•
Students see fractions as two
separate whole numbers 2/3 and 3/5 = 5/8

Students think 3/8 is bigger than 2/5 because
8 is greater than 5

Using region models, students often count
shaded pieces as whole numbers

Students may see 3/6 and 1/2 as equivalent
but they do not understand that 3/6 and 1/2
are identical numbers
Close to…
• Name a fraction close to 1 but not more
than 1.
• Name a fraction that is even closer to 1
than that.
• Why do you believe it is closer?
• Name a fraction that is even closer than
the previous fraction.
• Again…
Comparing Fractions

Game:
Getting Closer to 1




First player names a fraction that is close to
one (but does not go over one)
Next player names a fraction that is even
closer to one and explains why they think it
is closer
Continue for several rounds trying to get
closer to 1
Variations:
Try naming fractions closer to 0 or 1/2
Comparing Fractions
 Comparing
Fractions Worksheet
Compare each set of fractions. Decide
which fraction is greater without using
algorithms, cross multiplying, or
common denominators.
 Explain your reasoning.

What strategies did you use to solve
each problem and figure out which
fraction was greater?
Which fraction is larger?
1
4
or
1
8
Which fraction is larger?
3
8
or
5
8
Which fraction is larger?
3
10
or
3
8
Which fraction is larger?
3
6
or
5
8
Which fraction is larger?
3
4
or
5
6
Which fraction is larger?
5
8
or
8
5
Comparing Fractions
Strategies:
 Like denominators – When denominators are the
same, the fraction with the bigger numerator is
more because there are more of the same-sized
parts.
 Like numerators – When numerators are the
same, the fraction with the smaller denominator is
more because the size of the parts is larger.
 More or less than 0, 1/2, 1 – Fractions can be
compared by determining whether they are more
than or less than a benchmark number.
 Distance from 0, 1/2, 1 – Fractions can be
compared by finding the distance from a
benchmark number and then comparing that
distance.
Let’s Try It

Caution

Be cautious of
teaching rules or
algorithms for
comparing two
fractions. They require
no thought about the
size of the fractions.
Students must develop
number sense about
fraction size.
Fraction Games

Close to 1



Take turns rolling a number cube
Decide where you will place the number
(numerator, denominator, or Throw Away box)
Object: Try to get the fraction closest to 1
Throw Away
Close to 1
Partner 1
Partner 2
4
5
3
3
Throw Away
2
1
Throw Away
4
1
Fraction Games

Close to 1

Variations:
 Roll 2 number cubes and add the numbers
together before placing them in the boxes
 Roll 2 number cubes and multiply the numbers
together before placing them in the boxes
 Change the target number to 0 or ½
Throw Away
Caution

A fraction tells us only
about the relationship
between the part and
the whole. It does not
say anything about the
size of the whole or the
size of the parts.
Comparisons with any
model can be made only
if both fractions are
parts of the same
whole.
Set Models

Describe your table group with 3 – 4
fraction statements


One-half of the group is male
One-half of the group is wearing a hat
Set Models
 Whole
is the set of objects and
subsets of the whole make up
fractional parts
 Examples: counters, teddy bear
counters, candy, people
Set Model
Fractional Parts of a Whole (Set)

Take 24 chips.
 Divide them into thirds if you can.
 How
many in one third?
 two thirds?
 Divide
them into fourths if you can.
 How
many in one fourth?
 three fourths?
 Divide
 Why
 Into
them into fifths if you can.
can’t you divide them into fifths?
what other fractional parts can 24 be
divided?
Fractional Parts of a Whole (Set)
Caution:
When partitioning sets, children frequently confuse the
number of counters in a share with the number of
shares.
Example:
Say: Divide 12 counters into fourths:
(The child correctly makes four equal groups.)
Say: Show me three fourths of 12.
(Some children who correctly divided the set into four equal groups
above will now regroup the 12 chips into three groups of four.)
If you know a fractional part,
can you make the whole?
If this is two fifths of a set, make the whole
set.
OPERATIONS WITH
FRACTIONS
Adding/Subtracting
Build fractions from unit fractions by applying
and extending previous understandings of
operations on whole numbers.
4.NF.3 Understand a fraction a/b with a>1 as a sum of
fractions 1/b.
a. Understand addition and subtraction of fractions as
joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the
same denominator in more than one way, recording each
decomposition by an equation. Justify decompositions,
e.g., by using a visual fraction model. Examples: 3/8 =
1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8, 2 1/8 = 1 + 1 + 1/8
or 8/8 + 8/8 + 1/8.
Adding/Subtracting
Build fractions from unit fractions by applying
and extending previous understandings of
operations on whole numbers.
4.NF.3 Understand a fraction a/b with a>1 as a sum of
fractions 1/b.
c. Add and subtract mixed numbers with like
denominators, e.g., by replacing each mixed number
with an equivalent fraction, and/or by using properties of
operations and the relationship between addition and
subtraction.
d. Solve word problems involving addition and
subtraction of fractions referring to the same whole and
having like denominators, e.g., by using visual fraction
models and equations to represent the problems.
Addition and Subtraction
• If students have a solid understanding of fraction
concepts, they should be able to add and subtract
fractions with like denominators right away.
(e.g., 2/5 + 4/5, or 4 7/8 – 2 3/8)
• Understanding that the numerator is the count and
the denominator is what is counted makes adding
and subtracting like fractions the same process as
adding whole numbers. Two fifths + four fifths is
the same problem as adding two apples and four
apples.
• If children have problems with adding and
subtracting like fractions, they need further work
with fraction concept development before moving to
adding and subtracting unlike denominators.
Adding/Subtracting
Use equivalent fractions as a strategy to add and
subtract fractions.
5.NF.1 Add and subtract fractions with unlike denominators
(including mixed numbers) by replacing given fractions
with equivalent fractions in such a way as to produce an
equivalent sum or different for fractions with like
denominators.
5.NF.2 Solve word problems involving addition and subtraction
of fractions referring to the same whole, including cases of
unlike denominators, e.g., by using visual fraction models
or equations to represent the problem. Use benchmark
fractions and number sense of fractions to estimate
mentally and assess the reasonableness of answers. For
example, recognize an incorrect result 2/5 + 1/2 = 3/7, by
observing that 3/7 < 1/2.
Adding & Subtracting Fractions

Begin with simple contextual tasks.

Jack and Jill ordered two identical-sized
pizzas, one cheese and one pepperoni.
Jack
ate 4/6 of a pizza and Jill ate 1/3 of a pizza.
How much pizza did they eat together?
Explore with models.
 Connect the meaning of fraction computation
with whole-number computation.
 Have students estimate answers and check
for reasonableness of solutions.
 Develop strategies using estimation and
informal methods.

Addition and Subtraction
Begin with informal exploration

Paul and his brother were eating the same kind
of candy bar. Paul had 3/4 of his candy bar. His
brother still had 7/8 of a candy bar. How much
candy did the two boys have together?
Using nothing other than simple drawings, how would
you solve this problem without using an algorithm
and finding common denominators?
Try to think of two different methods.
Addition and Subtraction
Marie and Glen ordered two identical-sized
pizzas, one pepperoni and one veggie.
Marie ate 5/6 of a pizza and Glen ate ½ of a
pizza. How much pizza did they eat
altogether?
What model would you use for the whole?
How would you solve this problem without a
traditional algorithm?
Fraction Addition
How could you solve these problems without
finding a common denominator?
3+ 1
4 8
1+ 1
2 8
1
1 3
+
2 4
2+ 1
3 2
1
2 3
+
3 4
3
1
1 2
4
2
Addition and Subtraction
 Build
on informal explorations and
invented strategies to develop a
method for addition and subtraction
 Use
estimation strategies
 These
approaches help students see
that the common-denominator
approach – finding a common “family”
– is meaningful
Addition and Subtraction
Consider:
2+ 5
4 8
 Use
models.
 Key question: “How can we change this to a
problem with the parts the same? (like
“adding apples and apples”). In this case
fourths can be changed to eighths.
 Main
idea: 2/4 + 5/8 is the same problem as
4/8 + 5/8
Addition and Subtraction
Consider:
2 1
+
3 4
Focus attention on rewriting the problem
in a form that is “like adding apples and
apples” so that the parts of both the
fractions are the same.
 The new form is the same problem as the
old form.
 Demonstrate with models. (CD Fractions)

Common Denominator
As students work with modeling and
rewriting problems to make them
easy, they will come to understand
that the process of getting a common
denominator is actually one of finding
a way to change the statement of the
problem without changing the problem
itself.

From Van de Walle and Lovin, Teaching Student-Centered Mathematics, Grades 3-5
Using Visual Models

Fraction Tracks uses the number line as a visual
model.
http://illuminations.nctm.org
Using Estimation

Estimate the answer to 12/13 + 7/8
A.
B.
C.
D.
1
2
19
21
• Only 24% of 13 year olds answered correctly
• Equal numbers of students chose the other
answers
NAEP
Fractions in Balance Problems

Find the missing values.
x
1½
1¾
n
1¾
n
n
Figures that are the same size and shape must have the same value.
Adapted from Wheatley and Abshire, Developing Mathematical Fluency, Mathematical Learning, 2002
126
Multiplying Fractions
Build fractions from unit fractions by applying and
extending previous understandings of operations
on whole numbers.
4.NF.4 Apply and extend previous understandings of
multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For
example, use a visual fraction model to represent 5/4 as the
product 5 x (1/4), recording the conclusion by the equation
5/4 = 5 x (1/4).
b. 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 x
(2/5) as 6 x (1/5), recognizing this product as 6/5. In
general, n x (a/b) = (n x a)/b.)
Multiplying Fractions
Build fractions from unit fractions by applying and
extending previous understandings of operations
on whole numbers.
4.NF.4 Apply and extend previous understandings of
multiplication to multiply a fraction by a whole number.
c. Solve word problems involving multiplication of a
fraction by a whole number, e.g., by using visual
fraction models and equations to represent the
problem.
Multiplying Unit Fractions

Understand a fraction a/b as a multiple of 1/b
1
5

is the product of 5 x ( )
4
4
5
4
1
=5x
4
Multiplying Unit Fractions
Understand
a multiple of a/b as a multiple of 1/b,
and use this understanding to multiply a fraction by a
whole number
1
2
3 sets of
is the same as 6 sets of
5
5
Multiple Solution Strategies
 Solve
word problems involving
multiplication of a fraction by a whole
number
 At

your table, solve in 2 ways…
3
If each person at a party will eat 8 of a
pound of roast beef, and there will be 5
people at the party, how many pounds
of roast beef will be needed? Between
what two whole numbers does your
answer lie?
Multiplication with Fractions
Laura had $240. She spent 5/8 of it.
How money did she have left?
Eli had 18 cars in his toy car collection.
Two-thirds of the cars are blue. How
many blue cars does Eli have?
Solve each problem. How are these
problems different from the “roast
beef” problem?
Multiplying/Dividing
Apply and extend previous understandings of
multiplication and division to multiply and divide
fractions.
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. For example, interpret ¾ as the result of
dividing 3 by 4, noting that ¾ multiplied by 4 equals 3,
and that when 3 wholes are shared equally among 4
people each person has a share the size ¾. If 9 people
want to share a 50-pound sack of rice equally by weight,
how many pounds of rice should each person get?
Multiplying Fractions
Apply and extend previous understandings of
multiplication and division to multiply and divide
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. For example, use a visual fraction model
to show (2/3) x 4 = 8/3, and create a story context for this
equation. Do the same with (2/3) x (4/5) = 8/15. (In general,
(a/b) x (c/d) = ac/bd).
Multiplying Fractions
Apply and extend previous understandings of
multiplication and division to multiply and divide
fractions.
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. For example, create a
story context for (1/3) ÷ 4, and use a visual fraction model
to show the quotient. Use the relationship between
multiplication and division to explain that (1/3) ÷ 4 = 1/12
because (1/12) x 4 = 1/3.
Multiplying Fractions
5.NF.7
b. Interpret division of a whole number by a unit fraction,
and compute such quotients. For example, create a story
context for 4 ÷ (1/5), and use a visual fraction model to
show the quotient. Use the relationship between
multiplication and division to explain that 4 ÷ (1/5) = 20
because 20 x (1/5) = 4.
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. For
example, how much chocolate will each person get if 3
people share ½ lb of chocolate equally? How many 1/3-cup
servings are in 2 cups of raisins?
Fraction Operations

The meaning of each operation on
fractions is the same as the meaning for
the operations on whole numbers

For multiplication of fractions, it is useful
to recall that the denominator is a divisor.
This will allow students to find parts of the
other factor.

For division, it is useful to think of the
operation as partition and measurement
Ramseur Elementary School asked the fifth
grade students to help the art teacher
design some tile murals for the new art
room. The first mural is going to have ¾
of the design as red tiles and ½ of those
will have flowers on them. How many
tiles will have flowers on them?
How would you model and solve
this problem?
Multiplying Fractions
How would you model and solve without an
algorithm?

Raj had 2/3 of his bedroom left to paint.
After lunch, he painted 4/5 of what was
left. How much of the whole room did
Raj paint after lunch?
The type of model can impact students’
understanding of their solution.
Multiplication of Fractions

3/29/2016 • page 140
Multiplication of Fractions

3/29/2016 • page 141
Multiplication of Fractions

3/29/2016 • page 142
Multiplication of Fractions

3/29/2016 • page 143
Multiplying Fractions

Olivia was sharing a pitcher of lemonade
with her sister. Olivia drank 3/5 of the
pitcher; then her sister drank 3/4 of
what was left. How much of the pitcher
of lemonade did her sister drink?
Modeling the Process
3
3
x
5
4
means “3/5 of a set of ¾”
Make ¾, then take 3/5 of it.
Why does extending the lines (the dotted part) help?
Multiplying Fractions

You have 6/8 of a pizza left. If you give 1/6
of the leftover pizza to a friend, how much of
a whole pizza will be left for your brother?

Rita used 1/10 of a bottle of vanilla flavoring
for a cookie recipe, leaving 9/10 of the
bottle. If she then used 2/3 of what was left
in a cake recipe, how much of the whole
bottle did she use?
Challenges for Students…

Research shows that young students have
little trouble multiplying fractions; they can
easily multiply the top numbers and
bottoms numbers, but they have great
difficulty interpreting the meaning of the
solution

They interpret whole number multiplication
as repeated addition, but have no way to
interpret multiplication of fractions
Consider
 How
many fifths are in two wholes?
How would you begin to think about this
question?
 Create at least two representations to
show your solution

 What
operation is represented by
this problem?
Fractions and Brownies

You bake two pans of brownies. If 1/8 of a
pan equals 1 serving, how many servings
did you make?
Justify your response.
Fractions and Brownies

What are the big understandings that are
useful in being able to solve this problem?
2÷1/8=16
Connecting What We Know

Consider 1 ÷ ½ = ?

To determine how many of the unit
fractions of the divisor (1/2) are in the
dividend (1), think about it as:
How many one-halves are in 1?

How is this the same as thinking of 36÷9
as “How many nines are in 36?”
Words to Symbols

Write an equation for each situation:
A
grocer has 10 pounds of coffee
beans. If he sells the beans in ½
pound bags, how many will he have
to sell?
 If
you have a spool with 6 feet of
ribbon, and you need 1 ½ foot long
pieces for a craft project , how many
can you make?
Words to Symbols


A grocer has 10 pounds of coffee beans. If he sells the
beans in ½ pound bags, how many will he have to sell?
If you have a spool with 6 feet of ribbon, and you need 1
½ foot long pieces for a craft project, how many can you
make?
Is it easier for you to think about coffee
beans, lengths of ribbon, or symbols?
 Context gives meaning to the symbols
for numbers, operations, and their
relationships
 Students need the ability to
decontextualize and contextualize

Importance of Sense-Making
 Knowing
THAT an algorithm works is
not the same thing as knowing WHY
an algorithm works (or does not
work)
 Research
states that context gives
meaning to the symbols for
numbers, operations, and their
relationships
Division of Fractions
5÷⅓=?
3/29/2016 • page 155
Division of Fractions
5÷⅓=
3/29/2016 • page 156
Division of Fractions
5÷⅓=
1
2
4
3
10
11
12
5
6
7
13
14
15
8
9
During the carnival, Ms. Garcia
notices that there 5 bags of balloons.
She wants to give 1⁄2 a bag of
balloons to some of the volunteers.
How many volunteers can she give 1⁄2
a bag to?
Building Understanding
How many one-sixths are in 2?
2÷⅙ = ?
 How many one-halves are in 3?
3÷½ = ?
 How many one-fifths are in 2?
2÷⅕ = ?
 What patterns do you see?
 How might these patterns help develop
a method for dividing by fractions?

More Brownies
You have 1/3 of a pan of brownies left
after last night’s party. If you and four
friends share what is left of the brownies,
how much of the whole pan of brownies
will each of you get to eat?
Write an equation to solve this problem
 Solve the problem using models and share
your method with table partners

Division of Fractions
⅓÷5=
How is 1/3 ÷ 5
different?
• Use the relationship
between multiplication
and division to explain
that (1/3) ÷ 5 = 1/15
because (1/15) × 5 =
1/3.
• Create a story context.
3/29/2016 • page 162
During the carnival, Sheila found ½ of a
pizza. She wants to share it with her four
friends. How much of a whole pizza does
each friend get?
The popcorn booth serves tubs that weigh
¾ of a pound. I have a box that weighs 6
pounds. How many tubs can I make from
the box?
The face-painting booth has ¼ pint of paint.
There are five fifth graders who want their
faces painted. How much paint should we
use for each child?
Making Sense of Fractions
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
Fraction Hot Topic
Resources

Fourth & Fifth Grade Teachers



Fourth Grade


Lessons for Extending Fractions
by Marilyn Burns
Extending Children’s Mathematics
by Empson & Levi
Lessons for Introducing Fractions
by Marilyn Burns
Fifth Grade

Lessons for Multiplying and Dividing Fractions
by Marilyn Burns