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Day 1: Introduction
Fractions
 Research into student understanding of rational numbers indicates that
students who learn in a rich environment that requires problem solving,
conversations about mathematics, and higher cognitive demands learn well
(Lamon, 2001).
 Students who study rational numbers in an environment where learning is
by rote or limited meanings are less able to transfer or extend their learning.
 Fractions fall in a general category of reasoning called “Proportional
Reasoning” which has the components shown on the diagram below.
Figure 11: Components of Proportional Reasoning
 Researchers claim that it is impossible to teach proportional reasoning
directly because “… in short, the whole is greater than the sum of parts”
(Lemon, 1999).
 The development of proportional reasoning is not an all-or-nothing affair;
rather, competence in this kind of reasoning grows over a long period of time
and in several dimensions (Lamon, 1999).
 The operating theory for teaching proportional reasoning is that “by
providing children experiences with some of the critical components
proportional reasoning before proceeding to the more abstract, formal
presentations, we increase their chances of developing proportional
reasoning” (Lamon, 1999)
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Source: Lamon, S.J. (1999). Teaching Fractions and Ratios for Understanding.
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Interpretations of Fractions (example 4/5)2
Interpretation Explanation
Examples
Part-Whole
This interpretation involves partitioning a A circle partitioned into 5 equalwhole into equal parts
sized pieces. If 4 of the five parts
4
are shaded, then the of the
Area Model & Linear Model: —For
5
continuous quantities (e.g., line segment, circle is shown. The five tells the
circle, rectangle, cylinder) the partitioned number of equal-sized pieces into
pieces must have equal size.
which the whole is divided. The 4

tells the number
of pieces of
Set Model: — For discrete sets (a set of
interest.
individual objects), each partitioned sets
must contain the same number of objects. A set of 15 pattern blocks has 12
4
pieces circled. The refers to 4
5
of the 5 subsets.
4
Operator
An operator is a transformer that
of an amount can be found by
stretches or shrinks another value.
5
 amount by 5 and then
dividing the
multiplying it by 4 or multiplying
the amount by 4 then dividing by

4
5. For example of 55 can be
5
55
found by
 11 so 11 4  44
5
4
Ratio or Rate
A rational number may arise from
may
 represent 4 parts per 5
comparing quantities with like units (to
5
 or four dollars per 5
produce ratios) or quantities with unlike
parts (a ratio)

units (to produce rates)
candy bars (a rate).
Quotient
A rational number may be thought of as a 4
of a pizza is the amount each
division of the numerator by the 
5
denominator.
person receives when 5 people
share 4 pizzas. It is 4  5 .
Measure
A rational number may be thought of as a Four-fifths can be thought of as
 of
1
unit fraction (fraction with numerator
four measures of each.
1) repeated the number of times given in
5

the numerator.

2
Source: Rubestein, Beckmann, & Thompson. (2004). Teaching and Learning Middle
Grades Mathematics. Key College Publishing
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Area Model
Activity 1:
For this activity you are given five envelopes with three pieces of a square in the
envelope. Your task is to reassemble the five squares following these rules:
a. No talking is allowed.
b. You may give a piece to another student, and the only way you can obtain a
piece is from someone giving you his or hers.
c. After all the squares are completed, assign a fraction value to each piece. An
assembled square is one unit. Talking is allowed during this time. You should
not write on the pieces.
d. Sketch your squares and record the value of each piece below and explain
how you obtained this value.
#1
#2
#3
#4
#5
#6
#7
#8
#9
#10
#11
#12
#13
#14
#15
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Activity 2:
If this is the UNIT
This piece is:
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This piece is:
This piece is:
This piece is:
4
1
5

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2
5
Equivalent Fractions
a. Lay out the bar representing 1. Below it lay out the two halves, three thirds,
etc., as shown below, till you have used all the bars.
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i.
From your arrangement of fraction bars, list all of the fractions equal
1
to .
2

ii.
Use your arrangement to systematically list all other equivalent
1 2
fractions (e.g.,  )
3 6

iii.
What patterns do you see in the sets of equivalent fractions?
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b. Candy Bar Fractions: Meanings and Comparisons
A candy bar is sectioned into 12 equal-sized pieces as shown below.
i.
What fraction of the bar is one piece?
ii.
One row of the bar is what fraction of the bar? State this in two ways.
iii.
One column of the bar is what fraction of the bar? State this in two
ways.
iv.
How many pieces make
2
of the bar? Show that this amount is, in fact,
3
two measures of one-third each.

v.
3
of the bar? Show that this amount is, in fact,
4
three measures of one-fourth each.
How many pieces make

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vi.
Compare each pair of candy portions shown below. Insert symbols for
“less than” (<), “greater than” (>), or “equals” (=).
2
3
3
4

1
2

5
6
 vii.
viii.
1
2
3
2
3
7
12

1
2
3
10
6
5
6
2
3

9
4
1
3
4
2
9
2

to compare the fractions.

Explain strategies
you used
What mathematical concepts does this activity develop?
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c. Use pattern blocks to:
i.
Build a rectangle that is
1
green.
3

ii.
Build a rectangle that is

iii.

Build a rectangle that is

2
1
2
red, green and blue.
3
9
9

3
1
blue and green.
4
4

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iv.
Illustrate:
1 2
of
2 3

3 1
of
4 3

1 3
of
3 4

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Set Model
In an adult condominium complex, 2/3 of the men are married to 3/5 of the women.
Assuming the residents of the condominium are monogamous, find the fraction of
residents that are married.
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Ordering Fractions and Density Property of Fractions
Definition:
a
b
a b
Let and be any fractions (where c  0 ). Then,  if and only if a  b.
c
c
c c
Why does this work?





Examples: Insert the symbols <, >, or = appropriately:
1 2
3 18
4 5
2 3
4 24
5 6



8
9
4
5

Theorem: Cross Multiplication of Fraction Inequality
a
c
a c
Let and be any fractions (where b  0, and d  0). Then  if and only if
b
d
b d
ad  bc .



Why does this work?


Examples: Use this theorem to insert the symbols <, >, or = appropriately:
1
2

2
3
3
4

18
24
4
5

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6
8
9
4
5

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

Theorem:
a
c
a b
Let and be any fractions (where b  0, and d  0). Suppose  . Then
b
d
c c
a ac c

 .
b b d d


Examples: Find a fraction between each of the followingpairs of fractions
1 2
4 5
a.
b.
2 3
5 6


c.
8
9
4
5

d.
100
101
101
102

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