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→Introduction
== Introduction ==
== Introduction ==
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. 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. Instruction in fractions that focuses only on the mechanics of procedures and not on reasoning misses valuable opportunities to guide students in developing this core mathematical skill. This section explains the meaning of fractions, reviews some of the common difficulties in understanding the meaning of fractions, and describes how to use simple pictures to represent fractions. n this section, we define what we mean by a fraction is part of of an object, collection or quantity.
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 divisionInterpreting fractions
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 aboput 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?
A paper tape measure is a valuable illustration of different fractions. For example, learners can write on 1/2m, 0.50m and 50cm for their own portable equivalence chart.
Folding of paper also can illustrate fractions
I have ten bars of chocolate, and I share them equally between four people. How much will they each get?
We recommend that teachers explicitly use thelanguage 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’.
Commonly fractions are always approached by teaching it through
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
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.
These approaches to fraction teaching are discussed here.
Half
Half
(½) : The whole is divided into '''two
(½) : The whole is divided into '''two'''
equal '''parts.
equal '''parts.'''
One-Fourth
One-Fourth
(1/4) : The whole is divided into '''four
(1/4) : The whole is divided into '''four'''
equal '''parts.
equal '''parts.'''
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.'''
Two
Two
Fifth (2/5) : The whole is divided into '''five
Fifth (2/5) : The whole is divided into '''five'''
equal '''parts.
equal '''parts.'''
[[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.'''
Seven
Seven
tenth (7/10) : The whole is divided into '''ten
tenth (7/10) : The whole is divided into '''ten'''
equal '''parts.
equal '''parts.'''
'''Terms Numerator
'''Terms Numerator'''
and Denominator and their meaning'''
and Denominator and their meaning
Three
Three
Eight (3/8) The whole is divided into '''eight
Eight (3/8) The whole is divided into '''eight'''
equal '''parts.
equal '''parts.'''
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
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.
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.
'''''Materials and
'''''Materials and'''''
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]]
'''''Pre-requisites/
'''''Pre-requisites/'''''
Instructions Method
Do
Do
=== Activity 2: Proper and Improper Fractions ===
=== Activity 2: Proper and Improper Fractions ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials
'''''Materials'''''
and resources required
[[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]]
[[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_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_5518d221.jpg]]
[[Image:KOER%20Fractions_html_55f65a3d.gif]]
[[Image:KOER%20Fractions_html_55f65a3d.gif]]
'''Pre-requisites/
'''Pre-requisites/'''
Instructions Method '''
Instructions Method
=== Activity 3: Comparing Fractions ===
=== Activity 3: Comparing Fractions ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials
'''''Materials'''''
and resources required
'''Pre-requisites/
'''Pre-requisites/'''
Instructions Method'''
Instructions Method
# What should we do to make the sizes of the parts the same ?
# What should we do to make the sizes of the parts the same ?
=== Activity 4: Equivalent Fractions ===
=== Activity 4: Equivalent Fractions ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials
'''''Materials'''''
and resources required
'''Pre-requisites/
'''Pre-requisites/'''
Instructions Method'''
Instructions Method
== N-Distractors ==
== N-Distractors ==
# '''''Resistance to N-Distractor: '''''The student is completely free (conceptually and algorithmically) of N-Distractor errors.
# '''''Resistance to N-Distractor: '''''The student is completely free (conceptually and algorithmically) of N-Distractor errors.
== Enrichment Activities ==
== Enrichment Activities ==
== Addition and Subtraction ==
== Addition and Subtraction ==
== Multiplication ==
== Multiplication ==
* The whole rectangle also is ''1 square unit'', in terms of area.
* The whole rectangle also is ''1 square unit'', in terms of area.
== Division ==
== Division ==
=== Activity 1 Addition of Fractions ===
=== Activity 1 Addition of Fractions ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials
'''''Materials'''''
and resources required
'''''Pre-requisites/
'''''Pre-requisites/'''''
Instructions Method
=== Activity 2 Fraction Subtraction ===
=== Activity 2 Fraction Subtraction ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials and
'''''Materials and'''''
resources required
'''''Pre-requisites/
'''''Pre-requisites/'''''
Instructions Method
=== Activity 3 Multiplication of fractions ===
=== Activity 3 Multiplication of fractions ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials and
'''''Materials and'''''
resources required
'''''Pre-requisites/
'''''Pre-requisites/'''''
Instructions Method
=== Activity 4 Division by Fractions ===
=== Activity 4 Division by Fractions ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials and
'''''Materials and'''''
resources required
'''''Pre-requisites/
'''''Pre-requisites/'''''
Instructions Method
as
as
[[Image:KOER%20Fractions_html_m1369c56e.gif]]
[[Image:KOER%20Fractions_html_m1369c56e.gif]]
where '''10 is the numerator''' and the '''denominator is '''<u>'''always'''</u>'''
where '''10 is the numerator''' and the '''denominator is '''<u>'''always'''</u>
100'''. In this case 10 % of the cost of the book is
100'''. In this case 10 % of the cost of the book is '''
[[Image:KOER%20Fractions_html_m50e22a06.gif]].
[[Image:KOER%20Fractions_html_m50e22a06.gif]].
So you can buy the book for 200 – 20 = 180 rupees.
So you can buy the book for 200 – 20 = 180 rupees.
are a number of common ones that are useful to learn. Here is a table
are a number of common ones that are useful to learn. Here is a table
showing you the ones that you should learn.
showing you the ones that you should learn.
{| border="1"
{| border="1"
|
|
Percentage
Percentage
|
|
Fraction
Fraction
|-
|-
|
|
100%
100%
|
|
[[Image:KOER%20Fractions_html_m15ed765d.gif]]
[[Image:KOER%20Fractions_html_m15ed765d.gif]]
|-
|-
|
|
50%
50%
|
|
[[Image:KOER%20Fractions_html_df52f71.gif]]
[[Image:KOER%20Fractions_html_df52f71.gif]]
|-
|-
|
|
25%
25%
|
|
[[Image:KOER%20Fractions_html_m6c97abb.gif]]
[[Image:KOER%20Fractions_html_m6c97abb.gif]]
|-
|-
|
|
75%
75%
|
|
[[Image:KOER%20Fractions_html_m6cb13da4.gif]]
[[Image:KOER%20Fractions_html_m6cb13da4.gif]]
|-
|-
|
|
10%
10%
|
|
[[Image:KOER%20Fractions_html_26bc75d0.gif]]
[[Image:KOER%20Fractions_html_26bc75d0.gif]]
|-
|-
|
|
20%
20%
|
|
[[Image:KOER%20Fractions_html_m73e98509.gif]]
[[Image:KOER%20Fractions_html_m73e98509.gif]]
|-
|-
|
|
40%
40%
|
|
[[Image:KOER%20Fractions_html_m2dd64d0b.gif]]
[[Image:KOER%20Fractions_html_m2dd64d0b.gif]]
|}
|}
'''What
'''What'''
is ratio?'''
is ratio?
[[Image:KOER%20Fractions_html_m22cda036.gif]]
[[Image:KOER%20Fractions_html_m22cda036.gif]]
'''What
'''What'''
is Inverse Proportion ?'''
is Inverse Proportion ?
Study
Study
the table below, observe that as the speed increases time taken to
the table below, observe that as the speed increases time taken to
cover the distance decreases
cover the distance decreases
{| border="1"
{| border="1"
|-
|-
|
|
|
|
Walk
Walk
|
|
Run
Run
|
|
Cycle
Cycle
|
|
Bus
Bus
|-
|-
Speed
Speed
Km/Hr
Km/Hr
|
|
3
3
|
|
6
6
(walk speed *2)
(walk speed *2)
|
|
9
9
(walk speed *3)
(walk speed *3)
|
|
45
45
(walk speed *15)
(walk speed *15)
|-
|-
Time
Time
Taken (minutes)
Taken (minutes)
|
|
30
30
|
|
15
15
(walk Time * ½)
(walk Time * ½)
|
|
10
10
(walk Time * 1/3)
(walk Time * 1/3)
|
|
2
2
(walk Time * 1/15)
(walk Time * 1/15)
|}
|}
As
As
'''Moving from Additive Thinking to
'''Moving from Additive Thinking to'''
Multiplicative Thinking '''
Multiplicative Thinking
thinks that if you use 5 spoons of sugar to make 6 cups of tea, then
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
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
as the cups before. Avinash would be using an '''''additive'''''
transformation''''''''; '''he thinks that since we added 2 more
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
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
add to more spoons of sugar. What he does not know is that for it to
=== Proportional Reasoning ===
=== Proportional Reasoning ===
'''''Proportional
'''''Proportional'''''
thinking''''' involves the ability to understand and compare
thinking''''' involves the ability to understand and compare'''''
ratios, and to predict and produce equivalent ratios. It requires
ratios, and to predict and produce equivalent ratios. It requires
comparisons between quantities and also the relationships between
comparisons between quantities and also the relationships between
=== Activity 1 Fractions representation of decimal numbers ===
=== Activity 1 Fractions representation of decimal numbers ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials and
'''''Materials and'''''
resources required
'''''Pre-requisites/
'''''Pre-requisites/'''''
Instructions Method
# Write 0.45, 0.68, 0.05 in fraction form and represent as a fraction 100 square.
# Write 0.45, 0.68, 0.05 in fraction form and represent as a fraction 100 square.
=== Activity 2 Fraction representation and percentages ===
=== Activity 2 Fraction representation and percentages ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials and
'''''Materials and'''''
resources required
'''''Pre-requisites/
'''''Pre-requisites/'''''
Instructions Method
# What other way can we represent a fraction whose denominator is 100.
# What other way can we represent a fraction whose denominator is 100.
=== Activity 3 Fraction representation and rational and irrational numbers ===
=== Activity 3 Fraction representation and rational and irrational numbers ===
'''''Learning
'''''Learning'''''
Objectives
'''''Materials and
'''''Materials and'''''
resources required
'''''Pre-requisites/
'''''Pre-requisites/'''''
Instructions Method
# What is Pi ? Why is it a special number ?
# What is Pi ? Why is it a special number ?
