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Chapter 6 Notes
0. Fraction Strips
Fraction Strip Applet
http://www.arcytech.org/java/fractions/fractions.html
Fraction Strips
Brt. Pink
Brt. Yellow
Green
Purple
Pink
Yellow
Blue
a. When you divided your strip into two parts, what fraction of the whole strip did one strip represent?
b. When you divided your strip into four parts, what fraction of the whole strip did one strip represent?
c. What other fractions have the same value as 1/2?
d. What other fractions have the same value as 2/3?
e. What do you notice about the fractions that are the same as 1/2?
f. Can you identify other fractions (for which you don’t have a fraction strips) that are the same as 1/2 based on
this pattern?
Worksheet: Comparing and Ordering Fractions with Fraction Strips
Section 6.1 - Fractions and Rational Numbers
1. What is a fraction? Why is this type of number needed? The answers to these questions depend on who
answers them.
2. We want to develop a number system that allows closure for all 4 operations – Rational numbers
a
or
b
a/b indicating that some unit whole has been divided or partitioned into b equal sized parts and a of them are
under consideration. The a part of the fraction is called the numerator and the b part is called the
a
denominator. The meaning of
is relative to what unit whole is referenced. A rational number is a number
b
a
represented by two integers, a and b, b ≠ 0 using the notation
or a/b. The set of rational numbers is given by
b
a
a a
a
 a a
 {a / b | a  and b  , b  0} . For  ,  
and      .

b
b
b
b
 b b
Definition. A fraction is a number represented by two whole numbers, a and b, b ≠ 0 using the notation
3. There are three different perspectives from which to view fractions:
3
means 3 equal sized parts out of 4 equal parts of a unit whole. This
4
is the conceptualization with which most elementary students are introduced to fractions.
Parts of a whole: From this perspective
Quotient (Division Problem): From this perspective we view the fraction bar as another symbol representing
3
division, hence or 3  4 means 3 divided by 4 and the quotient of this division is represented. This is the
4
perspective we use when we teach students to convert fractions to their decimal representation.
Ratio and Rates: Ratios are a broader concept of which fractions are a special case. Ratios and rates will be
3
discussed further, but for now,
(or 3:4) means 3 units compared to 4 units of the same or different measure,
4
for example, “The ratio of boys to girls in my classroom is three to four”.
The ‘Parts of a Whole’ conceptualization is the one most easily supported by children’s understanding
of division and fair-sharing.
4. Modeling fractions: Three types of common models used to represent fractions are set models, area
models, and number line models. When choosing any model, it is important that students have been given some
a
guidance in how to interpret it. A key concept in understanding fractions is relating the representation
b
to its unit whole.
A set (discrete) model is appropriate when examples involving discrete objects are being used.
An example of a set model for ¾ might look like this:
Three-fourths of the happy faces are light.
An area model (colored regions) is another way that fractions are commonly represented for students.
An area model is appropriate when giving examples involving objects that can be subdivided into any number
of pieces. The model consists of some type of two-dimensional figure, such as a circle, square, rectangle, etc.
An example of an area model might look like this:
Three-fifths of the large rectangle is shaded.
A number line model is appropriate when using examples involving linear measurement. An
advantage of number lines is that most children are already familiar with this instructional aid
5
The fraction represented by this diagram is 1 .
6
0
1
2
3
4
When choosing any model, it is important that students have been given some guidance in how to
a
interpret it. A key concept in understanding fractions is relating the representation
to its unit whole.
b
5. Equivalent fractions:
Fractions that represent the same relative amount are called equivalent or equal fractions. Every
1
fraction has an infinite number of representations. For example, the fraction can be represented with the
3
following area model:
One third of the large rectangle is shaded. By dividing each of the one third sections into two equal sections, the
2
model can be changed to represent the fraction .
6
This process of dividing each of the sections into equal sized subsections can continue to represent an infinite
number of different fractions that all relate to the same relative amount.
10
. But all of the diagrams have the same sized rectangle (the unit whole), with
30
the same amount of the area of that rectangle shaded blue and the same amount of the area of the rectangle not
shaded. The only difference between the three diagrams is how many divisions of the area are shown. This is
the concept behind equivalent or equal fractions.
This model could represent
a
, for which
b
b ≠ 0, we can represent the same relative amount in an infinite number of ways by multiplying both the
numerator and denominator by the same natural number n, therefore
a an
a

. When
is such that a and b have no common factors greater than 1, then we say that the fraction is in
b bn
b
simplest form, reduced form, or in lowest terms.
Fundamental Law of Fractions: Given any fraction
Set Model Worksheet: Eggsactly Eggs
Fraction Bar Link: http://nlvm.usu.edu
Virtual Cuisenaire Rods Link: http://arcytech.org/java/integers/integers.html
Worksheet: Modeling Fractions with Cuisenaire Rods
Worksheet: Equivalent Fractions with Cuisenaire Rods
There are several methods for determining if two fractions are equivalent.
Method 1 – Use cross products (Only after students have learned to compare using common denominators, for
3 6
example  ).
4 8
a c
14
16
 if and only if ad  bc.
For example, determine if
and
are equal? Using cross
24
21
b d
products it can be shown that 14  24= 336 and 16  21= 336 , so the two fractions are representing the
same relative amount.
Method 2 – Simplify both fractions
14
16
and
are equal fractions is to simplify both fractions and
24
21
show they are equal to the same fraction.
14 2  7 2
2
16 2  8 2
2
14
16
2

 1  and

 1  . Since both
and
equal , they must equal each
21 3  7 3
3
24 3  8 3
3
24
3
21
other.
Another way to determine if
Method 3 – Convert both fractions to their decimal representations (Note: this method is not appropriate for
students who haven’t studied decimals)
a
The conceptualization that = a  b , leads to the use of the decimal representation of this
b
14
16
quotient to determine if the two fractions are equal. Since both
and
are equal to 0.6 , then by
24
21
the transitive property the two fractions must equal the same relative amount. (Decimal representations
of fractions will be discussed more in the next chapter.)
Click on “Geometry”, click on “Pattern Blocks” link: http://nlvm.usu.edu
Another pattern block link: http://www.arcytech.org/java/patterns/patterns_j.shtml
Worksheet: Pattern Block Relations
Worksheet: Modeling Fractions with Pattern Blocks
Section 6.2 - Adding and Subtracting Fractions
Adding and Subtracting Fractions by Modeling with Pattern Blocks
Adding and Subtracting Fractions by Modeling with Fraction Circles
Adding and Subtracting Fractions by Modeling with Cuisenaire Rods
What are the three ways to view fractions?
Virtual Cuisenaire Rods Link: http://arcytech.org/java/integers/integers.html
Example:
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How do you write as a mixed fraction? Explain this process illustrating with C-rods.
6
2
How do you write 3 as a improper fraction? Explain this process illustrating with C-rods.
3
Students need to be able to easily move between improper representations of fractions and mixed
b Ac  b
numeral representation of fractions A 
but need to see why this works. Another thing that needs to be
c
c
understood about the mixed numeral form is the implied addition between the whole number part of the answer
b
b
and the fraction part. A  A  .
c
c
,
1 1
a. 
6 3
1 2
b. 1 
6 3
1 1
c. 1 
6 2
1 1
d. 2  1
6 2
1 3
e. 1 
2 4
1 5
f. 1 
2 12
Section 6.3 – Multiplying and Dividing Fractions
Examples: Use the Area model to show:
a. 2 * 3
1
b. 2*
3
2 4
c. *
3 5
2
4
d. 1 * 2
3
5
Recall that “*” is read “groups of”, or simply “of”. So, “2 * 3” means “2 groups of 3” while “3 * 2” means
“3 groups of 2”.
Examples: Multiply Fractions by Using Cuisenaire Rods:
1
1
a. 2* vs. *2
3
3
1
1
b. 5* vs. *5
2
2
1 4
4 1
* vs. * (Interpret: Half of four thirds means “Break 4/3 into two pieces and take one.” Four
2 3
3 2
thirds of 1/2 means “Break 1/2 into three pieces and take 4 of them.” )
2 3
3 2
d. * vs. *
3 2
2 3
2 3
3 2
e. * vs. * (Interpret: Two-thirds of three-fourths means “Break 3/4 into 3 pieces and take two of
3 4
4 3
them.” Three-fourths of two-thirds means “Break 2/3 into four pieces and take three of them.)
c.
1
2
2 1
f. 1 * 2 vs. 2 *1
2
3
3 2
Multiplying Fractions by Using Pattern Blocks
1
1
a. 5* vs. *5
3
3
1 5
5 1
* vs. *
2 3
3 2
1 1
1 1
c. * vs. *
3 2
2 3
2 1
1 2
d. *1 vs. 1 *
3 2
2 3
1
2
2 1
e. 1 * 2 vs. 2 *1
2
3
3 2
b.
Multiplying Fractions by using Fraction Circles
1 3
a. *
2 4
2 5
b. *
3 4
1 1
c. 2 *1
4 2
a c
c
* : Break into b pieces and take a of those pieces.
b d
d
Recall that “  ” can be read “divided into groups of”. So, “12  3” represents “How many groups of 3 are
in 12?” or “How many are in each group if 12 objects are divided into 3 groups?”
Dividing Fractions by Using Cuisenaire Rods
1
a. 2 
(Explain what this means.)
6
2 1
b. 
3 6
10 5

c.
3 6
1 5
d. 2 
2 6
2
e. 3 
3
1 3
f. 2 
2 2
1
3
g.
2
a c
c
c
a
 : How many whole groups of and how many portions of are in ?
b d
b
d
d
Why does the invert and multiply rule work?
Dividing Fractions by Using Pattern Blocks
1
a. 3 
2
1
3
b.
2
2 1
c. 
3 6
5 2
d. 
3 3
2 1
e. 1  1
3 2
1
2
f.
3
Dividing Fractions by using Fraction Circles
1
a. 3 
4
1
4
b.
2
4 1
c. 
3 6
1 3
d. 1 
2 8
Ordering fractions
Traditional textbook methods
•Find a common denominator. This reflects the first of the intuitive techniques listed above. A representation
that gives both fractions a common denominator is found and the fraction with the larger numerator is the larger
fraction. For example, which is smaller 11/20 or 4/7?
11 77
80 4



20 140 140 7
Therefore, 11/20 < 4/7.

Convert to decimal. This method is not appropriate for students who haven’t studied decimals, but this is a
common method of comparing fractions once students understand the concept of fractions as representing
a
the division of the numerator by the denominator.  a  b
b
AN IMPORTANT PROPERTY
Fractions have an interesting property that natural numbers, whole numbers, and integers do not possess.
Consider this question. Name a whole number between 0 and 1? The answer, of course, is there are no whole
numbers between 0 and 1. But if the questions is changed to “Can you name a fraction between 0 and 1?”, the
answer is “yes”. What is amazing is that there are an infinite number of fractions between 0 and 1 and between
any two fractions in that interval, there are an infinite number of fractions. This concept is called the density
property.
Given any two numbers a and b in a set, if there exists another number c such that a < c < b, then the set of
numbers exhibits the density property.
A summary of concepts about fractions
•There are various ways to conceptualize and define fractions
–The most commonly used conceptualization in the K-8 curriculum is the part-whole
concept
•When using the part-whole definition of fractions, two things must be stressed.
–Students must understand what the ‘whole’ is
–Students must understand that the fraction must have equal sized parts of the ‘whole’
•There are an infinite number of representations for any single fraction amount – we call these
equivalent or equal fractions.
•A fraction is considered simplified if the numerator and denominators have no factors in
common, other than 1.
•There are intuitive and traditional methods for ordering fractions, both have value for students.
•Students need to be proficient with moving from improper fraction form to a mixed numeral
form.