Difference between revisions of "Fractions"

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(Created page with "'''Scope of this document''' <br> <br> The following is a background literature for teachers. It summarises the things to be known to a teacher to teach this topic more ...")
 
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'''Scope of this document''' 
 
<br>
 
<br>
 
  
 +
= Introduction =
 
   
 
   
The following is a
+
The following is a background literature for teachers. It
background literature for teachers. It summarises the things to be
+
summarises the various concepts, approaches to be known to a teacher  
known to a teacher to teach this topic more effectively . This
+
to teach this topic effectively . This literature is meant to be a
literature is meant to be a ready reference for the teacher to
+
ready reference for the teacher to develop the concepts, inculcate
develop the concepts, inculcate necessary skills, and impart
+
necessary skills, and impart knowledge in fractions from Class 6 to
knowledge in fractions from Class 6 to Class 10.
+
Class 10.
  
 
   
 
   
It is a well known fact
+
It is a well known fact that teaching and learning fractions is a
that teaching and learning fractions is a complicated process in
+
complicated process in primary and middle school. Although much of
primary and middle school. Although much of fractions is covered in
+
fractions is covered in the middle school, if the foundation is not
the middle school, if the foundation is not holistic and conceptual,
+
holistic and conceptual, then topics in high school mathematics
then topics in high school mathematics become very tough to grasp.
+
become very tough to grasp. Hence this documents is meant to
Hence this documents is meant to understand the research that has
+
understand the research that has been done towards simplifying and
been done towards simplifying and conceptually understanding topics
+
conceptually understanding topics of fractions.
of fractions.
+
 
 +
 
 +
 
 +
= Mind Map =
 +
 +
[[Image:KOER%20Fractions_html_m700917.png]]
  
 
   
 
   
It is also very common
+
= Different Models for interpreting and teaching-learning fractions =
for the school system to treat themes in a separate manner. Fractions
 
are taught as stand alone chapters. In this resource book an attempt
 
to connect it to other middle school topics such as Ratio Proportion,
 
Percentage and high school topics such as rational, irrational
 
numbers and inverse proportions are made. These other topics are not
 
discussed in detail themselves, but used to show how to link these
 
other topics with the already understood concepts of fractions.
 
  
 +
== Introduction ==
 
   
 
   
Also commonly fractions
+
Commonly fractions are always approached by teaching it through
are always approached by teaching it through one model or
+
one model or interpretation namely the '''part-whole '''model
interpretation namely the '''part-whole '''model
 
 
where the '''whole '''is
 
where the '''whole '''is
 
divided into equal parts and the fraction represents one or more
 
divided into equal parts and the fraction represents one or more
 
of the parts. The limitations of this method, especially in
 
of the parts. The limitations of this method, especially in
 
explaining mixed fractions, multiplication and division of fractions
 
explaining mixed fractions, multiplication and division of fractions
be fractions has led to educators using other interpretations such as
+
has led to educators using other interpretations such as '''equal
'''equal share''' and
+
share''' and '''measure'''.
'''measure'''. These
+
These approaches to fraction teaching are discussed here.
approaches to fraction teaching are discussed.
 
  
 
   
 
   
Also
+
== Objectives ==
a brief understanding of the common errors that children make when it
+
comes to fractions are addressed to enable teachers to understand the
+
The objective of this section is to
child's levels of conceptual understanding to address the
+
enable teachers to visualise and interpret fractions in different
misconceptions.
+
ways in order to clarify the concepts of fractions using multiple
 +
methods. The idea is for teachers to be able to select the
 +
appropriate method depending on the context, children and class they
 +
are teaching to effectively understand fractions.
  
 
   
 
   
<br>
+
== Part-whole ==
<br>
+
 
 
 
   
 
   
= Syllabus =
+
The
           
+
most commonly used model is the part whole model where where the
{| border="1"
+
'''whole '''is
|-
+
divided into <u>equal</u>
|
+
parts and the fraction represents one or more of the parts.
'''Class 6'''
 
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_78a5005.gif]]
'''Class 7'''
 
  
 
   
 
   
|-
+
Half
|
+
(½) : The whole is divided into '''two
Fractions:
+
equal '''parts.
  
 
   
 
   
Revision of what a fraction is, Fraction as a
+
One part is coloured, this part
part of whole, Representation of fractions (pictorially and on
+
represents the fraction ½.
number line), fraction as a division, proper, improper &amp; mixed
 
fractions, equivalent fractions, comparison of fractions, addition
 
and subtraction of
 
  
 
   
 
   
fractions
+
[[Image:KOER%20Fractions_html_6fbd7fa5.gif]]
  
 
   
 
   
<br>
+
One-Fourth
<br>
+
(1/4) : The whole is divided into '''four
 +
equal '''parts.
  
 
   
 
   
Review of the idea of a decimal fraction, place
+
One part is coloured, this part represents the fraction ¼.
value in the context of decimal fraction, inter conversion of
 
fractions and decimal fractions comparison of two decimal
 
fractions, addition and subtraction of decimal fractions upto
 
100th place.
 
  
 
   
 
   
<br>
 
<br>
 
  
 
Word problems involving addition and
 
subtraction of decimals (two operations together on money,mass,
 
length, temperature and time)
 
  
 
   
 
   
<br>
+
[[Image:KOER%20Fractions_html_43b75d3a.gif]]
  
 
   
 
   
|
+
One
'''Fractions and rational numbers: '''
+
(2/2 or 1) : The whole is divided into '''two
 +
equal '''parts.
  
 
   
 
   
<br>
+
'''Two'''
 +
part are coloured, this part represents the fraction 2/2
  
 
   
 
   
Multiplication of fractions ,Fraction as an operator
+
which is equal to the whole or 1.
,Reciprocal of a fraction
 
  
 
   
 
   
Division of fractions ,Word problems involving mixed fractions
+
[[Image:KOER%20Fractions_html_2faaf16a.gif]]
  
 
   
 
   
Introduction to rational numbers (with representation on number
+
Two
line)
+
Fifth (2/5) : The whole is divided into '''five
 +
equal '''parts.
  
 
   
 
   
Operations on rational numbers (all operations)
+
'''Two'''
 +
part are coloured, this part represents the fraction 2/5.
  
 
   
 
   
Representation of rational number as a decimal.
+
 
  
 
   
 
   
Word problems on rational numbers (all operations)
+
 
  
 
   
 
   
Multiplication and division of decimal fractions
 
  
 
Conversion of units (lengths &amp; mass)
 
  
 
   
 
   
Word problems (including all operations)
+
[[Image:KOER%20Fractions_html_9e5c77.gif]]Three
 +
Seventh (3/7) : The whole is divided into '''seven
 +
equal '''parts.
  
 
   
 
   
<br>
+
'''Three'''
 +
part are coloured, this part represents the fraction 3/7.
  
 
   
 
   
'''Percentage-'''
 
  
 
<br>
 
  
 
   
 
   
An introduction w.r.t life situation.
 
  
 
'''Understanding percentage as a fraction with denominator 100'''
 
  
 
   
 
   
Converting fractions and decimals into percentage and
+
[[Image:KOER%20Fractions_html_m30791851.gif]]
vice-versa.
 
  
 
   
 
   
Application to profit &amp; loss (single transaction only)
 
  
 
Application to simple interest (time period
 
  
 
   
 
   
in complete years)
+
Seven
 +
tenth (7/10) : The whole is divided into '''ten
 +
equal '''parts.
  
 
   
 
   
|}
+
'''Seven'''
<br>
+
part are coloured, this part represents the fraction 7/10 .
<br>
 
  
 
   
 
   
= Concept Map =
 
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m8e7238e.jpg]]<br>
 
<br>
 
  
 
 
= Theme Plan =
 
 
<br>
 
<br>
 
  
                                                                                                                             
 
{| border="1"
 
|-
 
|
 
<br>
 
  
 
   
 
   
|
+
 
'''THEME PLAN FOR THE TOPIC
 
FRACTIONS'''
 
  
 
   
 
   
|
+
'''Terms Numerator
<br>
+
and Denominator and their meaning'''
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_3bf1fc6d.gif]]
<br>
 
  
 
   
 
   
|-
+
Three
|
+
Eight (3/8) The whole is divided into '''eight
'''CLASS'''
+
equal '''parts.
  
 
   
 
   
|
 
'''SUBTOPIC'''
 
  
 
|
 
'''CONCEPT <br>
 
DEVELOPMENT'''
 
  
 
   
 
   
|
+
'''Three'''
'''KNOWLEDGE'''
+
part are coloured, this part represents the fraction 3/8 .
  
 
   
 
   
|
 
'''SKILL'''
 
  
 
|
 
'''ACTIVITY'''
 
  
 
   
 
   
|-
+
3/8 is also written as
|
+
numerator/denominator. Here the number above the line- numerator
6
+
tells us '''HOW MANY PARTS''' are involved. It 'enumerates' or
 +
counts the coloured parts.
  
 
   
 
   
|
+
The number '''BELOW''' the line tells – denominator tells us
Introduction to Fractions
+
'''WHAT KIND OF PARTS''' the whole is divided into. It 'denominates'
 +
or names the parts.
  
 
   
 
   
|
+
 
A fraction is a part of a whole,
+
 
when the whole is divided into equal parts. Understand what the
 
numerator represents and what the denominator represents in a
 
fraction
 
  
 
   
 
   
|
+
The important factor to note here is '''WHAT IS THE WHOLE . '''In
Terms - Numerator and Denominator.
+
both the figures below the fraction quantity is 1/4. In fig 1 one
 +
circle is the whole and in fig 2, 4 circles is the whole.
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_1683ac7.gif]]
To be able to Identify/specify
 
fraction quantities from any whole unit that has been divided.
 
Locate a fraction on a number line.
 
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_729297ef.gif]][[Image:KOER%20Fractions_html_4282c1e5.gif]][[Image:KOER%20Fractions_html_6fbd7fa5.gif]]
ACTIVITY1
+
 
 +
 
 +
 
 +
 
 +
 
  
 
   
 
   
|-
+
== Equal Share ==
|
+
6
+
In the equal share interpretation the fraction '''m/n''' denotes
 +
one share when '''m identical things''' are '''shared equally among
 +
n'''. The relationships between fractions are arrived at by logical
 +
reasoning (Streefland, 1993). For example ''' 5/6 '''is the share of
 +
one child when 5 rotis (disk-shaped handmade bread) are shared
 +
equally among 6 children. The sharing itself can be done in more than
 +
one way and each of them gives us a relation between fractions. If we
 +
first distribute 3 rotis by dividing each into two equal pieces and
 +
giving each child one piece each child gets 1⁄2 roti. Then the
 +
remaining 2 rotis can be distributed by dividing each into three
 +
equal pieces giving each child a piece. This gives us the relations
 +
 
 +
 
 +
[[Image:KOER%20Fractions_html_3176e16a.gif]]
  
 
   
 
   
|
+
 
Proper and Improper Fractions
+
 
  
 
   
 
   
|
+
The relations 3/6 = 1⁄2 and 2/6 = 1/3 also follow from the
The difference between Proper and
+
process of distribution. Another way of distributing the rotis would
Improper. Know that a fraction can be represented as an Improper
+
be to divide the first roti into 6 equal pieces give one piece each
or mixed but have the same value.
+
to the 6 children and continue this process with each of the
 +
remaining 4 rotis. Each child gets a share of rotis from each of the
 +
5 rotis giving us the relation
  
 
   
 
   
|
+
 
Terms – proper, improper or mixed
+
 
fractions
 
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_m39388388.gif]]
Differentiate between proper and
 
improper fraction. Method to convert fractions from improper to
 
mixed representation
 
  
 
   
 
   
|
 
ACTIVITY2
 
  
+
 
|-
 
|
 
6
 
  
 
   
 
   
|
+
It is important to note here that the fraction symbols on both
Comparing Fractions
+
sides of the equation have been arrived at simply by a repeated
 +
application of the share interpretation and not by appealing to prior
 +
notions one might have of these fraction symbols. In the share
 +
interpretation of fractions, unit fractions and improper fractions
 +
are not accorded a special place.
  
 
   
 
   
|
+
Also converting an improper fraction to a mixed fraction becomes
Why do we need the concept of LCM
+
automatic. 6/5 is the share that one child gets when 6 rotis are
for comparing fractions
+
shared equally among 5 children and one does this by first
 +
distributing one roti to each child and then sharing the remaining 1
 +
roti equally among 5 children giving us the relation
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_m799c1107.gif]]
Terms to learn – Like and Unlike
 
Fractions
 
  
 
   
 
   
|
+
Share interpretation does not provide a direct method to answer
Recognize/identify like /unlike
+
the question ‘how much is the given unknown quantity’. To say
fractions. Method/Algorithm to enable comparing fractions
+
that the given unknown quantity is 3⁄4 of the whole, one has figure
 +
out that four copies of the given quantity put together would make
 +
three wholes and hence is equal to one share when these three wholes
 +
are shared equally among 4. '''''Share interpretation is also the
 +
quotient interpretation of fractions in the sense that 3⁄4 = 3 ÷ 4
 +
and this is important for developing students’ ability to solve
 +
problems involving multiplicative and linear functional relations. '''''
  
 
   
 
   
|
+
 
ACTIVITY3
+
 
  
 
   
 
   
|-
+
To understand the
|
+
equal share model better, use the GeoGebra file explaining the equal
6
+
share model available on [[http://rmsa.karnatakaeducation.org]].
 +
See figure below. Move the sliders m and n and see how the equal
 +
share model is interpreted.
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_17655b73.png]]
Equivalent Fractions
+
 
  
 
   
 
   
|
 
Why are fractions equivalent and not
 
equal
 
  
 +
 +
 +
 +
== Measure Model ==
 
   
 
   
|
+
Measure interpretation defines the unit fraction ''1/n ''as the
Know the term Equivalent Fraction
+
measure of one part when one whole is divided into ''n ''equal
 +
parts. The ''composite fraction'' ''m/n '' is as the measure of
 +
m such parts. Thus ''5/6 '' is made of 5 piece units of size ''1/5
 +
''each and ''6/5 ''is made of 6 piece units of size ''1/5''
 +
each. Since 5 piece units of size make a whole, we get the relation
 +
6/5 = 1 + 1/5.
  
 
   
 
   
|
+
Significance of measure interpretation lies in the fact that it
Method/Algorithm to enable comparing
+
gives a direct approach to answer the ‘how much’ question and the
fractions
+
real task therefore is to figure out the appropriate n so that
 +
finitely many pieces of size will be equal to a given quantity. In a
 +
sense then, the measure interpretation already pushes one to think in
 +
terms of infinitesimal quantities. Measure interpretation is
 +
different from the part whole interpretation in the sense that for
 +
measure interpretation we fix a certain unit of measurement which is
 +
the whole and the unit fractions are sub-units of this whole. The
 +
unit of measurement could be, in principle, external to the object
 +
being measured.
  
 
   
 
   
|
 
ACTIVITY4
 
  
+
 
|-
 
|
 
6
 
  
 
   
 
   
|
+
=== Introducing Fractions Using Share and Measure Interpretations ===
Addition of Fractions
 
 
 
 
   
 
   
|
+
One of the major difficulties a child faces with fractions is
Why do we need LCM to add fractions.
+
making sense of the symbol ''m/n''. In order to facilitate
Understand Commutative law w.r.t. Fraction addition
+
students’ understanding of fractions, we need to use certain
 +
models. Typically we use the area model in both the measure and share
 +
interpretation and use a circle or a rectangle that can be
 +
partitioned into smaller pieces of equal size. Circular objects like
 +
roti that children eat every day have a more or less fixed size. Also
 +
since we divide the circle along the radius to make pieces, there is
 +
no scope for confusing a part with the whole. Therefore it is
 +
possible to avoid explicit mention of the whole when we use a
 +
circular model. Also, there is no need to address the issue that no
 +
matter how we divide the whole into n equal parts the parts will be
 +
equal. However, at least in the beginning we need to instruct
 +
children how to divide a circle into three or five equal parts and if
 +
we use the circular model for measure interpretation, we would need
 +
ready made teaching aids such as the circular fraction kit for
 +
repeated use.
  
 
   
 
   
|
+
Rectangular objects (like cake) do not come in the same size and
Fraction addition Algorithm
+
can be divided into n equal parts in more than one way. Therefore we
 +
need to address the issues (i) that the size of the whole should be
 +
fixed (ii) that all 1⁄2’s are equal– something that children do
 +
not see readily. The advantage of rectangular objects is that we
 +
could use paper models and fold or cut them into equal parts in
 +
different ways and hence it easy to demonstrate for example that 3/5
 +
= 6/10 using the measure interpretation .
  
 
   
 
   
|
+
Though we expose children to the use of both circles and
Applying the Algorithm and adding
+
rectangles, from our experience we feel circular objects are more
fractions. Solving simple word problems
+
useful when use the share interpretation as children can draw as many
 +
small circles as they need and since the emphasis not so much on the
 +
size as in the share, it does not matter if the drawings are not
 +
exact. Similarly rectangular objects would be more suited for measure
 +
interpretation for, in some sense one has in mind activities such as
 +
measuring the length or area for which a student has to make repeated
 +
use of the unit scale or its subunits.
  
 
   
 
   
|
 
ACTIVITY5
 
  
 
|-
 
|
 
6
 
  
 
|
 
Subtraction of Fractions
 
  
 
   
 
   
|
+
== Activities ==
Why we need LCM to subtract
 
fractions.
 
 
 
 
   
 
   
|
+
=== Activity1: Introduction to fractions ===
Fraction subtraction Algorithm
 
 
 
 
   
 
   
|
+
'''''Learning
Applying the Algorithm and adding
+
Objectives '''''
fractions. Solving simple word problems
 
  
 
   
 
   
|
+
Introduce
ACTIVITY6
+
fractions using the part-whole method
  
 
   
 
   
|-
+
'''''Materials and
|
+
resources required '''''
6
 
  
 
   
 
   
|
 
Linking Fractions with Decimal
 
Number Representation
 
  
 
|
 
The denominator of a fraction is
 
always 10 and powers of 10 when representing decimal numbers as
 
fractions
 
  
 
|
 
Difference between integers and
 
decimals. Algorithm to convert decimal to fraction and vice versa
 
  
 
   
 
   
|
+
# Write the Number Name and the number of the picture like the example  [[Image:KOER%20Fractions_html_m1d9c88a9.gif]]Number Name = One third  Number: [[Image:KOER%20Fractions_html_52332ca.gif]]
Represent decimal numbers on the
 
number line. How to convert simple decimal numbers into fractions
 
and vice versa
 
 
 
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_2625e655.gif]][[Image:KOER%20Fractions_html_m685ab2.gif]][[Image:KOER%20Fractions_html_55c6e68e.gif]][[Image:KOER%20Fractions_html_mfefecc5.gif]][[Image:KOER%20Fractions_html_m12e15e63.gif]]
ACTIVITY7
 
  
+
   
|-
+
Question:
|
+
What is the value of the numerator and denominator in the last figure
6
+
, the answer is [[Image:KOER%20Fractions_html_m2dc8c779.gif]]
  
 
   
 
   
|
 
(Linking to Fraction Topic) Ratio &amp;
 
Proportion
 
  
 
|
 
What does it mean to represent a
 
ratio in the form of a fraction. The relationship between the
 
numerator and denominator – proportion
 
  
 
|
 
Terms Ratio and Proportion and link
 
them to the fraction representation
 
  
 
   
 
   
|
+
# Colour the correct amount that represents the fractions
Transition from Additive Thinking to
 
Multiplicative Thinking
 
 
 
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_19408cb.gif]][[Image:KOER%20Fractions_html_m12e15e63.gif]][[Image:KOER%20Fractions_html_m6b49c523.gif]][[Image:KOER%20Fractions_html_m6f2fcb04.gif]]
ACTIVITY8
 
  
 
|-
 
|
 
7
 
  
+
 
|
+
7/10 3/8
Multiplication of Fractions
+
1/5 4/7
  
 
   
 
   
|
+
Question:
Visualise the quantities when a
+
Before colouring count the number of parts in each figure. What does
fraction is multiplied 1) whole number 2) fraction. Where is
+
it represent. Answer: Denominator
multiplication of fractions used
 
  
 
   
 
   
|
 
“of” Operator means
 
multiplication. Know the fraction multiplication algorithm
 
  
 
|
 
Apply the algorithm to multiply
 
fraction by fraction
 
  
 
|
 
ACTIVITY9
 
  
 
   
 
   
|-
+
# Divide the circle into fractions and colour the right amount to show the fraction
|
 
7
 
 
 
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
Division of Fractions
 
  
 
|
 
Visualise the quantities when a
 
fraction is divided 1) whole number 2) fraction .Where Division of
 
fractions would be used 3) why is the fraction reversed and
 
multiplied
 
  
 
   
 
   
|
 
Fraction division algorithm
 
  
 
|
 
Apply the algorithm to divide
 
fraction by fraction
 
  
 
|
 
ACTIVITY10
 
  
 
   
 
   
|-
 
|
 
7
 
  
+
 
|
+
 
Linking Fractions with Percentage
+
 
 +
3/5
 +
6/7 1/3 5/8 2/5
  
 
   
 
   
|
 
The denominator of a fraction is
 
always 100.
 
  
+
 
|
 
Convert from fraction to percentage
 
and vice versa
 
  
 
   
 
   
|
+
# Draw the Fraction and observe which is the greater fraction – observe that the parts are equal for each pair
Convert percentage
 
 
 
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
ACTIVITY11
 
  
 
   
 
   
|-
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
|
 
8
 
  
 
   
 
   
|
 
(Linking to Fraction Topic) Inverse
 
Proportion
 
  
 
|
 
The relationship between the
 
numerator and denominator – for both direct and inverse
 
proportion
 
  
 
   
 
   
|
 
Reciprocal of a fraction
 
  
 
|
 
Determine if the ratio is directly
 
proportional or inversely proportional in word problems
 
  
 
   
 
   
|
+
1/3 2/3 4/5 2/5
ACTIVITY12
+
3/7 4/7
  
 
   
 
   
|-
 
|
 
8
 
  
 
|
 
(Linking
 
to Fraction Topic)
 
  
 
   
 
   
Rational &amp; Irrational Numbers
 
  
 +
 +
 +
# Draw the Fraction and observe which is the greater fraction – Observe that the parts are different sizes for each pair.
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
The number line is fully populated
 
with natural numbers, integers and irrational and rational numbers
 
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
Learn to recognize irrational and
 
rational numbers. Learn about some naturally important
 
irrational numbers. Square roots of prime numbers are
 
irrational numbers
 
  
 
   
 
   
|
+
 
How to calculate the square roots of
 
a number. The position of an irrational number is definite
 
but cannot be determined accurately
 
  
 
   
 
   
|
+
 
ACTIVITY13
 
  
 
   
 
   
|}
 
<br>
 
<br>
 
  
 
<br>
 
<br>
 
  
 
   
 
   
= Curricular Objectives =
 
 
# Conceptualise and understand algorithms for basic operations (addition, subtraction, multiplication and division) on fractions.
 
# Apply the understanding of fractions as simple mathematics models.
 
# Understand the different mathematical terms associated with fractions.
 
# To be able to see multiple interpretations of fractions such as in measurement, ratio and proportion, quotient, representation of decimal numbers, percentages, understanding rational and irrational numbers.
 
 
= Different Models used for Learning Fractions =
 
 
== Part-Whole ==
 
 
The
 
most commonly used model is the part whole model where where the
 
'''whole '''is
 
divided into <u>equal</u>
 
parts and the fraction represents one or more of the parts.
 
  
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_78a5005.gif]]<br>
 
  
 
   
 
   
Half
+
1/3
(½) : The whole is divided into '''two
+
1/4 1/5 1/8
equal '''parts.
+
1/6 1/2
  
 
   
 
   
One part is coloured, this part represents the fraction ½.
 
  
 
<br>
 
  
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_6fbd7fa5.gif]]<br>
 
  
 
   
 
   
One-Fourth
+
# Solve these word problems by drawing
(1/4) : The whole is divided into '''four
+
## Amar divided an apple into 8 equal pieces. He ate 5 pieces. He put the a  other 3 in a box. What fraction did Amar eat?
equal '''parts.
+
## There are ten biscuits in the box. 3 are cream biscuits. 2 are salt biscuits. 4 are chocolate biscuits. 1 is a sugar biscuit. What fraction of the biscuits in the box are salt biscuits.
 
+
## Radha has 6 pencils. She gives one to Anil and he gives one to Anita. She keeps the  rest. What fraction of her pencils did she give away?
 +
#
 +
# The circles in the box represent the whole; colour the right amount to show the fraction [[Image:KOER%20Fractions_html_m78f3688a.gif]]''Hint: Half is 2 circles''   [[Image:KOER%20Fractions_html_m867c5c2.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
 
   
 
   
One part is coloured, this part represents the fraction ¼.
 
  
 
<br>
 
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_43b75d3a.gif]]<br>
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
  
 
One
 
(2/2 or 1) : The whole is divided into '''two
 
equal '''parts.
 
  
 
   
 
   
'''Two'''
 
part are coloured, this part represents the fraction 2/2
 
  
 
which is equal to the whole or 1.
 
  
 
<br>
 
  
 
   
 
   
<br>
 
  
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_2faaf16a.gif]]Two
 
Fifth (2/5) : The whole is divided into '''five
 
equal '''parts.
 
  
 
   
 
   
'''Two'''
+
'''''Pre-requisites/
part are coloured, this part represents the fraction 2/5.
+
Instructions Method '''''
  
 
   
 
   
<br>
 
  
 
<br>
 
  
 
   
 
   
<br>
+
Do
 
+
the six different sections given in the activity sheet. For each
 +
section there is a discussion point or question for a teacher to ask
 +
children.
 +
 
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_9e5c77.gif]]Three
+
After
Seventh (3/7) : The whole is divided into '''seven
+
the activity sheet is completed, please use the evaluation questions
equal '''parts.
+
to see if the child has understood the concept of fractions
  
 
   
 
   
'''Three'''
+
'''''Evaluation'''''
part are coloured, this part represents the fraction 3/7.
 
  
 +
 
 +
=== Activity 2: Proper and Improper Fractions ===
 
   
 
   
<br>
+
'''''Learning
 +
Objectives'''''
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m30791851.gif]]<br>
+
Proper and Improper Fractions
  
 
   
 
   
Seven
+
'''''Materials
tenth (7/10) : The whole is divided into '''ten
+
and resources required '''''
equal '''parts.
 
  
 
   
 
   
'''Seven'''
+
# [[Image:KOER%20Fractions_html_5518d221.jpg]]If you want to understand proper fraction , example 5/6. In the equal share model , 5/6 represents the share that each child gets when 5 rotis are divided among 6 children '''equally.'''
part are coloured, this part represents the fraction 7/10 .
+
 +
[[Image:KOER%20Fractions_html_5e906d5b.jpg]][[Image:KOER%20Fractions_html_5518d221.jpg]]
  
 
<br>
 
<br>
 
  
 
   
 
   
<br>
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
  
 
'''Terms Numerator and Denominator and their meaning'''
 
  
 
   
 
   
<br>
 
  
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_3bf1fc6d.gif]]Three
 
Eight (3/8) The whole is divided into '''eight
 
equal '''parts.
 
  
 
<br>
 
  
 
   
 
   
'''Three'''
+
[[Image:KOER%20Fractions_html_5518d221.jpg]][[Image:KOER%20Fractions_html_5e906d5b.jpg]]
part are coloured, this part represents the fraction 3/8 .
 
  
 
<br>
 
  
 
   
 
   
3/8
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
is also written as '''numerator/denominator.
 
'''Here
 
the number above the line- numerator tells us '''HOW
 
MANY PARTS '''are
 
involved. It 'enumerates' or counts the coloured parts.
 
  
 
The number BELOW the
 
line tells – denominator tells us '''WHAT KIND OF PARTS '''the
 
whole is divided into. It 'denominates' or names the parts.
 
  
 
   
 
   
<br>
+
[[Image:KOER%20Fractions_html_5518d221.jpg]]
 +
 
  
 
   
 
   
== Equal Share ==
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]]
 +
 
 +
 
 
   
 
   
<br>
+
 
<br>
+
 
  
 
   
 
   
In the equal share
 
interpretation the fraction '''m/n''' denotes one share when '''m
 
identical things''' are '''shared equally among n'''. The
 
relationships between fractions are arrived at by logical reasoning
 
(Streefland, 1993). For example ''' 5/6 '''is the share of one child
 
when 5 rotis (disk-shaped handmade bread) are shared equally among 6
 
children. The sharing itself can be done in more than one way and
 
each of them gives us a relation between fractions. If we first
 
distribute 3 rotis by dividing each into two equal pieces and giving
 
each child one piece each child gets 1⁄2 roti. Then the remaining 2
 
rotis can be distributed by dividing each into three equal pieces
 
giving each child a piece. This gives us the relations
 
  
 
+
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_3176e16a.gif]]
 
  
 
   
 
   
<br>
+
 
<br>
+
 
  
 
   
 
   
The relations 3/6 = 1⁄2
+
# If you want to understand improper fraction , example 8/3. In the equal share model , 8/3 represents the share that each child gets when 8 rotis are divided among 3 children equally. The child in this case will usually distribute 2 full rotis to each child and then try to divide the remaining rotis. At this point you can show the mixed fraction representation as 2 2/3
and 2/6 = 1/3 also follow from the process of distribution. Another
+
way of distributing the rotis would be to divide the first roti into
+
 
6 equal pieces give one piece each to the 6 children and continue
+
 
this process with each of the remaining 4 rotis. Each child gets a
 
share of rotis from each of the 5 rotis giving us the relation
 
  
 
   
 
   
<br>
+
 
<br>
+
 
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m39388388.gif]]
+
 
 +
 
  
 
   
 
   
<br>
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_5e906d5b.jpg]][[Image:KOER%20Fractions_html_55f65a3d.gif]]
<br>
+
 
  
 
   
 
   
It is important to note
+
[[Image:KOER%20Fractions_html_5518d221.jpg]]
here that the fraction symbols on both sides of the equation have
+
 
been arrived at simply by a repeated application of the share
 
interpretation and not by appealing to prior notions one might have
 
of these fraction symbols. In the share interpretation of fractions,
 
unit fractions and improper fractions are not accorded a special
 
place.
 
  
 
   
 
   
Also converting an
+
 
improper fraction to a mixed fraction becomes automatic. 6/5 is the
+
 
share that one child gets when 6 rotis are shared equally among 5
 
children and one does this by first distributing one roti to each
 
child and then sharing the remaining 1 roti equally among 5 children
 
giving us the relation
 
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m799c1107.gif]]
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_5518d221.jpg]]
 +
 
  
 
   
 
   
Share interpretation
+
[[Image:KOER%20Fractions_html_55f65a3d.gif]]
does not provide a direct method to answer the question ‘how much
+
 
is the given unknown quantity’. To say that the given unknown
 
quantity is 3⁄4 of the whole, one has figure out that four copies
 
of the given quantity put together would make three wholes and hence
 
is equal to one share when these three wholes are shared equally
 
among 4. '''''Share ''''''''''interpretation is also the
 
quotient interpretation of fractions in the sense that 3⁄4 = 3 ÷ 4
 
and this is important for developing students’ ability to solve
 
problems involving multiplicative and linear functional relations. '''''
 
  
 
   
 
   
<br>
 
<br>
 
  
 
'''Introducing Fractions
 
Using Share and Measure Interpretations '''
 
  
 
One of the major
 
difficulties a child faces with fractions is making sense of the
 
symbol ''m/n''. In order to facilitate students’ understanding
 
of fractions, we need to use certain models. Typically we use the
 
area model in both the measure and share interpretation and use a
 
circle or a rectangle that can be partitioned into smaller pieces of
 
equal size. Circular objects like roti that children eat every day
 
have a more or less fixed size. Also since we divide the circle along
 
the radius to make pieces, there is no scope for confusing a part
 
with the whole. Therefore it is possible to avoid explicit mention of
 
the whole when we use a circular model. Also, there is no need to
 
address the issue that no matter how we divide the whole into n
 
equal parts the parts will be equal. However, at least in the
 
beginning we need to instruct children how to divide a circle into
 
three or five equal parts and if we use the circular model for
 
measure interpretation, we would need ready made teaching aids such
 
as the circular fraction kit for repeated use.
 
  
 
   
 
   
Rectangular objects
+
'''Pre-requisites/
(like cake) do not come in the same size and can be divided into n
+
Instructions Method '''
equal parts in more than one way. Therefore we need to address the
 
issues (i) that the size of the whole should be fixed (ii) that all
 
1⁄2’s are equal– something that children do not see readily.
 
The advantage of rectangular objects is that we could use paper
 
models and fold or cut them into equal parts in different ways and
 
hence it easy to demonstrate for example that 3/5 = 6/10 using the
 
measure interpretation .
 
  
 
   
 
   
Though we expose
+
Examples of Proper and improper
children to the use of both circles and rectangles, from our
+
fractions are given. The round disks in the figure represent rotis
experience we feel circular objects are more useful when use the
+
and the children figures represent children. Cut each roti and each
share interpretation as children can draw as many small circles as
+
child figure and make the children fold, tear and equally divide the
they need and since the emphasis not so much on the size as in the
+
roits so that each child figure gets equal share of roti.
share, it does not matter if the drawings are not exact. Similarly
 
rectangular objects would be more suited for measure interpretation
 
for, in some sense one has in mind activities such as measuring the
 
length or area for which a student has to make repeated use of the
 
unit scale or its subunits.
 
  
 
   
 
   
== Measure Model ==
+
 
 
Measure interpretation
 
defines the unit fraction ''1/n ''as the measure of one part when
 
one whole is divided into ''n ''equal parts. The ''composite
 
fraction'' ''m/n '' is as the measure of m such parts. Thus ''5/6
 
'' is made of 5 piece units of size ''1/5 ''each and ''6/5 ''is
 
made of 6 piece units of size ''1/5'' each. Since 5 piece units of
 
size make a whole, we get the relation 6/5 = 1 + 1/5.
 
  
 
   
 
   
Significance of measure
+
'''''Evaluation'''''
interpretation lies in the fact that it gives a direct approach to
 
answer the ‘how much’ question and the real task therefore is to
 
figure out the appropriate n so that finitely many pieces of size
 
will be equal to a given quantity. In a sense then, the measure
 
interpretation already pushes one to think in terms of infinitesimal
 
quantities. Measure interpretation is different from the part whole
 
interpretation in the sense that for measure interpretation we fix a
 
certain unit of measurement which is the whole and the unit fractions
 
are sub-units of this whole. The unit of measurement could be, in
 
principle, external to the object being measured.
 
  
 
   
 
   
== Key vocabulary: ==
+
# What happens when the numerator and denominator are the same, why ?
 +
# What happens when the numerator is greater than the denominator why ? How can we represent this in two ways ?
 
   
 
   
<br>
+
 
<br>
+
 
  
 
   
 
   
# 1. (a) A '''fraction''' is a number representing a part of a whole. The whole may be a single object or a group of objects.  (b) When expressing a situation of counting parts to write a fraction, it must be ensured that all parts are equal.
+
=== Activity 3: Comparing Fractions ===
# In [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m2988e86b.gif]], 5 is called the '''numerator''' and 7 is called the '''denominator'''.
 
# Fractions can be shown on a number line. Every fraction has a point associated with it on the number line.
 
# In a '''proper fraction''', the numerator is less than the denominator. The fractions, where the numerator is greater than the denominator are called '''improper fractions'''.
 
# An improper fraction can be written as a combination of a whole and a part, and such fraction then called '''mixed fractions'''.
 
# Each proper or improper fraction has many '''equivalent fractions'''. To find an equivalent fraction of a given fraction, we may multiply or divide both the numerator and the denominator of the given fraction by the same number.
 
# A fraction is said to be in the simplest (or lowest) form if its numerator and the denominator have no common factor except 1.
 
 
   
 
   
== Additional resources : ==
+
'''''Learning
 +
Objectives'''''
 +
 
 
   
 
   
<br>
+
Comparing-Fractions
<br>
 
  
 
   
 
   
# [[http://vimeo.com/22238434]] Video on teaching fractions using the equal share method made by Eklavya an NGO based in Madhya Pradesh, India
+
'''''Materials
# [[http://mathedu.hbcse.tifr.res.in/]] Mathematics resources from Homi Baba Centre for Science Education
+
and resources required '''''
 +
 
 
   
 
   
<br>
+
 
<br>
+
 
  
 
   
 
   
<br>
+
 
<br>
+
 
  
 
   
 
   
= Errors with fractions =
+
[[http://www.superteacherworksheets.com/fractions/comparing-fractions.pdf]]
 +
 
 
   
 
   
When
+
[[http://www.superteacherworksheets.com/fractions/comparing-fractions2.pdf]]
fractions are operated erroneously like natural numbers, i.e.
 
treating the numerator and the denominators separately and not
 
considering the relationship between the numerator and the
 
denominator is termed as N-Distracter. For example 1/3 + ¼ are
 
added to result in 2/7. Here 2 units of the numerator are added and 3
 
&amp; four units of the denominator are added. This completely
 
ignores the relationship between the numerator and denominator of
 
each of the fractions. Streefland (1993) noted this challenge as
 
N-distrators and a slow-down of learning when moving from the
 
'''concrete level to the abstract level'''.
 
  
 
   
 
   
<br>
+
'''Pre-requisites/
 +
Instructions Method'''
  
 
   
 
   
The
+
Print the
five levels of resistance to N-Distracters that a child develops are:
+
document and work out the
 +
activity sheet
  
 
   
 
   
<br>
+
'''''Evaluation'''''
  
 
   
 
   
# '''''Absence of cognitive conflict:''''' The child is unable to recognize the error even when she sees the same operation performed resulting in a correct answer. The child thinks both the answers are the same in spite of different results. Eg. ½ + ½ she erroneously calculated as 2/4. But when the child by some other method, say, through manipulatives (concrete) sees ½ + ½ = 1 does not recognize the conflict.
+
# Does the child know the symbols '''&gt;, &lt;''' and '''='''
# '''''Cognitive conflict takes place: '''''The student sees a conflict when she encounters the situation described in level 1 and rejects the ½+1/2 = 2/4 solution and recognizes it as incorrect. She might still not have a method to arrive at the correct solution.
+
# What happens to the size of the part when the denominator is different ?
# '''''Spontaneous refutation of N-Distracter errors:''''' The student may still make N-Distracter errors, but is able to detect the error for herself. This detection of the error may be followed by just rejection or explaining the rejection or even by a correct solution.
+
# Does it decrease or increase when the denominator becomes larger ?
# '''''Free of N-Distracter: '''''The written work is free of N-Distracters. This could mean a thorough understanding of the methods/algorithms of manipulating fractions.
+
# Can we compare quantities when the parts are different sizes ?
# '''''Resistance to N-Distracter: '''''The student is completely free (conceptually and algorithmically) of N-Distracter errors.
+
# What should we do to make the sizes of the parts the same ?
 
   
 
   
<br>
 
  
 +
 +
 +
 +
=== Activity 4: Equivalent Fractions ===
 
   
 
   
<br>
+
'''''Learning
 +
Objectives'''''
  
 
   
 
   
== Key vocabulary: ==
+
To understand Equivalent Fractions
 +
 
 
   
 
   
<br>
+
'''''Materials
<br>
+
and resources required'''''
  
 
   
 
   
# '''N-Distractor''': as defined above.
+
[[http://www.superteacherworksheets.com/fractions/fractions-matching-game.pdf]]
 +
 
 
   
 
   
== Additional resources: ==
+
 
 +
 
 +
 
 
   
 
   
<br>
+
'''Pre-requisites/
<br>
+
Instructions Method'''
  
 
   
 
   
# [[www.merga.net.au/publications/counter.php?pub=pub_conf&id=1410]] A PDF Research paper titled Probing Whole Number Dominance with Fractions.
+
Print 10 copies
# [[www.merga.net.au/documents/RP512004.pdf]] A PDF research paper titled “Why You Have to Probe to Discover What Year 8 Students Really Think About Fractions ”
+
of the document from pages 2 to 5 fractions-matching-game
# [[http://books.google.com/books?id=Y5Skj-EA2_AC&pg=PA251&lpg=PA251&dq=streefland+fractions&source=bl&ots=aabKa]][[ciwrA&sig=DcM0mi7r1GJlTUbZVq9J0l53Lrc&hl=en&ei=0xJBTvLjJ8bRrQfC3JmyBw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false]] A google book Fractions in realistic mathematics education: a paradigm of developmental By Leen Streefland
+
Cut the each fraction part. Play memory game as described in
+
the document in groups of 4 children.
= Operations on Fractions =
 
 
== Addition and Subtraction ==
 
 
<br>
 
  
 
   
 
   
Adding and subtracting like fractions is simple. It must be
+
'''''Evaluation'''''
emphasised thought even during this process that the parts are equal
 
in size or quantity because the denominator is the same and hence for
 
the result we keep the common denominator and add the numerators.
 
  
 
   
 
   
<br>
+
# What is reducing a fraction to the simplest form ?
 
+
# What is GCF – Greatest Common Factor ?
 +
# Use the document [[simplifying-fractions.pdf]]
 +
# Why are fractions called equivalent and not equal.
 
   
 
   
Adding and subtracting unlike fractions requires the child to
+
== Evaluation ==
visually understand that the parts of each of the fractions are
 
differing in size and therefore we need to find a way of dividing the
 
whole into equal parts so that the parts of all of the fractions
 
look equal. Once this concept is established, the terms LCM and the
 
methods of determining them may be introduced.
 
 
 
 
   
 
   
<br>
+
== Self-Evaluation ==
 
 
 
   
 
   
<br>
+
== Further Exploration ==
 
+
 
 +
== Enrichment Activities ==
 
   
 
   
== Multiplication ==
+
= Errors with fractions =
 
   
 
   
<br>
+
== Introduction ==
 
 
 
   
 
   
Multiplying a fraction by a whole number: Here the repeated addition
+
A brief
logic of multiplying whole numbers is still valid. 1/6 multiplied by
+
understanding of the common errors that children make when it comes
4 is 4 times 1/6 which is equal to 4/6.
+
to fractions are addressed to enable teachers to understand the
 +
child's levels of conceptual understanding to address the
 +
misconceptions.
  
 
   
 
   
<br>
+
== Objectives ==
 
 
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_714bce28.gif]]
+
When fractions are operated erroneously
 +
like natural numbers, i.e. treating the numerator and the
 +
denominators separately and not considering the relationship between
 +
the numerator and the denominator is termed as N-Distractor. For
 +
example 1/3 + ¼ are added to result in 2/7. Here 2 units of the
 +
numerator are added and 3 &amp; four units of the denominator are
 +
added. This completely ignores the relationship between the numerator
 +
and denominator of each of the fractions. Streefland (1993) noted
 +
this challenge as N-distractors and a slow-down of learning when
 +
moving from the '''concrete level to the abstract level'''.
  
 
   
 
   
<br>
 
  
 
Multiplying a fraction by a fraction: In this case the child is
 
confused as repeated addition does not make sense. To make a child
 
understand the ''of operator ''we
 
can use the language and demonstrate it using the measure model and
 
the area of a rectangle.
 
  
 
   
 
   
<br>
 
  
 
The area of a rectangle is found by
 
multiplying side length by side length. For example, in the rectangle
 
below, the sides are 3 units and 9 units, and the area is 27 square
 
units.
 
  
 
   
 
   
<br>
+
== N-Distractors ==
 
 
 
   
 
   
<br>
 
  
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m66ce78ea.gif]]<br>
 
  
 
   
 
   
<br>
+
The five levels of resistance to
 +
N-Distractors that a child develops are:
  
 
   
 
   
<br>
 
  
 
<br>
 
  
 
   
 
   
<br>
+
# '''''Absence of cognitive conflict:''''' The child is unable to recognize the error even when she sees the same operation performed resulting in a correct answer. The child thinks both the answers are the same in spite of different results. Eg. ½ + ½ she erroneously calculated as 2/4. But when the child by some other method, say, through manipulatives (concrete) sees ½ + ½ = 1 does not recognize the conflict.
 
+
# '''''Cognitive conflict takes place: '''''The student sees a conflict when she encounters the situation described in level 1 and rejects the ½+1/2 = 2/4 solution and recognizes it as incorrect. She might still not have a method to arrive at the correct solution.
 +
# '''''Spontaneous refutation of N-Distractor errors:''''' The student may still make N-Distractor errors, but is able to detect the error for herself. This detection of the error may be followed by just rejection or explaining the rejection or even by a correct solution.
 +
# '''''Free of N-Distractor: '''''The written work is free of N-Distractors. This could mean a thorough understanding of the methods/algorithms of manipulating fractions.
 +
# '''''Resistance to N-Distractor: '''''The student is completely free (conceptually and algorithmically) of N-Distractor errors.
 
   
 
   
<br>
 
  
 
<br>
 
  
 
   
 
   
<br>
+
== Activities ==
 
 
 
   
 
   
We can apply that idea to fractions, too.
+
== Evaluation ==
 
 
 
   
 
   
* The one side of the rectangle is 1 unit (in terms of length).
+
== Self-Evaluation ==
* The other side is 1 unit also.
 
* The whole rectangle also is ''1 square unit'', in terms of area.
 
 
   
 
   
<br>
+
== Further Exploration ==
 
+
 +
# [[www.merga.net.au/publications/counter.php?pub=pub_conf&id=1410]] A PDF Research paper titled Probing Whole Number Dominance with Fractions.
 +
# [[www.merga.net.au/documents/RP512004.pdf]] A PDF research paper titled “Why You Have to Probe to Discover What Year 8 Students Really Think About Fractions ”
 +
# [[http://books.google.com/books?id=Y5Skj-EA2_AC&pg=PA251&lpg=PA251&dq=streefland+fractions&source=bl&ots=aabKaciwrA&sig=DcM0mi7r1GJlTUbZVq9J0l53Lrc&hl=en&ei=0xJBTvLjJ8bRrQfC3JmyBw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false]] A google book Fractions in realistic mathematics education: a paradigm of developmental ...By Leen Streefland
 +
 +
== Enrichment Activities ==
 
   
 
   
See figure below to see how the following multiplication can be
 
shown.
 
  
 
<br>
 
  
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6c9f1742.gif]]
 
  
 
   
 
   
<br>
+
= Operations on Fractions =
 
 
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_753005a4.gif]]
+
== Introduction ==
 
 
 
   
 
   
<br>
+
This topic introduces the different operations on fractions. When
 +
learners move from whole numbers to fractions, many of the operations
 +
are counter intuitive. This section aims to clarify the concepts
 +
behind each of the operations.
  
 
   
 
   
<br>
+
== Objectives ==
 
 
 
   
 
   
<br>
+
The aim of this section is to visualise and conceptually
 +
understand each of the operations on fractions.
  
 
   
 
   
<br>
+
== Addition and Subtraction ==
 
 
 
   
 
   
<br>
 
  
 
<br>
 
  
 
   
 
   
<br>
+
Adding and
 +
subtracting like fractions is simple. It must be emphasised thought
 +
even during this process that the parts are equal in size or quantity
 +
because the denominator is the same and hence for the result we keep
 +
the common denominator and add the numerators.
  
 
   
 
   
<br>
 
  
 
<br>
 
  
 
   
 
   
<br>
+
Adding and
 +
subtracting unlike fractions requires the child to visually
 +
understand that the parts of each of the fractions are differing in
 +
size and therefore we need to find a way of dividing the whole into
 +
equal parts so that the parts of all of the fractions look equal.
 +
Once this concept is established, the terms LCM and the methods of
 +
determining them may be introduced.
  
 
   
 
   
<br>
 
  
 
'''Remember:
 
'''The two fractions to multiply
 
represent the length of the sides, and the answer fraction represents
 
area.
 
  
 
   
 
   
<br>
 
  
 
<br>
 
  
 
   
 
   
== Division ==
+
== Multiplication ==
 
   
 
   
<br>
+
 
  
 
   
 
   
Dividing
+
Multiplying a
a fraction by a whole number can be demonstrated just like division
+
fraction by a whole number: Here the repeated addition logic of
of whole numbers. When we divide 3/4 by 2 we can visualise it as
+
multiplying whole numbers is still valid. 1/6 multiplied by 4 is 4
dividing 3 parts of a whole roti among 4 people.
+
times 1/6 which is equal to 4/6.
  
 
   
 
   
<br>
+
 
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_1f617ac8.gif]]
+
[[Image:KOER%20Fractions_html_714bce28.gif]]
  
 
   
 
   
Here
+
 
3/4 is divided between two people. One fourth piece is split into
 
two.<br>
 
Each person gets 1/4 and 1/8.
 
  
 
   
 
   
<br>
+
Multiplying a
 +
fraction by a fraction: In this case the child is confused as
 +
repeated addition does not make sense. To make a child understand the
 +
''of operator ''we can use the
 +
language and demonstrate it using the measure model and the area of
 +
a rectangle.
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m5f26c0a.gif]]
 
  
 
<br>
 
  
 
   
 
   
OR
+
The
 +
area of a rectangle is found by multiplying side length by side
 +
length. For example, in the rectangle below, the sides are 3 units
 +
and 9 units, and the area is 27 square units.
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m25efcc2e.gif]]<br>
 
  
 
Another
 
way of solving the same problem is to split each fourth piece into 2.
 
  
 
   
 
   
This
 
means we change the 3/4 into 6/8.
 
  
 
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_7ed8164a.gif]]
 
  
 
   
 
   
<br>
+
[[Image:KOER%20Fractions_html_m66ce78ea.gif]]
  
 
   
 
   
<br>
 
  
 
<br>
 
  
 
   
 
   
When
 
dividing a fraction by a fraction, we use the measure interpretation.
 
  
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m3192e02b.gif]]<br>
 
  
 
   
 
   
When
+
 
we divide 2 by ¼ we ask how many times does ¼ fit into 2
 
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m257a1863.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m257a1863.gif]]<br>
+
 
  
 
   
 
   
<br>
 
  
 
 
<br>
 
  
 
   
 
   
<br>
+
 
  
 
   
 
   
It
+
 
fits in 4 times in each roti, so totally 8 times.
 
  
 
   
 
   
<br>
+
We can apply that
 +
idea to fractions, too.
  
 
   
 
   
We
+
* The one side of the rectangle is 1 unit (in terms of length).
write it as
+
* The other side is 1 unit also.
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m390fcce6.gif]]
+
* The whole rectangle also is ''1 square unit'', in terms of area.
 +
 +
 
  
 
   
 
   
<br>
+
See figure below
<br>
+
to see how the following multiplication can be shown.
  
 
   
 
   
<br>
+
 
<br>
 
  
 
   
 
   
== Key vocabulary: ==
+
[[Image:KOER%20Fractions_html_m6c9f1742.gif]]
 +
 
 
   
 
   
<br>
+
 
<br>
 
  
 
   
 
   
# '''Least Common Multiple: '''In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers ''a'' and ''b'', usually denoted by LCM(''a'', ''b''), is the smallest positive integer that is a multiple of both ''a'' and ''b''. It is familiar from grade-school arithmetic as the "lowest common denominator" that must be determined before two fractions can be added.
+
[[Image:KOER%20Fractions_html_753005a4.gif]]
 +
 
 
   
 
   
# '''Greatest Common Divisor:''' In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.
+
 
 +
 
 
   
 
   
== Additional resources: ==
+
 
 +
 
 
   
 
   
<br>
+
 
<br>
 
  
 
   
 
   
# [[http://www.youtube.com/watch?v=41FYaniy5f8]] detailed conceptual understanding of division by fractions
+
 
# [[http://www.homeschoolmath.net/teaching/f/understanding_fractions.php]] understanding fractions
+
 
# [[http://www.geogebra.com]] Understand how to use Geogebra a mathematical computer aided tool
 
# [[http://www.superteacherworksheets.com]] Worksheets in mathematics for teachers to use
 
 
   
 
   
= Linking Fractions to other Topics =
+
'''Remember: '''The
 +
two fractions to multiply represent the length of the sides, and the
 +
answer fraction represents area.
 +
 
 
   
 
   
== Decimal Numbers ==
+
 
 +
 
 
   
 
   
“Decimal”
 
comes from the Latin root '''''decem''''',
 
which simply means ten. The number system we use is called the
 
decimal number system, because the place value units go in tens: you
 
have
 
ones, tens, hundreds, thousands, and so on, each unit being 10 times
 
the previous one.
 
  
 +
 +
 +
== Division ==
 
   
 
   
In
+
 
common language, the word “decimal number” has come to mean
 
numbers which have digits after the decimal point, such as 5.8 or
 
9.302. But in reality, any number within the decimal number system
 
could be termed a decimal number, including whole numbers such as 12
 
or 381.
 
  
 
   
 
   
<br>
+
Dividing a fraction by a whole number
<br>
+
can be demonstrated just like division of whole numbers. When we
 +
divide 3/4 by 2 we can visualise it as dividing 3 parts of a whole
 +
roti among 4 people.
  
 
   
 
   
The
+
[[Image:KOER%20Fractions_html_1f617ac8.gif]]
simplest way to link or connect fractions to the decimal number
 
system is with the number line representation. Any scale that a
 
child uses is also very good for this purpose, as seen in the figure
 
below.
 
  
 
   
 
   
<br>
+
Here 3/4 is divided between two
<br>
+
people. One fourth piece is split into two. Each person gets
 +
1/4 and 1/8.
  
 
   
 
   
The
+
 
number line between 0 and 1 is divided into ten parts. Each of these
 
ten parts is '''1/10''', a '''tenth'''.
 
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_3d7b669f.gif]]<br>
+
[[Image:KOER%20Fractions_html_m5f26c0a.gif]]
<br>
 
  
 
   
 
   
Under
+
 
the tick marks you see decimal numbers such as 0.1, 0.2, 0.3, and so
+
 
on. These are the same numbers as the fractions 1/10, 2/10, 3/10 and
+
so on.
+
OR
  
 
   
 
   
We
+
[[Image:KOER%20Fractions_html_m25efcc2e.gif]]
can write any fraction with '''tenths (denominator 10) '''using the
 
decimal point. Simply write after the decimal point how many tenths
 
the number has. 0.6 means six tenths or 1/6. 1.5 means 1 whole and 5
 
tenths or
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m7f1d448c.gif]]
 
  
 
   
 
   
Note:
+
Another way of solving the same
A common error one sees is 0.7 is written as 1 /7. It is seven
+
problem is to split each fourth piece into 2.
tenths and not one seventh. That the denominator is always 10 has to
 
be stressed. To reinforce this one can use a simple rectangle divided
 
into 10 parts , the same that was used to understand place value in
 
whole numbers.
 
  
 
   
 
   
The
+
This means we change the 3/4
coloured portion represents 0.6 or 6/10 and the whole block
+
into 6/8.
represents 1.
 
  
 
    
 
    
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_1cf72869.gif]] <br>
+
[[Image:KOER%20Fractions_html_7ed8164a.gif]]
<br>
 
  
 
   
 
   
<br>
+
 
<br>
 
  
 
   
 
   
== Percentages ==
+
When dividing a fraction by a fraction,
+
we use the measure interpretation.
Fractions and
 
percentages are different ways of writing the same thing. When we
 
say that a book costs Rs. 200 and the shopkeeper is giving a 10 %
 
discount. Then we can represent the 10% as a fraction as
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m1369c56e.gif]]
 
where '''10 is the numerator''' and the '''denominator is '''<u>'''always'''</u>'''
 
100'''. In this case 10 % of the cost of the book is
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m50e22a06.gif]].
 
So you can buy the book for 200 – 20 = 180 rupees.
 
  
 
   
 
   
<br>
+
[[Image:KOER%20Fractions_html_m3192e02b.gif]]
<br>
 
  
 
   
 
   
<br>
+
When we divide 2 by ¼ we ask how many
<br>
+
times does ¼
  
 
   
 
   
There
 
are a number of common ones that are useful to learn. Here is a table
 
showing you the ones that you should learn.
 
  
                                     
 
{| border="1"
 
|-
 
|
 
Percentage
 
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_m257a1863.gif]][[Image:KOER%20Fractions_html_m257a1863.gif]]
Fraction
+
 
 +
 +
'''fit into 2'''.
  
 
   
 
   
|-
+
 
|
 
100%
 
  
 
   
 
   
|
+
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m15ed765d.gif]]
 
  
 
   
 
   
|-
+
It fits in 4 times in each roti, so
|
+
totally 8 times.
50%
 
  
 
   
 
   
|
+
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_df52f71.gif]]
 
  
 
   
 
   
|-
+
We write it as
|
+
[[Image:KOER%20Fractions_html_m390fcce6.gif]]
25%
 
  
 
   
 
   
|
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6c97abb.gif]]
 
  
 +
 +
 +
 +
== Activities ==
 +
 +
=== Activity 1 Addition of Fractions ===
 
   
 
   
|-
+
'''''Learning
|
+
Objectives'''''
75%
 
  
 
   
 
   
|
+
Understand Addition of Fractions
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6cb13da4.gif]]
 
  
 
   
 
   
|-
+
'''''Materials
|
+
and resources required '''''
10%
+
 
 +
 +
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]]
  
 
   
 
   
|
+
'''''Pre-requisites/
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_26bc75d0.gif]]
+
Instructions Method '''''
  
 
   
 
   
|-
+
Open link
|
+
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]]
20%
 
  
 
   
 
   
|
+
[[Image:KOER%20Fractions_html_m3dd8c669.png]]
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m73e98509.gif]]
+
 
  
 
   
 
   
|-
+
Move the sliders
|
+
Numerator1 and Denominator1 to set Fraction 1
40%
 
  
 
   
 
   
|
+
Move the sliders
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m2dd64d0b.gif]]
+
Numerator2 and Denominator2 to set Fraction 2
  
 
   
 
   
|} 
+
See the last bar
<br>
+
to see the result of adding fraction 1 and fraction 2
<br>
 
  
 
   
 
   
<br>
+
When you move
<br>
+
the sliders ask children to
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m60c76c68.gif]]To
+
Observe and
see 40 % visually see the figure :
+
describe what happens when the denominator is changed.
  
 
   
 
   
You
+
Observe and
can see that if the shape is divided into 5 equal parts, then 2 of
+
describe what happens when denominator changes
those parts are shaded.
 
  
 
   
 
   
If
+
Observe and
the shape is divided into 100 equal parts, then 40 parts are shaded.
+
describe the values of the numerator and denominator and relate it to
 +
the third result fraction.
  
 
   
 
   
These
+
Discuss LCM and
are equivalent fractions as in both cases the same amount has been
+
GCF
shaded.
+
 
 +
 +
'''''Evaluation'''''
  
 +
 
 +
=== Activity 2 Fraction Subtraction ===
 
   
 
   
<br>
+
'''''Learning
<br>
+
Objectives '''''
  
 
   
 
   
== Ratio and Proportion ==
+
Understand Fraction Subtraction
 +
 
 
   
 
   
It
+
'''''Materials and
is important to understand that fractions also can be interpreted as
+
resources required'''''
ratio's. Stressing that a fraction can be interpreted in many ways is
 
of vital importance. Here briefly I describe the linkages that must
 
be established between Ratio and Proportion and the fraction
 
representation. Connecting multiplication of fractions is key to
 
understanding ratio and proportion.
 
  
 
   
 
   
<br>
+
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
<br>
+
 
 +
 
 +
'''''Pre-requisites/
 +
Instructions Method '''''
  
 
   
 
   
'''What
+
Open link
is ratio?'''
+
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
  
 
   
 
   
Ratio
+
 
is a way of comparing amounts of something. It shows how much bigger
+
 
one thing is than another. For example:
 
  
 
   
 
   
* Use 1 measure detergent (soap) to 10 measures water
+
[[Image:KOER%20Fractions_html_481d8c4.png]]
* Use 1 shovel (bucket) of cement to 3 shovels (buckets) of sand
+
 
* Use 3 parts blue paint to 1 part white
+
 
 
   
 
   
Ratio
+
Move the sliders Numerator1 and Denominator1 to set Fraction 1
is the number of '''parts''' to a mix. The paint mix is 4
 
parts, with 3 parts blue and 1 part white.
 
  
 
   
 
   
The
+
Move the sliders Numerator2 and Denominator2 to set Fraction 2
order in which a ratio is stated is important. For example, the ratio
 
of soap to water is 1:10. This means for every 1 measure of soap
 
there are 10 measures of water.
 
  
 
   
 
   
Mixing
+
See the last bar to see the result of subtracting fraction 1 and
paint in the ratio 3:1 (3 parts blue paint to 1 part white paint)
+
fraction 2
means 3 + 1 = 4 parts in all.
 
  
 
   
 
   
3
+
 
parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white
+
 
paint.
 
  
 
   
 
   
<br>
+
When you move the sliders ask children to
<br>
 
  
 
   
 
   
Cost
+
observe and describe what happens when the denominator is
of a pen is Rs 10 and cost of a pencil is Rs 2. How many times the
+
changed.
cost of a pencil is the cost of a pen? Obviously it is five times.
 
This can be written as
 
  
 
   
 
   
<br>
+
observe and describe what happens when denominator changes
<br>
 
  
 
   
 
   
The
+
observe and describe the values of the numerator and denominator
ratio of the cost of a pen to the cost of a pencil =
+
and relate it to the third result fraction.
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m762fb047.gif]]
 
  
 
   
 
   
<br>
+
Discuss LCM and GCF
<br>
 
  
 
   
 
   
What
+
'''''Evaluation'''''
is Direct Proportion ?
+
 
 +
 
 +
 
  
 
Two
 
quantities are in direct proportion when they increase or decrease in
 
the same ratio. For example you could increase something by doubling
 
it or decrease it by halving. If we look at the example of mixing
 
paint the ratio is 3 pots blue to 1 pot white, or 3:1.
 
  
 
   
 
   
Paint
+
=== Activity 3  Multiplication of fractions ===
pots in a ratio of 3:1
+
 +
'''''Learning
 +
Objectives '''''
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m22cda036.gif]]<br>
+
Understand Multiplication of fractions
<br>
 
  
 
   
 
   
<br>
+
'''''Materials and
<br>
+
resources required'''''
  
 
   
 
   
<br>
+
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
<br>
 
  
 
   
 
   
<br>
+
'''''Pre-requisites/
<br>
+
Instructions Method '''''
  
 
   
 
   
<br>
+
Open link
<br>
+
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
  
 
   
 
   
<br>
+
 
<br>
+
 
  
 
   
 
   
But
+
[[Image:KOER%20Fractions_html_12818756.png]]
this amount of paint will only decorate two walls of a room. What if
+
 
you wanted to decorate the whole room, four walls? You have to double
 
the amount of paint and increase it in the same ratio.
 
  
 
   
 
   
If
+
Move the sliders Numerator1 and Denominator1 to set Fraction 1
we double the amount of blue paint we need 6 pots.
 
  
 
   
 
   
If
+
Move the sliders Numerator2 and Denominator2 to set Fraction 2
we double the amount of white paint we need 2 pots.
 
  
 
   
 
   
Six
+
On the right hand side see the result of multiplying fraction 1
paint pots in a ratio of 3:1
+
and fraction 2
  
 
   
 
   
<br>
+
'''Material/Activity Sheet'''
<br>
 
  
 
   
 
   
The
+
Please open
amount of blue and white paint we need increase in direct proportion
+
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
to each other. Look at the table to see how as you use more blue
+
in Firefox and follow the process
paint you need more white paint:
 
  
 
   
 
   
Pots
+
When you move the sliders ask children to
of blue paint 3 6 9 12
 
  
 
   
 
   
Pots
+
observe and describe what happens when the denominator is
of white paint 1 2 3 4
+
changed.
  
 
   
 
   
<br>
+
observe and describe what happens when denominator changes
<br>
 
  
 
   
 
   
<br>
+
One unit will be the large square border-in blue solid lines
<br>
 
  
 
   
 
   
Two
+
A sub-unit is in dashed lines within one square unit.
quantities which are in direct proportion will always produce a graph
 
where all the points can be joined to form a straight line.
 
  
 
   
 
   
<br>
+
The thick red lines represent the fraction 1 and 2 and also the
<br>
+
side of the quadrilateral
  
 
   
 
   
'''What
+
The product represents the area of the the quadrilateral
is Inverse Proportion ?'''
 
  
 
   
 
   
Two
+
'''''Evaluation'''''
quantities may change in such a manner that if one quantity increases
 
the the quantity decreases and vice-versa. For example if we are
 
building a room, the time taken to finish decreases as the number of
 
workers increase. Similarly when the speed increases the time to
 
cover a distance decreases. Zaheeda can go to school in 4 different
 
ways. She can walk, run, cycle or go by bus.
 
  
 
   
 
   
Study
+
When
the table below, observe that as the speed increases time taken to
+
two fractions are multiplied
cover the distance decreases
+
is the product larger or smaller that the multiplicands – why ?
  
 
   
 
   
<br>
 
<br>
 
  
                           
+
 
{| border="1"
 
|-
 
|
 
<br>
 
  
 
   
 
   
|
+
 
Walk
+
 
  
 
   
 
   
|
+
=== Activity 4 Division by Fractions ===
Run
+
 +
'''''Learning
 +
Objectives '''''
  
 
   
 
   
|
+
Understand Division by Fractions
Cycle
 
  
 
   
 
   
|
+
'''''Materials and
Bus
+
resources required'''''
  
 
   
 
   
|-
+
[[http://www.superteacherworksheets.com/fractions/fractionstrips_TWQWF.pdf]]
|
 
Speed
 
Km/Hr
 
  
 
   
 
   
|
+
Crayons/ colour
3
+
pencils, Scissors, glue
  
 
   
 
   
|
+
'''''Pre-requisites/
6
+
Instructions Method '''''
(walk speed *2)
 
  
 
   
 
   
|
+
Print out the pdf
9
+
[[http://www.superteacherworksheets.com/fractions/fractionstrips_TWQWF.pdf]]
(walk speed *3)
 
  
 
   
 
   
|
+
Colour each of the unit fractions in different colours. Keep the
45
+
whole unit (1) white.
(walk speed *15)
 
  
 
   
 
   
|-
+
Cut out each unit fraction piece.
|
+
 
Time
+
Taken (minutes)
+
Give examples
 +
[[Image:KOER%20Fractions_html_m282c9b3f.gif]]
  
 
   
 
   
|
+
For example if we try the first one,
30
+
[[Image:KOER%20Fractions_html_21ce4d27.gif]]
 +
See how many
 +
[[Image:KOER%20Fractions_html_m31bd6afb.gif]]strips
 +
will fit exactly onto whole unit strip.
  
 
   
 
   
|
+
 
15
+
 
(walk Time * ½)
 
  
 
   
 
   
|
+
'''''Evaluation'''''
10
 
(walk Time * 1/3)
 
  
 
   
 
   
|
+
When
2
+
we divide by a fraction is the result larger or smaller why ?
(walk Time * 1/15)
 
  
 
   
 
   
|}
+
 
<br>
+
 
<br>
 
  
 
   
 
   
As
+
== Evaluation ==
Zaheeda doubles her speed by running, time reduces to half. As she
+
increases her speed to three times by cycling, time decreases to one
+
== Self-Evaluation ==
third. Similarly, as she increases her speed to 15 times, time
+
decreases to one fifteenth. (Or, in other words the ratio by which
+
== Further Exploration ==
time decreases is inverse of the ratio by which the corresponding
+
speed increases). We can say that speed and time change inversely in
+
# [[http://www.youtube.com/watch?v=41FYaniy5f8]] detailed conceptual understanding of division by fractions
proportion.
+
# [[http://www.homeschoolmath.net/teaching/f/understanding_fractions.php]] understanding fractions
 
+
# [[http://www.superteacherworksheets.com]] Worksheets in mathematics for teachers to use
 +
 +
= Linking Fractions to other Topics =
 
   
 
   
=== Moving from Additive Thinking to Multiplicative Thinking ===
+
== Introduction ==
 
   
 
   
Avinash
+
It is also very common for the school system to treat themes in a
thinks that if you use 5 spoons of sugar to make 6 cups of tea, then
+
separate manner. Fractions are taught as stand alone chapters. In
you would need 7 spoons of sugar to make 8 cups of tea just as sweet
+
this resource book an attempt to connect it to other middle school
as the cups before. Avinash would be using an '''''additive
+
topics such as Ratio Proportion, Percentage and high school topics
transformation''''''''; '''he thinks that since we added 2 more
+
such as rational and irrational numbers, inverse proportions are
cups of tea from 6 to 8. To keep it just as sweet he would need to
+
made. These other topics are not discussed in detail themselves, but
add to more spoons of sugar. What he does not know is that for it to
+
used to show how to link these other topics with the already
taste just as sweet he would need to preserve the ratio of sugar to
+
understood concepts of fractions.
tea cup and use '''multiplicative thinking'''. He is unable to
 
detect the ratio.
 
  
 
   
 
   
=== Proportional Reasoning ===
 
 
'''''Proportional
 
thinking''''' involves the ability to understand and compare
 
ratios, and to predict and produce equivalent ratios. It requires
 
comparisons between quantities and also the relationships between
 
quantities. It involves quantitative thinking as well as qualitative
 
thinking. A feature of proportional thinking is the multiplicative
 
relationship among the quantities and being able to recognize this
 
relationship. The relationship may be direct (divide), i.e. when one
 
quantity increases, the other also increases. The relationship is
 
inverse (multiply), when an increase in one quantity implies a
 
decrease in the other, in both cases the ratio or the rate of change
 
remains a constant.
 
  
+
 
<br>
 
  
 
   
 
   
The
+
== Objectives ==
process of adding involved situations such as adding, joining,
 
subtracting, removing actions which involves the just the two
 
quantities that are being joined, while proportional thinking is
 
associated with shrinking, enlarging, scaling , fair sharing etc. The
 
process involves multiplication. To be able to recognize, analyse and
 
reason these concepts is '''''multiplicative thinking/reasoning'''''.
 
Here the student must be able to understand the third quantity which
 
is the ratio of the two quantities. The preservation of the ratio is
 
important in the multiplicative transformation.
 
 
 
 
   
 
   
<br>
+
Explicitly link the other
<br>
+
topics in school mathematics that use fractions.
  
 
   
 
   
== Rational & Irrational Numbers ==
+
== Decimal Numbers ==
 
   
 
   
After
+
“Decimal”
the number line was populated with natural numbers, zero and the
+
comes from the Latin root '''''decem''''',
negative integers, we discovered that it was full of gaps. We
+
which simply means ten. The number system we use is called the
discovered that there were numbers in between the whole numbers -
+
decimal number system, because the place value units go in tens: you
fractions we called them.
+
have ones, tens, hundreds, thousands, and so on, each unit being 10
 +
times the previous one.
  
 
   
 
   
But,
+
In
soon we discovered numbers that could not be expressed as a fraction.
+
common language, the word “decimal number” has come to mean
These numbers could not be represented as a simple fraction. These
+
numbers which have digits after the decimal point, such as 5.8 or
were called irrational numbers. The ones that can be represented by a
+
9.302. But in reality, any number within the decimal number system
simple fraction are called rational numbers. They h ad a very
+
could be termed a decimal number, including whole numbers such as 12
definite place in the number line but all that could be said was that
+
or 381.
square root of 2 is between 1.414 and 1.415. These numbers were very
 
common. If you constructed a square, the diagonal was an irrational
 
number. The idea of an irrational number caused a lot of agony to
 
the Greeks. Legend has it that Pythagoras was deeply troubled by
 
this discovery made by a fellow scholar and had him killed because
 
this discovery went against the Greek idea that numbers were perfect.
 
  
 
   
 
   
How
+
 
can we be sure that an irrational number cannot be expressed as a
+
 
fraction? This can be proven algebraic manipulation. Once these
 
"irrational numbers" came to be identified, the numbers
 
that can be expressed of the form p/q where defined as rational
 
numbers.
 
  
 
   
 
   
There
+
The
is another subset called transcendental numbers which have now been
+
simplest way to link or connect fractions to the decimal number
discovered. These numbers cannot be expressed as the solution of an
+
system is with the number line representation. Any scale that a
algebraic polynomial. "pi" and "e" are such
+
child uses is also very good for this purpose, as seen in the figure
numbers.
+
below.
  
 
   
 
   
== Vocabulary ==
+
 
+
 
Decimal
 
Numbers, Percentages, Ratio, Direct Proportion, Inverse Proportion,
 
Rational Numbers, Irrational Numbers
 
  
 
   
 
   
<br>
+
The
<br>
+
number line between 0 and 1 is divided into ten parts. Each of these
 +
ten parts is '''1/10''', a '''tenth'''.
  
 
   
 
   
== Additional Resources ==
+
[[Image:KOER%20Fractions_html_3d7b669f.gif]]
 
[[http://www.bbc.co.uk/skillswise/numbers/wholenumbers/]]
 
  
 
[[http://en.wikipedia.org/wiki/Koch_snowflake]]
 
  
 
   
 
   
[[http://www.realmagick.com/5552/bringing-it-down-to-earth-a-fractal-approach/]]
+
Under
 +
the tick marks you see decimal numbers such as 0.1, 0.2, 0.3, and so
 +
on. These are the same numbers as the fractions 1/10, 2/10, 3/10 and
 +
so on.
  
 
   
 
   
= Activities : =
+
We
+
can write any fraction with '''tenths (denominator 10) '''using the
== Activity1: Introduction to fractions ==
+
decimal point. Simply write after the decimal point how many tenths
+
the number has. 0.6 means six tenths or 1/6. 1.5 means 1 whole and 5
=== Objective: ===
+
tenths or
+
[[Image:KOER%20Fractions_html_m7f1d448c.gif]]
Introduce
 
fractions using the part-whole method
 
  
 
   
 
   
=== Procedure: ===
+
Note: A common error one sees is 0.7 is written as 1 /7. It is
 +
seven tenths and not one seventh. That the denominator is always 10
 +
has to be stressed. To reinforce this one can use a simple rectangle
 +
divided into 10 parts , the same that was used to understand place
 +
value in whole numbers.
 +
 
 
   
 
   
Do
+
The
the six different sections given in the activity sheet. For each
+
coloured portion represents 0.6 or 6/10 and the whole block
section there is a discussion point or question for a teacher to ask
+
represents 1.
children.
+
 
 +
 
 +
[[Image:KOER%20Fractions_html_1cf72869.gif]]
 +
 
  
 +
 
 +
== Percentages ==
 
   
 
   
After
+
Fractions and percentages are different ways of writing the same
the activity sheet is completed, please use the evaluation questions
+
thing. When we say that a book costs Rs. 200 and the shopkeeper is
to see if the child has understood the concept of fractions
+
giving a 10 % discount. Then we can represent the 10% as a fraction
 +
as
 +
[[Image:KOER%20Fractions_html_m1369c56e.gif]]
 +
where '''10 is the numerator''' and the '''denominator is '''<u>'''always'''</u>'''
 +
100'''. In this case 10 % of the cost of the book is
 +
[[Image:KOER%20Fractions_html_m50e22a06.gif]].
 +
So you can buy the book for 200 – 20 = 180 rupees.
  
 
   
 
   
<br>
 
<br>
 
  
+
 
'''Material/Activity
 
Sheet'''
 
  
 
   
 
   
# Write the Number Name and the number of the picture like the example  [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m1d9c88a9.gif]]Number Name = One third  Number: [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_52332ca.gif]]
 
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_2625e655.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m685ab2.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55c6e68e.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_mfefecc5.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m12e15e63.gif]]
 
  
   
+
 
Question:
 
What is the value of the numerator and denominator in the last figure
 
, the answer is [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m2dc8c779.gif]]
 
  
 
   
 
   
# Colour the correct amount that represents the fractions
+
There
+
are a number of common ones that are useful to learn. Here is a table
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_19408cb.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m12e15e63.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6b49c523.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6f2fcb04.gif]]<br>
+
showing you the ones that you should learn.
<br>
 
  
 
+
                                     
7/10 3/8
+
{| border="1"
1/5 4/7
+
|-
 +
|
 +
Percentage
  
 
   
 
   
Question:
+
|
Before colouring count the number of parts in each figure. What does
+
Fraction
it represent. Answer: Denominator
 
  
 
   
 
   
<br>
+
|-
<br>
+
|
 +
100%
  
 
   
 
   
# Divide the circle into fractions and colour the right amount to show the fraction
+
|
+
[[Image:KOER%20Fractions_html_m15ed765d.gif]]
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
 
<br>
 
  
 
   
 
   
<br>
+
|-
<br>
+
|
 +
50%
  
 
   
 
   
<br>
+
|
<br>
+
[[Image:KOER%20Fractions_html_df52f71.gif]]
 
 
 
 
3/5
 
6/7 1/3 5/8 2/5
 
  
 
   
 
   
<br>
+
|-
<br>
+
|
 +
25%
  
 
   
 
   
# Draw the Fraction and observe which is the greater fraction – observe that the parts are equal for each pair
+
|
+
[[Image:KOER%20Fractions_html_m6c97abb.gif]]
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
 
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
+
|-
 +
|
 +
75%
  
 
   
 
   
<br>
+
|
 +
[[Image:KOER%20Fractions_html_m6cb13da4.gif]]
  
 
   
 
   
<br>
+
|-
 +
|
 +
10%
  
 
   
 
   
1/3 2/3 4/5 2/5
+
|
3/7 4/7
+
[[Image:KOER%20Fractions_html_26bc75d0.gif]]
  
 
   
 
   
<br>
+
|-
 +
|
 +
20%
  
 
   
 
   
<br>
+
|
 +
[[Image:KOER%20Fractions_html_m73e98509.gif]]
  
 
   
 
   
# Draw the Fraction and observe which is the greater fraction – Observe that the parts are different sizes for each pair.
+
|-
 +
|
 +
40%
 +
 
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
+
|
 +
[[Image:KOER%20Fractions_html_m2dd64d0b.gif]]
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
+
|} 
 +
 
 +
 
  
 
   
 
   
<br>
+
 
 +
 
  
 
   
 
   
<br>
+
[[Image:KOER%20Fractions_html_m60c76c68.gif]]To
 +
see 40 % visually see the figure :
  
 
   
 
   
<br>
+
You
 +
can see that if the shape is divided into 5 equal parts, then 2 of
 +
those parts are shaded.
  
 
   
 
   
<br>
+
If
 +
the shape is divided into 100 equal parts, then 40 parts are shaded.
  
 
   
 
   
1/3 1/4 1/5 1/8
+
These
1/6 1/2
+
are equivalent fractions as in both cases the same amount has been
 +
shaded.
  
 
   
 
   
<br>
+
 
<br>
+
 
  
 
   
 
   
# Solve these word problems by drawing
+
== Ratio and Proportion ==
## Amar divided an apple into 8 equal pieces. He ate 5 pieces. He put the a   other 3 in a box. What fraction did Amar eat?
+
## There are ten biscuits in the box. 3 are cream biscuits. 2 are salt biscuits. 4 are chocolate biscuits. 1 is a sugar biscuit. What fraction of the biscuits in the box are salt biscuits.
+
It
## Radha has 6 pencils. She gives one to Anil and he gives one to Anita. She keeps the   rest. What fraction of her pencils did she give away?
+
is important to understand that fractions also can be interpreted as
#
+
ratio's. Stressing that a fraction can be interpreted in many ways is
 
+
of vital importance. Here briefly I describe the linkages that must
=== Evaluation Questions ===
+
be established between Ratio and Proportion and the fraction
 
+
representation. Connecting multiplication of fractions is key to
== Activity 2: Proper and Improper Fractions ==
+
understanding ratio and proportion.
 
=== Objective: ===
 
 
Proper and Improper
 
Fractions
 
  
 
   
 
   
=== Procedure: ===
 
 
Examples
 
of Proper and improper fractions are given. The round disks in the
 
figure represent rotis and the children figures represent children.
 
Cut each roti and each child figure and make the children fold, tear
 
and equally divide the roits so that each child figure gets equal
 
share of roti.
 
  
 
Material/Activity
 
Sheet
 
  
 
# [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]]If you want to understand proper fraction , example 5/6. In the equal share model , 5/6 represents the share that each child gets when 5 rotis are divided among 6 children '''equally.'''
 
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5e906d5b.jpg]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]]<br>
 
<br>
 
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
+
'''What
<br>
+
is ratio?'''
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5e906d5b.jpg]]<br>
+
Ratio
<br>
+
is a way of comparing amounts of something. It shows how much bigger
 +
one thing is than another. For example:
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]]<br>
+
* Use 1 measure detergent (soap) to 10 measures water
<br>
+
* Use 1 shovel (bucket) of cement to 3 shovels (buckets) of sand
 
+
* Use 3 parts blue paint to 1 part white
 
   
 
   
<br>
+
Ratio
<br>
+
is the number of '''parts''' to a mix. The paint mix is 4
 +
parts, with 3 parts blue and 1 part white.
  
 
   
 
   
<br>
+
The
<br>
+
order in which a ratio is stated is important. For example, the ratio
 +
of soap to water is 1:10. This means for every 1 measure of soap
 +
there are 10 measures of water.
  
 
   
 
   
# If you want to understand improper fraction , example 8/3. In the equal share model , 8/3 represents the share that each child gets when 8 rotis are divided among 3 children equally. The child in this case will usually distribute 2 full rotis to each child and then try to divide the remaining rotis. At this point you can show the mixed fraction representation as 2 2/3
+
Mixing
 +
paint in the ratio 3:1 (3 parts blue paint to 1 part white paint)
 +
means 3 + 1 = 4 parts in all.
 +
 
 
   
 
   
<br>
+
3
<br>
+
parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white
 +
paint.
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5e906d5b.jpg]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
+
 
<br>
+
 
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]]<br>
+
Cost
<br>
+
of a pen is Rs 10 and cost of a pencil is Rs 2. How many times the
 +
cost of a pencil is the cost of a pen? Obviously it is five times.
 +
This can be written as
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
+
 
<br>
+
 
  
 
   
 
   
<br>
+
The
<br>
+
ratio of the cost of a pen to the cost of a pencil =
 +
[[Image:KOER%20Fractions_html_m762fb047.gif]]
  
 
   
 
   
<br>
+
 
<br>
+
 
  
 
   
 
   
=== Evaluation Question ===
+
What
 +
is Direct Proportion ?
 +
 
 
   
 
   
# What happens when the numerator and denominator are the same, why ?
+
Two
# What happens when the numerator is greater than the denominator why ? How can we represent this in two ways ?
+
quantities are in direct proportion when they increase or decrease in
 +
the same ratio. For example you could increase something by doubling
 +
it or decrease it by halving. If we look at the example of mixing
 +
paint the ratio is 3 pots blue to 1 pot white, or 3:1.
 +
 
 
   
 
   
<br>
+
Paint
<br>
+
pots in a ratio of 3:1
  
 
   
 
   
== Activity 3: Comparing Fractions ==
+
[[Image:KOER%20Fractions_html_m22cda036.gif]]
+
 
=== Objective: ===
 
 
Comparing-Fractions
 
  
 
   
 
   
=== Procedure: ===
 
 
Print
 
the document '''Comparing-Fractions.pdf ''' and'''
 
Comparing-Fractions2 a'''nd work
 
out the activity sheet
 
  
 
<br>
 
<br>
 
  
 
'''Material/
 
Activity Sheet'''
 
  
 
   
 
   
[[Comparing-Fractions.pdf]]
 
  
 
[[Comparing-Fractions2.pdf]]
 
  
 
<br>
 
<br>
 
  
 
   
 
   
=== Evaluation Question ===
 
 
# Does the child know the symbols '''&gt;, &lt;''' and '''='''
 
# What happens to the size of the part when the denominator is different ?
 
# Does it decrease or increase when the denominator becomes larger ?
 
# Can we compare quantities when the parts are different sizes ?
 
# What should we do to make the sizes of the parts the same ?
 
 
== Activity 4: Equivalent Fractions ==
 
 
<br>
 
<br>
 
  
 
=== Objective: ===
 
 
To understand Equivalent
 
Fractions
 
  
 
=== Procedure: ===
 
 
Print
 
10 copies of the document from pages 2 to 5
 
'''fractions-matching-game.pdf'''
 
  
 
   
 
   
Cut
 
the each fraction part
 
  
 
Play
 
memory game as described in the document in groups of 4 children.
 
  
 
'''Activity
 
Sheet'''
 
  
 
   
 
   
[[fractions-matching-game.pdf]]
+
 
 +
 
  
 
   
 
   
=== Evaluation Question ===
+
But
 +
this amount of paint will only decorate two walls of a room. What if
 +
you wanted to decorate the whole room, four walls? You have to double
 +
the amount of paint and increase it in the same ratio.
 +
 
 
   
 
   
# What is reducing a fraction to the simplest form ?
+
If
# What is GCF – Greatest Common Factor ?
+
we double the amount of blue paint we need 6 pots.
# Use the document [[simplifying-fractions.pdf]]
 
# Why are fractions called equivalent and not equal.
 
 
== Activity 5: Fraction Addition ==
 
 
=== Objective: ===
 
 
Understand Addition of
 
Fractions
 
  
 
   
 
   
=== Procedure: ===
+
If
+
we double the amount of white paint we need 2 pots.
<br>
 
<br>
 
  
 
   
 
   
Open
+
Six
Geogebra applications
+
paint pots in a ratio of 3:1
  
 
   
 
   
Open
 
link
 
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]]
 
  
 
Move
 
the sliders Numerator1 and Denominator1 to set Fraction 1
 
  
 
Move
 
the sliders Numerator2 and Denominator2 to set Fraction 2
 
  
 
   
 
   
See
+
The
the last bar to see the result of adding fraction 1 and fraction 2
+
amount of blue and white paint we need increase in direct proportion
 +
to each other. Look at the table to see how as you use more blue
 +
paint you need more white paint:
  
 
   
 
   
'''Activity
+
Pots
Sheet'''
+
of blue paint 3 6 9 12
  
 
   
 
   
Please
+
Pots
open
+
of white paint 1 2 3 4
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]]
 
in Firefox and follow the process
 
  
 
   
 
   
When
 
you move the sliders ask children to
 
  
 
Observe
 
and describe what happens when the denominator is changed.
 
  
 
Observe
 
and describe what happens when denominator changes
 
  
 
   
 
   
Observe
 
and describe the values of the numerator and denominator and relate
 
it to the third result fraction. Discuss LCM and GCF
 
  
 
<br>
 
<br>
 
  
 
=== Evaluation Question ===
 
 
 
== Activity 6: Fraction Subtraction ==
 
 
=== Objective: ===
 
 
Understand Fraction
 
Subtraction
 
  
 
   
 
   
=== Procedure: ===
+
Two
+
quantities which are in direct proportion will always produce a graph
Open Geogebra
+
where all the points can be joined to form a straight line.
applications
 
  
 
   
 
   
Open link
 
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
 
  
 
Move the sliders
 
Numerator1 and Denominator1 to set Fraction 1
 
  
 
Move the sliders
 
Numerator2 and Denominator2 to set Fraction 2
 
  
 
   
 
   
See the last bar to see
+
'''What
the result of subtracting fraction 1 and fraction 2
+
is Inverse Proportion ?'''
  
 
   
 
   
<br>
+
Two
<br>
+
quantities may change in such a manner that if one quantity increases
 
+
the the quantity decreases and vice-versa. For example if we are
+
building a room, the time taken to finish decreases as the number of
'''Material/Activity
+
workers increase. Similarly when the speed increases the time to
Sheet'''
+
cover a distance decreases. Zaheeda can go to school in 4 different
 +
ways. She can walk, run, cycle or go by bus.
  
 
   
 
   
Please open link
+
Study
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
+
the table below, observe that as the speed increases time taken to
in Firefox and follow the process
+
cover the distance decreases
  
 
   
 
   
When you move the
 
sliders ask children to
 
  
 
observe and describe
 
what happens when the denominator is changed.
 
  
 
observe and describe
 
what happens when denominator changes
 
  
+
                           
observe and describe
+
{| border="1"
the values of the numerator and denominator and relate it to the
+
|-
third result fraction. Discuss LCM and GCF
+
|
  
 
=== Evaluation Question ===
 
 
 
== Activity 7: Linking to Decimals ==
 
 
=== Objective: ===
 
 
Fractions
 
representation of decimal numbers
 
  
 
   
 
   
=== Procedure: ===
+
|
+
Walk
<br>
 
<br>
 
  
 
   
 
   
Make copies of the
+
|
worksheets decimal-tenths-squares.pdf and
+
Run
decimal-hundreths-tenths.pdf
 
  
 
   
 
   
<br>
+
|
<br>
+
Cycle
  
 
   
 
   
'''Activity Sheet'''
+
|
 +
Bus
  
 
   
 
   
decimal-tenths-squares.pdf
+
|-
 +
|
 +
Speed
 +
Km/Hr
  
 
   
 
   
decimal-hundreths-tenths.pdf
+
|
 +
3
  
 
   
 
   
<br>
+
|
<br>
+
6
 +
(walk speed *2)
  
 
   
 
   
=== Evaluation Question ===
+
|
+
9
# Draw a number line and name the fraction and decimal numbers on the number line. Take a print of the document '''decimal-number-lines-1.pdf . '''Ask students to place any fraction and decimal numbers between between 0 and 10 on the number line
+
(walk speed *3)
# Write 0.45, 0.68, 0.05 in fraction form and represent as a fraction 100 square.
 
 
== Activity 8: Ratio and Proportion ==
 
 
=== Objective: ===
 
 
Linking fractional
 
representation and Ratio and Proportion
 
  
 
   
 
   
=== Procedure: ===
+
|
+
45
Use
+
(walk speed *15)
the NCERT Class 6 mathematics textbook chapter 12 and work out
 
Exercise 12.1
 
  
 
   
 
   
<br>
+
|-
<br>
+
|
 +
Time
 +
Taken (minutes)
  
 
   
 
   
'''Activity Sheet'''
+
|
 +
30
  
 
   
 
   
NCERT [[Class6 Chapter 12 RatioProportion.pdf]] Exercise 12.1
+
|
 +
15
 +
(walk Time * ½)
  
 
   
 
   
<br>
+
|
<br>
+
10
 +
(walk Time * 1/3)
  
 
   
 
   
=== Evaluation Question ===
+
|
 +
2
 +
(walk Time * 1/15)
 +
 
 
   
 
   
# Explain what the numerator means in the word problem
+
|}
# Explain what the denominator means
+
 
# Finally describe the whole fraction in words in terms of ratio and proportion.
+
 
 +
 
 
   
 
   
<br>
+
As
<br>
+
Zaheeda doubles her speed by running, time reduces to half. As she
 +
increases her speed to three times by cycling, time decreases to one
 +
third. Similarly, as she increases her speed to 15 times, time
 +
decreases to one fifteenth. (Or, in other words the ratio by which
 +
time decreases is inverse of the ratio by which the corresponding
 +
speed increases). We can say that speed and time change inversely in
 +
proportion.
  
 
   
 
   
== Activity 9: Fraction Multiplication ==
+
 
 +
 
 +
 
 
   
 
   
=== Objective: ===
+
'''Moving from Additive Thinking to
 +
Multiplicative Thinking '''
 +
 
 
   
 
   
Understand
 
Multiplication of fractions
 
  
 
=== Procedure: ===
 
 
Open Geogebra
 
applications
 
  
 
   
 
   
Open link
+
Avinash
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
+
thinks that if you use 5 spoons of sugar to make 6 cups of tea, then
 +
you would need 7 spoons of sugar to make 8 cups of tea just as sweet
 +
as the cups before. Avinash would be using an '''''additive
 +
transformation''''''''; '''he thinks that since we added 2 more
 +
cups of tea from 6 to 8. To keep it just as sweet he would need to
 +
add to more spoons of sugar. What he does not know is that for it to
 +
taste just as sweet he would need to preserve the ratio of sugar to
 +
tea cup and use '''multiplicative thinking'''. He is unable to
 +
detect the ratio.
  
 
   
 
   
Move the sliders
+
=== Proportional Reasoning ===
Numerator1 and Denominator1 to set Fraction 1
 
 
 
 
   
 
   
Move the sliders
+
'''''Proportional
Numerator2 and Denominator2 to set Fraction 2
+
thinking''''' involves the ability to understand and compare
 
+
ratios, and to predict and produce equivalent ratios. It requires
+
comparisons between quantities and also the relationships between
On the right hand side
+
quantities. It involves quantitative thinking as well as qualitative
see the result of multiplying fraction 1 and fraction 2
+
thinking. A feature of proportional thinking is the multiplicative
 +
relationship among the quantities and being able to recognize this
 +
relationship. The relationship may be direct (divide), i.e. when one
 +
quantity increases, the other also increases. The relationship is
 +
inverse (multiply), when an increase in one quantity implies a
 +
decrease in the other, in both cases the ratio or the rate of change
 +
remains a constant.
  
 
   
 
   
'''Material/Activity
 
Sheet'''
 
  
 
Please open
 
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/F]][[raction_MultiplyArea.html]]
 
in Firefox and follow the process
 
  
 
   
 
   
When you move the
+
The
sliders ask children to
+
process of adding involved situations such as adding, joining,
 +
subtracting, removing actions which involves the just the two
 +
quantities that are being joined, while proportional thinking is
 +
associated with shrinking, enlarging, scaling , fair sharing etc. The
 +
process involves multiplication. To be able to recognize, analyse and
 +
reason these concepts is '''''multiplicative thinking/reasoning'''''.
 +
Here the student must be able to understand the third quantity which
 +
is the ratio of the two quantities. The preservation of the ratio is
 +
important in the multiplicative transformation.
  
 
   
 
   
observe and describe
 
what happens when the denominator is changed.
 
  
 
observe and describe
 
what happens when denominator changes
 
  
 
One unit will be the
 
large square border-in blue solid lines
 
  
 
   
 
   
A sub-unit is in
+
== Rational & Irrational Numbers ==
dashed lines within one square unit.
 
 
 
 
   
 
   
The thick red lines
+
After
represent the fraction 1 and 2 and also the side of the quadrilateral
+
the number line was populated with natural numbers, zero and the
 +
negative integers, we discovered that it was full of gaps. We
 +
discovered that there were numbers in between the whole numbers -
 +
fractions we called them.
  
 
   
 
   
The product represents
+
But,
the area of the the quadrilateral
+
soon we discovered numbers that could not be expressed as a fraction.
 +
These numbers could not be represented as a simple fraction. These
 +
were called irrational numbers. The ones that can be represented by a
 +
simple fraction are called rational numbers. They h ad a very
 +
definite place in the number line but all that could be said was that
 +
square root of 2 is between 1.414 and 1.415. These numbers were very
 +
common. If you constructed a square, the diagonal was an irrational
 +
number. The idea of an irrational number caused a lot of agony to
 +
the Greeks. Legend has it that Pythagoras was deeply troubled by
 +
this discovery made by a fellow scholar and had him killed because
 +
this discovery went against the Greek idea that numbers were perfect.
  
 
   
 
   
=== Evaluation Question ===
+
How
+
can we be sure that an irrational number cannot be expressed as a
When
+
fraction? This can be proven algebraic manipulation. Once these
two fractions are multiplied is the product larger or smaller that
+
&quot;irrational numbers&quot; came to be identified, the numbers
the multiplicands – why ?
+
that can be expressed of the form p/q where defined as rational
 +
numbers.
  
 
   
 
   
<br>
+
There
<br>
+
is another subset called transcendental numbers which have now been
 +
discovered. These numbers cannot be expressed as the solution of an
 +
algebraic polynomial. &quot;pi&quot; and &quot;e&quot; are such
 +
numbers.
  
 
   
 
   
== Activity 10: Division of fractions ==
+
== Activities ==
 
   
 
   
=== Objective: ===
+
=== Activity 1  Fractions representation of decimal numbers ===
 
   
 
   
Understand Diviion by
+
'''''Learning
Fractions
+
Objectives '''''
  
 
   
 
   
=== Procedure: ===
+
Fractions representation of decimal
 +
numbers
 +
 
 
   
 
   
<br>
+
'''''Materials and
<br>
+
resources required'''''
  
 
   
 
   
Print out the
+
[[http://www.superteacherworksheets.com/decimals/decimal-tenths-squares_TWDWQ.pdf]]
[[fractionsStrips.pdf]]
 
  
 
   
 
   
Colour each of the unit
+
[[http://www.superteacherworksheets.com/decimals/decimal-hundredths-tenths.pdf]]
fractions in different colours. Keep the whole unit (1) white.
 
  
 
   
 
   
Cut out each unit
+
 
fraction piece.
+
 
  
 
   
 
   
Give examples
+
'''''Pre-requisites/
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m282c9b3f.gif]]
+
Instructions Method '''''
  
 
   
 
   
For example if we try
+
Make copies of the worksheets
the first one,
+
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_21ce4d27.gif]]
+
See how many
+
[[http://www.superteacherworksheets.com/decimals/decimal-tenths-squares_TWDWQ.pdf]]
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m31bd6afb.gif]]strips
 
will fit exactly onto whole unit strip.
 
  
 
   
 
   
<br>
+
[[http://www.superteacherworksheets.com/decimals/decimal-hundredths-tenths.pdf]]
<br>
 
  
 
   
 
   
'''Material /Activity
 
Sheet'''
 
  
+
 
[[fractionsStrips.pdf]]
 
, Crayons, Scissors, glue
 
  
 
   
 
   
<br>
+
'''''Evaluation'''''
<br>
 
  
 
   
 
   
=== Evaluation Question ===
+
# Draw a number line and name the fraction and decimal numbers on the number line. Take a print of the document [[http://www.superteacherworksheets.com/decimals/decimal-number-lines-tenths.pdf]] ''' . '''Ask students to place any fraction and decimal numbers between between 0 and 10 on the number line
 +
# Write 0.45, 0.68, 0.05 in fraction form and represent as a fraction 100 square.
 
   
 
   
When
+
 
we divide by a fraction is the result larger or smaller why ?
+
 
  
 
   
 
   
== Activity 11: Percentages ==
+
 
 +
 
 +
 
 
   
 
   
=== Objective: ===
+
=== Activity 2 Fraction representation and percentages ===
 
   
 
   
Understand fraction
+
'''''Learning
representation and percentages
+
Objectives '''''
  
 
   
 
   
<br>
+
Understand fraction representation and percentages
<br>
 
  
 
   
 
   
=== Procedure: ===
+
 
+
 
<br>
 
<br>
 
  
 
   
 
   
Please print copies of the 2 activity sheets [[percentage-basics-1.pdf]]
+
'''''Materials and
and [[percentage-basics-2.pdf]]
+
resources required'''''
and discuss the various percentage quantities with the various
 
shapes.
 
  
 
   
 
   
Then print a copy each of [[spider-percentages.pdf]]
+
[[http://www.superteacherworksheets.com/percents/converting-fractions-decimals-percents_EASIE.pdf]]
and make the children do this activity
 
  
 
   
 
   
<br>
+
[[http://www.superteacherworksheets.com/percents/basic-percentages1.pdf]]
<br>
 
  
 
   
 
   
'''Activity Sheet'''
+
'''''Pre-requisites/
 +
Instructions Method '''''
  
 
   
 
   
Print
+
Please print
out [[spider-percentages.pdf]]
+
copies of the 2 activity sheets
 +
[[http://www.superteacherworksheets.com/percents/converting-fractions-decimals-percents_EASIE.pdf]] and
 +
[[http://www.superteacherworksheets.com/percents/basic-percentages1.pdf]] and discuss the various percentage quantities with
 +
the various shapes.
  
 
   
 
   
<br>
 
<br>
 
  
 
=== Evaluation Question ===
 
 
What
 
value is the denominator when we represent percentage as fraction ?
 
  
 
   
 
   
What
+
Then print a copy
does the numerator represent ?
+
each of [[spider-percentages.pdf]]
 +
and make the children do this activity
  
 
   
 
   
What
 
does the whole fraction represent ?
 
  
 
What
 
other way can we represent a fraction whoose denominator is 100.
 
  
 
<br>
 
  
 
   
 
   
<br>
+
'''''Evaluation'''''
<br>
 
  
 
   
 
   
== Activity 12: Inverse Proportion ==
+
# What value is the denominator when we represent percentage as fraction ?
 +
# What does the numerator represent ?
 +
# What does the whole fraction represent ?
 +
# What other way can we represent a fraction whose denominator is 100.
 
   
 
   
=== Objective: ===
 
 
Understand fraction
 
representation and Inverse Proportion.
 
  
 
 
=== Procedure: ===
 
 
<br>
 
<br>
 
  
 
   
 
   
Use
 
the NCERT Class 8 mathematics textbook chapter 13 and work out
 
Exercise 13.1
 
  
 
<br>
 
<br>
 
  
 
'''Activity Sheet'''
 
  
 
   
 
   
[[NCERT Class 8 Chapter 13 InverseProportion.pdf]] Exercise 13.1
+
=== Activity 3 ===
 
 
 
   
 
   
<br>
+
'''''Learning
<br>
+
Objectives '''''
  
 
   
 
   
'''Evaluation Question'''
+
Understand fraction representation and rational and irrational
 +
numbers
  
 
   
 
   
1. Given a set of
+
'''''Materials and
fractions are they directly proportional or inversely proportional ?
+
resources required'''''
  
 
   
 
   
2.
+
Thread
In the word problem, identify the numerator, identify the denominator
+
of a certain length.
and explain what the fraction means in terms of Inverse proportions
 
 
 
 
<br>
 
  
 
   
 
   
== Activity 13: Rational and Irrational Numbers ==
+
'''''Pre-requisites/
+
Instructions Method '''''
=== Objective: ===
 
 
Understand fraction
 
representation and rational and irrational numbers
 
 
 
 
=== Procedure: ===
 
 
<br>
 
<br>
 
  
 
   
 
   
Line 2,680: Line 2,114:
  
 
   
 
   
<br>
+
 
  
 
   
 
   
Line 2,688: Line 2,122:
  
 
   
 
   
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m1a6bd0d0.gif]]<br>
+
[[Image:KOER%20Fractions_html_m1a6bd0d0.gif]]
  
 
   
 
   
Line 2,696: Line 2,130:
  
 
   
 
   
<br>
 
 
 
<br>
 
 
 
<br>
 
  
 
<br>
 
  
 
   
 
   
 
Identify
 
Identify
the various places where pi, "e" and the golden ratio
+
the various places where pi, &quot;e&quot; and the golden ratio
 
occur
 
occur
  
 
   
 
   
'''Material'''
 
  
 
Thread
 
of a certain length.
 
  
 
<br>
 
<br>
 
  
 
   
 
   
=== Evaluation Question ===
 
 
How
 
many numbers can I represent on a number line between 1 and 2.
 
  
 
What
 
is the difference between a rational and irrational number, give an
 
example ?
 
  
 
What
 
is Pi ? Why is it a special number ?
 
  
 
   
 
   
<br>
+
'''''Evaluation'''''
<br>
 
  
 
   
 
   
= Interesting Facts =
+
# How many numbers can I represent on a number line between 1 and 2.
 +
# What is the difference between a rational and irrational number, give an example ?
 +
# What is Pi ? Why is it a special number ?
 
   
 
   
In this article we will
 
look into the history of the fractions, and we’ll find out what the
 
heck that line in a fraction is called anyway.
 
  
 
<br>
 
<br>
 
 
 
Nearly everybody uses,
 
or has used, fractions for some reason or another. But most people
 
have no idea of the origin, and almost none of them have any idea
 
what that line is even called. Most know ways to express verbally
 
that it is present (e.g. “x over y-3,” or “x divided by y-3″),
 
but frankly, it HAS to have a name. To figure out the name, we must
 
also investigate the history of fractions.
 
 
 
 
The concept of fractions
 
can be traced back to the Babylonians, who used a place-value, or
 
positional, system to indicate fractions. On an ancient Babylonian
 
tablet, the number
 
  
 
[[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_636d55c5.gif]]<br>
 
<br>
 
  
 
   
 
   
, appears, which
+
== Evaluation ==
indicates the square root of two. The symbols are 1, 24, 51, and 10.
 
Because the Babylonians used a base 60, or sexagesimal, system, this
 
number is (1 * 60 0 ) + (24 * 60-1 ) + (51 * 60-2 ) + (10 * 60-3 ),
 
or about 1.414222. A fairly complex figure for what is now indicated
 
by √2.
 
 
 
 
   
 
   
<br>
+
== Self-Evaluation ==
<br>
 
 
 
 
   
 
   
In early Egyptian and
+
== Further Exploration ==
Greek mathematics, unit fractions were generally the only ones
 
present. This meant that the only numerator they could use was the
 
number 1. The notation was a mark above or to the right of a number
 
to indicate that it was the denominator of the number 1.
 
 
 
 
   
 
   
<br>
+
# Percentage and Fractions, [[http://www.bbc.co.uk/skillswise/numbers/wholenumbers/]]
<br>
+
# A mathematical curve Koch snowflake, [[http://en.wikipedia.org/wiki/Koch_snowflake]]
 
+
# Bringing it Down to Earth: A Fractal Approach, [[http://www.realmagick.com/5552/bringing-it-down-to-earth-a-fractal-approach/]]
 
   
 
   
The Romans used a system
+
= See Also =
of words indicating parts of a whole. A unit of weight in ancient
 
Rome was the as, which was made of 12 uncias. It was from this that
 
the Romans derived a fraction system based on the number 12. For
 
example, 1/12 was uncia, and thus 11/12 was indicated by deunx (for
 
de uncia) or 1/12 taken away. Other fractions were indicated as :
 
 
 
 
   
 
   
<br>
+
# Video on teaching fractions using the equal share method made by Eklavya an NGO based in Madhya Pradesh, India [[http://vimeo.com/22238434]]
<br>
+
# Mathematics resources from Homi Baba Centre for Science Education , [[http://mathedu.hbcse.tifr.res.in/]]
 
+
# [[http://www.geogebra.com]] Understand how to use Geogebra a mathematical computer aided tool
 
10/12 dextans (for de
 
sextans),
 
 
 
 
3/12 quadrans (for
 
quadran as)
 
 
 
 
9/12 dodrans (for de
 
quadrans),
 
 
 
 
2/12 or 1/6 sextans (for
 
sextan as)
 
 
 
 
8/12 bes (for bi as)
 
also duae partes (2/3)
 
 
 
 
1/24 semuncia (for semi
 
uncia)
 
 
 
 
7/12 septunx (for septem
 
unciae)
 
 
 
 
1/48 sicilicus
 
 
 
 
6/12 or 1/2 semis (for
 
semi as)
 
 
 
 
1/72 scriptulum
 
 
 
 
5/12 quincunx (for
 
quinque unciae)
 
 
 
 
1/144 scripulum
 
 
 
 
4/12 or 1/3 triens (for
 
trien as)
 
 
 
 
1/288 scrupulum
 
 
 
 
<br>
 
<br>
 
 
 
 
This system was quite
 
cumbersome, yet effective in indicating fractions beyond mere unit
 
fractions.
 
 
 
 
The Hindus are believed
 
to be the first group to indicate fractions with numbers rather than
 
words. Brahmagupta (c. 628) and Bhaskara (c. 1150) were early Hindu
 
mathematicians who wrote fractions as we do today, but without the
 
bar. They wrote one number above the other to indicate a fraction.
 
 
 
 
<br>
 
<br>
 
 
 
 
The next step in the
 
evolution of fraction notation was the addition of the horizontal
 
fraction bar. This is generally credited to the Arabs who used the
 
Hindu notation, then improved on it by inserting this bar in between
 
the numerator and denominator. It was at this point that it gained a
 
name, vinculum. Later on, Fibonacci (c.1175-1250), the first European
 
mathematician to use the fraction bar as it is used today, chose the
 
Latin word virga for the bar.
 
 
 
 
<br>
 
<br>
 
 
 
 
The most recent addition
 
to fraction notation, the diagonal fraction bar, was introduced in
 
the 1700s. This was solely due to the fact that, typographically, the
 
horizontal bar was difficult to use, being as it took three lines of
 
text to be properly represented. This was a mess to deal with at a
 
printing press, and so came, what was originally a short-hand, the
 
diagonal fraction bar. The earliest known usage of a diagonal
 
fraction bar occurs in a hand-written document. This document is
 
Thomas Twining’s Ledger of 1718, where quantities of tea and coffee
 
transactions are listed (e.g. 1/4 pound green tea). The earliest
 
known printed instance of a diagonal fraction bar was in 1784, when a
 
curved line resembling the sign of integration was used in the
 
Gazetas de Mexico by Manuel Antonio Valdes.
 
 
 
 
 
When the diagonal
 
fraction bar became popularly used, it was given two names : virgule,
 
derived from Fibonacci’s virga; and solidus, which originated from
 
the Roman gold coin of the same name (the ancestor of the shilling,
 
of the French sol or sou, etc.). But these are not the only names for
 
this diagonal fraction bar.
 
 
 
 
According to the Austin
 
Public Library’s website, “The oblique stroke (/) is called a
 
separatrix, slant, slash, solidus, virgule, shilling, or diagonal.
 
Thus, it has multiple names.
 
 
 
 
A related symbol,
 
commonly used, but for the most part nameless to the general public,
 
is the “division symbol,” or ÷ . This symbol is called an
 
obelus. Though this symbol is generally not used in print or writing
 
to indicate fractions, it is familiar to most people due to the use
 
of it on calculators to indicate division and/or fractions.
 
 
 
 
Fractions are now
 
commonly used in recipes, carpentry, clothing manufacture, and
 
multiple other places, including mathematics study; and the notation
 
is simple. Most people begin learning fractions as young as 1st or
 
2nd grade. The grand majority of them don’t even realize that
 
fractions could have possibly been as complicated as they used to be,
 
and thus, don’t really appreciate them for their current
 
simplicity.
 
 
 
 
= ANNEXURE A – List of activity sheets attached =
 
 
comparing-fractions.pdf
 
 
 
 
comparing-fractions2.pdf
 
 
 
 
   
 
   
fractions-matching-game.pdf
+
= Teachers Corner =
 
 
 
   
 
   
fractionstrips.pdf
+
= Books =
 
 
 
   
 
   
NCERT Class6 Chapter 12
+
# [[http://books.google.com/books?id=Y5Skj-EA2_AC&pg=PA251&lpg=PA251&dq=streefland+fractions&source=bl&ots=aabKaciwrA&sig=DcM0mi7r1GJlTUbZVq9J0l53Lrc&hl=en&ei=0xJBTvLjJ8bRrQfC3JmyBw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false]] A google book Fractions in realistic mathematics education: a paradigm of developmental ...By Leen Streefland
RatioProportion.pdf
 
 
 
 
   
 
   
NCERT Class8 Chapter 13
+
= References =
DirectInverseProportion.pdf
 
 
 
 
   
 
   
percentage-basics-1.pdf
 
  
 
percentage-basics-2.pdf
 
 
 
simplifying-fractions.pdf
 
  
 
   
 
   
spider-percentages.pdf
+
# Introducing Fractions Using Share and Measure Interpretations: A Report from Classroom Trials . Jayasree Subramanian, Eklavya, Hoshangabad, India and Brijesh Verma,Muskan, Bhopal, India
 +
# Streefland, L. (1993). “Fractions: A Realistic Approach.” In Carpenter, T.P., Fennema, E., Romberg, T.A. (Eds.), Rational Numbers: an Integration of Research. (289-325). New Jersey: Lawrence, Erlbaum Associates, Publishers.

Revision as of 12:51, 7 January 2013

Introduction

The following is a background literature for teachers. It summarises the various concepts, approaches to be known to a teacher to teach this topic effectively . This literature is meant to be a ready reference for the teacher to develop the concepts, inculcate necessary skills, and impart knowledge in fractions from Class 6 to Class 10.


It is a well known fact that teaching and learning fractions is a complicated process in primary and middle school. Although much of fractions is covered in the middle school, if the foundation is not holistic and conceptual, then topics in high school mathematics become very tough to grasp. Hence this documents is meant to understand the research that has been done towards simplifying and conceptually understanding topics of fractions.


Mind Map

KOER Fractions html m700917.png


Different Models for interpreting and teaching-learning fractions

Introduction

Commonly fractions are always approached by teaching it through one model or interpretation namely the part-whole model where the whole is divided into equal parts and the fraction represents one or more of the parts. The limitations of this method, especially in explaining mixed fractions, multiplication and division of fractions has led to educators using other interpretations such as equal share and measure. These approaches to fraction teaching are discussed here.


Objectives

The objective of this section is to enable teachers to visualise and interpret fractions in different ways in order to clarify the concepts of fractions using multiple methods. The idea is for teachers to be able to select the appropriate method depending on the context, children and class they are teaching to effectively understand fractions.


Part-whole

The most commonly used model is the part whole model where where the whole is divided into equal parts and the fraction represents one or more of the parts.


KOER Fractions html 78a5005.gif


Half (½) : The whole is divided into two equal parts.


One part is coloured, this part represents the fraction ½.


KOER Fractions html 6fbd7fa5.gif


One-Fourth (1/4) : The whole is divided into four equal parts.


One part is coloured, this part represents the fraction ¼.



KOER Fractions html 43b75d3a.gif


One (2/2 or 1) : The whole is divided into two equal parts.


Two part are coloured, this part represents the fraction 2/2


which is equal to the whole or 1.


KOER Fractions html 2faaf16a.gif


Two Fifth (2/5) : The whole is divided into five equal parts.


Two part are coloured, this part represents the fraction 2/5.






KOER Fractions html 9e5c77.gifThree Seventh (3/7) : The whole is divided into seven equal parts.


Three part are coloured, this part represents the fraction 3/7.





KOER Fractions html m30791851.gif



Seven tenth (7/10) : The whole is divided into ten equal parts.


Seven part are coloured, this part represents the fraction 7/10 .





Terms Numerator and Denominator and their meaning


KOER Fractions html 3bf1fc6d.gif


Three Eight (3/8) The whole is divided into eight equal parts.



Three part are coloured, this part represents the fraction 3/8 .



3/8 is also written as numerator/denominator. Here the number above the line- numerator tells us HOW MANY PARTS are involved. It 'enumerates' or counts the coloured parts.


The number BELOW the line tells – denominator tells us WHAT KIND OF PARTS the whole is divided into. It 'denominates' or names the parts.




The important factor to note here is WHAT IS THE WHOLE . In both the figures below the fraction quantity is 1/4. In fig 1 one circle is the whole and in fig 2, 4 circles is the whole.


KOER Fractions html 1683ac7.gif


KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 729297ef.gifKOER Fractions html 4282c1e5.gifKOER Fractions html 6fbd7fa5.gif




Equal Share

In the equal share interpretation the fraction m/n denotes one share when m identical things are shared equally among n. The relationships between fractions are arrived at by logical reasoning (Streefland, 1993). For example 5/6 is the share of one child when 5 rotis (disk-shaped handmade bread) are shared equally among 6 children. The sharing itself can be done in more than one way and each of them gives us a relation between fractions. If we first distribute 3 rotis by dividing each into two equal pieces and giving each child one piece each child gets 1⁄2 roti. Then the remaining 2 rotis can be distributed by dividing each into three equal pieces giving each child a piece. This gives us the relations


KOER Fractions html 3176e16a.gif




The relations 3/6 = 1⁄2 and 2/6 = 1/3 also follow from the process of distribution. Another way of distributing the rotis would be to divide the first roti into 6 equal pieces give one piece each to the 6 children and continue this process with each of the remaining 4 rotis. Each child gets a share of rotis from each of the 5 rotis giving us the relation




KOER Fractions html m39388388.gif




It is important to note here that the fraction symbols on both sides of the equation have been arrived at simply by a repeated application of the share interpretation and not by appealing to prior notions one might have of these fraction symbols. In the share interpretation of fractions, unit fractions and improper fractions are not accorded a special place.


Also converting an improper fraction to a mixed fraction becomes automatic. 6/5 is the share that one child gets when 6 rotis are shared equally among 5 children and one does this by first distributing one roti to each child and then sharing the remaining 1 roti equally among 5 children giving us the relation


KOER Fractions html m799c1107.gif


Share interpretation does not provide a direct method to answer the question ‘how much is the given unknown quantity’. To say that the given unknown quantity is 3⁄4 of the whole, one has figure out that four copies of the given quantity put together would make three wholes and hence is equal to one share when these three wholes are shared equally among 4. Share interpretation is also the quotient interpretation of fractions in the sense that 3⁄4 = 3 ÷ 4 and this is important for developing students’ ability to solve problems involving multiplicative and linear functional relations.




To understand the equal share model better, use the GeoGebra file explaining the equal share model available on [[1]]. See figure below. Move the sliders m and n and see how the equal share model is interpreted.


KOER Fractions html 17655b73.png




Measure Model

Measure interpretation defines the unit fraction 1/n as the measure of one part when one whole is divided into n equal parts. The composite fraction m/n is as the measure of m such parts. Thus 5/6 is made of 5 piece units of size 1/5 each and 6/5 is made of 6 piece units of size 1/5 each. Since 5 piece units of size make a whole, we get the relation 6/5 = 1 + 1/5.


Significance of measure interpretation lies in the fact that it gives a direct approach to answer the ‘how much’ question and the real task therefore is to figure out the appropriate n so that finitely many pieces of size will be equal to a given quantity. In a sense then, the measure interpretation already pushes one to think in terms of infinitesimal quantities. Measure interpretation is different from the part whole interpretation in the sense that for measure interpretation we fix a certain unit of measurement which is the whole and the unit fractions are sub-units of this whole. The unit of measurement could be, in principle, external to the object being measured.




Introducing Fractions Using Share and Measure Interpretations

One of the major difficulties a child faces with fractions is making sense of the symbol m/n. In order to facilitate students’ understanding of fractions, we need to use certain models. Typically we use the area model in both the measure and share interpretation and use a circle or a rectangle that can be partitioned into smaller pieces of equal size. Circular objects like roti that children eat every day have a more or less fixed size. Also since we divide the circle along the radius to make pieces, there is no scope for confusing a part with the whole. Therefore it is possible to avoid explicit mention of the whole when we use a circular model. Also, there is no need to address the issue that no matter how we divide the whole into n equal parts the parts will be equal. However, at least in the beginning we need to instruct children how to divide a circle into three or five equal parts and if we use the circular model for measure interpretation, we would need ready made teaching aids such as the circular fraction kit for repeated use.


Rectangular objects (like cake) do not come in the same size and can be divided into n equal parts in more than one way. Therefore we need to address the issues (i) that the size of the whole should be fixed (ii) that all 1⁄2’s are equal– something that children do not see readily. The advantage of rectangular objects is that we could use paper models and fold or cut them into equal parts in different ways and hence it easy to demonstrate for example that 3/5 = 6/10 using the measure interpretation .


Though we expose children to the use of both circles and rectangles, from our experience we feel circular objects are more useful when use the share interpretation as children can draw as many small circles as they need and since the emphasis not so much on the size as in the share, it does not matter if the drawings are not exact. Similarly rectangular objects would be more suited for measure interpretation for, in some sense one has in mind activities such as measuring the length or area for which a student has to make repeated use of the unit scale or its subunits.




Activities

Activity1: Introduction to fractions

Learning Objectives


Introduce fractions using the part-whole method


Materials and resources required




  1. Write the Number Name and the number of the picture like the example KOER Fractions html m1d9c88a9.gifNumber Name = One third Number: KOER Fractions html 52332ca.gif

KOER Fractions html 2625e655.gifKOER Fractions html m685ab2.gifKOER Fractions html 55c6e68e.gifKOER Fractions html mfefecc5.gifKOER Fractions html m12e15e63.gif


Question: What is the value of the numerator and denominator in the last figure , the answer is KOER Fractions html m2dc8c779.gif




  1. Colour the correct amount that represents the fractions

KOER Fractions html 19408cb.gifKOER Fractions html m12e15e63.gifKOER Fractions html m6b49c523.gifKOER Fractions html m6f2fcb04.gif


7/10 3/8 1/5 4/7


Question: Before colouring count the number of parts in each figure. What does it represent. Answer: Denominator




  1. Divide the circle into fractions and colour the right amount to show the fraction

KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gif






3/5 6/7 1/3 5/8 2/5




  1. Draw the Fraction and observe which is the greater fraction – observe that the parts are equal for each pair

KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gif


KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gif





1/3 2/3 4/5 2/5 3/7 4/7





  1. Draw the Fraction and observe which is the greater fraction – Observe that the parts are different sizes for each pair.

KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gif


KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gif








1/3 1/4 1/5 1/8 1/6 1/2




  1. Solve these word problems by drawing
    1. Amar divided an apple into 8 equal pieces. He ate 5 pieces. He put the a other 3 in a box. What fraction did Amar eat?
    2. There are ten biscuits in the box. 3 are cream biscuits. 2 are salt biscuits. 4 are chocolate biscuits. 1 is a sugar biscuit. What fraction of the biscuits in the box are salt biscuits.
    3. Radha has 6 pencils. She gives one to Anil and he gives one to Anita. She keeps the rest. What fraction of her pencils did she give away?
  2. The circles in the box represent the whole; colour the right amount to show the fraction KOER Fractions html m78f3688a.gifHint: Half is 2 circles KOER Fractions html m867c5c2.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gif



KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gif






Pre-requisites/ Instructions Method



Do the six different sections given in the activity sheet. For each section there is a discussion point or question for a teacher to ask children.


After the activity sheet is completed, please use the evaluation questions to see if the child has understood the concept of fractions


Evaluation


Activity 2: Proper and Improper Fractions

Learning Objectives


Proper and Improper Fractions


Materials and resources required


  1. KOER Fractions html 5518d221.jpgIf you want to understand proper fraction , example 5/6. In the equal share model , 5/6 represents the share that each child gets when 5 rotis are divided among 6 children equally.

KOER Fractions html 5e906d5b.jpgKOER Fractions html 5518d221.jpg


KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gif




KOER Fractions html 5518d221.jpgKOER Fractions html 5e906d5b.jpg


KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gif


KOER Fractions html 5518d221.jpg


KOER Fractions html 55f65a3d.gif








  1. If you want to understand improper fraction , example 8/3. In the equal share model , 8/3 represents the share that each child gets when 8 rotis are divided among 3 children equally. The child in this case will usually distribute 2 full rotis to each child and then try to divide the remaining rotis. At this point you can show the mixed fraction representation as 2 2/3







KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 5e906d5b.jpgKOER Fractions html 55f65a3d.gif


KOER Fractions html 5518d221.jpg




KOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 55f65a3d.gifKOER Fractions html 5518d221.jpg


KOER Fractions html 55f65a3d.gif




Pre-requisites/ Instructions Method


Examples of Proper and improper fractions are given. The round disks in the figure represent rotis and the children figures represent children. Cut each roti and each child figure and make the children fold, tear and equally divide the roits so that each child figure gets equal share of roti.



Evaluation


  1. What happens when the numerator and denominator are the same, why ?
  2. What happens when the numerator is greater than the denominator why ? How can we represent this in two ways ?



Activity 3: Comparing Fractions

Learning Objectives


Comparing-Fractions


Materials and resources required






[[2]]


[[3]]


Pre-requisites/ Instructions Method


Print the document and work out the activity sheet


Evaluation


  1. Does the child know the symbols >, < and =
  2. What happens to the size of the part when the denominator is different ?
  3. Does it decrease or increase when the denominator becomes larger ?
  4. Can we compare quantities when the parts are different sizes ?
  5. What should we do to make the sizes of the parts the same ?



Activity 4: Equivalent Fractions

Learning Objectives


To understand Equivalent Fractions


Materials and resources required


[[4]]




Pre-requisites/ Instructions Method


Print 10 copies of the document from pages 2 to 5 fractions-matching-game Cut the each fraction part. Play memory game as described in the document in groups of 4 children.


Evaluation


  1. What is reducing a fraction to the simplest form ?
  2. What is GCF – Greatest Common Factor ?
  3. Use the document simplifying-fractions.pdf
  4. Why are fractions called equivalent and not equal.

Evaluation

Self-Evaluation

Further Exploration

Enrichment Activities

Errors with fractions

Introduction

A brief understanding of the common errors that children make when it comes to fractions are addressed to enable teachers to understand the child's levels of conceptual understanding to address the misconceptions.


Objectives

When fractions are operated erroneously like natural numbers, i.e. treating the numerator and the denominators separately and not considering the relationship between the numerator and the denominator is termed as N-Distractor. For example 1/3 + ¼ are added to result in 2/7. Here 2 units of the numerator are added and 3 & four units of the denominator are added. This completely ignores the relationship between the numerator and denominator of each of the fractions. Streefland (1993) noted this challenge as N-distractors and a slow-down of learning when moving from the concrete level to the abstract level.





N-Distractors

The five levels of resistance to N-Distractors that a child develops are:



  1. Absence of cognitive conflict: The child is unable to recognize the error even when she sees the same operation performed resulting in a correct answer. The child thinks both the answers are the same in spite of different results. Eg. ½ + ½ she erroneously calculated as 2/4. But when the child by some other method, say, through manipulatives (concrete) sees ½ + ½ = 1 does not recognize the conflict.
  2. Cognitive conflict takes place: The student sees a conflict when she encounters the situation described in level 1 and rejects the ½+1/2 = 2/4 solution and recognizes it as incorrect. She might still not have a method to arrive at the correct solution.
  3. Spontaneous refutation of N-Distractor errors: The student may still make N-Distractor errors, but is able to detect the error for herself. This detection of the error may be followed by just rejection or explaining the rejection or even by a correct solution.
  4. Free of N-Distractor: The written work is free of N-Distractors. This could mean a thorough understanding of the methods/algorithms of manipulating fractions.
  5. Resistance to N-Distractor: The student is completely free (conceptually and algorithmically) of N-Distractor errors.



Activities

Evaluation

Self-Evaluation

Further Exploration

  1. www.merga.net.au/publications/counter.php?pub=pub_conf&id=1410 A PDF Research paper titled Probing Whole Number Dominance with Fractions.
  2. www.merga.net.au/documents/RP512004.pdf A PDF research paper titled “Why You Have to Probe to Discover What Year 8 Students Really Think About Fractions ”
  3. [[5]] A google book Fractions in realistic mathematics education: a paradigm of developmental ...By Leen Streefland

Enrichment Activities

Operations on Fractions

Introduction

This topic introduces the different operations on fractions. When learners move from whole numbers to fractions, many of the operations are counter intuitive. This section aims to clarify the concepts behind each of the operations.


Objectives

The aim of this section is to visualise and conceptually understand each of the operations on fractions.


Addition and Subtraction

Adding and subtracting like fractions is simple. It must be emphasised thought even during this process that the parts are equal in size or quantity because the denominator is the same and hence for the result we keep the common denominator and add the numerators.



Adding and subtracting unlike fractions requires the child to visually understand that the parts of each of the fractions are differing in size and therefore we need to find a way of dividing the whole into equal parts so that the parts of all of the fractions look equal. Once this concept is established, the terms LCM and the methods of determining them may be introduced.





Multiplication

Multiplying a fraction by a whole number: Here the repeated addition logic of multiplying whole numbers is still valid. 1/6 multiplied by 4 is 4 times 1/6 which is equal to 4/6.



KOER Fractions html 714bce28.gif



Multiplying a fraction by a fraction: In this case the child is confused as repeated addition does not make sense. To make a child understand the of operator we can use the language and demonstrate it using the measure model and the area of a rectangle.



The area of a rectangle is found by multiplying side length by side length. For example, in the rectangle below, the sides are 3 units and 9 units, and the area is 27 square units.





KOER Fractions html m66ce78ea.gif












We can apply that idea to fractions, too.


  • The one side of the rectangle is 1 unit (in terms of length).
  • The other side is 1 unit also.
  • The whole rectangle also is 1 square unit, in terms of area.



See figure below to see how the following multiplication can be shown.



KOER Fractions html m6c9f1742.gif



KOER Fractions html 753005a4.gif








Remember: The two fractions to multiply represent the length of the sides, and the answer fraction represents area.





Division

Dividing a fraction by a whole number can be demonstrated just like division of whole numbers. When we divide 3/4 by 2 we can visualise it as dividing 3 parts of a whole roti among 4 people.


KOER Fractions html 1f617ac8.gif


Here 3/4 is divided between two people. One fourth piece is split into two. Each person gets 1/4 and 1/8.



KOER Fractions html m5f26c0a.gif



OR


KOER Fractions html m25efcc2e.gif


Another way of solving the same problem is to split each fourth piece into 2.


This means we change the 3/4 into 6/8.


KOER Fractions html 7ed8164a.gif



When dividing a fraction by a fraction, we use the measure interpretation.


KOER Fractions html m3192e02b.gif


When we divide 2 by ¼ we ask how many times does ¼



KOER Fractions html m257a1863.gifKOER Fractions html m257a1863.gif


fit into 2.





It fits in 4 times in each roti, so totally 8 times.



We write it as KOER Fractions html m390fcce6.gif




Activities

Activity 1 Addition of Fractions

Learning Objectives


Understand Addition of Fractions


Materials and resources required


[[6]]


Pre-requisites/ Instructions Method


Open link [[7]]


KOER Fractions html m3dd8c669.png


Move the sliders Numerator1 and Denominator1 to set Fraction 1


Move the sliders Numerator2 and Denominator2 to set Fraction 2


See the last bar to see the result of adding fraction 1 and fraction 2


When you move the sliders ask children to


Observe and describe what happens when the denominator is changed.


Observe and describe what happens when denominator changes


Observe and describe the values of the numerator and denominator and relate it to the third result fraction.


Discuss LCM and GCF


Evaluation


Activity 2 Fraction Subtraction

Learning Objectives


Understand Fraction Subtraction


Materials and resources required


[[8]]


Pre-requisites/ Instructions Method


Open link [[9]]




KOER Fractions html 481d8c4.png


Move the sliders Numerator1 and Denominator1 to set Fraction 1


Move the sliders Numerator2 and Denominator2 to set Fraction 2


See the last bar to see the result of subtracting fraction 1 and fraction 2




When you move the sliders ask children to


observe and describe what happens when the denominator is changed.


observe and describe what happens when denominator changes


observe and describe the values of the numerator and denominator and relate it to the third result fraction.


Discuss LCM and GCF


Evaluation




Activity 3 Multiplication of fractions

Learning Objectives


Understand Multiplication of fractions


Materials and resources required


[[10]]


Pre-requisites/ Instructions Method


Open link [[11]]




KOER Fractions html 12818756.png


Move the sliders Numerator1 and Denominator1 to set Fraction 1


Move the sliders Numerator2 and Denominator2 to set Fraction 2


On the right hand side see the result of multiplying fraction 1 and fraction 2


Material/Activity Sheet


Please open [[12]] in Firefox and follow the process


When you move the sliders ask children to


observe and describe what happens when the denominator is changed.


observe and describe what happens when denominator changes


One unit will be the large square border-in blue solid lines


A sub-unit is in dashed lines within one square unit.


The thick red lines represent the fraction 1 and 2 and also the side of the quadrilateral


The product represents the area of the the quadrilateral


Evaluation


When two fractions are multiplied is the product larger or smaller that the multiplicands – why ?






Activity 4 Division by Fractions

Learning Objectives


Understand Division by Fractions


Materials and resources required


[[13]]


Crayons/ colour pencils, Scissors, glue


Pre-requisites/ Instructions Method


Print out the pdf [[14]]


Colour each of the unit fractions in different colours. Keep the whole unit (1) white.


Cut out each unit fraction piece.


Give examples KOER Fractions html m282c9b3f.gif


For example if we try the first one, KOER Fractions html 21ce4d27.gif See how many KOER Fractions html m31bd6afb.gifstrips will fit exactly onto whole unit strip.




Evaluation


When we divide by a fraction is the result larger or smaller why ?




Evaluation

Self-Evaluation

Further Exploration

  1. [[15]] detailed conceptual understanding of division by fractions
  2. [[16]] understanding fractions
  3. [[17]] Worksheets in mathematics for teachers to use

Linking Fractions to other Topics

Introduction

It is also very common for the school system to treat themes in a separate manner. Fractions are taught as stand alone chapters. In this resource book an attempt to connect it to other middle school topics such as Ratio Proportion, Percentage and high school topics such as rational and irrational numbers, inverse proportions are made. These other topics are not discussed in detail themselves, but used to show how to link these other topics with the already understood concepts of fractions.




Objectives

Explicitly link the other topics in school mathematics that use fractions.


Decimal Numbers

“Decimal” comes from the Latin root decem, which simply means ten. The number system we use is called the decimal number system, because the place value units go in tens: you have ones, tens, hundreds, thousands, and so on, each unit being 10 times the previous one.


In common language, the word “decimal number” has come to mean numbers which have digits after the decimal point, such as 5.8 or 9.302. But in reality, any number within the decimal number system could be termed a decimal number, including whole numbers such as 12 or 381.




The simplest way to link or connect fractions to the decimal number system is with the number line representation. Any scale that a child uses is also very good for this purpose, as seen in the figure below.




The number line between 0 and 1 is divided into ten parts. Each of these ten parts is 1/10, a tenth.


KOER Fractions html 3d7b669f.gif


Under the tick marks you see decimal numbers such as 0.1, 0.2, 0.3, and so on. These are the same numbers as the fractions 1/10, 2/10, 3/10 and so on.


We can write any fraction with tenths (denominator 10) using the decimal point. Simply write after the decimal point how many tenths the number has. 0.6 means six tenths or 1/6. 1.5 means 1 whole and 5 tenths or KOER Fractions html m7f1d448c.gif


Note: A common error one sees is 0.7 is written as 1 /7. It is seven tenths and not one seventh. That the denominator is always 10 has to be stressed. To reinforce this one can use a simple rectangle divided into 10 parts , the same that was used to understand place value in whole numbers.


The coloured portion represents 0.6 or 6/10 and the whole block represents 1.


KOER Fractions html 1cf72869.gif


Percentages

Fractions and percentages are different ways of writing the same thing. When we say that a book costs Rs. 200 and the shopkeeper is giving a 10 % discount. Then we can represent the 10% as a fraction as KOER Fractions html m1369c56e.gif where 10 is the numerator and the denominator is always 100. In this case 10 % of the cost of the book is KOER Fractions html m50e22a06.gif. So you can buy the book for 200 – 20 = 180 rupees.






There are a number of common ones that are useful to learn. Here is a table showing you the ones that you should learn.


Percentage


Fraction


100%


KOER Fractions html m15ed765d.gif


50%


KOER Fractions html df52f71.gif


25%


KOER Fractions html m6c97abb.gif


75%


KOER Fractions html m6cb13da4.gif


10%


KOER Fractions html 26bc75d0.gif


20%


KOER Fractions html m73e98509.gif


40%


KOER Fractions html m2dd64d0b.gif






KOER Fractions html m60c76c68.gifTo see 40 % visually see the figure :


You can see that if the shape is divided into 5 equal parts, then 2 of those parts are shaded.


If the shape is divided into 100 equal parts, then 40 parts are shaded.


These are equivalent fractions as in both cases the same amount has been shaded.




Ratio and Proportion

It is important to understand that fractions also can be interpreted as ratio's. Stressing that a fraction can be interpreted in many ways is of vital importance. Here briefly I describe the linkages that must be established between Ratio and Proportion and the fraction representation. Connecting multiplication of fractions is key to understanding ratio and proportion.




What is ratio?


Ratio is a way of comparing amounts of something. It shows how much bigger one thing is than another. For example:


  • Use 1 measure detergent (soap) to 10 measures water
  • Use 1 shovel (bucket) of cement to 3 shovels (buckets) of sand
  • Use 3 parts blue paint to 1 part white

Ratio is the number of parts to a mix. The paint mix is 4 parts, with 3 parts blue and 1 part white.


The order in which a ratio is stated is important. For example, the ratio of soap to water is 1:10. This means for every 1 measure of soap there are 10 measures of water.


Mixing paint in the ratio 3:1 (3 parts blue paint to 1 part white paint) means 3 + 1 = 4 parts in all.


3 parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white paint.




Cost of a pen is Rs 10 and cost of a pencil is Rs 2. How many times the cost of a pencil is the cost of a pen? Obviously it is five times. This can be written as




The ratio of the cost of a pen to the cost of a pencil = KOER Fractions html m762fb047.gif




What is Direct Proportion ?


Two quantities are in direct proportion when they increase or decrease in the same ratio. For example you could increase something by doubling it or decrease it by halving. If we look at the example of mixing paint the ratio is 3 pots blue to 1 pot white, or 3:1.


Paint pots in a ratio of 3:1


KOER Fractions html m22cda036.gif












But this amount of paint will only decorate two walls of a room. What if you wanted to decorate the whole room, four walls? You have to double the amount of paint and increase it in the same ratio.


If we double the amount of blue paint we need 6 pots.


If we double the amount of white paint we need 2 pots.


Six paint pots in a ratio of 3:1




The amount of blue and white paint we need increase in direct proportion to each other. Look at the table to see how as you use more blue paint you need more white paint:


Pots of blue paint 3 6 9 12


Pots of white paint 1 2 3 4






Two quantities which are in direct proportion will always produce a graph where all the points can be joined to form a straight line.




What is Inverse Proportion ?


Two quantities may change in such a manner that if one quantity increases the the quantity decreases and vice-versa. For example if we are building a room, the time taken to finish decreases as the number of workers increase. Similarly when the speed increases the time to cover a distance decreases. Zaheeda can go to school in 4 different ways. She can walk, run, cycle or go by bus.


Study the table below, observe that as the speed increases time taken to cover the distance decreases





Walk


Run


Cycle


Bus


Speed Km/Hr


3


6 (walk speed *2)


9 (walk speed *3)


45 (walk speed *15)


Time Taken (minutes)


30


15 (walk Time * ½)


10 (walk Time * 1/3)


2 (walk Time * 1/15)




As Zaheeda doubles her speed by running, time reduces to half. As she increases her speed to three times by cycling, time decreases to one third. Similarly, as she increases her speed to 15 times, time decreases to one fifteenth. (Or, in other words the ratio by which time decreases is inverse of the ratio by which the corresponding speed increases). We can say that speed and time change inversely in proportion.




Moving from Additive Thinking to Multiplicative Thinking



Avinash thinks that if you use 5 spoons of sugar to make 6 cups of tea, then you would need 7 spoons of sugar to make 8 cups of tea just as sweet as the cups before. Avinash would be using an additive transformation'''; he thinks that since we added 2 more cups of tea from 6 to 8. To keep it just as sweet he would need to add to more spoons of sugar. What he does not know is that for it to taste just as sweet he would need to preserve the ratio of sugar to tea cup and use multiplicative thinking. He is unable to detect the ratio.


Proportional Reasoning

Proportional thinking involves the ability to understand and compare ratios, and to predict and produce equivalent ratios. It requires comparisons between quantities and also the relationships between quantities. It involves quantitative thinking as well as qualitative thinking. A feature of proportional thinking is the multiplicative relationship among the quantities and being able to recognize this relationship. The relationship may be direct (divide), i.e. when one quantity increases, the other also increases. The relationship is inverse (multiply), when an increase in one quantity implies a decrease in the other, in both cases the ratio or the rate of change remains a constant.



The process of adding involved situations such as adding, joining, subtracting, removing actions which involves the just the two quantities that are being joined, while proportional thinking is associated with shrinking, enlarging, scaling , fair sharing etc. The process involves multiplication. To be able to recognize, analyse and reason these concepts is multiplicative thinking/reasoning. Here the student must be able to understand the third quantity which is the ratio of the two quantities. The preservation of the ratio is important in the multiplicative transformation.




Rational & Irrational Numbers

After the number line was populated with natural numbers, zero and the negative integers, we discovered that it was full of gaps. We discovered that there were numbers in between the whole numbers - fractions we called them.


But, soon we discovered numbers that could not be expressed as a fraction. These numbers could not be represented as a simple fraction. These were called irrational numbers. The ones that can be represented by a simple fraction are called rational numbers. They h ad a very definite place in the number line but all that could be said was that square root of 2 is between 1.414 and 1.415. These numbers were very common. If you constructed a square, the diagonal was an irrational number. The idea of an irrational number caused a lot of agony to the Greeks. Legend has it that Pythagoras was deeply troubled by this discovery made by a fellow scholar and had him killed because this discovery went against the Greek idea that numbers were perfect.


How can we be sure that an irrational number cannot be expressed as a fraction? This can be proven algebraic manipulation. Once these "irrational numbers" came to be identified, the numbers that can be expressed of the form p/q where defined as rational numbers.


There is another subset called transcendental numbers which have now been discovered. These numbers cannot be expressed as the solution of an algebraic polynomial. "pi" and "e" are such numbers.


Activities

Activity 1 Fractions representation of decimal numbers

Learning Objectives


Fractions representation of decimal numbers


Materials and resources required


[[18]]


[[19]]




Pre-requisites/ Instructions Method


Make copies of the worksheets


[[20]]


[[21]]




Evaluation


  1. Draw a number line and name the fraction and decimal numbers on the number line. Take a print of the document [[22]] . Ask students to place any fraction and decimal numbers between between 0 and 10 on the number line
  2. Write 0.45, 0.68, 0.05 in fraction form and represent as a fraction 100 square.





Activity 2 Fraction representation and percentages

Learning Objectives


Understand fraction representation and percentages




Materials and resources required


[[23]]


[[24]]


Pre-requisites/ Instructions Method


Please print copies of the 2 activity sheets [[25]] and [[26]] and discuss the various percentage quantities with the various shapes.



Then print a copy each of spider-percentages.pdf and make the children do this activity




Evaluation


  1. What value is the denominator when we represent percentage as fraction ?
  2. What does the numerator represent ?
  3. What does the whole fraction represent ?
  4. What other way can we represent a fraction whose denominator is 100.





Activity 3

Learning Objectives


Understand fraction representation and rational and irrational numbers


Materials and resources required


Thread of a certain length.


Pre-requisites/ Instructions Method


Construct Koch's snowflakes .



Start with a thread of a certain length (perimeter) and using the same thread construct the following shapes (see Figure).


KOER Fractions html m1a6bd0d0.gif


See how the shapes can continue to emerge but cannot be identified definitely with the same perimeter (length of the thread).



Identify the various places where pi, "e" and the golden ratio occur






Evaluation


  1. How many numbers can I represent on a number line between 1 and 2.
  2. What is the difference between a rational and irrational number, give an example ?
  3. What is Pi ? Why is it a special number ?



Evaluation

Self-Evaluation

Further Exploration

  1. Percentage and Fractions, [[27]]
  2. A mathematical curve Koch snowflake, [[28]]
  3. Bringing it Down to Earth: A Fractal Approach, [[29]]

See Also

  1. Video on teaching fractions using the equal share method made by Eklavya an NGO based in Madhya Pradesh, India [[30]]
  2. Mathematics resources from Homi Baba Centre for Science Education , [[31]]
  3. [[32]] Understand how to use Geogebra a mathematical computer aided tool

Teachers Corner

Books

  1. [[33]] A google book Fractions in realistic mathematics education: a paradigm of developmental ...By Leen Streefland

References

  1. Introducing Fractions Using Share and Measure Interpretations: A Report from Classroom Trials . Jayasree Subramanian, Eklavya, Hoshangabad, India and Brijesh Verma,Muskan, Bhopal, India
  2. Streefland, L. (1993). “Fractions: A Realistic Approach.” In Carpenter, T.P., Fennema, E., Romberg, T.A. (Eds.), Rational Numbers: an Integration of Research. (289-325). New Jersey: Lawrence, Erlbaum Associates, Publishers.