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Fractions (view source)

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<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#ffffff; vertical-align:top; text-align:center; padding:5px;">
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''[http://karnatakaeducation.org.in/KOER/index.php//೧೦ನೇ_ತರಗತಿಯ_ಭಿನ್ನರಾಶಿಗಳು ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ]''</div>
 
= Introduction =
 
= Introduction =
 
   
 
   
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ready reference for the teacher to develop the concepts, inculcate
 
ready reference for the teacher to develop the concepts, inculcate
 
necessary skills, and impart knowledge in fractions from Class 6 to
 
necessary skills, and impart knowledge in fractions from Class 6 to
−
Class 10.
+
Class X
    
   
 
   
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conceptually understanding topics of fractions.
 
conceptually understanding topics of fractions.
    +
This can be used as part of the bridge course material alongwith Number Systems
   −
 
   
= Mind Map =
 
= Mind Map =
 
   
 
   
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== Introduction ==
 
== Introduction ==
 
   
 
   
−
Commonly fractions are always approached by teaching it through
+
Fractions  are defined in relation to a whole—or unit amount—by dividing the whole into equal parts. The notion of dividing into equal parts may seem simple, but it can be problematic. Commonly fractions are always approached by teaching it through
 
one model or interpretation namely the '''part-whole '''model
 
one model or interpretation namely the '''part-whole '''model
 
where the '''whole '''is
 
where the '''whole '''is
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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
−
has led to educators using other interpretations such as '''equal
+
has led to educators using other interpretations such as '''equal'''
−
share''' and '''measure'''.
+
share''' and '''measure'''.''' 
−
These approaches to fraction teaching are discussed here.
+
 
 +
Although we use pairs of numbers to represent fractions, a fraction stands for a single number, and as such, has a location on the number line. Number lines provide an excellent way to represent improper fractions, which represent an amount that is more than the related whole. 
 +
 
 +
Given their different representations, and the way they sometimes refer to a number and sometimes an operation, it is important to be able to discuss fractions in the many ways they appear. A multiple representation activity, including different numerical and visual representations, is one way of doing this. Sharing food is a good way to introduce various concepts about fractions. For example, using a chocolate bar and dividing it into pieces. This can be highly motivating if learners can eat it afterwards.  A clock face shows clearly what halves and quarters look like, and can be extended to other fractions with discussion about why some are easier to show than others. We can find a third of an hour, but what about a fifth? 
 +
 
 +
The five meanings listed below serve as conceptual models or tools for thinking about and working with fractions and serve as a framework for designing teaching activities that engage students in sense making as they construct knowledge about fractions.
 +
 
 +
1.Part of a whole 2.Part of a group/set 3.Measure (name for point on number line) 4.Ratio 5.Indicated division
 +
 
 +
We recommend that teachers explicitly use the language of fractions in other parts of the curriculum for reinforcement. For example, when looking at shapes, talk about ‘half a square’ and ‘third of a circle’.
   −
+
The various approaches to fraction teaching are discussed here.
 
== Objectives ==
 
== Objectives ==
 
   
 
   
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Half
 
Half
−
(½) : The whole is divided into '''two
+
(½) : The whole is divided into '''two'''
−
equal '''parts.
+
equal '''parts.'''
    
   
 
   
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One-Fourth
 
One-Fourth
−
(1/4) : The whole is divided into '''four
+
(1/4) : The whole is divided into '''four'''
−
equal '''parts.
+
equal '''parts.'''
    
   
 
   
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−
  −
   
   
 
   
 
[[Image:KOER%20Fractions_html_43b75d3a.gif]]
 
[[Image:KOER%20Fractions_html_43b75d3a.gif]]
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One
 
One
−
(2/2 or 1) : The whole is divided into '''two
+
(2/2 or 1) : The whole is divided into '''two'''
−
equal '''parts.
+
equal '''parts.'''
    
   
 
   
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Two
 
Two
−
Fifth (2/5) : The whole is divided into '''five
+
Fifth (2/5) : The whole is divided into '''five'''
−
equal '''parts.
+
equal '''parts.'''
    
   
 
   
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part are coloured, this part represents the fraction 2/5.
 
part are coloured, this part represents the fraction 2/5.
   −
+
 
−
 
  −
 
  −
  −
 
  −
 
  −
  −
 
  −
 
   
   
 
   
 
[[Image:KOER%20Fractions_html_9e5c77.gif]]Three
 
[[Image:KOER%20Fractions_html_9e5c77.gif]]Three
−
Seventh (3/7) : The whole is divided into '''seven
+
Seventh (3/7) : The whole is divided into '''seven'''
−
equal '''parts.
+
equal '''parts.'''
    
   
 
   
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part are coloured, this part represents the fraction 3/7.
 
part are coloured, this part represents the fraction 3/7.
   −
+
 
−
 
  −
 
  −
  −
 
  −
 
   
   
 
   
 
[[Image:KOER%20Fractions_html_m30791851.gif]]
 
[[Image:KOER%20Fractions_html_m30791851.gif]]
    
   
 
   
−
  −
   
   
 
   
 
Seven
 
Seven
−
tenth (7/10) : The whole is divided into '''ten
+
tenth (7/10) : The whole is divided into '''ten'''
−
equal '''parts.
+
equal '''parts.'''
    
   
 
   
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part are coloured, this part represents the fraction 7/10 .
 
part are coloured, this part represents the fraction 7/10 .
    +
 
 
   
 
   
−
 
+
'''Terms Numerator'''
−
 
+
and Denominator and their meaning
−
 
  −
  −
 
  −
 
  −
  −
'''Terms Numerator
  −
and Denominator and their meaning'''
      
   
 
   
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Three
 
Three
−
Eight (3/8) The whole is divided into '''eight
+
Eight (3/8) The whole is divided into '''eight'''
−
equal '''parts.
+
equal '''parts.'''
    
   
 
   
−
  −
   
   
 
   
 
'''Three'''
 
'''Three'''
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In the equal share interpretation the fraction '''m/n''' denotes
 
In the equal share interpretation the fraction '''m/n''' denotes
−
one share when '''m identical things''' are '''shared equally among
+
one share when '''m identical things''' are '''shared equally among'''
−
n'''. The relationships between fractions are arrived at by logical
+
n'''. The relationships between fractions are arrived at by logical'''
 
reasoning (Streefland, 1993). For example ''' 5/6 '''is the share of
 
reasoning (Streefland, 1993). For example ''' 5/6 '''is the share of
 
one child when 5 rotis (disk-shaped handmade bread) are shared
 
one child when 5 rotis (disk-shaped handmade bread) are shared
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out that four copies of the given quantity put together would make
 
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
 
three wholes and hence is equal to one share when these three wholes
−
are shared equally among 4. '''''Share interpretation is also the
+
are shared equally among 4. '''''Share interpretation is also the'''''
 
quotient interpretation of fractions in the sense that 3⁄4 = 3 ÷ 4
 
quotient interpretation of fractions in the sense that 3⁄4 = 3 ÷ 4
 
and this is important for developing students’ ability to solve
 
and this is important for developing students’ ability to solve
−
problems involving multiplicative and linear functional relations. '''''
+
problems involving multiplicative and linear functional relations.  
      Line 269: Line 254:  
measure of one part when one whole is divided into ''n ''equal
 
measure of one part when one whole is divided into ''n ''equal
 
parts. The ''composite fraction'' ''m/n '' is as the measure of
 
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
+
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 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
 
each. Since 5 piece units of size make a whole, we get the relation
 
6/5 = 1 + 1/5.
 
6/5 = 1 + 1/5.
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−
  −
  −
   
   
 
   
 
=== Introducing Fractions Using Share and Measure Interpretations ===
 
=== Introducing Fractions Using Share and Measure Interpretations ===
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−
  −
  −
   
   
 
   
 
== Activities ==
 
== Activities ==
 
   
 
   
 
=== Activity1: Introduction to fractions ===
 
=== Activity1: Introduction to fractions ===
 +
This video helps to know the basic information about fraction.
 +
 +
{{#widget:YouTube|id=n0FZhQ_GkKw}}
 
   
 
   
 +
 
'''''Learning Objectives '''''
 
'''''Learning Objectives '''''
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−
'''''Materials and
+
'''''Materials and'''''
−
resources required '''''
+
resources required
 
   
 
   
 
# 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]]
 
# 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]]
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'''''Pre-requisites/
+
'''''Pre-requisites/'''''
−
Instructions Method '''''
+
Instructions Method
 
   
 
   
 
Do
 
Do
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=== Activity 2: Proper and Improper Fractions ===
 
=== Activity 2: Proper and Improper Fractions ===
 
   
 
   
−
'''''Learning
+
'''''Learning'''''
−
Objectives'''''
+
Objectives
    
   
 
   
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−
'''''Materials
+
'''''Materials'''''
−
and resources required '''''
+
and resources required
    
   
 
   
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[[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_5518d221.jpg]][[Image:KOER%20Fractions_html_5e906d5b.jpg]]
 
[[Image:KOER%20Fractions_html_5518d221.jpg]][[Image:KOER%20Fractions_html_5e906d5b.jpg]]
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[[Image:KOER%20Fractions_html_55f65a3d.gif]]
 
[[Image:KOER%20Fractions_html_55f65a3d.gif]]
   −
 
+
 
 +
 +
# 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
 +
 +
 
 
   
 
   
 +
[[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]]
       +
 +
[[Image:KOER%20Fractions_html_5518d221.jpg]]
    
   
 
   
 +
 +
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_5518d221.jpg]]
       +
 +
[[Image:KOER%20Fractions_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.
    
   
 
   
−
# 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
   
   
 
   
 +
'''''Evaluation'''''
    +
# 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 ?
    +
=== Activity 3: Comparing Fractions ===
 +
 +
'''''Learning'''''
 +
Objectives
    
   
 
   
 +
Comparing-Fractions
    +
 +
'''''Materials'''''
 +
and resources required
    +
 
 +
 +
[[http://www.superteacherworksheets.com/fractions/comparing-fractions.pdf]]
    
   
 
   
 +
[[http://www.superteacherworksheets.com/fractions/comparing-fractions2.pdf]]
    +
 +
'''Pre-requisites/'''
 +
Instructions Method
    +
 +
Print the
 +
document and work out the
 +
activity sheet
    
   
 
   
−
[[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]]
+
'''''Evaluation'''''
    +
 +
# 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 ?
 +
    
   
 
   
−
[[Image:KOER%20Fractions_html_5518d221.jpg]]
+
=== Activity 4: Equivalent Fractions ===
−
 
+
 +
'''''Learning'''''
 +
Objectives
    
   
 
   
−
 
+
To understand Equivalent Fractions
−
 
      
   
 
   
−
[[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_5518d221.jpg]]
+
'''''Materials'''''
−
 
+
and resources required
    
   
 
   
−
[[Image:KOER%20Fractions_html_55f65a3d.gif]]
+
[[http://www.superteacherworksheets.com/fractions/fractions-matching-game.pdf]]
−
 
      
   
 
   
−
  −
  −
   
   
 
   
−
'''Pre-requisites/
+
'''Pre-requisites/'''
−
Instructions Method '''
+
Instructions Method
    
   
 
   
−
Examples of Proper and improper
+
Print 10 copies
−
fractions are given. The round disks in the figure represent rotis
+
of the document from pages 2 to 5 fractions-matching-game
−
and the children figures represent children. Cut each roti and each
+
Cut the each fraction part. Play memory game as described in
−
child figure and make the children fold, tear and equally divide the
+
the document in groups of 4 children.
−
roits so that each child figure gets equal share of roti.
  −
 
  −
  −
 
      
   
 
   
 
'''''Evaluation'''''
 
'''''Evaluation'''''
   −
# 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 ?
  −
  −
=== Activity 3: Comparing Fractions ===
   
   
 
   
−
'''''Learning
+
# What is reducing a fraction to the simplest form ?
−
Objectives'''''
+
# What is GCF – Greatest Common Factor ?
−
 
+
# Use the document [[simplifying-fractions.pdf]]
 +
# Why are fractions called equivalent and not equal.
 
   
 
   
−
Comparing-Fractions
+
== Evaluation ==
   −
+
== Self-Evaluation ==
−
'''''Materials
+
This '''PhET simulation''', lets you
−
and resources required '''''
+
* Find matching fractions using numbers and pictures <br>
 +
* Make the same fractions using different numbers <br>
 +
* Match fractions in different picture patterns <br>
 +
* Compare fractions using numbers and patterns <br>
 +
 
 +
[https://phet.colorado.edu/sims/html/fraction-matcher/latest/fraction-matcher_en.html Fraction Matcher]
    +
== Further Exploration ==
 +
 
 +
== Enrichment Activities ==
 
   
 
   
−
 
+
= Errors with fractions =
−
 
  −
 
   
   
 
   
−
 
+
== Introduction ==
−
 
  −
 
   
   
 
   
−
[[http://www.superteacherworksheets.com/fractions/comparing-fractions.pdf]]
+
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.
    
   
 
   
−
[[http://www.superteacherworksheets.com/fractions/comparing-fractions2.pdf]]
+
== Objectives ==
−
 
   
   
 
   
−
'''Pre-requisites/
+
When fractions are operated erroneously
−
Instructions Method'''
+
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'''.
    +
 
 
   
 
   
−
Print the
+
== N-Distractors ==
−
document and work out the
  −
activity sheet
  −
 
   
   
 
   
−
'''''Evaluation'''''
      
   
 
   
−
# Does the child know the symbols '''&gt;, &lt;''' and '''='''
+
The five levels of resistance to
−
# What happens to the size of the part when the denominator is different ?
+
N-Distractors that a child develops are:
−
# 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 ===
   
   
 
   
−
'''''Learning
+
# '''''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.
−
Objectives'''''
+
# '''''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.
 
   
 
   
−
To understand Equivalent Fractions
      
   
 
   
−
'''''Materials
+
== Activities ==
−
and resources required'''''
  −
 
   
   
 
   
−
[[http://www.superteacherworksheets.com/fractions/fractions-matching-game.pdf]]
+
== Evaluation ==
−
 
+
 +
== Self-Evaluation ==
 
   
 
   
−
 
+
== Further Exploration ==
−
 
  −
 
   
   
 
   
−
'''Pre-requisites/
+
# [[www.merga.net.au/publications/counter.php?pub=pub_conf&id=1410]] A PDF Research paper titled Probing Whole Number Dominance with Fractions.
−
Instructions Method'''
+
# [[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
 
   
 
   
−
Print 10 copies
+
== Enrichment Activities ==
−
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'''''
      
   
 
   
−
# What is reducing a fraction to the simplest form ?
+
= Operations on Fractions =
−
# What is GCF – Greatest Common Factor ?
  −
# Use the document [[simplifying-fractions.pdf]]
  −
# Why are fractions called equivalent and not equal.
   
   
 
   
−
== Evaluation ==
+
== Introduction ==
 
   
 
   
−
== Self-Evaluation ==
+
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.
 +
 
 
   
 
   
−
== Further Exploration ==
+
== Objectives ==
−
 
+
−
== Enrichment Activities ==
+
The aim of this section is to visualise and conceptually
 +
understand each of the operations on fractions.
 +
 
 
   
 
   
−
= Errors with fractions =
+
== Addition and Subtraction ==
 
   
 
   
−
== Introduction ==
+
 
 
   
 
   
−
A brief
+
Adding and
−
understanding of the common errors that children make when it comes
+
subtracting like fractions is simple. It must be emphasised thought
−
to fractions are addressed to enable teachers to understand the
+
even during this process that the parts are equal in size or quantity
−
child's levels of conceptual understanding to address the
+
because the denominator is the same and hence for the result we keep
−
misconceptions.
+
the common denominator and add the numerators.
    
   
 
   
−
== Objectives ==
   
   
 
   
−
When fractions are operated erroneously
+
Adding and
−
like natural numbers, i.e. treating the numerator and the
+
subtracting unlike fractions requires the child to visually
−
denominators separately and not considering the relationship between
+
understand that the parts of each of the fractions are differing in
−
the numerator and the denominator is termed as N-Distractor. For
+
size and therefore we need to find a way of dividing the whole into
−
example 1/3 + ¼ are added to result in 2/7. Here 2 units of the
+
equal parts so that the parts of all of the fractions look equal.
−
numerator are added and 3 &amp; four units of the denominator are
+
Once this concept is established, the terms LCM and the methods of
−
added. This completely ignores the relationship between the numerator
+
determining them may be introduced.
−
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'''.
      +
 
 +
 +
== 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.
    
   
 
   
−
== N-Distractors ==
   
   
 
   
−
 
+
[[Image:KOER%20Fractions_html_714bce28.gif]]
    
   
 
   
−
The five levels of resistance to
  −
N-Distractors that a child develops are:
  −
   
   
 
   
−
 
+
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.
    
   
 
   
−
# '''''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.
   
   
 
   
 +
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:KOER%20Fractions_html_m66ce78ea.gif]]
    +
     
 
   
 
   
−
== Activities ==
+
We can apply that
 +
idea to fractions, too.
 +
 
 
   
 
   
−
== Evaluation ==
+
* 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.
 +
 +
 
 
   
 
   
−
== Self-Evaluation ==
+
See figure below
 +
to see how the following multiplication can be shown.
 +
 
 
   
 
   
−
== 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.
+
[[Image:KOER%20Fractions_html_m6c9f1742.gif]]
−
# [[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 ==
   
   
 
   
 +
[[Image:KOER%20Fractions_html_753005a4.gif]]
    +
   
 +
 +
'''Remember: '''The
 +
two fractions to multiply represent the length of the sides, and the
 +
answer fraction represents area.
   −
 
+
 
 
   
 
   
−
= Operations on Fractions =
+
== Division ==
 
   
 
   
−
== Introduction ==
+
 
 
   
 
   
−
This topic introduces the different operations on fractions. When
+
Dividing a fraction by a whole number
−
learners move from whole numbers to fractions, many of the operations
+
can be demonstrated just like division of whole numbers. When we
−
are counter intuitive. This section aims to clarify the concepts
+
divide 3/4 by 2 we can visualise it as dividing 3 parts of a whole
−
behind each of the operations.
+
roti among 4 people.
    
   
 
   
−
== Objectives ==
+
[[Image:KOER%20Fractions_html_1f617ac8.gif]]
 +
 
 
   
 
   
−
The aim of this section is to visualise and conceptually
+
Here 3/4 is divided between two
−
understand each of the operations on fractions.
+
people. One fourth piece is split into two. Each person gets
 +
1/4 and 1/8.
    
   
 
   
−
== Addition and Subtraction ==
   
   
 
   
 +
[[Image:KOER%20Fractions_html_m5f26c0a.gif]]
   −
   
   
 
   
−
Adding and
+
−
subtracting like fractions is simple. It must be emphasised thought
+
OR
−
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.
      
   
 
   
−
 
+
[[Image:KOER%20Fractions_html_m25efcc2e.gif]]
    
   
 
   
−
Adding and
+
Another way of solving the same
−
subtracting unlike fractions requires the child to visually
+
problem is to split each fourth piece into 2.
−
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.
      
   
 
   
 +
This means we change the 3/4
 +
into 6/8.
    +
 
 +
[[Image:KOER%20Fractions_html_7ed8164a.gif]]
    
   
 
   
 +
 +
When dividing a fraction by a fraction,
 +
we use the measure interpretation.
    +
 +
[[Image:KOER%20Fractions_html_m3192e02b.gif]]
    
   
 
   
−
== Multiplication ==
+
When we divide 2 by ¼ we ask how many
 +
times does ¼
 +
 
 
   
 
   
−
  −
   
   
 
   
−
Multiplying a
+
[[Image:KOER%20Fractions_html_m257a1863.gif]][[Image:KOER%20Fractions_html_m257a1863.gif]]
−
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.
      
   
 
   
 +
'''fit into 2'''.
    +
 
 +
 +
It fits in 4 times in each roti, so
 +
totally 8 times.
    
   
 
   
−
[[Image:KOER%20Fractions_html_714bce28.gif]]
+
 +
We write it as
 +
[[Image:KOER%20Fractions_html_m390fcce6.gif]]
    
   
 
   
 +
 +
== Activities ==
 +
 +
=== Activity 1 Addition of Fractions ===
 +
 +
'''''Learning'''''
 +
Objectives
    +
 +
Understand Addition of Fractions
    
   
 
   
−
Multiplying a
+
'''''Materials'''''
−
fraction by a fraction: In this case the child is confused as
+
and resources required
−
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.
      
   
 
   
−
 
+
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]]
    
   
 
   
−
The
+
'''''Pre-requisites/'''''
−
area of a rectangle is found by multiplying side length by side
+
Instructions Method
−
length. For example, in the rectangle below, the sides are 3 units
  −
and 9 units, and the area is 27 square units.
      
   
 
   
−
 
+
Open link
 +
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]]
    
   
 
   
 +
[[Image:KOER%20Fractions_html_m3dd8c669a.png]]
       
   
 
   
−
[[Image:KOER%20Fractions_html_m66ce78ea.gif]]
+
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
    
   
 
   
−
We can apply that
+
'''''Materials and'''''
−
idea to fractions, too.
+
resources required
    
   
 
   
−
* The one side of the rectangle is 1 unit (in terms of length).
+
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
−
* The other side is 1 unit also.
  −
* The whole rectangle also is ''1 square unit'', in terms of area.
  −
      +
 
 +
'''''Pre-requisites/'''''
 +
Instructions Method
    
   
 
   
−
See figure below
+
Open link
−
to see how the following multiplication can be shown.
+
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
    
   
 
   
 +
 +
[[Image:KOER%20Fractions_html_481d8c4.png|600px]]
       
   
 
   
−
[[Image:KOER%20Fractions_html_m6c9f1742.gif]]
+
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
    
   
 
   
−
[[Image:KOER%20Fractions_html_753005a4.gif]]
  −
   
   
 
   
−
 
+
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
    
   
 
   
−
'''Remember: '''The
+
Understand Multiplication of fractions
−
two fractions to multiply represent the length of the sides, and the
  −
answer fraction represents area.
      
   
 
   
 +
'''''Materials and'''''
 +
resources required
    +
 +
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
    
   
 
   
 +
'''''Pre-requisites/'''''
 +
Instructions Method
    +
 +
Open link
 +
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
    
   
 
   
−
== Division ==
   
   
 
   
 +
[[Image:KOER%20Fractions_html_12818756.png|600px]]
       
   
 
   
−
Dividing a fraction by a whole number
+
Move the sliders Numerator1 and Denominator1 to set Fraction 1
−
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.
      
   
 
   
−
[[Image:KOER%20Fractions_html_1f617ac8.gif]]
+
Move the sliders Numerator2 and Denominator2 to set Fraction 2
    
   
 
   
−
Here 3/4 is divided between two
+
On the right hand side see the result of multiplying fraction 1
−
people. One fourth piece is split into two. Each person gets
+
and fraction 2
−
1/4 and 1/8.
      
   
 
   
 +
'''Material/Activity Sheet'''
    +
 +
Please open
 +
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
 +
in Firefox and follow the process
    
   
 
   
−
[[Image:KOER%20Fractions_html_m5f26c0a.gif]]
+
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
    
   
 
   
−
OR
+
One unit will be the large square border-in blue solid lines
    
   
 
   
−
[[Image:KOER%20Fractions_html_m25efcc2e.gif]]
+
A sub-unit is in dashed lines within one square unit.
    
   
 
   
−
Another way of solving the same
+
The thick red lines represent the fraction 1 and 2 and also the
−
problem is to split each fourth piece into 2.
+
side of the quadrilateral
    
   
 
   
−
This means we change the 3/4
+
The product represents the area of the the quadrilateral
−
into 6/8.
     −
 
+
−
[[Image:KOER%20Fractions_html_7ed8164a.gif]]
+
'''''Evaluation'''''
    
   
 
   
 +
When
 +
two fractions are multiplied
 +
is the product larger or smaller that the multiplicands – why ?
    +
=== Activity 4 Division by Fractions ===
 +
 +
'''''Learning'''''
 +
Objectives
    
   
 
   
−
When dividing a fraction by a fraction,
+
Understand Division by Fractions
−
we use the measure interpretation.
      
   
 
   
−
[[Image:KOER%20Fractions_html_m3192e02b.gif]]
+
'''''Materials and'''''
 +
resources required
    
   
 
   
−
When we divide 2 by ¼ we ask how many
+
[[http://www.superteacherworksheets.com/fractions/fractionstrips_TWQWF.pdf]]
−
times does ¼
      
   
 
   
−
 
+
Crayons/ colour
 +
pencils, Scissors, glue
    
   
 
   
−
[[Image:KOER%20Fractions_html_m257a1863.gif]][[Image:KOER%20Fractions_html_m257a1863.gif]]
+
'''''Pre-requisites/'''''
 +
Instructions Method
    
   
 
   
−
'''fit into 2'''.
+
Print out the pdf
 +
[[http://www.superteacherworksheets.com/fractions/fractionstrips_TWQWF.pdf]]
    
   
 
   
−
 
+
Colour each of the unit fractions in different colours. Keep the
 +
whole unit (1) white.
    
   
 
   
−
 
+
Cut out each unit fraction piece.
    
   
 
   
−
It fits in 4 times in each roti, so
+
Give examples
−
totally 8 times.
+
[[Image:KOER%20Fractions_html_m282c9b3f.gif]]
    
   
 
   
 +
For example if we try the first one,
 +
[[Image:KOER%20Fractions_html_21ce4d27.gif]]
 +
See how many
 +
[[Image:KOER%20Fractions_html_m31bd6afb.gif]]strips
 +
will fit exactly onto whole unit strip.
   −
   
   
 
   
−
We write it as
+
−
[[Image:KOER%20Fractions_html_m390fcce6.gif]]
+
== Evaluation ==
 +
When
 +
we divide by a fraction is the result larger or smaller why ?
   −
+
== Self-Evaluation ==
 +
This '''PhET simulation''' enables you to
 +
*Predict and explain how changing the numerator or denominator of a fraction affects the fraction's value. <br>
 +
* Make equivalent fractions using different numbers. <br>
 +
* Match fractions in different picture patterns. <br>
 +
*Find matching fractions using numbers and pictures. <br>
 +
* Compare fractions using numbers and patterns. <br>
    +
[https://phet.colorado.edu/en/simulation/legacy/fractions-intro Fractions-intro]
    +
Software requirement: Sun Java 1.5.0_15 or later version
    +
== Further Exploration ==
 
   
 
   
−
== Activities ==
+
# [[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.superteacherworksheets.com]] Worksheets in mathematics for teachers to use
 
   
 
   
−
=== Activity 1 Addition of Fractions ===
+
= Linking Fractions to other Topics =
 
   
 
   
−
'''''Learning
+
== Introduction ==
−
Objectives'''''
  −
 
   
   
 
   
−
Understand Addition of Fractions
+
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.
    
   
 
   
−
'''''Materials
+
== Decimal Numbers ==
−
and resources required '''''
+
 +
“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.
    
   
 
   
−
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]]
+
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.
    
   
 
   
−
'''''Pre-requisites/
  −
Instructions Method '''''
  −
   
   
 
   
−
Open link  
+
The
−
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]]
+
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.
    
   
 
   
−
[[Image:KOER%20Fractions_html_m3dd8c669a.png]]
  −
  −
   
   
 
   
−
Move the sliders
+
The
−
Numerator1 and Denominator1 to set Fraction 1
+
number line between 0 and 1 is divided into ten parts. Each of these
 +
ten parts is '''1/10''', a '''tenth'''.
    
   
 
   
−
Move the sliders
+
[[Image:KOER%20Fractions_html_3d7b669f.gif]]
−
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
+
Under
−
the sliders ask children to
+
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.
    
   
 
   
−
Observe and
+
We
−
describe what happens when the denominator is changed.
+
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:KOER%20Fractions_html_m7f1d448c.gif]]
    
   
 
   
−
Observe and
+
Note: A common error one sees is 0.7 is written as 1 /7. It is
−
describe what happens when denominator changes
+
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.
    
   
 
   
−
Observe and
+
The
−
describe the values of the numerator and denominator and relate it to
+
coloured portion represents 0.6 or 6/10 and the whole block
−
the third result fraction.
+
represents 1.
   −
+
 
−
Discuss LCM and
+
[[Image:KOER%20Fractions_html_1cf72869.gif]]
−
GCF
     −
  −
'''''Evaluation'''''
     −
=== Activity 2 Fraction Subtraction ===
+
 
 +
== Percentages ==
 
   
 
   
−
'''''Learning
+
Fractions and percentages are different ways of writing the same
−
Objectives '''''
+
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: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.
    +
 
 
   
 
   
−
Understand Fraction Subtraction
+
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
 
   
 
   
−
'''''Materials and
+
|
−
resources required'''''
+
Fraction
−
 
   
   
 
   
−
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
+
|-
−
 
+
|
−
 
+
100%
−
'''''Pre-requisites/
  −
Instructions Method '''''
  −
 
   
   
 
   
−
Open link
+
|
−
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
+
[[Image:KOER%20Fractions_html_m15ed765d.gif]]
−
 
   
   
 
   
−
 
+
|-
−
 
+
|
−
 
+
50%
 
   
 
   
−
[[Image:KOER%20Fractions_html_481d8c4.png|600px]]
+
|
−
 
+
[[Image:KOER%20Fractions_html_df52f71.gif]]
−
 
   
   
 
   
−
Move the sliders Numerator1 and Denominator1 to set Fraction 1
+
|-
−
 
+
|
 +
25%
 
   
 
   
−
Move the sliders Numerator2 and Denominator2 to set Fraction 2
+
|
−
 
+
[[Image:KOER%20Fractions_html_m6c97abb.gif]]
 
   
 
   
−
See the last bar to see the result of subtracting fraction 1 and
+
|-
−
fraction 2
+
|
−
 
+
75%
 
   
 
   
−
 
+
|
−
 
+
[[Image:KOER%20Fractions_html_m6cb13da4.gif]]
−
 
   
   
 
   
−
When you move the sliders ask children to
+
|-
−
 
+
|
 +
10%
 
   
 
   
−
observe and describe what happens when the denominator is
+
|
−
changed.
+
[[Image:KOER%20Fractions_html_26bc75d0.gif]]
−
 
   
   
 
   
−
observe and describe what happens when denominator changes
+
|-
−
 
+
|
 +
20%
 
   
 
   
−
observe and describe the values of the numerator and denominator
+
|
−
and relate it to the third result fraction.
+
[[Image:KOER%20Fractions_html_m73e98509.gif]]
−
 
   
   
 
   
−
Discuss LCM and GCF
+
|-
−
 
+
|
 +
40%
 
   
 
   
−
'''''Evaluation'''''
+
|
−
 
+
[[Image:KOER%20Fractions_html_m2dd64d0b.gif]]
−
=== Activity 3  Multiplication of fractions ===
   
   
 
   
−
'''''Learning
+
|} 
−
Objectives '''''
      
   
 
   
−
Understand Multiplication of fractions
  −
   
   
 
   
−
'''''Materials and
+
[[Image:KOER%20Fractions_html_m60c76c68.gif]]To
−
resources required'''''
+
see 40 % visually see the figure :
    
   
 
   
−
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
+
You
 +
can see that if the shape is divided into 5 equal parts, then 2 of
 +
those parts are shaded.
    
   
 
   
−
'''''Pre-requisites/
+
If
−
Instructions Method '''''
+
the shape is divided into 100 equal parts, then 40 parts are shaded.
    
   
 
   
−
Open link
+
These
−
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
+
are equivalent fractions as in both cases the same amount has been
 +
shaded.
    
   
 
   
−
  −
  −
   
   
 
   
−
[[Image:KOER%20Fractions_html_12818756.png|600px]]
+
== Ratio and Proportion ==
−
 
  −
 
   
   
 
   
−
Move the sliders Numerator1 and Denominator1 to set Fraction 1
+
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.
    
   
 
   
−
Move the sliders Numerator2 and Denominator2 to set Fraction 2
  −
   
   
 
   
−
On the right hand side see the result of multiplying fraction 1
+
'''What'''
−
and fraction 2
+
is ratio?
    
   
 
   
−
'''Material/Activity Sheet'''
+
Ratio
 +
is a way of comparing amounts of something. It shows how much bigger
 +
one thing is than another. For example:
    
   
 
   
−
Please open
+
* Use 1 measure detergent (soap) to 10 measures water
−
[[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
+
* Use 1 shovel (bucket) of cement to 3 shovels (buckets) of sand
−
in Firefox and follow the process
+
* Use 3 parts blue paint to 1 part white
−
 
   
   
 
   
−
When you move the sliders ask children to
+
Ratio
 +
is the number of '''parts''' to a mix. The paint mix is 4
 +
parts, with 3 parts blue and 1 part white.
    
   
 
   
−
observe and describe what happens when the denominator is
+
The
−
changed.
+
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.
    
   
 
   
−
observe and describe what happens when denominator changes
+
Mixing
 +
paint in the ratio 3:1 (3 parts blue paint to 1 part white paint)
 +
means 3 + 1 = 4 parts in all.
    
   
 
   
−
One unit will be the large square border-in blue solid lines
+
3
 +
parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white
 +
paint.
    
   
 
   
−
A sub-unit is in dashed lines within one square unit.
  −
   
   
 
   
−
The thick red lines represent the fraction 1 and 2 and also the
+
Cost
−
side of the quadrilateral
+
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 product represents the area of the the quadrilateral
+
 +
The
 +
ratio of the cost of a pen to the cost of a pencil =
 +
[[Image:KOER%20Fractions_html_m762fb047.gif]]
    
   
 
   
−
'''''Evaluation'''''
  −
   
   
 
   
−
When
+
What
−
two fractions are multiplied
+
is Direct Proportion ?
−
is the product larger or smaller that the multiplicands – why ?
     −
=== Activity 4 Division by Fractions ===
   
   
 
   
−
'''''Learning
+
Two
−
Objectives '''''
+
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.
    
   
 
   
−
Understand Division by Fractions
+
Paint
 +
pots in a ratio of 3:1
    
   
 
   
−
'''''Materials and
+
[[Image:KOER%20Fractions_html_m22cda036.gif]]
−
resources required'''''
      +
   
 
   
 
   
−
[[http://www.superteacherworksheets.com/fractions/fractionstrips_TWQWF.pdf]]
+
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.
    
   
 
   
−
Crayons/ colour
+
If
−
pencils, Scissors, glue
+
we double the amount of blue paint we need 6 pots.
    
   
 
   
−
'''''Pre-requisites/
+
If
−
Instructions Method '''''
+
we double the amount of white paint we need 2 pots.
    
   
 
   
−
Print out the pdf
+
Six
−
[[http://www.superteacherworksheets.com/fractions/fractionstrips_TWQWF.pdf]]
+
paint pots in a ratio of 3:1
    
   
 
   
−
Colour each of the unit fractions in different colours. Keep the
+
−
whole unit (1) white.
+
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:
    
   
 
   
−
Cut out each unit fraction piece.
+
Pots
 +
of blue paint 3 6 9 12
    
   
 
   
−
Give examples
+
Pots
−
[[Image:KOER%20Fractions_html_m282c9b3f.gif]]
+
of white paint 1 2 3 4
    +
 
 
   
 
   
−
For example if we try the first one,
+
Two
−
[[Image:KOER%20Fractions_html_21ce4d27.gif]]
+
quantities which are in direct proportion will always produce a graph
−
See how many
+
where all the points can be joined to form a straight line.
−
[[Image:KOER%20Fractions_html_m31bd6afb.gif]]strips
  −
will fit exactly onto whole unit strip.
      
   
 
   
−
  −
  −
   
   
 
   
−
'''''Evaluation'''''
+
'''What'''
 +
is Inverse Proportion ?
    
   
 
   
−
When
+
Two
−
we divide by a fraction is the result larger or smaller why ?
+
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
 +
                           
 +
{| border="1"
 +
|-
 +
|
 
   
 
   
−
== Evaluation ==
+
|
 +
Walk
 +
 +
|
 +
Run
 
   
 
   
−
== Self-Evaluation ==
+
|
 +
Cycle
 
   
 
   
−
== Further Exploration ==
+
|
 +
Bus
 
   
 
   
−
# [[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.superteacherworksheets.com]] Worksheets in mathematics for teachers to use
+
Speed
 +
Km/Hr
 
   
 
   
−
= Linking Fractions to other Topics =
+
|
 +
3
 
   
 
   
−
== Introduction ==
+
|
 +
6
 +
(walk speed *2)
 
   
 
   
−
It is also very common for the school system to treat themes in a
+
|
−
separate manner. Fractions are taught as stand alone chapters. In
+
9
−
this resource book an attempt to connect it to other middle school
+
(walk speed *3)
−
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.
  −
 
   
   
 
   
−
 
+
|
−
 
+
45
−
 
+
(walk speed *15)
 
   
 
   
−
== Objectives ==
+
|-
 +
|
 +
Time
 +
Taken (minutes)
 
   
 
   
−
Explicitly link the other
+
|
−
topics in school mathematics that use fractions.
+
30
−
 
   
   
 
   
−
== Decimal Numbers ==
+
|
 +
15
 +
(walk Time * ½)
 
   
 
   
−
“Decimal”
+
|
−
comes from the Latin root '''''decem''''',
+
10
−
which simply means ten. The number system we use is called the
+
(walk Time * 1/3)
−
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
+
2
−
numbers which have digits after the decimal point, such as 5.8 or
+
(walk Time * 1/15)
−
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
+
As
−
simplest way to link or connect fractions to the decimal number
+
Zaheeda doubles her speed by running, time reduces to half. As she
−
system is with the number line representation. Any scale that a
+
increases her speed to three times by cycling, time decreases to one
−
child uses is also very good for this purpose, as seen in the figure
+
third. Similarly, as she increases her speed to 15 times, time
−
below.
+
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
    
   
 
   
−
The
  −
number line between 0 and 1 is divided into ten parts. Each of these
  −
ten parts is '''1/10''', a '''tenth'''.
  −
   
   
 
   
−
[[Image:KOER%20Fractions_html_3d7b669f.gif]]
+
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<nowiki>'''</nowiki>'''''; '''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.
    
   
 
   
−
Under
+
=== Proportional Reasoning ===
−
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
+
'''''Proportional'''''
−
can write any fraction with '''tenths (denominator 10) '''using the
+
thinking''''' involves the ability to understand and compare'''''
−
decimal point. Simply write after the decimal point how many tenths
+
ratios, and to predict and produce equivalent ratios. It requires
−
the number has. 0.6 means six tenths or 1/6. 1.5 means 1 whole and 5
+
comparisons between quantities and also the relationships between
−
tenths or  
+
quantities. It involves quantitative thinking as well as qualitative
−
[[Image:KOER%20Fractions_html_m7f1d448c.gif]]
+
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.
    
   
 
   
−
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
 
The
−
coloured portion represents 0.6 or 6/10 and the whole block
+
process of adding involved situations such as adding, joining,
−
represents 1.
+
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.
   −
 
  −
[[Image:KOER%20Fractions_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
  −
[[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.
  −
   
   
 
   
−
 
+
== 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
 +
&quot;irrational numbers&quot; came to be identified, the numbers
 +
that can be expressed of the form p/q where defined as rational
 +
numbers.
    
   
 
   
 
There
 
There
−
are a number of common ones that are useful to learn. Here is a table
+
is another subset called transcendental numbers which have now been
−
showing you the ones that you should learn.
+
discovered. These numbers cannot be expressed as the solution of an
−
 
+
algebraic polynomial. &quot;pi&quot; and &quot;e&quot; are such
−
                                     
+
numbers.
−
{| border="1"
  −
|-
  −
|
  −
Percentage
      
   
 
   
−
|
+
== Activities ==
−
Fraction
+
−
 
+
=== Activity 1  Fractions representation of decimal numbers ===
 
   
 
   
−
|-
+
'''''Learning'''''
−
|
+
Objectives
−
100%
  −
 
   
   
 
   
−
|
+
Fractions representation of decimal
−
[[Image:KOER%20Fractions_html_m15ed765d.gif]]
+
numbers
−
 
   
   
 
   
−
|-
+
'''''Materials and'''''
−
|
+
resources required
−
50%
  −
 
   
   
 
   
−
|
+
[[http://www.superteacherworksheets.com/decimals/decimal-tenths-squares_TWDWQ.pdf Decimals: Tenths]]
−
[[Image:KOER%20Fractions_html_df52f71.gif]]
+
−
 
+
[[http://www.superteacherworksheets.com/decimals/decimal-hundredths-tenths.pdf Decimal: Hundredths and Tenths]]
 +
 +
 +
'''''Pre-requisites/'''''
 +
Instructions Method
 
   
 
   
−
|-
+
Make copies of the above given resources.
−
|
  −
25%
     −
+
'''''Evaluation'''''
−
|
+
# 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
−
[[Image:KOER%20Fractions_html_m6c97abb.gif]]
+
# 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
−
75%
     −
+
Understand fraction representation and percentages
−
|
  −
[[Image:KOER%20Fractions_html_m6cb13da4.gif]]
     −
+
'''''Materials and'''''
−
|-
+
resources required
−
|
  −
10%
      +
[[http://www.superteacherworksheets.com/percents/converting-fractions-decimals-percents_EASIE.pdf Converting fractions, decimals and percents]]<br>
 +
[[http://www.superteacherworksheets.com/percents/basic-percentages1.pdf Percentage]]
 
   
 
   
−
|
+
'''''Pre-requisites/Instructions Method'''''
−
[[Image:KOER%20Fractions_html_26bc75d0.gif]]
     −
+
Please print copies of the above given activity sheets and discuss the various percentage quantities with various shapes.
−
|-
  −
|
  −
20%
     −
+
Then print a copy each of [[spider-percentages.pdf]] and make the children do this activity
−
|
  −
[[Image:KOER%20Fractions_html_m73e98509.gif]]
     −
+
'''''Evaluation'''''
−
|-
+
# What value is the denominator when we represent percentage as fraction ?
−
|
+
# What does the numerator represent ?
−
40%
+
# What does the whole fraction represent ?
 +
# What other way can we represent a fraction whose denominator is 100.
    +
=== Activity 3 Fraction representation and rational and irrational numbers ===
 
   
 
   
−
|
+
'''''Learning Objectives'''''
−
[[Image:KOER%20Fractions_html_m2dd64d0b.gif]]
      +
Understand fraction representation and rational and irrational numbers <br>
 +
'''''Materials and resources required'''''
 
   
 
   
−
|} 
+
Thread of a certain length.
−
 
  −
 
  −
 
   
   
 
   
−
 
+
'''''Pre-requisites/Instructions Method'''''
−
 
  −
 
   
   
 
   
−
[[Image:KOER%20Fractions_html_m60c76c68.gif]]To
+
Construct Koch's snowflakes .  
−
see 40 % visually see the figure :
  −
 
   
   
 
   
−
You
+
Start
−
can see that if the shape is divided into 5 equal parts, then 2 of
+
with a thread of a certain length (perimeter) and using the same
−
those parts are shaded.
+
thread construct the following shapes (see Figure).
−
 
+
 +
[[Image:KOER%20Fractions_html_m1a6bd0d0.gif]]
 
   
 
   
−
If
+
See
−
the shape is divided into 100 equal parts, then 40 parts are shaded.
+
how the shapes can continue to emerge but cannot be identified
−
 
+
definitely with the same perimeter (length of the thread).  
 
   
 
   
−
These
+
Identify
−
are equivalent fractions as in both cases the same amount has been
+
the various places where pi, &quot;e&quot; and the golden ratio occur
−
shaded.
      +
== Evaluation ==
 +
# 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 ?
 
   
 
   
 +
== Self-Evaluation ==
    +
== Worksheets ==
 +
# fraction addition worksheet [http://karnatakaeducation.org.in/KOER/en/images/1/11/Fractionaddition.odt Fraction simple addition]
 +
# fraction addition worksheet  Fraction simple addition]
 +
# Fraction multiplication worksheet [http://karnatakaeducation.org.in/KOER/en/images/8/83/Multiplication.odt multiplication]
 +
# Fraction Division worksheet [http://karnatakaeducation.org.in/KOER/en/images/c/ca/Division.odt Division]
 +
# Fraction Subtraction worksheet[http://karnatakaeducation.org.in/KOER/en/images/1/1d/Subtraction.odt Subtraction]
   −
 
+
== Further Exploration ==
 
   
 
   
−
== Ratio and Proportion ==
+
# Percentage and Fractions, [[http://www.bbc.co.uk/skillswise/numbers/wholenumbers/]]
 +
# 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/]]
 
   
 
   
−
It
+
= See Also =
−
is important to understand that fractions also can be interpreted as
+
# At Right Angles December 2012 Fractions Pullout [[http://www.teachersofindia.org/en/article/atria-pullout-section-december-2012]]
−
ratio's. Stressing that a fraction can be interpreted in many ways is
+
# Video on teaching fractions using the equal share method made by Eklavya an NGO based in Madhya Pradesh, India [[http://vimeo.com/22238434]]
−
of vital importance. Here briefly I describe the linkages that must
+
# Mathematics resources from Homi Baba Centre for Science Education [[http://mathedu.hbcse.tifr.res.in/]]
−
be established between Ratio and Proportion and the fraction
+
# Understand how to use Geogebra a mathematical computer aided tool [[http://www.geogebra.com]]  <br>
−
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 =
  −
[[Image:KOER%20Fractions_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
  −
 
  −
  −
[[Image:KOER%20Fractions_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
  −
 
  −
  −
 
  −
 
  −
 
  −
                           
  −
{| border="1"
  −
|-
  −
|
  −
 
  −
 
  −
  −
|
  −
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
  −
&quot;irrational numbers&quot; 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. &quot;pi&quot; and &quot;e&quot; are such
  −
numbers.
  −
 
  −
  −
== Activities ==
  −
  −
=== Activity 1  Fractions representation of decimal numbers ===
  −
  −
'''''Learning
  −
Objectives '''''
  −
 
  −
  −
Fractions representation of decimal
  −
numbers
  −
 
  −
  −
'''''Materials and
  −
resources required'''''
  −
 
  −
  −
[[http://www.superteacherworksheets.com/decimals/decimal-tenths-squares_TWDWQ.pdf]]
  −
 
  −
  −
[[http://www.superteacherworksheets.com/decimals/decimal-hundredths-tenths.pdf]]
  −
 
  −
  −
 
  −
 
  −
 
  −
  −
'''''Pre-requisites/
  −
Instructions Method '''''
  −
 
  −
  −
Make copies of the worksheets
  −
 
  −
  −
[[http://www.superteacherworksheets.com/decimals/decimal-tenths-squares_TWDWQ.pdf]]
  −
 
  −
  −
[[http://www.superteacherworksheets.com/decimals/decimal-hundredths-tenths.pdf]]
  −
 
  −
  −
 
  −
 
  −
 
  −
  −
'''''Evaluation'''''
  −
 
  −
  −
# 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.
  −
  −
 
  −
 
  −
 
  −
  −
 
  −
 
  −
 
  −
  −
=== Activity 2 Fraction representation and percentages ===
  −
  −
'''''Learning
  −
Objectives '''''
  −
 
  −
  −
Understand fraction representation and percentages
  −
 
  −
  −
 
  −
 
  −
 
  −
  −
'''''Materials and
  −
resources required'''''
  −
 
  −
  −
[[http://www.superteacherworksheets.com/percents/converting-fractions-decimals-percents_EASIE.pdf]]
  −
 
  −
  −
[[http://www.superteacherworksheets.com/percents/basic-percentages1.pdf]]
  −
 
  −
  −
'''''Pre-requisites/
  −
Instructions Method '''''
  −
 
  −
  −
Please print
  −
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.
  −
 
  −
  −
 
  −
 
  −
   
  −
Then print a copy
  −
each of [[spider-percentages.pdf]]
  −
and make the children do this activity
  −
 
  −
  −
 
  −
 
  −
 
  −
  −
'''''Evaluation'''''
  −
 
  −
  −
# 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.
  −
  −
 
  −
 
  −
  −
 
  −
 
  −
 
  −
  −
=== Activity 3 Fraction representation and rational and irrational numbers ===
  −
  −
'''''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).
  −
 
  −
  −
[[Image:KOER%20Fractions_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, &quot;e&quot; and the golden ratio
  −
occur
  −
 
  −
  −
 
  −
 
  −
 
  −
  −
 
      +
= Teachers Corner =
   −
   
+
GOVT  HIGH SCHOOL, DOMLUR<br><br>
−
'''''Evaluation'''''
     −
   
+
Our 8th std students  are learning about fraction using projector . <br>
−
# How many numbers can I represent on a number line between 1 and 2.
+
[[File:1.jpg|400px]]<br>
−
# What is the difference between a rational and irrational number, give an example ?
+
Students are actively participating in the activity.<br>
−
# What is Pi ? Why is it a special number ?
  −
      +
They are learning about  meaning of the fractions, equivalent fractions and addition of fractions using paper cuttings.<br>
 +
[[File:2..jpg|400px]][[File:3.jpg|400px]][[File:4.jpg|400px]] <br><br>
 +
A student is showing 1/4=? 1/6+1/12<br>
 +
[[File:5.jpg|400px]]<br>
 +
[[File:6..jpg|400px]]<br>
    +
Showing 1/3=1/4+1/12<br>
   −
+
They started to solve the problems easily <br>
−
== Evaluation ==
+
[[File:7.jpg|400px]]<br>[[File:8.jpg|400px]]
−
  −
== Self-Evaluation ==
  −
  −
== Further Exploration ==
  −
  −
# Percentage and Fractions, [[http://www.bbc.co.uk/skillswise/numbers/wholenumbers/]]
  −
# 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/]]
  −
  −
= See Also =
  −
# At Eight Angles December 2012 Fractions Pullout [[http://www.teachersofindia.org/en/article/atria-pullout-section-december-2012]]
  −
# Video on teaching fractions using the equal share method made by Eklavya an NGO based in Madhya Pradesh, India [[http://vimeo.com/22238434]]
  −
# Mathematics resources from Homi Baba Centre for Science Education [[http://mathedu.hbcse.tifr.res.in/]]
  −
# Understand how to use Geogebra a mathematical computer aided tool [[http://www.geogebra.com]] <br>
     −
= Teachers Corner =
  −
   
= Books =
 
= Books =
 
   
 
   
−
# [[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
+
# [[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 Lee Streefland
−
+
 
 
= References =
 
= References =
 
   
 
   
Line 2,045: Line 1,680:  
# Introducing Fractions Using Share and Measure Interpretations: A Report from Classroom Trials . Jayasree Subramanian, Eklavya, Hoshangabad, India and Brijesh Verma,Muskan, Bhopal, India
 
# 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.
 
# 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.
 +
 +
[[Category:Fractions]]
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