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