CLASS
VI
CBSE-i
PARTS OF WHOLES
FRACTIONS & DECIMALS
UNIT
4
Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India
CBSE-i
PARTS OF WHOLES
FRACTIONS & DECIMALS
UNIT
4
CLASS - VI
Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India
The CBSE-International is grateful for permission to reproduce and/or
translate copyright material used in this publication.
The
acknowledgements have been included wherever appropriate and
sources from where the material has been taken duly mentioned. In
case anything has been missed out, the Board will be pleased to rectify
the error at the earliest possible opportunity.
All Rights of these documents are reserved. No part of this publication
may be reproduced, printed or transmitted in any form without the
prior permission of the CBSE-i. This material is meant for the use of
schools who are a part of the CBSE-International only.
Preface
The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making
the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a
fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the
learning process in harmony with the existing personal, social and cultural ethos.
The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It
has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been
conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain
elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged.
The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in
view.
The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to
nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand,
appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations
and additions wherever and whenever necessary.
The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The
speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink
their approaches for knowledge processing by their learners. In this context, it has become imperative for them to
incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to
upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant
life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of
these requirements.
The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and
creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and
media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all
subject areas to cater to the different pace of learners.
The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now
introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is
to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous
and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some nonevaluative components in the curriculum which would be commented upon by the teachers and the school. The objective
of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal
knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives,
SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'.
The Core skills are the most significant aspects of a learner's holistic growth and learning curve.
The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework
(NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to
millions of learners, many of whom are now global citizens.
The Board does not interpret this development as an alternative to other curricula existing at the international level, but as
an exercise in providing the much needed Indian leadership for global education at the school level. The International
Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The
Board while addressing the issues of empowerment with the help of the schools' administering this system strongly
recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to
their peers through the interactive platforms provided by the Board.
I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr.
Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the
development and implementation of this material.
The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion
forums provided on the portal. Any further suggestions are welcome.
Vineet Joshi
Chairman
Acknowledgements
Advisory
Shri Vineet Joshi, Chairman, CBSE
Shri Shashi Bhushan, Director(Academic), CBSE
Ideators
Ms. Aditi Misra
Ms. Amita Mishra
Ms. Anita Sharma
Ms. Anita Makkar
Dr. Anju Srivastava
Dr. Indu Khetarpal
Ms. Vandana Kumar
Ms. Anju Chauhan
Ms. Deepti Verma
Ms. Ritu Batra
Conceptual Framework
Shri G. Balasubramanian, Former Director (Acad), CBSE
Ms. Abha Adams, Consultant, Step-by-Step School, Noida
Dr. Sadhana Parashar, Head (I & R),CBSE
Ms. Anuradha Sen
Ms. Jaishree Srivastava
Ms. Archana Sagar
Dr. Kamla Menon
Ms. Geeta Varshney
Dr. Meena Dhami
Ms. Guneet Ohri
Ms. Neelima Sharma
Dr. Indu Khetrapal
Dr. N. K. Sehgal
Material Production Group: Classes I-V
Ms. Rupa Chakravarty
Ms. Anita Makkar
Ms. Anuradha Mathur
Ms. Kalpana Mattoo
Ms. Savinder Kaur Rooprai
Ms. Monika Thakur
Ms. Seema Choudhary
Mr. Bijo Thomas
Ms. Kalyani Voleti
Dr. Rajesh Hassija
Ms. Rupa Chakravarty
Ms. Sarita Manuja
Ms. Himani Asija
Dr. Uma Chaudhry
Ms. Nandita Mathur
Ms. Seema Chowdhary
Ms. Ruba Chakarvarty
Ms. Mahua Bhattacharya
Material Production Groups: Classes VI-VIII
Mathematics :
Ms. Seema Rawat
Ms. N. Vidya
Ms. Mamta Goyal
Ms. Chhavi Raheja
Political Science:
Ms. Kanu Chopra
Ms. Shilpi Anand
English :
Ms. Rachna Pandit
Ms. Neha Sharma
Ms. Sonia Jain
Ms. Dipinder Kaur
Ms. Sarita Ahuja
Science :
Dr. Meena Dhami
Mr. Saroj Kumar
Ms. Rashmi Ramsinghaney
Ms. Seema kapoor
Ms. Priyanka Sen
Dr. Kavita Khanna
Ms. Keya Gupta
English :
Ms. Sarita Manuja
Ms. Renu Anand
Ms. Gayatri Khanna
Ms. P. Rajeshwary
Ms. Neha Sharma
Ms. Sarabjit Kaur
Ms. Ruchika Sachdev
Geography:
Ms. Deepa Kapoor
Ms. Bharti Dave
Ms. Bhagirathi
Ms. Archana Sagar
Ms. Manjari Rattan
Mathematics :
Dr. K.P. Chinda
Mr. J.C. Nijhawan
Ms. Rashmi Kathuria
Ms. Reemu Verma
Science :
Ms. Charu Maini
Ms. S. Anjum
Ms. Meenambika Menon
Ms. Novita Chopra
Ms. Neeta Rastogi
Ms. Pooja Sareen
Political Science:
Ms. Sharmila Bakshi
Ms. Srelekha Mukherjee
Economics:
Ms. Mridula Pant
Mr. Pankaj Bhanwani
Ms. Ambica Gulati
Geography:
Ms. Suparna Sharma
Ms. Leela Grewal
History :
Ms. Leeza Dutta
Ms. Kalpana Pant
Material Production Groups: Classes IX-X
Dr. Sadhana Parashar,
Head (I and R)
Shri R. P. Sharma, Consultant
Ms. Seema Lakra, S O
History :
Ms. Jayshree Srivastava
Ms. M. Bose
Ms. A. Venkatachalam
Ms. Smita Bhattacharya
Coordinators:
Ms. Sugandh Sharma,
Dr. Srijata Das,
Dr. Rashmi Sethi,
E O (Com)
E O (Maths)
E O (Science)
Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO
Ms. Preeti Hans, Proof Reader
Preface
CONTENT
Acknowledgment
1. Syllabus
1
2. Scope Document
2
3. Teacher’s Support Material
8
2
Teacher’s Note
9
2
Activity Skill Matrix
18
2
Warm up W1 : Importance of Numbers
20
2
Warm up W2 : Tangrams
20
2
Pre Content Worksheet P1
21
A Quality as a part of Whole
2
Pre Content Worksheet P2
22
What is it any way
2
Content Worksheet C1
22
Understanding and Representing
Fractions and its Components
2
Content Worksheet C2
23
Locating Fractions on the Number Line
2
Content Worksheet C3
25
Introducing Decimals
2
Content Worksheet C4
25
Placing Decimals on the Number Line
2
Content Worksheet C5
26
Equivalent Fractions
2
Content Worksheet C6
27
Converting Fractions to Decimals
2
Content Worksheet C7
28
Simplification of Fractions
2
Content Worksheet C8
29
Independent Practice
Content Worksheet C9
2
Place Value Chart of Decimals
29
2
Content Worksheet C10
30
CONTENT
Comparison of Fractions Decimals
2
Content Worksheet C11
31
Addition and Subtraction of Fractions
2
Content Worksheet C12
32
Addition and Subtraction of Fractions
2
Content Worksheet C13
33
Multiplication and Division as Repeated
Addition and Subtraction
2
Content Worksheet C14
33
Word Problems Involving Fractions and Decimals
2
Content Worksheet C15
34
Rounding off and Estimation
2
Content Worksheet C16
35
Estimation and Significance
2
Content Worksheet C17
35
Ratio and Proportion
2
Post Content Worksheet P1
2
Post Content Worksheet P2
2
Post Content Worksheet P3
2
Post Content Worksheet P4
4. Assessment Guidance Plan
37
5. Study Material
41
6. Student’s Support Material
SW 1 : Warm up W1 : Importance of Numbers
2
126
SW 2 : Warm up W2 : Tangrams
2
128
SW 3 : Pre Content Worksheet P1 : A Quality as a Part of a Whole
2
131
SW 4 : Pre Content Worksheet P2 : What is it any ways
2
133
SW 5 : Content Worksheet C1 : Understanding and Representing
2
136
Fractions and its Components
CONTENT
2
SW 6 : Content Worksheet C2 : Locating Fractions on the
137
Number Line
2
SW 7 : Content Worksheet C3 : Introducing Decimals
147
2
SW 8 : Content Worksheet C4 : Placing Decimals on a Number Line 151
2
SW 9 : Content Worksheet C5 : Equivalent Fractions
153
2
SW 10 : Content Worksheet C6 : Converting Fractions to Decimals
155
2
SW 11 : Content Worksheet C7 : Simplification of Fractions
158
2
SW 12 : Content Worksheet C8 : Independent Practice
161
2
SW 13 : Content Worksheet C9 : Place Value Chart of Decimals
166
2
SW 14 : Content Worksheet C10 : Comparison of
169
Fractions/Decimals
2
SW 15 : Content Worksheet C11 : Addition and Subtraction
174
of Fractions
2
SW 16 : Content Worksheet C12 : Addition and Subtraction
179
of Fractions
SW 17 : Content Worksheet C13 : Multiplication and Division as
2
180
Repeated Addition and Subtraction
SW 18 : Content Worksheet C14 : Word Problems Involving
2
184
Fractions and Decimals
SW 19 : Content Worksheet C15 : Rounding Off and Estimation
2
188
SW 20 : Content Worksheet C16 : Estimation and Significance
2
191
SW 21 : Content Worksheet C17 : Ratio and Proportion
2
193
SW 22 : Post Content Worksheet P1
2
197
SW 23 : Post Content Worksheet P2
2
198
SW 24 : Post Content Worksheet P3
2
201
SW 25 : Post Content Worksheet P43
2
202
Acknowledgments
2
205
Suggested Videos/Links/PPT’s
2
205
SYLLABUS - Unit 1
FRACTIONS AND DECIMALS
Fractions
• Understanding and representing fractions
• Components of a fraction
• Representing fraction on number line
• Types of fractions
• Simplifying a fraction
• Comparing and ordering fractions
• Addition and subtraction of fractions and extension to
word problems
• The fraction of a quantity
• Division and multiplication of a fraction by a whole
number
• Extension of BODMAS to fractions
• Ratio and proportion
• Applications to real life
Decimals
• Converting a fraction to decimal
• Representing a decimal on a place value chart
• Expanding a decimal number
• Components of a decimal number
• Representing decimal on a number line
• Comparing and ordering decimal numbers (extension of
fractions and place value),
• Basic operations on decimals and extension to word
problems,
• Extending
addition
and
subtraction
of
decimals
multiplication and division by a whole number,
• Extension of BODMAS to decimals,
• Estimation and significance
1
to
SCOPE DOCUMENT
FRACTIONS AND DECIMALS
Prerequisite: Knowledge of Factors and multiples, skill of multiplication and division of
two numbers, HCF and LCM, BODMAS on integers, simple word problems
extending the concept of the four operations, Estimation.
Concepts:
•
Understanding and representing fractions
•
Placing fractions on a number line
•
Understanding decimals and converting fractions into decimals
•
Representing decimals on number line
•
Equivalent fractions
•
Comparing and ordering fractions and decimals
•
Adding and subtracting fractions and decimals
•
Multiplying and dividing fractions and decimals by a whole number.
•
BODMAS with reference to fractions and decimals
•
Estimation with fractions and decimals
•
Applications of fractions - Ratio and Proportion
•
Application of decimals - Scientific notations, concept of significant figures
Learning objectives
At the end of this lesson student will be able to
•
Define fractions, its components and types of fractions
•
Represent fractions on the number line
•
Define decimal and equivalent decimals.
•
Identify through given pictures the whole number and fractional parts of a decimal
and write decimals in expanded form.
•
Represent decimal numbers on the number line
2
•
Differentiate between more than 1 and less than 1 fractions.
•
Define equivalent fractions and convert fractions to decimals and vice-versa.
•
Know about the place values of a decimal number, and apply it in various situations.
•
Compare and order two or more fractions/ decimals.
•
Perform addition and subtraction on fractions and decimal numbers and extend it to
solving word problems
•
Extend repeated addition and subtraction to multiplication and division of fractions
and decimals.
•
Give estimations with the significance of number specified.
•
Apply the knowledge of ratio and proportion to real life problems.
Extension activities:
1. Write an article for the school magazine on how decimal system is used for classification
of books in the library.
2. Research on Golden ratio / number and its patterns. Where do you see the golden
number used in real life?
Project activity:
Using Ms Excel, divide two natural numbers chosen randomly and find out when the
quotient comes out to be a proper fraction, an improper fraction or a natural number. You
may choose numbers between 1 and 20.
Refer to the following images to generate the random numbers.
3
4
t numberr format in the
t column
n C is choseen as fractio
on upto 2 diigits.
Note that the
Now in th
he 4th column D, div
vide column A by co
olumn B. Now the number
n
forrmat is
‘general’ which
w
takess in decimall format. Compare
C
th
he fractionss obtained in
i column 3 to the
decimals obtained
o
in column 4. Record you
ur observattions.
5
Cross currricular liinks
ENGLISH
H:
Students would
w
resea
arch and write
w
about how decim
mal system
m is useful in
i classificaation of
books in a library and
d write an article
a
for th
he school magazine.
m
SCIENCE
E:
Measure th
he height an
nd weightss of 5 friend
ds. Find ou
ut which of them havee their heigh
hts and
weights as natural nu
umbers and
d which of th
hem have it
i as decimaals.
ART:
Mathematiics of goldeen ratio exissts and is used
u
extensiively in all types of arrt. The mov
vie’ “Da
Vinci Code”, the pain
nting of Mona Lisa and
a many architectura
a
al wonders all have reeference
to some asspect of gollden ratio and
a proporttions.
http://ww
ww.markwa
ahl.com/index.php?id
d=22
6
Use fractions to make mosaic designs and represent by different colours used in fraction
form.
Fraction Dialogue Read a dialogue on fractions at
http://nrich.maths.org/1510
Technology help:
Web links for reference, research and remedial
Power point presentations
You tube to deliver content
7
Teacher’s
Support
Material
8
TEACHER’S NOTE
The teaching of Mathematics should enhance the child’s resources to think and reason, to
visualise and handle abstractions, to formulate and solve problems. As per NCF 2005, the
vision for school Mathematics include :
1.
Children learn to enjoy mathematics rather than fear it.
2.
Children see mathematics as something to talk about, to communicate through, to
discuss among themselves, to work together on.
3.
Children pose and solve meaningful problems.
4.
Children use abstractions to perceive relation-ships, to see structures, to reason out
things, to argue the truth or falsity of statements.
5.
Children understand the basic structure of Mathematics: Arithmetic, algebra, geometry
and trigonometry, the basic content areas of school Mathematics, all offer a
methodology for abstraction, structuration and generalisation.
6.
Teachers engage every child in class with the conviction that everyone can learn
mathematics.
Students should be encouraged to solve problems through different methods like
abstraction, quantification, analogy, case analysis, reduction to simpler situations, even
guess-and-verify exercises during different stages of school. This will enrich the students and
help them to understand that a problem can be approached by a variety of methods for
solving it. School mathematics should also play an important role in developing the useful
skill of estimation of quantities and approximating solutions. Development of visualisation
and representations skills should be integral to Mathematics teaching. There is also a need to
make connections between Mathematics and other subjects of study. When children learn to
draw a graph, they should be encouraged to perceive the importance of graph in the
teaching of Science, Social Science and other areas of study. Mathematics should help in
developing the reasoning skills of students. Proof is a process which encourages systematic
way of argumentation. The aim should be to develop arguments, to evaluate arguments, to
make conjunctures and understand that there are various methods of reasoning. Students
should be made to understand that mathematical communication is precise, employs
unambiguous use of language and rigour in formulation. Children should be encouraged to
appreciate its significance.
At the upper primary stage, students get the first taste of power of Mathematics through the
application of powerful abstract concepts like Algebra, Number System, Geometry etc.
9
Revisiting of the previous knowledge and consolidating basic concepts and skills learnt at
the Primary Stage helps the child to appreciate the abstract nature of Mathematics. Whether
it is Number system or algebra or Geometry, these topics should be introduced by relating it
to the child’s every day experience and taking it forward to abstraction so that the child can
appreciate the importance of study of these topics.
The mathematics curriculum during the preschool, elementary school and the middle school
years has many components. But at the heart of mathematics in those years are concepts of
number and operations with numbers. Proficiency with the numbers and numerical
operations is an important foundation for further education in mathematics and in fields that
use mathematics.
Fractions, Ratio and Proportion and Decimals are important concepts in the middle school
curriculum, but their development and understanding takes place at the elementary school.
The students should be made to understand that the key to success in mathematics is their
belief that they can make sense of the mathematics they study. Unfortunately, many
children lose this belief when they begin to work with fractions and decimals. If they believe
that Mathematics is a set of rules to memorize and apply, it becomes very difficult for them
to learn.
An informal introduction to fractions begins long before the concept is introduced formally
as a part of mathematics. When a child wants to share half of his chocolate with his friend or
distributes his set of candies equally amongst three of his friends, he is unknowingly talking
about fractions.
The comparision of fractions can also be informally introduced by paper cutting before
introducing it formally. For eg. If we are to compare 5/9 and ½, to show that 5/9 is bigger,
take 5 strips and divide each strip into half.
Now if these 10 strips have to be given to 9 friends, 1 part of the strip so still left out,
showing that 5/9 is greater than ½
Take another eg of 5/7 and 1/3.
10
Again since a couple of parts are left over after giving out one part to each of the 7 friends,
5/7 is greater than 1/3.
The students may be introduced to equivalent fractions by paper folding methods or
overlapping methods before they are formally exposed to the mathematical procedure.
The study of fractions should build on the students’ prior knowledge of whole number
concepts and skills and their applications in everyday life. For example, students may use
fractions and decimals to report measurements, to indicate scale factors and compare
responses from samples of unequal sizes. The teacher may help the students deepen their
understanding of fractions by encouraging flexible thinking and justifying their answers.
Eg. 1. Given 3/4th of a strip, draw its ½.
If the figure below is 3/4th of a given strip, draw half of the given strip
2. Given a points 1 and 1 located on the number line, locate
,2
etc.
The area model may be used to represent a part of a whole by shading a portion of whole
area to find fractional parts and even equivalent fractions.
It may further be extended to the introduction of decimals
11
For eg. Using a 10x10 grid, shading 3/4th of the grid or 0.75 of the grid mean the same.
This gives them an idea how ¾, 75/100, .75, 15/20 etc. mean the same.
The comaprsion of frations can be introduced pictorially. Eg.
Shaded portion represents 7/8 and
Shaded portion represents ¾
The comparison is clear through the figure above that 7/8 is a grater fraction than ¾
although both have one piece left out of the whole. This reasoning may further be extended
by mathematical explanations.
12
Multiplicative reasoning as a transition from Additive reasoning
The teacher should emphasize on the development of multiplicative reasoning as a transition
from the additive reasoning. This may be clarified by an example. If an investment of $5
gives me $15 back after one year, and another investment of $20 gives me $30 back after one
year, then which one is a better investment? An obvious answer the teacher might get is that
both are same as both earn me $10. This was additive reasoning. This is where the concept
of ratio needs to be developed. The teacher now may convert the investments into simple
comparable numbers such as $1 giving $5 in the first investment and $8 giving me $18 in the
second investment. Now which one is better? Obviously I would like to invest in the first
one as it is earning me $5 on an investment of $1 each time. So if I invest $8 in the first
investment, I can earn $40! This is the development of multiplicative reasoning. This may
give the teacher a base line to develop the concept of ratio.
Proportional reasoning
Proportional reasoning is fundamental to the understanding of the mathematics of the
secondary school curriculum. Thus, it is important that the students, at this stage develop
the ability to use the proportional reasoning before they go into more advanced mathematics.
If the students can make sense of fractions, ratios and proportions and can reason in a logical
way, the transition to the secondary school curricula will be a smooth one.
Understanding proportionality should not only mean setting two ratios equal and solving
for a missing terms; it means recognizing quantities in tables , graphs etc. and find out their
relationship.
To be taken care of…
Teachers need to be attentive to the conceptual obstacles that the students may face as they
make a transition from operations with whole numbers. Multiplying and dividing fractions
and decimals can be challenging for many because of the problems that are primarily
conceptual rather than procedural. From their experience with whole numbers, many
students develop a thinking that multiplication makes a number greater and division makes
a number smaller. Teacher should check to see of his/her students has this misconception
and should take steps to build their understanding.
While teaching addition of fractions, the teacher may encourage the students to use bench
marks and estimations. For example : A very common error of the type
can be
avoided if the students are1 taught the method of estimation using bench marks:
13
3 1
4 1
so the sum ou
ught to be greater than 1! A solid conceptua
al foundation
> and >
5 2
5 2
using estimations leads to less numb
ber of mistakes.
Since
The division of fractions may be intrroduced as repeated subtraction; just as it is introduced
i
in
whole numbers.
For eg. Divide 7 by 3/5 can be refram
med as:
What is the number of pieces of striings obtained when a length of 3/5m is cut from
f
a string
of length 7
Now there are 11 pieces of length 3/
/5 m and 2/5 th part of the 3/5m is left out; which
w
is
2
3
2
2 /5
m ie
m of string is left out. So
o, 7 ÷ = 11
3
3 /5
5
3
The understanding of the propertiies of associatively and commutatively can
n be used to
7 4
simplify some of the computations in fractions. For eg. computation of 3 × × can be
2 3
7
4
simplified by using the associativee property as × (3 × ) . Similar other pro
operties used
2
3
intelligently can simplify the operations on fractions.
The unit Fractions and Decimals ha
as activities where the students may talk abo
out the use of
fractions and decimals in their life; they can find and speak about the insttances where
d around us.
fractions and decimals can be found
14
The students may be encouraged to use and find the use of fractions in their daily lives.
They may be asked to
•
Make
tables
of
equivalent
ratios
relating
quantities
with
whole-number
measurements, find missing values in the tables, and plot the pairs of values on the
coordinate plane. Use tables to compare ratios.
•
Solve unit rate problems including those involving unit pricing and constant
speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many
lawns could be mowed in 35 hours? At what rate were lawns being mowed?
•
Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
•
Find the use of fractions and ratio while cooking a dish. ( the amount of ingredients
used)
•
Use of fractions and ratio while building something.
15
COMMON ERRORS
Type of error
Error made
Conversion to and
from the mixed
fractions
1
8
4
6
Addition of fractions
with different
denominators
4
Correction
4
6
5
14
4
4
1
8
3
3
16
24
19
24
3
24
Thumb rule: MAD: multiply, add,
denominators same
etc
4
=
Thumb rule: QRD
14
3
Addition Of mixed
fractions
2
Multiplying and
dividing the mixed
fractions
3
8
3
4
5
2
5
=5
Multiplying and
dividing proper
fratcions
3
4
2
16
21
48
168
3
)
6
=5
Cancel the common factors in the
numerator and denominator
Conver the mixed fractions into
improper fractions, cancel the
common factors if any and then
multiply the numerators and
denominators and convert the
fraction into a mixed fraction.
2
3
3
6
2
16
1
4
3
1
3
9
4
10
3
15
2
7
1
2
Overview of the students’ worksheets
The first warm up (W1) activity motivates the students to write or speak about the number
system that they have already studied. They shall recall the division of symmetrical figures.
In the second warm up (W2) activity, the student shall use the famous tangram puzzles and
its parts to introduce a fraction. They may also be encouraged to speak about the use of
fractions in their daily life.
In the first pre content activity (P1), the students shall be talking about the fractions
informally. They may be motivated to generalise the problem given to them for n friends.
The focus on the warm up an pre content activities shall be to refresh the previous
knowledge of the students so that they can comfortably build up the new topic. The pre
content activities act as a bridge between the previously learnt concepts and the new
concepts to be studied.
In the 2nd pre content activity (P2), the students shall cut various shapes given to them,
colour them, and answer the given set of questions leading to the development of the
concept of fraction, numerator and denominator.
The content worksheets from C1 through C17 aim at achieving the above stated learning
objectives. Not only shall the students learn the basic concepts of fractions and decimals,
they shall be encouraged to apply them to their daily lives. The teacher may give examples
where the students can find fractions and decimals and may find the ratio of the like
quantities. The teacher may encourage them to make projects where they find fractions while
dealing with a merchant or a milkman or even in the library.
Further the post content activities are designed to assess the students’ understanding of the
concepts learnt in the unit.
17
ACTIVITY – SKILL MATRIX
Activity
Warm up (W1)
Name of activity
Importance of numbers
Skills learnt
Previous knowledge of numbers
and division of a symmetrical
object into equal parts
Warm up (W2)
Tangrams
Relating mathematics with art
puzzles
Pre content (P1)
Fraction as a part of a whole Memory,
knowledge,
understanding
Pre content (P2)
Identifying the parts of a Memory,
fraction
Content worksheet (C1)
knowledge,
understanding
Understanding of a fraction Comprehends
and its components
multiple
instructions, recalls and follows
the given instruction
Content worksheet (C2)
Locating
fractions
on
a Drawing skill, Thinking skill
number line
Content Worksheet (C3)
Introducing decimals
Synthesis skill
Content Worksheet (C4)
Placing decimals on the Drawing skill, Thinking skill
number line
Content worksheet (C5)
Equivalent fractions
Diagrammatic and mathematical
understanding
Content Worksheet (C6)
Content Worksheet (C7)
Converting
fractions
to Diagrammatic and mathematical
decimals
understanding
Simplification of fractions
Diagrammatic and mathematical
understanding
Content Worksheet (C8)
Independent practice
Content Worksheet (C9)
Place
value
chart
decimals
Knowledge, Memory
of Understanding and computing
skills
18
Content Worksheet (C10)
Comparison
of
fractions Perception and application skills
and decimals
Content Worksheet (C11)
Addition and subtraction of Rational
fractions
comparison,
Computational,
language
comprehension and application
Content Worksheet (C12)
Addition and subtraction of Understanding , thinking and
decimals
Content Worksheet (C13)
Extend
rational comparison
multiplication Comparison
and
application
and division as repeated skill, Higher order thinking skill
addition and subtraction
Content Worksheet (C14)
Word problems involving Application,
fractions and decimals
comprehension
language
and
problems
solving
Content Worksheet (C15)
Rounding
off
and Application skill, estimation skill
estimation
Content Worksheet (C16)
Content Worksheet (C17)
Estimation
and Application
significance
thinking.
Ratio and Proportion
Knowledge,
skill,
critical
understanding,
critical thinking
Post Content Worksheet
Knowledge and self learning
(PC1)
Post Content Worksheet
Knowledge and self learning
(PC2)
Post Content Worksheet
Computational
(PC3)
solving
Post Content Worksheet
Computational
(PC4)
solving
19
and
problem
and
problem
ACTIVITY 1- WARM UP (W1), IMPORTANCE OF NUMBERS
Specific Objective:
To discuss importance of natural numbers and whole numbers
Description: The students shall see the video clip 1 to understand how the knowledge of
fractions is useful in real life situations. Questionnaire will also allow the students to think
about how and where such numbers are used.
Execution: Teacher will make a questionnaire which talks about different types of numbers
and their importance, numbers which are neither whole nor integers.
Parameters for Assessment:
Students will be able to:
•
Recall different type of numbers they have come across or they have used.
•
Divide a quantity in equal parts.
ACTIVITY 2- WARM UP (W2), TANGRAMS
Specific Objective:
To review and recall the concept of natural numbers and whole numbers
Description: The students shall be told about tangram puzzle set. They would then
complete the table given in the task .They shall be helped to relate it to part of a whole.
Execution: Teacher may take print outs of the sheet and ask the students to write in the
given space. Teacher may encourage them to make their own tangram puzzle set and use it
to make a shape of their own choice.
Parameters for Assessment:
Students will be able to:
•
•
Compare quantities using concept of division
Represent a quantity as a part of a whole
20
ACTIVITY 3- PRE CONTENT (P1), A QUANTITY AS A
PARTS OF A WHOLE
Specific Objective:
To represent a quantity as a part of a whole
(Through hands on activity)
Material Required -. Pattern blocks handout and worksheet for each child.
Pre preparation - Teacher introduces the topic by reviewing the use of terms such as ½ in
daily life.
•
Students quote examples from daily life:
o
I walked ½ km today;
o
I ate ½ of my brother’s share of chocolate.
Teacher encourages the students to provide more examples of ½ and other fractions familiar
to them.
Description: The students shall cut out the shapes and through colouring activity they will
be able to understand the meaning of the word ‘whole’ and ‘parts of a whole’ as well the
meanings of the words ‘numerator’ and ‘denominator’.
Execution: The Teacher distributes the hand out to the students. Different shapes other
than a circle are taken to install the idea of a fraction as a part of a whole.
Parameters for Assessment:
Students will be able to:
•
•
Comprehend given problem situation
Represent a quantity as a part of a whole
21
ACTIVITY 4- PRE CONTENT (P2), WHAT IS IT ANYWAY
Specific Objective:
To represent a quantity as a part of a whole
Description: The students shall interpret the given problem and write their answers in
the given space.
Execution: Teacher may take print outs of the sheet and ask the students to write in the
given space. Teacher may ask them about notion of ‘one-third’, ‘two-fifth’ etc.
Parameters for Assessment:
Students will be able to:
•
•
•
Comprehend given problem situation
Represent quantities using concept of division
Represent a quantity as a part of a whole
ACTIVITY 5- CONTENT WORKSHEET (C1), UNDERSTANDING
AND REPRESENTING FACTORS AND ITS COMPONENTS
Specific Objective
To understand a fraction and its components
Description: The students shall interpret the given problem and write their answers in the
worksheet. The students may be divided into groups to do this activity.
Execution: Teacher checks for understanding of the meaning of the words numerator,
denominator and fraction through a worksheet.
Parameters for Assessment:
Students will be able to:
22
•
•
•
Hass knowledg
ge of variou
us types of fractions.
f
Relaate the giveen problem to fractionss.
Can
n speak crea
atively abou
ut fractionss and their need
n
and usage.
u
Extra Reaading:
http://w
www.youtube.com/waatch?v=gJgusNWTIkA
A&feature=
=related
http://w
www.amath
hsdictionary
yforkids.com/
ACTIVIT
TY 6- CO
ONTENT
T WORK
KSHEET
T (C2), LO
OCATIN
NG
FRAC
CTIONS ON
O TEH
H NUMBE
ER LINE
E
O
e:
Specific Objective
To represent fractionss on a numb
ber line
Activity 1 & 3:
Descriptiion: The concept
c
of parts
p
of fracctions is to be used to introduce representaation of
fractions on a number line. The students sh
hall be told
d about how
w to represeent the fracction on
the numbeer line with
h help of video clip 4 by
b making as many markings
m
ass the denom
minator
of the given fraction.
Activity 2:
2
Materialss needed:: Gems for each
e
child, Cloth’s pin
ns, Clotheslline with 0 and
a 1 mark
ked.
23
Pre preparation: Gems packets, a clothesline with markings 0 to 1 and clothes pins.
Description – After distributing a packet of gems to each child, teacher asks each child to
randomly pick up 10 gems from the packet and then follow the directions given in the
instruction sheet. Teacher helps students to model.
Teacher will allow student to come and demonstrate his/her knowledge on the board. Since
each child has a different favourite colour hence will get to represent different fraction on the
number line. This activity will give a visual representation to a fractional number.]
Note: At this point teacher may want to extend the knowledge of number line and
ask students to arrange the fractions representing different colours in the ascending or
the descending order.
Execution: Students follow the directions and complete the activity as well as some more
questions based on knowledge acquired.
Parameters for Assessment:
Students will be able to:
•
•
•
Exhibit knowledge of parts of fraction.
Divide a unit distance into as many equal parts as the denominator.
Represent fractions on a number line.
24
ACTIVITY 7- CONTENT WORKSHEET (C3),
INTRODUCING DECIMALS
Specific Objective:
To introduce decimals and represent quantities using decimal
Description: The concept of parts of fractions is to be used to introduce decimal as a special
case of a fraction with denominator in powers of 10.
Execution: Activities given can be given in form of worksheet or practical examples to the
students. Teacher may ask the students to do conversion from centimetre into metre etc.
Parameters for Assessment:
Students will be able to:
•
•
Exhibit knowledge of fraction and parts of fraction.
Represent fraction with denominator in powers of 10 using concept of decimal.
•
Represent quantities using decimal.
Extra Reading:
http://home.avvanta.com/~math/FDU1.HTM
ACTIVITY 8 - CONTENT WORKSHEET (C4),
PLACING DECIMALS ON THE NUMBER LINE
Specific Objective:
To represent decimal on a number line
Description: The concept of representation of fractions on number line is to be used to
introduce representation of decimal on number line, as a special case of a fraction with
denominator in powers of 10. Students shall write their answers in the given space and then
represent decimal on number line accordingly.
Execution: Activity given to be given in form of worksheet to the students. Teacher may
ask the students to represent decimal on a number line after viewing video clip 5.
25
Parameters for Assessment:
Students will be able to:
•
•
•
Represent fractions on a number line.
Divide a unit distance into 10 equal parts and represent related decimal on it.
Represent decimal on a number line.
ACTIVITY 9- CONTENT WORKSHEET (C5),
EQUIVALENT FRACTIONS
Specific Objective:
To demonstrate, communicate and explain equivalent fractions.
Description: The students shall use the hands out (cut outs) of congruent rectangles to
represent fractions diagrammatically. . This worksheet will help students to represent same
fraction in many ways hence understand equivalent fractions.
Execution: Teacher will show Video clip 7 to the students to explain the concept visually.
The teacher prepares a comprehensive worksheet for students to practice the concepts learnt.
Parameters for Assessment:
Students will be able to:
•
•
Represent fractions diagrammatically
Understand and find equivalent fractions to a given fraction
Extra Reading:
www.helpingwithmath.com/by-subject/fractions/fractions.htm
26
ACTIVITY 10- CONTENT WORKSHEET (C6),
CONVERTING FRACTIONS TO DECIMALS
Specific Objective:
To convert fractions to decimals
Description: This worksheet is based on equivalent fractions. The fractions are converted
to an equivalent fraction with denominator 10 or powers of 10. Teacher induces the topic
by showing the graphics where a figure is divided into number of parts. Students shall
answer the questions based on the graphics. The concept and discussion is extended to the
same shapes cut into 10 equal parts and then 100 equal parts.
Execution: Teacher shows the video clip 8 again to show conversion of equivalent fractions
to reinforce learning. A worksheet may be given to the students in which they try to convert
fractions to decimals based on what they have learnt.
Parameters for Assessment:
Students will be able to:
•
•
•
Represent fractions diagrammatically
Understand equivalent fractions
Convert fractions to decimals
Extra Reading:
www.resources/games/fraction-games4/equivalent01.html
27
ACTIVITY 11 - CONTENT WORKSHEET (C7),
SIMPLIFICATION OF FRACTIONS
Specific Objective:
To demonstrate simplification of fractions
Description: Students view video clips 9 to understand the simplification of fractions
visually. After the explanation and video clip students are allowed to work on the situations
independently.
Execution: This worksheet also deals with equivalent fractions but instead of finding
multiples, teacher guides the students to divide by a common factor and arrive at a fraction
whose numerator and denominator have no factor other than 1 common.
Parameters for Assessment:
Students will be able to:
•
•
•
Represent fractions diagrammatically
Knowledge of equivalent fractions
Understanding of simplification of fractions
Extra Reading:
http://www.funbrain.com/cgi-bin/fob.cgi?A1=s&A2=0
http://www.math-aids.com/Fractions/
28
ACTIVITY 12- CONTENT WORKSHEET (C8),
INDEPENDENT PRACTICES
Specific Objective:
To find Equivalent fractions and simplify fractions
Description: Students recall Equivalent fractions and simplification of fractions and are
allowed to work on the situations independently.
Execution: Teacher may prepare a comprehensive worksheet to test student’s
understanding of classifying fraction, simplify fractions and converting fractions to decimals,
using concept of Equivalent fractions and simplification of fractions.
Parameters for Assessment:
Students will be able to:
•
•
•
Has knowledge of various types of fraction
Is able to relate the given number to different kind of fractions
Can speak creatively about numbers and their need and usage
ACTIVITY 13- CONTENT WORKSHEET (C9),
PLACE VALUE CHART OF DECIMALS
Specific Objective:
To learn to read and use Place value chart of decimals
Description: Students watch video clip 3 and after a brief recall of the place value chart of
numbers, they then answer questions based on what they saw in the video.
Execution: Teacher asks questions to test understanding of the place value of each digit
and then asks students to give the expanded form / the place value of a particular digit/
number name etc.
Parameters for Assessment:
29
Students will be able to:
•
•
Understands the place value chart for decimals
Read and write decimals in expanded form
ACTIVITY 14- CONTENT WORKSHEET (C10),
COMPARISON OF FRACTIONS/DECIMALS
Specific Objective:
To visualise and compare fractions/decimals using manipulative and to define like and
unlike fractions
Description: In activity 1, Students to cut out fractional parts 1, 1/2 , ¾ , 1/6 , 2/5 , 1/10 ,
1/12.Now arrange them on your desk from the smallest to greatest to the smallest.
In activity 2- Like and unlike fractions
Material Required: Poster depicting the graphic.
With the help of the questions based on the above graphics teacher explains the concept of
like and unlike fractions. Just as we cannot say that tower T2 is taller than tower T7, because
they are not at the same level, we cannot compare the fractions of the type 2
5
and 1
4
.
The teacher sums up the argument and concludes that for anything to be compared they
should be on the same levels. Similarly in order to compare fractions, we need to put them
on the same level (same denominator)
30
In activity 2 & 3, students shall write their answers in the given space.
Execution: In activity 1, teacher will distribute fraction strips to all the students. Teacher
will give instructions to students which will help them to cut out fractional parts 1, ½,
1
6 ,
2
5 ,
¾,
1
1
10 , 12 and arrange them on their desk from smallest to greatest or greatest to
the smallest.
In activity 2 & 3, Through modelling and questioning, teacher will extend the knowledge of
comparison of like fractions to unlike fractions as well as to decimal numbers.
Parameters for Assessment:
Students will be able to:
•
Is able to understand fraction strip and use it creatively
•
Is able to understand like and unlike fractions
•
Is able to compare decimals
Extra Reading:
http://www.funbrain.com/cgi-bin/fob.cgi?A1=s&A2=0
ACTIVITY 15- CONTENT WORKSHEET (C11),
ADDITION AND SUBTRACTION OF FRACTIONS
Specific Objective:
To understand addition and subtraction of fractions
Description: In activity 1 – DONINO ACTIVITY
students shall arrange cutouts of
dominoes so that the matching expressions are equivalent and form a continuous loop. The
outcome shall be:
31
Execution
n: This iss a guided
d worksheet where students will
w
learn how to use
u
the
manipulatiive and solve the worksheet at th
he same tim
me. Teacherr will show video clip 12 and
video clip 13 to reinfo
orce additio
on and subttraction of fractions.
f
Parameteers for Assessmentt:
Students will be ablle to:
•
•
•
Has knowledge off various ty
ypes of fracttions
Is able to
t add and
d subtract frractions
Can sollve related problems on
o fractionss
Extra Reaading:
www.help
pingwithma
ath.com/by
y-subject/frractions/fraactions.htm
m
http://ww
ww.mathsneet.net/gcsee/worksheeets.html
ACTIV
VITY 16- CONTE
ENT WO
ORKSHE
EET (C12
2),
ADDITIION AND
D SUBT
TRACTIO
ON OF FRACTIO
F
ONS
Specific Objective
O
e:
To understtand addition and sub
btraction of decimals
Descriptiion: This worksheet deals withh the addiition and subtraction
s
n of decimaals and
extended to
t real life word prob
blems. In acctivity 1, sttudents apply the con
ncept learn
nt using
manipulatiive by solviing the worrksheet.
32
Students shall write their answers in the given space.
Execution: Teacher will show video clip 14 to the students to reinforce the
concept. This
is followed by the worksheet which is done by the students independently.
Parameters for Assessment:
Students will be able to:
•
•
•
Has knowledge of decimal
Add and subtract decimals
Can solve related problems on decimals
Extra Reading:
http://www.sometests.com/tests/Math/MathsQuizFractionsFor6Class.html
ACTIVITY 17- CONTENT WORKSHEET (C13), MULTIPLICATION
AND DIVISION AS REPEATED ADDITION AND SUBTRACTION
Specific Objective:
To extend multiplication and division as repeated addition and subtraction
Description: This worksheet helps students to understand the fact that multiplication is
repeated addition and similarly division is repeated subtraction. Students shall write their
answers in the given space.
Execution: Teacher will prepare questions where students will understand the concept and
have questions to practice the concept as well. Teacher ensures that students follow all
instructions and complete the worksheet.
Parameters for Assessment:
Students will be able to:
•
•
Understand multiplication as repeated addition
Understand division as repeated subtraction
33
ACTIVITY 18- CONTENT WORKSHEET (C14), WORD PROBLEMS
INVOLVING FRACTIONS AND DECIMALS
Specific Objective:
To test understanding of fractions and decimals as a concept and
application
Description: Students test their understanding and apply skills of addition and subtraction
of fractions and decimals to a game The Match Maker.
Material needed required - Clue cards
Teacher prepares cards for each student of the class. Cards are distributed to the students.
Teacher will guide them to find a match for their card. There are two clues one on the front
side and one at the back side of the card. This is just played as a game in the class. Students
will record their observations on the worksheet. Students get 10 minutes to complete the
whole chain of clues.
Execution: Teacher will prepare a comprehensive worksheet for the students to test their
understanding of fractions and decimals as a concept and application.
Parameters for Assessment:
Students will be able to:
•
•
Has knowledge of fractions and decimals
Can solve related problems on decimals and fractions
ACTIVITY 19- CONTENT WORKSHEET (C15),
ROUNDING OFF AND ESTIMATION
Specific Objective:
To explain rounding off and estimation
Material Needed: Measuring tape.
34
Description: Students shall discuss rounding off and its purpose and shall view video clip
no 15 to understand the process of rounding off. After watching the video students complete
the task given in the worksheet.
Execution: - Teacher will prepare a set of questions through which the students will
understand the need of estimation in real life. Teacher will ask students to pair up and
answer the same set of questions for their partners. Teacher will guide students to analyse
and discuss the possible reasons for the variations in the two answers. She/he will pose
questions to stimulate thinking.
Parameters for Assessment:
Students will be able to:
•
•
•
Understand the need of estimation in real life
Can analyse the possible reasons for the variations in the two answers
Understand the process of rounding off
ACTIVITY 20- CONTENT WORKSHEET (C16),
ESTIMATION AND SIGNIFICANCE
Specific Objective:
To understand significant figures and learn the rules for the significant figures
Description: A guided worksheet which has questions explaining the concept of significant
figures is prepared and is given to the students. Student answers the question and
understand the appropriate significant digit to which the answer must be given. As
reinforcement teacher shows Watch video clip 16 and write the rules of giving answers to
the correct accuracy.
Execution: Teacher ensures that students understand and follow instructions and record all
discussions and then test their knowledge by solving questions on significant figures.
Parameters for Assessment:
Students will be able to:
35
•
•
Understand the rules for the significant figures.
Solving questions on significant figures.
ACTIVITY 21- CONTENT WORKSHEET (C17),
RATIO AND PROPORTION
Specific Objective:
To understand and apply the knowledge of ratio and proportion to real life problems
Description: Activity 1: Ratio
Teacher to start with a mental starter where she/he asks questions related to daily life where
student already has an idea about the terminology or may meaning of the word ratio.
Teacher introduces ratio as a way of comparing
amounts of something.
For example: Use 3 parts blue paint to 1 part white
This means , mixing paint in the ratio 3 :1 (3 parts blue paint to 1 part white paint) means 3 +
1 = 4 parts in all Teacher relate lesson to fraction by showing 3 parts blue paint to 1 part
white paint = is ¾ blue paint to ¼ white paint.
Activity 2: Proportion
In this activity teacher develops the lesson on proportion and to make students understand
that how is ratio different from proportion.
Activity3: Its Maggi, maggi maggi.
Materials Required: Maggi packets and the home science lab to make the maggi.
Teacher instructs students to pick a slip from the bowl kept on the table. The slip contains the
number of maggi cakes the student is required to prepare. Teacher asks one student to read
the instructions from the packet given to prepare maggi. Its a hands on activity on how
proportion and ratio are useful in day to day life situations.
Execution: Students follow the directions given by the teacher and do some work
independently after the discussions are over.
36
Parameters for Assessment:
Students will be able to:
•
•
relate the given problem to ratio and solve them
relate the given problem to proportion and solve them
Extra Reading:
www.helpingwithmath.com/by-subject/fractions/fractions.htm
http://www.mathsnet.net/gcse/worksheets.html
http://www.math-aids.com/Fractions/
37
SAMPLE RUBRIC (ASSESSMENT)
FRACTIONS AND DECIMALS
0
1
2
3
4
Ability to
Can not
With guidance can
Can demonstrate
With 90%
Can
recognize
define
follow instructions
diagrammatically
accuracy can
demonstrate
and define a
fraction
and represent
the
demonstrate
understandin
fraction and
and
fractions
representation of
understandin
g of fraction
a decimal
decimal .
diagrammatically
a fraction and
g of fraction
and decimal
using correct
but fails to connect
decimal and has
and decimal
and
notations
it to abstract as well
clarity about the
communicate
communicate
as extend it to a
representation of
with
with
decimal
fraction and a
appropriate
appropriate
representation
decimal, but is
notations
notations
unable to express
with, as well
with, as well
it independently
as without
as without
without a
Manipulative.
Manipulative,
Parameter
manipulative.
in various
contexts.
Ability to
Cannot
Can
Can
Can
Can
represent
relate,
plot fractions and
plot fractions
-plot and
-Plot and
fractions and
visualize
decimals on a
and decimals on a order
number line, but
number line,
decimals and
decimals and
on a
with a lot of
independently
fractions on
fractions, in
number
guidance.
but takes a lot of
number line
various
time to follow
with 90%
contexts, on a
instructions
accuracy
number line.
decimals on a or work
number line
line.
Locate
is able extend
it to order
them and
38
explain the
relationship
of
Decimal to
division and
fraction.
Ability to
Can
Can classify
Can classify
Can classify
Can classify
classify
recall
fractions. Can
fractions, identify
fractions,
fractions,
fractions ,
definition
simplify fractions
equivalent
identify
identify
find
s of types
and complete
fractions through
equivalent
equivalent
equivalent
of
patterns but is
patterns only
fractions
fractions
fractions and
fractions
unable to relate a
.Can simplify if
through
through
simplify
but
decimal to division
guided ,can
patterns as
patterns as
fractions.
cannot
of a fraction.
understand link
well as frame
well as frame
Can juggle
classify
between fraction
a rule .Can
a rule in
between
them.
and decimal but
simplify
various
lacks accuracy
efficiently
contexts. Can
with
understand
simplify
fractions and
link between
efficiently in
decimals
fraction and
various
decimal
contexts. Can
accurately,
understand
and is able to
link between
extend it to
fraction and
ordering with
decimal
guidance
accurately,
problems
and is able to
extend it to
ordering .
Ability to
Cannot
Knows the theory
Can compare and
Can order
With a high
compare and
make any
but can put the
orders fractions
fractions and
degree of
39
orders
sense of
concepts into
and decimals but
decimals. Can
effectiveness
fractions and
the order
practice under lot
is unable to work
juggle
can compare
decimals ,
of
of supervision.
in an integrated
between the
and
manner
two and
order
and
integrate
fractions
decimals
learning . Is
and decimals
.
able to
and is able to
communicate
integrate
his thoughts.
learning
fractions
relate to
various
contexts.
Ability to
Cannot
Can make and
Can make and
Can make
Can make
Model and
work
carry out
carry out a plan
and carry out
and carry out
perform the
with
a plan to pose and
to pose and solve
a plan to pose
a plan to pose
operations
fractions
solve problems
problems with
and
and
and
with
decimals and
solve
solve
decimals
decimals and
fractions, using
problems
problems
in any
fractions applying
some strategies
with
with
context
the skills , using a
usually resulting
decimals and
decimals and
limited range of
in a partially
fractions,
fractions
appropriate
accurate
using
accurately,
strategies;
solution
appropriate
using
which
strategies of
appropriate
rarely results in an
the skills
strategies of
accurate solution
learnt
the skills that
usually
may offer an
resulting in
innovative
an accurate
approach
solution
40
Ability to
Cannot
Can perform
Can perform
Can perform Can perform
extend skill
relate
through repeated
multiplication
multiplication multiplication
of addition
and has
addition and
and division, as
and division
and division in
and
not been
subtractions only.
repeated
in various
various
subtraction
able to
additions but
contexts with contexts
to
master
makes errors. Has
90% accuracy efficiently and
multiplicatio
the
not mastered the
helps peers in
n and
addition
process yet.
understanding
division of
and
fractions and
subtracti
decimals
on skills
Ability to
Cannot
Can apply
With some
With
With 100%
Apply to real
apply
knowledge of
effectiveness can
considerable
accuracy can
life
knowled
fraction to ratios
apply knowledge
effectiveness
apply
Ability to
ge of any
and solve real life
of fraction to ratio
can apply
knowledge of
strategies
concept
problems or gives
though with a lot
knowledge of
fraction to
and apply
the answer to the
of guidance
fraction to
ratio , comes
the skill of
correct significance
contains lots of
ratio coming
up with
ratio and
very rarely.
errors in
up correct
alternate
proportion to
estimation and
answers to
strategies to
problems
scientific
the specified
solve
and coming
notations
degree of
problems
up with
accuracy in
posed and is
solutions
estimation
always
corrected to
and scientific
specific to the
the specified
notations
degree of
skills.
accuracy
degree of
accuracy.
41
Study
Material
41
FRACTIONS AND DECIMALS
Introduction
Earlier, we discussed counting numbers (also called natural numbers) 1, 2,
3,… and whole numbers 0,1,2,3,…. and operations (+,−, ×, ÷) on these
numbers. We also discussed some properties of there numbers. In fact
natural numbers help us in the process of quantification of objects.
However, there are certain situations which can be quantified by natural
numbers as such.
For example, when an object is divided or broken into two equal parts, will natural
numbers help to tell what one part represents? To deal with such cases, we introduce some
other type of numbers called fractions.
In this unit we shall discuss fractions and some other related concepts like decimals, ratio
and proportion and their use in daily life.
What is a fraction?
Suppose we are to distribute equally 14 apples between 2
children. Clearly each child will get 2 apples. If we had 2
apples, then each child will get 1 apple. What will each
child get it we have only 1 apple?
In this situation, we have to cut the apple into two equal parts and give one part to each
child. We say that each child is getting one part out of two equal parts.
This is symbolically expressed as
_
One apple
Two equal parts
½ is known as a fraction
Thus, we can say that a fraction is a part of a whole
In this case, “Whole” is an apple.
42
Further, suppose the same apples to be distributed equally among 4 children.
Again in this case, we have to cut the apple into 4 equal parts and each child will get part.
This is symbolically expressed as
One apple
Four equal parts
Here again, a whole (an apple) is divided into 4 equal parts and one part represents the
fraction ¼
To understand more about fractions, look at the following Figures:
In (i) A rectangle (whole) is divided into 2 equal parts. One part is shaded. The shaded part
is represented by the fraction
. Similarly unshaded part is also represented by
43
(why?)
In (ii) A circle (whole) is divided into 4 equal parts here the shaded part is represented by
and unshaded by
In (iii) Shaded portion is represented by the fraction
In (iv) Shaded portion is represented by
and unshaded by
and unshaded by
In all the above cases, whole was a single object such as an apple a rectangle or a circle
However, a whole can also be a collection of objects.
For example, in case of distributing 4 apples between 2 children, the whole was a collection
of 4 apples, and in this case each child got 2 apples.
Divide the collection by a line into two equal parts. Obviously, each part is one half (1/2)
of the whole collection (4 apples)
It is written
as
Similarly, its 5 totters are distributed equally among 2 children, then each will get
and so on.
Thus we can also have fractions such as
, , ,
Components of Fractions
In the fraction
, 1 is called the numerator and
2 is called the denominator of the fraction.
44
,
, , etc.
totters
ator and deenominatorr are separaated by a baar (--------)
Note that the numera
Similarly, in , 2 is th
he numerattor and 3 is the denom
minator.
and in
2 is the num
merator an
nd 4 is the denominato
d
or.
is read as
a 1 by 2 orr 1 upon 2 or
o divided by
b 2 Similaarly,
is reaad as 2 by 3 or 2 upon
n 3 or 2
divided by
y 3 ad so on
n.
•
In words,
w
is read
r
as onee half
•
iss read as tw
wo thirds
•
iss read as thrree- fourthss and so on
n.
Examplee 1: Correesponding to the shaaded portioons write the
t
fraction
n in each of the
following if possible:
Solution
n
(i)
(as 2 paarts taken frrom 3 equaal parts)
45
1
4
(ii)
(iii)
(iv)
(v)
No fraction, as parts are not equal
(vi)
No fraction, as parts are not equal
Example 2: Write fractions having
(i)
3 as a numerator and 5 as the denominator
(ii)
12 as the denominator and 7 as the numerator
(iii)
1 as numerator and 8 as the denominator
Solution:
(i)
3
5
(ii)
12
7
(iii)
1
8
Example 3: Which of the following fractions which have numerator 2?
, , , , ,
Solution: Fractions
and
have numerator 2
Example 4: Which of the following fractions have denominator 5?
, , , ,
46
Solution: Fractions
and
have numerator 5
Representation of fractions on the number line
We have learnt how to represent whole numbers on a number line (fee figure below):
We also know that a number to the right of a number is greater then the other. In a similar
way, we can represent fractions on a nummular line.
•
Let us represent the fraction
We know that
on the number line
is a part a whole in which out 2 equal parts, one part is taken
So we divide one unit (the distance between 0 and 1) into two equal parts by the point A
(see figure below)
Point A represents the fraction
•
Let us now represent the fraction
on the number line.
is a part a whole is write 2 parts have been taken out of 3 equal parts.
So, we divide unit distance, 0 to 1 into three equal parts by point A and B as shown in
the figure below:
Second point B represents the fraction
.
Cleary the first point A represents the fraction
Let us now represent the fraction
.
on the number line.
47
Method 1
Divide the unit distance 0 to 1 in 4 equal parts by the points A, B and C (see figure below).
Point A represents
.
Now, we move 5 steps (each equal to OA) towards right of O as shown in the figure and
reach at the point D
5
The point D represents the fraction .
4
Method 2
Take 5 units from O i.e. , the distance 0 to 5 on the number line.
Divide this distance into 4 equal parts by the points A, B and C as shown below.
The point A will represent
Example 5: Represent the following fractions on the number line:
(i)
3
8
(ii)
5
2
Solution:
(i)
Divide unit distance 0 to 1 into 8 equal points. find point from 0 will represent
3
8
0
1
48
(ii)
Method 1: Divide unit distance 0 to 1 into 2 equal parts. The fast point of
division will represent . Move 5 steps to the right each step being .
Point B will represent
.
Method 2: Divide the distance 0 to 5 into two equal parts. The point of division will
represent
The point B represents
Types of fractions.
Proper and improper fractions
Look at two groups of fraction A and B given below
Group A:
, , , ,
Group B:
, , , , ,
,
,
,
, ,
Compare numerators and denominators of fraction in each group. What do you find?
In Group A, the numerator of each fraction is less than the denominator.
In Group B, the numerator of each fraction is greater or equal to the denominator.
So, on the basis of values of numerators and denominators fractions can the classified as
follows:
Fractions with numerators smaller than denominators are called proper fractions
49
Fractions with numerator greater or equal to denominators are called
improper fractions
Thus, fractions in group A are all proper fractions and fractions in group B are all improper
fractions
Unit fractions
Fractions like
, ,
, ,
,
, etc
in which numerator is 1, are called unit fractions.
, , , , etc not unit fractions (why?)
Clearly,
Like and Unlike fractions
Look at the denominators of the following fractions:
, , ,
,
what do you observe?
The denominators of all the fractions are the same i.e. 2 such fractions are called like
fractions.
,
Thus,
,
,
Similarly,
,
,
,
,
,
,
,
are also like fractions.
, , are also like fractions. What about fractions
?
Clearly, they are not like fractions as their denominators are not same
(though their numerators may be same)
Fractions which are not like fractions are called unlike fractions.
For example,
Similarly,
,
, , ,
,
, , , ,
are unlike fractions
are also unlike fraction because, all of them do not have the same
denominator
50
Mixed fraactions (or mixed
m
num
mbers.)
gure there is a collecttion of 2 objects
o
(circcles). Each
h object hass been
In the folllowing fig
divided in
nto two equ
ual parts.
We choose 3 parts
We see on
ne circle and
d circle sh
haded
Thus
and
an
nd is briefly
y expressed
d as 1
Numbers of the typee 1 are calleed mixed frractions (orr mixed num
mbers).
Improper fractiion
Some morre exampless of mixed fractions arre: 2
ccombinatio
on of a whole
n
number
(1) and a prop
per
1
f
fraction
2
3
2
11
5
etc.
Examplee 6: Which of theses arre proper and
a which are
a impropeer fractionss?
51
Solution:
, ,
Proper fractions
, (numerator is less thus denominator)
, ,
Improper Fractions:
, , (numerator is greater thus or equal to denominator)
Example 7: Which of the following are unit fractions?
2 1 3 8 1 1 3
, , , , ,
,
3 4 1 7 10 101 11
Solution
Unit fractions are
,
,
, (Numerator is 1)
Example 8: Which of the following are mixed fractions? Which are improper fractions?
, 3 , 2 , , 11 , , 5
Solution:
Mixed fraction: 3
1
, 11
5
,5
Improper fractions , , ,
3
Note: 2 is neither a mixed fraction nor an improper fraction (why?)
2
Example 9: Which the following fractions are like and which are unlike?
(i)
5 11 2 16
, , ,
3 8 3 8
(ii)
2 2 2 2
, , ,
5 11 9 61
(iii)
5 16 19 21 101
, , , ,
11 11 11 11 11
(iv)
1 4 16 18
, , ,
3 9 9 9
52
Solution
(i)
Not
like
fractions.
Hence,
they
are
unlike
fraction
because,
denominators are not the same.
(ii)
They are unlike fractions, though their numerators are the same.
(iii)
They are unlike fractions
(iv)
They are unlike fractions.
Expressing an improper fraction as a mixed fraction
Consider improper fractions say
(i)
,
4
4
is an improper fractions
i.e
(ii)
contains 4 whole and
Similarly,
i.e.
5
5
more
=5 ,
i.e. 5 +
28
25
3
, is a mixed fraction
Alternatively:
And
4
is an improper fraction
contains 5 whole and
Now 5
more , i.e 4+
17
16
1
,
,=
,
16
,+
4
,
5
4
4
5
So, we can always express an improper fraction as a mixed fraction.
Expressing a mixed fraction as an improper fraction
Consider mixed fractions, say 2
(i)
2
, 7 29
3
3
3
is a mixed fraction and 2 = 2 + =
4
4
4
53
+
3
4
11
(improper fraction )
4
Here, you can see that 2
So, we can directly write 2
(ii)
3
as
4
7
7
(improper fractions)
Here again, 7
In general, mixed fractions: a
,
+
can be easily writen as
can be expressed as an improper fraction
=
Example 10: Express the following improper fractious into mixed fractions:
(i)
(ii)
(iii)
(iv)
2
Solutions:
(i)
3
=2
5
(ii)
=8
(iii)
=5
(iv)
=1
5
So an improper fraction of the type
4
8
13
10
3
, ,
35
32
3
etc cannot be expressed as mixed fractions.
Example 11: Express the following mixed fractions into improper fractions:
(i)
7
1
4
(ii)
11
(iii)
27
54
9
10
(iv)
Solution
(i)
7
(ii)
11
(iii)
27
=
=
=
=
=
Equivalent Fractions
Look at the shaded portion of the following figure
What do you observe?
Shaded portion in (i) represents fraction
Shaded portion in (ii) represents fraction
Shaded portion in (iii) represents fraction
Shaded portion in (iv) represents fraction
55
Note that the shaded portion of each figure represents the same equal part of the given
rectangle so, we can say that correcting fractions
, , , and
are equal.
Such fractions
are called equivalent fractions.
If you represent these fractions namely
, , ,
and
on the number line,
you will find that all these fractions are represented by the same point. Check
it!!
Using the above process of shading suitable number of parts of congruent rectangles
(triangles of same shape and size) to represent
, , , , you can see that the fractions , , , , are also equivalent fractions
Obtaining equivalent fraction
We have seen that
, , , ,
Are equivalent fractions
Let us consider
, and
We see that
Thus
has been obtained from
by multiplying its numerator and denominator by the
same number (here it is 2)
Similarly
=
Here also, equivalent fraction
denominator of the fraction
has been obtained by multiplying the numerator
by the same number 3.
Multiplying the numerator and denominator by 4
Thus, when the numerator and the denominator of a fraction are both multiplied by the same
number (other than0), we obtain an equivalent fraction
56
Again consider
and
See that
similarly,
Thus, when the numerator and the denominator of a fraction are both divided by the same
number (other them 0), we obtain an equivalent fraction.
To check whether two fractions are equivalent are equivalent or not.
We see that
and
are equivalent fractions
i.e. We thus that 1 ( numerator of first ) x 4 (Numerator of second ) = 4
Also, 2(denominator of the first) x 2(numerator of the second)=4
thus 1
4
2
2
Similarly for equivalent fractions
,
And
3
8
6
,2
4
6
and
24
4
3
12 and so an
57
Thus in a pair of equivalent fraction
Numerator of the first x denominator of second
= Denominator of first x numerator of second
Example 12: Are the following two fraction equivalent?
,
(i)
,
(ii)
(iii)
,
(iv)
,
Solutions:
(i)
since 1
5
2
So,
(iii)
15
3 = 15.
and
So,
(ii)
15
8
are equivalent
16, 7
3
21
16
21,
and are not equivalent
25
40
20
5
1000
100
Since 25 40 20 5
Therefore
and
are not equivalent
58
(iv)
10 22=220
11 20=220
since 10
Therefore,
Example 13:
22
11
20,
and
are equivalent
fill in the blanks:
(i)
(ii)
(iii)
Solution:
135
(i)
120
(ii)
90
(iii)
45 therefore 2
4
5
45
30 therefore, 90
18
90
30
, 216
3
18
18
Example 14: find the equivalent fraction of whose
(i)
Denominator is 12
(ii)
Numerator is 21.
Solution :
(i)
Since denominator is 12 and 12 = 4 x 3, therefore required equivalent fraction
=
(ii)
Since denominator is 21 and 21=3 7, therefore required equivalent fraction
59
=
Fractions in the simplest form
Look at the following fractions:
4 12 2 20 6
, , , ,
6 18 3 30 9
Cheek that all these are equivalent fractions in the fraction , 2 and 3 are co prime that is,
there is no common factor between 2 and 3 other than 1. Such a fraction is called a fraction
is the simplest or lowest form or lowest terms.
is not is the simplest form as 4 and 6 have a common factor 2 other than 1 Similarly
is
also not is the simplest form.
Further
is again a fraction is the simplest tent because, 9 and 16 have no common factor other
then 1.
Reducing a fraction in the simplest form
Consider the fraction
Since 12 and 18 have common factors 3 , 2 and 6 besides 1, so,
is not in simplest form.
To reduce it into its simplest form, we divide both numerator and denominator by their
HCF. Here H C F (12, 18) =6
So,
Similarly
simplest form
is also not is the simplest form as 36 and 162 have factors 2, 3, 6, 9, 18 (other
than 1)
So
Note that H C F 36 ,162
Which is now is the simplest form.
60
18
Example 15: Reduce the following fraction is their simplest form:
(ii)
(i)
(iii)
Solution: (i)
20, 48
22,25
(ii)
1
68,86
(iii)
4
2
Example 16 : Which of the following fraction are in the simplest form?
(i)
(ii)
(iii)
(iv)
Solution:
(i)
is in the simplest form as HCF (16, 39) =1
(ii)
is not is the simplest form as HCF 218,242
(iii)
is also not in the simplest form (why?)
(iv)
is in the simplest form (why?)
61
2
Comparison of fractions
Comparing like fractions
Look at the shaded portions of each of the figure given below:
Obviously, the shaded portion which represents
represents
is larger than that of portion which
.
So,
Similarly, it can the seen that
Thus
For two like fractions, the fraction having greater numerator is
greater than the other.
Comparing unlike fractions
For comparing unlike fractions, we first convert them into like fractions and then compare
them by comparing their numerators. For example to compare fractions
Note that 30 is the L C M of the denominators 10 and 15
62
and
As 16
Since
9
So,
Another Method of comparing two fractions.
,
Consider:
3
15
45
10
8
80
And 80
10
45
80
So,
10
8.
8
,
in the same way, for fraction
5
9
And 45
,
45 , 6 4=24
24 ,
,
Example 17: Find the larger (greater) fraction in each pair
(i)
,
(ii)
;
,
(iii)
(iv)
,
Solution
(i)
Because denominator is the same and numerator of
numerator of
.
(ii)
(iii)
35 is LCM of 7 and 5
63
is greater than
So,
,
Alternatively:
since 21
4
5
20, 7
3
21
20,
So,
(iii)
11
15
165
18
13
234
As 234
165, Therefore
17
(iv)
49
833
As 945 > 833, Therefore,
Example 18: Compare the fraction
4
17
4
25
Solution
4
25
100
17 4=68
Since 100 > 68, therefore,
Note that amongst the fractions that have the same numerator; the fraction with smaller denominator
is greater.
Example 19: Arrange , , , is ascending order.
64
Solution: These are unlike fractions (Why)
They can be expressed as like fractions as following.
1
LCM = 2 x 3 = 6
1
3
1
3
2
2
2
6
1
2
1
3
2
2
2
6
1
0
1
6
Now
So given fractions in the ascending order are
, ,
Note: In this case, numerator is the same So, we can compare the fractions by comparing
their denominators.
Example 20: Arrange the following is descending order:
4 3 5 11
, , ,
7 8 14 18
65
LCM = 2
2
2
3
3
7 =504
Solution:
=
hence, the given fractions in descending order are
, , ,
Addition of fractions
Adding like fractions
The sum of two or more like fraction =
For example,
=
17
Similarly
66
Adding unlike fraction
To add two or more unlike fractions convert them in to like fractions and then
add
For example,
=
=
(45 is the lcm of 9 and 15)
25 6
45
31
45
Example 21: Find the following sums:
(i)
(ii)
(iii)
1
(iv)
5
3
Solution
(i)
1
Unlike fractions are converted into equivalent like fractions and then
addition is carried out
67
(i)
=
1
=
(ii)
1
(iii)
5
3
4
(LCM of 7,9,14)
Subtraction of fractions
o Subtracting like fractions
The difference between two like fractions
=
For example,
68
Subtracting Unlike Fractions
To find the difference between two unlike fractions, convert
them to like fractions and the subtract.
For example,
6
=
4
12
Example 22: Find
(i)
(ii)
4
2
21
3
2
7
(iii)
Solutions:
(i)
(ii)
4
3
(iii)
Unlike fraction are converted into equivalent like fractions and then
subtraction is carried out
69
Example 23:
Simplify
3
(i)
(ii)
3
(iii)
1
1
9
20
4
37
15
1
2
3
1
Solutions :
3
(i)
(ii)
1
3
3
4
=
6
Alternatively, we can also simplify in the following way:
Step 1: Calculating two whole number part
It is : 3
1
4
3
4
1
6
Step 2: Calculating two fractional part
It is:
Combing the two parts we get
3
(iii)
1
1
2
3
1
1
5
4
1
2
6
29
30
29
30
1
=
(1)
6
1
Multiplication of a fraction by whole number.
Observe the following figures:
70
If
Represents 1
Then
Represents
And
Represents .
Please make equal divisions
It also represent
two times =
x2
2
Thus,
3
Similarly,
4
4
2
3
4 2
3
8
3
And so on
We may say that :
The product of a fraction and a whole number is also a fraction whose
numerator is the product of the numerator of the given fraction and
whole number and the denominator is the same as that of the given
fraction.
71
Fraction of a quantity
Look at collection given below
It has 6 balls. The dotted lines have divided the collection of 6 balls into three equal parts
and each small collection is one third of the whole collection. There are 2 balls in each small
collection. It means that one third of 6 balls is 2 balls, i.e,
6
6
Similarly
and
2
6
6
4
or
6
2
or,
6
4
or
6
6
2
5
50
2
5
In the same way,
50
=
=Rs 20
72
2
5
50
Similarly,
1
3
and
147
540
1
3
540
540
3
180
147
=
km = 84 km
Thus, When we are to find a fraction of a quantity we multiply the fraction with the
given quantity,
In other words, the word ‘of ‘means ‘multiply’
Division of a fraction by a whole number
Let us find
4
number,
Fraction
Observe the following figure:
In the figure (i) shaded portion represents . To divided
by 4, we divide the shaded
portion into 4 equal parts by drawing two diameters as shown in figure
(i).
Each of these 4 parts represents 1/2
(ii)
represents
4 and also each part in figure
.
73
4
So,
=
Similarly, it can be seen that
6
2
And
and so on.
Thus
A fraction
whole number (other then 0)
=
Example 24 :
10
(i)
20
(ii)
(iii)
4
2
12.
(i)
(ii)
13
(iii)
0
17
26
Solution
(i)
10
(ii)
12
8
2
74
20
17
26
(iii)
(iv)
13
13 17
26
221
26
17
2
13
1
8
2
We can also do as:
=
13
(v)
4
(vi)
0
2
1
4
9
9
4
8
4 9
4
1
=
9
0
Example 25: Calculate
(i)
(ii)
50
280
(ii)
242
132
(vi)
Solution
(i)
50
50
(ii)
280
(iii)
242
33
280
5
40
200
4 x 22
1
242
88
(iv)
132
Example 26 : Find: (i)
(iii)
14
1
6
132
5
1
3
1 x 22
1
18
99
75
22
Solutions :
(i)
5
(ii)
18
(iii)
14
(iv)
99
Simplification using BODMAS
You know bow to simplify expressions involving whole numbers using BODMAS. This is
now extended in simplifying fractions including whole numbers.
Example 27 : simplify
2
(i)
(iii)
1
2
(ii)
2
4
7
6
18
3
7
4
35
Solution:
2
(i)
6
=
=
=
4
76
=
4
=
=
(i)
2
48
12
6
1
=
-
=
=
=
3
Another way to simplify:
2
= 2 13
1
1
5
7
4
7
2
21
=
=
=
=
=
3
77
(ii)
2
4
4
3
7
18
=
4
35
18
=
= 44
3
7
4
35
= 44
15
35
4
35
= 44
11
35
=
44
15 4
35
44
11
35
44
44
44
Example 28 : simplify
(i)
3
1
2
(ii)
9
7
15
2
3
16
71
2
3
2
3
16
2
1
2
3
x9
27
4x2
Solution:(i)
3
1
2
2
=
16
1
2
x9
3
x9
=
=
15
=
15
=
15
78
=
=
(ii)
9
7
35
18
2
3
71
=
71
=
71
=
+
=
x2
=
x2
=
27
4x2
27
18
4x2
4x2
x2
9
9
Word Problems on fractions
Example 29: Amit travelled a distance of 13 km in the morning on his bike and a
distance of 8 km in the evening find the distance travelled by him altogether?
Solution: Total distance travelled =13 km+8 km
= 13
8
km
km
=
=
km
km
79
=
km
=22 14 km
Note: we can also proceed as
13
1
2
8
= 13
8
=21
5
4
=21
1
=22
3
4
1
2
13
1
2
3
4
3
4
8
2
4
21
3
4
1
4
1
4
1
Rita buys a ribbon of length 5 , meters. Sheela buys a ribbon of length
3
1
8 meters. Who has the lager ribbon and by how much? What is the total length of
7
ribbons?
Example 30:
Solution:Clearly the length
8
1
7
5
1
3
?
So, Mena’s ribbon is lager than Rita’s
Now we find:
8
1
7
5
=
=
=
=
So, Mena’s ribbon is longer by
of2
17
meters
2
80
5
Total longer of two ribbons =
=
=
meters
=
meters
=
meters
=13
You can also find 8
1
7
5
1
3
10
21
1
7
by writing as 8
1
3
5
13
10
21
Example 31:
3
1 m of cloth is needed for preparing a shirt. How much cloth will be required to prepare
4
6 such 6 shirts ?
3
Solution: Cloth for one shirt = 1 4 m
Cloth for 6 shirts =
=
7
m
4
x6 m
m
=
m
10
m
3
A box contains 1 kg. of sweats. It is equally distributed among 40
5
children of a class. How many sweets will each child get?
Example 32:
3
Solution: Weight of sweets in the box = 1 5kg = kg
No. of children =40
So, amount of sweets received by each child=
81
40
.
=
1
25
kg
Example 33: Geoff spent of his salary on food,
16
on children’s education and
other items. If his monthly salary is $5000, what is his monthly savings?
Solution: Monthly Salary of Geoff = $ 5000
$ 5000
Expenditure on food =
5000
=$
=$
= $ 2000
$ 5000
Expenditure on children’s education =
5000
=$
= $ 500
Expenditure on other items
$ 5000
=
=$
1
8
5000
= $ 625
Total expenditure = $ 2000
$ 500
$ 625
Total monthly saving of Geoff = $ 5000
$ 3125
$ 3125
Alternative method
Total expenditure (in fraction)
=
82
$ 1875
on
=
=
=
Savings = 1
5
8
8
8
Thus monthly savings
2
8
5
8
=
8 5
8
3
8
$ 5000
5000
=$
=$
=$ 1875
Ratio And Proportion
Ratio
A car costs Rs 200000 and a motorbike costs Rs 50000.
We may say that price of car is Rs 200000 - Rs 50000 i.e. Rs 150000 more than that of
motorbike. This is called comparison by subtraction.
Another way is that the car’s price is
times that of the motorbike
i.e. 4 times of the price of motorbike.
This way of comparison is known as comparison by division and is also known as a ratio.
Thus the ratio of price of car and price of motorbike =
It is also written as 4:1 and read as 4 to 1 or 4 ratio 1. the sign “:” is read as “is to”
Example 34: In a class there are 30 girls and 24 boys. What is the ratio of number of girls
to that of the boys? And what is the ratio of number of boys to that girls ?
Solution: Number of girls = 30
Number of boys = 24
So, the ratio of number of girls to number of boys
83
=30: 24
(i)
Next, the ratio of number of boys to the number of girls
= 24: 30
(ii)
You can see that 30: 24 is different from 24:30
Thus, ratio a : b different from b : a, if a
0. Ratio 30: 24, can also be written as
.
It means 30: 24 is the same as 5:4
In ratio 30:24, 30 is called the first term of antecedent of the ratio and 24 is called the second
term of consequent of the ratio.
Example 35:
(i)
Length of a room is 15 m and its breadth is 10 m. What is the ratio of length to
breadth of the room?
(ii)
Cost of a toffee is 75 cents and cost of a chocolate is $ 3 find the ratio of their prices.
(iii)
Length of a cylinder is 25 cm and its diameter is 30 mm what is the ratio of the length
and diameter of the cylinder?
(iv)
Weight of a box is 20 kg and its length is 60 cm. What is the ratio of the weight of the
box to the length of the box?
Solution:
(i)
Ratio of length to breadth = 15 : 10 or 5 :2
(ii)
Can we say that the ratio of the cost of toffee to the cost of chocolate is 75 : 3 i.e. 25:1 ?
Which means cost of toffee is more then is more cost of chocolate. But is it so?
No!
Here cost of toffee is in cents and that of chocolate in $.
84
To get true comparison, of two costs, they must be in the same unit either in cents or
dollars.
It the cost is in cents, then cost of chocolate is 300 cents and required ratio
= 75 : 300
3
=
12
In the cost are in dollars, then cost of toffee in Rs
:3
Then required ratio
=
x
3
12
Two quantities can be companied only if they are in the same units.
(iii)
Length of cylinder
25
Diameter 30
So, required ratio = 250:30
= 25:3
(iv)
Weight of box in kg = 20
Height in cm
250
= 60
Note that the two quantities ‘kg’ and ‘cm’ are not in the same unit.
Can you convert kg into cm or cm to kg?
It is not possible!!
So, in this case, we can not find the ratio.
85
Ratio and fraction
You have a aleady seen that a ratio can be written as a fraction and a fraction can also be
written as a ratio. For example, in the above case, the ratio 5:2 =
etc. Similarly,
2
3,
4
As
,75:300 =
9 etc.
While discussing fractions, you have seen that fractions like
Equivalent Fractions
Equivalent Ratios
25:1 =
, ,
are
In the same very, we can say that the ratio 2:3, 4:6, 6:9 etc are
is the simplest form of the fraction
,
etc in the same way,
2:3 is the simplest form of the ratio 4:6, 6:9 etc
Example 36: Write the fraction
Solution :
24: 36
in the form of a ratio in the simplest form.
2: 3
Example 37: Express the ratio 15:35 in the form of a fraction in the simplest form
Solution: 15:35 =
•
Proportion
Consider the following situation:
Cost of 1 liter of petrol =Rs. 60
Cost of 1 liter of petrol will be Rs 60
5
300
Cost of 12 liters of petrol will be Rs 60 x 12 = Rs 720
What is the ratio of quantity of petrol?
It is 5: 12 or
What in the ratio of their costs?
It is 300: 720 =
Thus, we obese that:
86
The ratio of quantities of petrol (5:12) = The ratio of their costs (300: 720)
i.e 5:12=300:720
Such an equality of two ratios is called proportion. Sometimes, the symbol “::” is
used to denote the equality of two ratio. Thus, for the present situation, we may write
5:12:: 300:720
Here four numbers are involved which are called the respective terms of the
proportion.
Extreme terms
or Extremes
Middle terms or
Means
Further note that in the proportion
5: 12 : : 300 : 720
Product of first and fourth terms = 5
Product of second and third terms
720
12
3600
300
3600
So, product of first and fourth terms = Product of second and third terms.
Thus,
Four numbers a, b c and d are said to form a proportion or are in proportion
if product of extremes = product of means (middle terms)
i.e. a x d = b x c
87
Example 38 : Find whither the numbers are in proportion
(i)
(ii)
(iii)
3, 6, 21, 42,
2,5,10,20
7,9,56,81
Solution
(i)
Product of extremes terms = 3 x 42 =126
Product of middle terms = 6 x 21 = 126
Since product of extreme terms = product of middle terms therefore, the numbers 3,
6, 21, 42 are in proportion.
(ii)
Product of extreme terms = 2 x 20 = 40
Product of middle terms = 5 x 10 = 50
Since 40 50, so the number 2, 5, 10, 20are not in proportion.
(iii)
Product of extreme terms = 7 x 81 = 567
Product of middle terms = 9 x 56 = 506
Since 567
506,
So, the numbers 7, 9, 56, 81 are not is proportion.
Example 39: Determine if the following ratios form a proportion
(i)
440m:2km and 55cm:3m
(ii)
250 ml:100 ml and Rs.75 : Rs. 30
Solution
(i)
440
: 2 km = 440 m : 2 x 1000 m
11
=
50
and 55 cm : 3 m = 55 cm : 3 x 100 cm
=55
=
3
55
3 100
11: 60
Since 11:50 11 : 60,
therefore, the given ratios do not form a proportion.
88
(i)
250 ml : 100 ml =
Rs 75: Rs 30 =
5: 2
5 2
Here, the ratio 250 ml: 100 ml and Rs 75: Rs 30 form a proportion.
Note. Unit used in the first ratio is ml while unit used in second ratio is Rs. So, we
may have different unit is different ratios but we can not have different unit in the
same ratio.
Example 40: A train covers a distance of 200 km speed of 50 km/ hr in 4 hours and
covers the same distance at 40 km /hr. in 5 hours. Do the number 50, 40, 4, 5 make a
proportion?
Solution: The numbers will be in proportion if 50 x 5 = 40 x 4 or 250 = 160, which is not
true
So, the numbers 50, 40, 4, 5 are not is a proportion.
Example 41: If two places are 8 cm apart on a map with a scale of 1: 40000. What is the
actual distance between them?
Solution: 1 cm on the map represents 4000 cm in reality So, 8 cm in the map represents
8
2
cm on the map represents
5
=
40000 cm in reality
=
4000
=
= 336000
=
=
3
89
Example 42:
A bullock cart travels 36 km in 4 hours and a car travels 120 km in 2 hours. Using the
relation
, find the ratio of their speeds.
speed =
Solution:
Speed of bullock cart =
Speed of car =
60
9
/
/
So, the ratio of speed of bullock cart and car = 9 :60 = 3 :20
Example 43: In example 42 find the ratio of distance travelled by the bullock cart and car
and also the ratio or the time taken by them. Are the two ratios forming a proportion?
Solution: Ratio of distance travelled by the bullock cart and car
= 36:120
= 3:10
Ratio of time taken by them (in hrs.) by them
=4:2
= 2:1
Since 3:10
2:1,
So, the two ratio do not form a proportion
Example 44: Nelson buys 6 note books for Rs. 90 and Jacob buys 25 notebooks for Rs 375.
Do the ratio of number of note looks and their costs form a proportion?
Solution: Ratio of the number of note books
= 6: 25 =
Ratio of cost of note books
= 90: 375 =
So, the two ratios are in proportion.
90
Example 45: To make 12 buns, Sarita needs:
(i)
240g flour
(ii)
60g margarine
(iii)
24g sugar
(iv)
75 ml milk
(v)
12g salt
To make 18 buns, Nitu needs
(i)
360g flour
(ii)
90g margarine
(iii)
30g sugar
(iv)
100 ml milk
(v)
16g salt
Find the ratio of:
(a)
Number of buns made by Sarita and Nitu
(b)
Quantity of flour used by both
(c)
Quantity of margarine used by both
(d)
Quantity of sugar used by both
(e)
Quantity of milk used by both
(f)
Quantity of salt used by both
Which two ratios form a proportion?
Solution:
(a)
12:18 = 2:3
(b)
240:360 = 2:3
91
(c)
60:90 = 2:3
(d)
24:30 = 4:5
(e)
75:100 = 3:4
(f)
12:16 = 3:4
We find that (a) and (b), (b) and (c), (a) and (c) form a proportion. Also, (e) and (f) form a
proportion. But (a) and (d), (b) and (d), (c) and (d), (d) and (e), (d) and (f) do not form a
proportion.
Example 46: Check whether the numbers 3, 9, 9, 27 in proportion?
Solution:
Product of extremes = 3 × 27 = 81
Product of means = 9 x 9 = 81
Clearly 81 = 81
Hence, the numbers 3, 9, 9, 27 are in proportion.
Note that in the proportion, 9 is repeated at:
2nd and 3rd places or
2nd and 3rd terms are the same.
3, 9, 9, 27
9 is repeated
In such cases, we say that 3, 9, 27 are in continued proportion or simply that 3, 9, 27 are in
proportion.
In such cases, it is understood that middle terms are the same.
Thus, by saying that a, b, c are in continued proportion, we
mean a:b = b:c or
or b2=ac
92
DECIMALS
Concept of a Decimal
Consider the fractions:
, , ,
,
,
,
,
,
,
,
Out of these fractions, the fractions
,
,
,
,
Are such whose denominators are 10, 100, 1000 etc. Such fractions whose
denominators are 10, 100, 1000,........ are called decimal fractions.
Some more examples of decimal fractions are
,
,
,
,
etc.
Recall that
means 7 parts taken out of 10 equal parts of a
whole.
We may call each part as one tenth
So,
means 7 tenths and
2
.
or 2
93
7
10
2 and 7 tenths
represents 25 parts out of 100 equal parts of a whole (See fig. above)
We may call each part as one hundredth
1
Now
Similarly,
1 and 19 hundredths
represents 21 parts out of 1000 equal parts of a whole
We may call each part as one thousand th
So,
.
means 21 thousandths.
We write the fraction
as 0.7 (read as zero point seven)
as 2.7 (read as two point seven)
as 0.25 (read as zero point two five or decimal two five)
as 1.19 (read as one point one nine) and so on...
The point (.) in each of the numbers 0.7, 2.7, 0.25, 1.19 etc. is called a decimal point.
94
And the number itself is called a decimal number or simply a decimal.
Like whole numbers we can also write decimal numbers on a place value chart as
given below.
Representing a decimal on a place value chart as given below
Decimal
Decimal Thousands Hundreds Tens Ones Decimal Tenths
Number (x1000)
(x100)
(x10) (x1)
•
x
2351.84
219.135
17.09
0.7
2.7
1.19
0.124
2
3
2
5
1
1
1
9
7
0
2
1
0
•
•
•
•
•
•
•
Hundred
x
8
1
0
7
7
1
1
4
3
9
9
2
In the above table:
Value at the hundreds (100) place is
1
i.e. 100
1000
10
Similarly value at the tens (10) place is
1
i.e. 10
x 1000
10
Proceeding in this way,
Value at the ones (1) place is
1
i.e. 1
x 10
10
Value at the tenths
i.e.
of the value at hundred (100) place
of the value at tens (10) place
of the value at ones (1) place
1
Value at hundredths
i.e.
place is
of the value at thousand (1000) place
place is
of the value at tenths
and so on.
95
place
Thousand
1
1000
5
4
Thus, the value at each place is of the value of place just left to it as was in
the case of place value chart for whole numbers.
Expanded from of a decimal number
Look at the face value and place value of each digit in the number 2351.84 give in the table
above
Face
value
2
3
5
1
8
Place
Place value
2
Thousands
Hundreds
Tens
Ones
Tenths
1
10
8
4
hundredths
4x
1000
3 100
5 10
1 1
2000
300
50
1
8
or 0.8
10
1
100
4
or 0.04
100
Adding all the place values, we get
2000 + 300 + 50 + 1 + 0.8 + 0.04 = 2351.84
Left hand side is called the expanded form of the decimal number 2351.84 on the right.
In the decimal number 2351.84, 2351 is called the integral part or whole number part.
and 84 is called the decimal part.
Two parts are separated by a dot (.) called decimal point.
Similarly expanded form of 219.135 is
200 + 10 + 9 + 0.1 + 0.03 +0.005
and expanded form of the decimal number 2059.101
0
is 2 × 1000 + 0 × 100 + 5 × 10 + 9 × 1 + 1 ×
= 2000 + 50 + 9 + 0.1 + 0.001
96
1
Example 47: Write the following numbers in the place value chart.
(i)
20.3
(ii) 3.4
(iii) 89.1
Solution,
Tens
Ones
.
Tenths
2
0
.
3
(i)
20.3
(ii)
3.4
3
.
4
(iii)
89.1 8
9
.
1
Example 48: Write the following as decimals
(i)
(ii)
Two ones and four tenths
Twenty and six tenths
Solution:
(i)
(ii)
Two ones and four tenths = 2.4
Twenty and six tenths = 20.6
Example 49:
Write as decimals
3 8 6 23 34
, , ,
,
10 10 10 100 1000
Solution:
0.3
0.8
0.6
0.23
0.034
97
Example 50:
Write as fractions
0.2, 0.5, 0.9, 0.67
Solution:
0.2
0.5
0.9
0.67
Example 51: Read and write the number name, first one is done for you.
(i)
(ii)
(iii)
(iv)
(v)
7.5
9.4
18.2
225.71
1087.31
Solution:
(i)
seven and five tenths
(ii)
Nine and 4 tenths
(iii)
Eighteen and two tenths
(iv)
Two hundred twenty five and seventy one hundreds
(iii)
One thousand eighty seven and thirty one hundreds
Example 52: Write in decimal form
(i)
Eight tenths
(ii)
Eighty two and three tenths
(iii)
Two hundred twenty seven and seven tenths
(iv)
One hundred and twenty two hundredths
98
Solution:
(i) 0.8
(ii) 82.3
(iii) 227.7
(iv) 100.22
Example 53: Write the following decimal numbers in expanded form.
(i) 25.197
(ii) 0.401
Solution:
9
(i)
25.197 = 2 × 10 + 5 × 1 + 1 ×
(ii)
= 20 + 5 + 0.1 + 0.09 + 0.007
1
1
0.401 = 4
0
1
10
100
= 0.4 + 0.001
7
1
1000
Example 54: Write the decimal number of each of the following expansion:
(i)
3
1000
(ii)
1
10
2
7
100
1
0
5
1
10
10
1
1
1 1 8
10
1
1
5
100
1000
5
1
100
Solution:
(i)
Required decimal number
= 3000 + 200 + 50 + 1 + 0.8 + 0.05
= 3251.85
(ii)
Required decimal number
= 10 + 7 + 0 + 0.01 + 0.005
= 17.015
Representing decimals on the number line
We already discussed representation of whole numbers and fractions on a number line.
Recall that for representing a fraction say
on a number line we divided the unit distance 0
to 1 into 5 equal parts and the second point of division represented the fraction .
As decimals are also fractions (with denominators 10, 100, 1000 etc.), we can also represent
decimals on the number line.
For example to represent a decimal say 0.4 on the number line, divide unit 0 to 1 into 10
equal parts and the 4th point of division represents
99
i.e. 0.4 as shown below.
To represeent say 1.2 on the num
mber line, divide
d
each
h unit into 10
1 equal paarts and maark the
twelfth po
oint of divission startin
ng from 0. This
T
point will
w represen
nt
i.e. 1.2.
vely, since 1.2
1 = 1 + 0.2,
0 therefore, divide the
t unit disstance betw
ween 1 and 2 into
Alternativ
ten equal parts and mark
m
the seecond pointt of division
n.
This pointt will represent 1 + 0.22 i.e. 1.2 on the number line.
Fraction
ns as Decim
mals
We have already
a
seeen how a frraction with
h denominaator 10, 1000, 1000 etc. can be con
nverted
to decimaal.
Let us con
nvert the fra
action say into decim
mal.
Thu
us,
Sim
milarly,
and
d
an
nd so on
100
Decimals as Fractions
We can also convert decimals to fractions.
For example, 1.2
1
2
10
10
10
2
10
12
10
6
5
We can directly write
1.2
(Remove the decimal point. In the denominator place one
zero after 1 as the number of digits after the decimal
point in 1.2 is one)
=
Similarly 6.35
6
Here, also we can directly write
6.35
+
Here also we can directly write
(Remove the decimal point. In the denominator place one
zero after 1 as the number of digits after the decimal
point in 1.2 is one)
6.35 =
.
In the same way
.
11.309 =
Example 55: Convert the following fractions to decimals
(i)
(ii)
101
Solution:
0.6
(i)
0.95
(ii)
Note: To convert a fraction into a decimal, convert the denominator as 10, 100, 1000
etc.
Example 56: Convert the following decimals to fractions
(i)
9.25
(ii) 628.375
Solution
(i)
9.25
(ii)
628.375
.
Comparing Decimal Numbers
Like whole numbers, we can also compare decimal numbers using the place value concept.
For example to compare 0.05 and 0.1, we first look at the whole number parts of the two
numbers and compare them as whole number.
In 0.05, whole number part = 0
In 0.1, whole number part = 0
So, we then compare, the tenths digit of the two numbers.
In 0.05, the tenths digit is 0
In 0.1 the tenths digit is 1
As 1 > 0, so, 0.1 > 0.05
Consider one more example; let us compare 12.959 and 12.961
In these two numbers, again compare whole number parts. Here is each case it is 12.
So, we compare tenths digit of the two numbers. Again, they are the same each being 9.
102
Then, we compare hundredths digit in the two numbers.
In 12.959, hundredths digit is 5
In 12.961, hundredths digit is 6
Since 6 > 5, so 12.961 > 12.959
In the same way, it the hundredths digits are the same, then compare the thousandths
digits and so on.
Example 57: Compare the numbers 0.4 and 0.09.
Solution:
Step 1: Compare whole number parts. They are equal as each in 0.
Step 2: Compare tenths digit of both the number.
In 0.4, tenths digit is 4
In 0.09, tenths digit is 0
Since 0 < 4, therefore, 0.09 < 0.4
Example 58: Compare 18.191 and 9.999
Solution:
Step 1: Compare whole number parts
In 18.191, it is 18
In 9.999, it is 9
As 18 > 9, therefore 18.191 > 9.999
Example 59: Compare the numbers 4.8307 and 4.8316.
Solution:
Step 1: Compare whole number parts.
It is 4 in each case.
Step 2: Compare the tenths digits.
It is 8 in each case.
103
Step 3: Compare the hundredths digits.
It is 3 in each case.
Step 4: Compare the thousandths digits.
In 4.8307, it is 0
In 4.8316, it is 1
As 0 < 1, so 4.8307 < 4.8316
Example 60: Arrange in ascending order.
6.78, 7.77, 9.43, 3.78, 7.79
Solution:
Compare who number parts
3<6<7<9
So, 3.78 < 6.78 < 7.77 or 7.79 < 9.43
Now let us compare 7.77 and 7.79
Here again, whole number parts are same,
tenths digit of both numbers are also same,
so, we compare the hundredths digits.
In 7.79, hundredths digit is 9
In 7.77, hundredths digit is 7
Since, 9 > 7, so 7.79 > 7.77
i.e. 7.77 < 7.79
Hence, required ascending order is
3.78, 6.78, 7.77, 7.79, 9.43
104
Operations on Decimals
We have already learnt about decimals as fraction with denominators 10, 100, 1000 and so
on. We have also represented decimals using a place value chart using this place value
chart. We can perform different basic operation on decimals as we do for whole numbers.
We shall explain it through examples.
Addition:
Example 61: Find the following sum
(i)
25.38 + 31.51
(ii)
31.47 + 5.21
Solution:
(i)
We write the decimals as follows:
25 . 38 (Decimal points in the same column,
+ 31 . 51 tenth digits in the same column, etc.)
Sum 56 . 89
(ii)
We write the decimals as follows:
31.47 (Decimal points in the same column,
+ 05.21 tenth digits in the same column,
Sum 36.68 there is no tens digit in the number 5.21)
Here, the two decimals have been added without carrying over as it is done in whole
numbers. Decimal can be added with carrying over in the same way as in the case of whole
numbers.
Example 62: Find the following sum:
(i)
47.38 + 35.24
(ii)
659.45 + 42.35
Solution:
(i)
We again write the decimals such that decimal points are in the same column and
digits of different places (tens, units, tenths, etc.) are in the same column as follows.
105
1
1
← Carry over
47.38
+35.24
Sum 8 2 . 6 2
(ii)
We write the decimals as follows:
11
659.
+0 4 2 .
Sum 7 0 1 .
1 ← Carry over
45
(There is no digit in the hundreds
35
place in the second number.)
80
Example 63: Find the following sum.
(i)
27.824 + 261.16
(ii)
733.5 + 24.369
Solution:
(i)
In the first decimal, there are three digits 8, 2 and 4 after the decimal point and two
digits
1 and 6 after the decimal point.
To make the number of digits equal in both the decimal, we put a `0’ in the end in
the second decimal.
Now, we write these decimals as follows:
27.824
+261.160 ← (0 added)
Sum 288.984
(ii)
Here also, we add two `0’ in the first decimal to make the number of digits in the two
decimals after the decimal point the same.
Now, we write as follows:
733.500 ← (Two `0’ have been added)
+ 24.369
Sum 757.869
Example 64: Find the following sum.
(i)
246.59 + 29.347
(ii)
578.348 + 28.7
106
Solution: (i) We write as follows:
1 1
← Carry over
2 4 6 . 5 9 0 ← (One `0’ as added)
+ 29.347
Sum 2 7 5 . 9 3 7
(ii)
We write as follows:
1 1 1 ← Carry
578.348
+ 2 8 . 7 0 0 ← (Two `0’ are added)
Sum 6 0 7 . 0 4 8
We can add three or more decimals in the same manner.
Example 65: Find 22.17 + 242.2 + 1.289 + 87.4315
Solution: We write the decimals as follows:
111 11
022.1700
+242.2000
+001.2890
+087.4315
Sum 353.0905
← Carry
← (Two `0’ have been added)
← (Three `0’ have been added)
← (One `0’ has been added)
Subtraction:
Example 66: Find the following difference:
(i)
(ii)
56.89 – 25.38
36.68 – 5.21
Solution:
(i)
We write the decimals as follows:
56.89 (Decimal points are in the same column)
– 25.38
Difference→31.51
(ii)
We write the decimals as follows:
36.68
– 05.21
Difference→31.47
107
Comparing (i) and (ii) of example 61 with (i) and (ii) of example 66, can you see that
subtraction is the inverse operation of addition in the case of decimals also?
In the above example, we were able to subtract one decimal from the other without any
borrowing. But due to the place value chart, we can perform the subtraction with
borrowing, if necessary, as shown in the following examples.
Example: Find the following difference.
(i)
(ii)
93.73 – 53.34
601.85 – 45.36
Solution: (i) We write as follows:
6
9 3 . 7 13
–53.3 4
← Borrow
(1 has been borrowed from 7 to make it 6 and 3 as 13)
(Note: 1 tenth = 10 hundredths)
Difference→ 4 0 . 3 9
(ii)
We write as follows:
5 9
7
← Borrow
1
1
6 0 1 . 8 5 (1 hundred = 10 tens, 1 ten = 10 units,
– 0 4 5 . 3 6 1 tenth = 10 hundredths)
Difference→ 5 5 6 . 4 9
Example 67: Find the following difference:
(i)
(ii)
388.96 – 29.824
659.359 – 9.78
Solution: (i) There are two digits after the decimal point in the first decimal and there are
three digits in the decimal part of the second decimal.
So, we add one 0 to the first decimal and write as follows:
(i)
7
5
← Borrow
1
1
(One 0 added)
3 8 8.9 6 0
–0 2 9.8 2 4
(One hundredth = 10 thousandth, one ten = 10 units)
Difference→ 3 5 9 . 1 3 6
108
(ii)
4 18 12
← Borrow
1
659.3 5 9
–009.7 8 0
(One 0 added)
Difference→ 6 4 9 . 5 7 9
Mixed Operations
If we have an expression involving both the operations addition and subtraction, we
combine the decimals involving + sign and – sign separately and then simplify.
Example 68: Simplify:
(i)
(ii)
24.327 – 4.003 – 12.389 + 112.309
228.37 – 29.516 + 12.059 – 1.09
Solution:
(i)
24.327 – 4.003
– 12.389 + 112.309
= 24.327 + 112.309 – 4.003 – 12.389
=(24.327 + 112.309) – (4.003 + 12.389)
I
II
Note that we have used brackets in the same way as we did in the case of whole numbers.
Now, we simplify the two brackets as follows.
I
II
1
1
24. 3 2 7
14.003
+12.389
+112. 3 0 9
136. 6 3 6
26.392
Now, we subtract II from I (why)?
5
1 3 6 . 6 13 6
–0 2 6 . 3 9 2
110.2 44
(ii)
228.37 – 29.516 + 12.059 – 1.09
= 228.37 + 12.059 – 29.516 – 1.09
= (228.37 + 12.059) – (29.516 – 1.09)
I
II
I
II
1 1
1 1
109
228.370
(One 0 added)
29.516
+012.059
+01.090
240.429
30.606
Now, we subtract II from I as follows:
39
2 4 0 . 14 2 9
–0 3 0 . 6 0 6
209. 823
(One 0 added)
Word Problems
We shall explain the process of solving word problems on addition and subtraction of
decimals through some examples.
Example 69: A contractor collected 143.75 liters of milk from his first milk vendor, 138.08
liters from the second vendor and 165.5 liters of milk from the third vendor. How much
milk was collected by him?
Solution: Total collection = (143.75 + 138.08 + 165.5) liters
= (143.75 + 138.08 + 165.50) liters
Now, we add as follows:
111 1
143. 75
+ 138.08
+ 165.50
447.33
Thus, total milk collected = 447.33 liters, which in terms of liters and milliliters can be
written as 447 liters 330 milliliters.
(Recall that 1 liter = 1000 milliliters)
Example 70: Sum of two numbers is 27.57. If one number is 14.38, find the other number.
Solution: Two other number = 27.57 – 14.38
Thus, we have:
4
110
27. 5 17
– 14. 3 8
13. 1 9
So, the required number is 13.19.
Example 71: In a school fete, the collections at four counters were £585.75, £410, £375.50
and £333.25. Find the total collection of the four counters. Also find the difference between
the highest and lowest collection.
Solution: Total collection = £585.75 + £410 + £ 375.50 + £333.25
We add them as follows:
211 1
585.75
410.00
(Two 0 are added)
375.50
333.25
1704.50
Thus, total collection was £1704.50
Now, highest collection = £585.75
Lowest collection = £333.25
We find their difference as follows:
585.75
-333.25
252.50
Thus, required difference is £252.50
Example 72: From a barrel containing 453.76 kg of rice, a cook used 28.985 kg of rice in
the first three days of the week. How much rice left in the barrel?
Solution: Rice left in barrel = 453.76 kg – 28.985 kg
111
We find it as follows:
12
16 15
4 4 5 3 . 7 6 10 kg
(One 0 is added)
0 2 8 . 9 8 5 kg
4 2 4 . 7 7 5 kg
Thus, rice left in the barrel is 424.775 kg.
Example 73: Kanta purchased 3m25cm cloth for her shirt and 2m5cm for her trouser. How
much cloth was purchased by her in all?
Solution: Cloth for shirt = 3m25cm, which can be expressed as 3.25 m (Recall 1m =100
cm)
Cloth for trouser = 2m5cm = 2.05 m (Recall 1m = 100 cm)
So, total cloth purchased = 3.25m + 2.05m
Now, we add as follows:
1
3.25 m
+ 2.05 m
5.30 m
Thus, total cloth purchased by Kanta is 5.30m or 5m30cm.
Multiplication and Division by a Whole Number
We know that decimal fractions are fractions with denominator 10, 100, 1000 etc. We have
also discussed the multiplication and division of fraction by whole numbers in this unit.
Therefore, we can multiply and divide decimals by whole numbers by treating them as
decimal fractions. We explain the process through examples.
Multiplication
Example 74: Find the following products.
(i) 3.75 × 10 (ii) 22.109 × 100
(iii) 325.1 × 1000
112
Solution:
(i)
3.75 × 10
10 [3.75 is written as a decimal fraction
=
=
]
[Recall how we multiply a fraction by a whole number]
=
= 37.50
= 37.5 [Note that 37.50 and 37.5 are the same (why?). Also, note that decimal point
has been shifted to the right by one place]
(ii)
22.109 × 100
100 [22.109 is written as a decimal fraction]
=
=
= 2210.900
= 2210.9 [Note that decimal point has been shifted to the right by two places]
(iii)
325.1 × 100
1000
=
=
3251000
= 325100.0 [Note that decimal point has been shifted to the right by three places]
= 325100
Observe that on multiplying a decimal by 10, 100, 1000 etc. the decimal point is respectively
shifted to the right by one place, two places, three places, etc. in the product.
Example 75: Find the following products:
(i)
23.6 × 3
(ii)
4.375 × 7
(iii)
147.35 × 14
113
Solution:
(i)
23.6 × 3
=
3 [23.6 is written as a decimal fraction]
=
[Recall how a fraction is multiplied by a whole number]
=
708
= 70.8 [Decimal fraction is written as a decimal]
(ii)
4.375 × 7
7
=
=
=
1000
[4.375 is written as decimal fraction]
30625
= 30.625 [Decimal fraction written as a decimal]
(iii)
147.35 × 14
14 [Decimal number is written as a decimal fraction]
=
=
=
100
206290
= 2062.90
= 2062.9 [Decimal fraction written as a decimal]
Observe that in the multiplication of a decimal by a whole number, the decimal is written
in decimal fraction and finally in the product the decimal fraction is converted in a decimal.
114
Division
Example 76: Find the following quotients:
(i)
37.59 ÷ 10
(ii)
442.921 ÷ 100
(iii)
3895.23 ÷ 1000
Solution:
(i)
37.59 ÷ 10
=
10 [Decimal is written as a decimal fraction]
=
[Recall how a fraction is divided by a whole number]
=
= 3.759 [Note that decimal print has been shifted to the left by one place]
(ii)
442.921 ÷ 100
100 [Decimal is written as a decimal fraction]
=
=
=
442921
= 4.42921 [Note that decimal point has been shifted to the left by two places]
(iii)
3895.23 ÷ 100
1000 [Decimal is written as a decimal fraction]
=
=
=
100
100000
= 3.89523 [Note that decimal point has been shifted to the left by three places]
Observe that on dividing a decimal by 10, 100, 1000, etc. the decimal point is shifted to the
left respectively by one place, two places, three places, etc. in the quotient.
115
Example 77: Find the following quotients.
(i)
46.5 ÷ 4
(ii) 112.56 ÷ 3
(iii)
28.305 ÷ 15
Solution:
(i)
46.5 ÷ 4
4 [Decimal is written as a decimal fraction]
=
=
[Recall how a fraction is divided by a whole number]
=
40
[465 is divided by 40 as whole numbers, using long division]
= 11.625
Alternative solution: After obtaining
=
465
, we proceed as follows:
40
=
[Numerator and denominator are multiplied and divided by
25 to obtain 1000in denominator]
=
.625
= 11.625
(ii)
112.56 ÷ 3
3
=
=
=
=
=
100
11256
[ or divide 11256 by 300 by long division]
[Using by division]
= 37.52
116
Alternative solution: After obtaining
11256
, we can proceed as follows:
=
[Numerator and denominator have been divided by 3 to obtain 100 in the denominator]
=
=
100
= 37.52
(iii)
28.305 ÷ 15
=
28305
15
=
= 28305
Alternative solution: After obtaining
=
=
, we can proceed as follows:
15000
28305
[Numerator and denominator have been divided by 15 to obtain 1000
in the denominators]
=
= 1.887
Word Problems
Example 78: Weight of one watermelon is 5 kg 215g. Find the weight of five such
watermelons.
Solution: Weight of one watermelon = 5 kg 215g
= 5.215 kg
So, weight of 5 such watermelons:
= 5.215 kg × 5
117
= 5.215 × 5 kg
5 kg
=
=
kg
= 26.075 kg
= 26 kg 75g
Example 79: A train travels a distance of 375 km 15m in 3 hours. Find the distance
travelled by the train in one hour.
Solution: Distance travelled in 3 hours:
= 375 km 15m
= 375.015 km
So, distance travelled by train in 1 hour
= 375.015 km ÷ 3
= 375.015 ÷ 3 km
=
=
=
=
=
1000
÷ 3 km
375015
375015
km
km
km
3000
1000
km
= 125.005 km
= 125 km 5m
Example 80: Cost of one book is $29.57. Find the cost of 3 such books.
Solution: Cost of 1 book = $29.57
118
So, cost of 3 books = $29.57 × 3
3
=$
=$
= $88.71
Example 81: 3 kg 275g of sweets were distributed among 25 students of class. Find the
sweets received by each student.
Solution:
Total sweets = 3 kg 275g
= 3.275 kg
Number of students = 25
So, sweets received by one student:
= 3.275 kg ÷ 25
= 3.275 ÷ 25 kg
=
25 kg
=
kg
kg
=
=
=
25000
1000
kg
kg
= 0.131 kg or 131g
119
BODMAS in Decimals
Simplification of expressions involving decimals using BODMAS can be done in the same
way as was done in whole numbers
Example 82: Simplify the following expressions:
(i)
2.5 × 7 – 3.5 ÷ 10 × 6 – (3.75 + 1.15)
(ii)
3.75 × 1000 + 2.85 × 8 + (2.6 ÷ 2 × 10 – 5.2)
Solution:
(i)
2.5 × 7 – 3.5 ÷ 10 × 6 – (3.75 + 1.15)
=
0.35
6
4.90
4.90
=
= 17.5
4.90
= 17.5 – 2.10 – 4.90
= 17.5 – (2.10 + 4.90)
= 17.5 – (7.00)
= 17.5 – 7.0
= 10.5
(ii)
3.75 × 1000 + 2.85 × 8 + (2.6 ÷ 2 × 10 – 5.2)
= 3750
285 8
100
= 3750
2280
100
= 3750
22.80
= 3750
22.80
26
10
2
26
10 2
5.2
5.2
26 10
5.2
20
260
20
10
5.2
= 3750 + 22.80 + (13 – 5.2)
120
= 3750 + 22.80 + (13.0 – 5.2)
= 3772.80 + 7.8
= 3772.8 + 7.8
= 3780.6
Estimation
Recall that there are a number of situations in which we do not require to know the exact
quantity but need to know only reasonable guess or estimate about it. For example, while
presenting a budget the Finance Minister gives some budget estimates of amount of money
to be spent on the Education Programmes of the country. We have already discussed some
methods of estimation for whole numbers in earlier units. There we have discussed the
process of estimating the outcomes of the operations on whole numbers, using the
technique of rounding off numbers. The same process can be applied to estimate the
outcomes on operations on decimals. The main purpose of such estimation is to check the
reasonableness of the answers obtained. The rules of rounding off a decimal to a certain
place are the same as for whole numbers.
For example, we want to round off 465.275 to the nearest tenth place (say), then we look for
the digit in the `hundredth place’. It is 7 here (it is more than 5). So, we shall round off
465.275 to 465.3. (Note that tenth place digit 2 has been increased to 3).
Suppose, we want to round off 465.275 to the nearest hundredth place, then we shall look
for the next place, to thousandth place. Here the digit is 5.
So, we round off 465.275 to the nearest hundredth place as 465.28 (Note that digit 7 has
been increased to 8).
Suppose we want to round off 26.895472 say fifth decimal place.
For this, we look for the sixth decimal place. Here the digit is 2.
So, 26.895472 shall be rounded off to the fifth decimal place as 26.89547 (Note that the digit
7 has not been changed because 2 < 5)
Suppose we want to round off 26.895472 to the nearest fourth decimal place, than we shall
look for the next place, namely the fifth decimal place. Here the digit is 7.
So, 26.895472 is rounded off to the nearest fourth place (i.e. ten thousandths place) as
26.8955. (Note that 4 has been increased to 5 as next digit 7 > 5).
Suppose, we want to round off 26.895472 to the nearest thousandth place. Now, we should
look for the ten thousandth place. Here, the digit is 4, which is less than 5.
121
So, 26.895472 is rounded off to the nearest thousandth place as 26.895. (Note that 5 remains
the same here as 4 < 5).
Suppose, we want to round off 26.895472 to the nearest hundredth place. We shall look for
the next place, i.e. thousandth place. Here, the digit is 5.
So, the digit at the hundredth place (which is 9 here) is to be increased by 1. It will become
9+1=10. Thus, in 26.895472, digit 9 will become 0 and digit 8 will become 8+1=9.
Thus, 26.895472 is round off to the nearest hundredth place as 26.90.
Clearly, if we want to round off 26.895472 to the nearest tenth place, it will be 26.9.
Now, we shall explain the estimation of outcomes of the operations on decimals through
examples.
Example 83: Estimate the outcomes of the following.
(i) 12.27 + 3.42
(ii) 583.21 + 2.534 (iii) 9.005 + 25.655 + 2.509
Solution:
(i)
(ii)
(iii)
In this case, we may round off 12.27 and 3.42 each to the nearest tenth
place and than estimate the operation (+). We have:
12.27 = 12.3 and
3.42 = 3.4
So, estimation of (12.27 + 3.42) = 12.3 + 3.4 = 15.7
Here, we may round off 583.21 and 2.534 each to the nearest hundredth place.
So, we have:
583.21 + 2.534 = 583.21 + 2.53 = 585.74
Here, we may round off 9.005, 25.655 and 2.509 each to the nearest hundredth place.
So, we have:
9.005 + 25.655 + 2.509 = 9.01 + 25.66 + 2.51 = 37.18
Recall that there is no hard and fast rule of rounding off different numbers involved
in the operations, regarding to the nearest places, they must be rounded off.
For example, we may round off each of the decimals 9.005, 25.655 and 2.509 to the
nearest tenth place.
Thus, we have the following estimation:
9.005 + 25.655 + 2.509 = 9.0 + 25.7 + 2.5 = 37.2
122
Note that we may also round off 9.005 and 2.509 to the nearest hundredth place and
25.655 to nearest tenth place and obtain the following estimation:
9.005 + 25.655 + 2.509 = 9.01 + 25.7 + 2.51 = 9.01 + 25.70 + 2.51 = 37.22
Example 84: Estimate each of the following difference.
(i)
14.372 – 4.005
(ii) 36.38 – 17.846
(iii) 122.50 – 31.2395
Solution:
(i)
We may round off 14.372 and 4.005 each to the nearest hundredth place and obtain
the following estimation:
= 14.372 – 4.005 = 14.37 – 4.01 = 10.36
We may also round off each of these decimals to the nearest tenth place and obtain
the estimation as follows:
= 14.372 – 4.005 = 14.4 – 4.0 = 10.4
(ii)
We may round off 17.846 to the nearest hundredth place, without any change in first
decimal 36.38.
Thus, we obtain the estimation as:
= 36.38 – 17.846 = 36.38 – 17.85 = 18.53
We may also round off each decimal to the tenth place and obtain the estimation:
= 36.38 – 17.846 = 36.4 – 17.8 = 18.6
(iii)
We may round off 31.2395 to the nearest hundredth place and keep other decimal
122.50 as it is. Thus, we have the following estimation:
= 122.50 – 31.2395 = 122.50 – 31.24 = 91.26
In this case, we may also round off each decimal to the tenth place and obtain the
estimation as:
= 122.50 – 31.2395 = 122.5 – 31.2 = 91.3
Significant Figures
We can write any number using only ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 under the
base ten system of numeration. For example, we have numbers like:
435860709
273201008
123
509381092, and so on,
Using each of the digits of the above numbers, we can read the numbers with some
meaning in each case.
Now, let us consider the following numbers:
008507219
092168308
00070958621, and so on,
If we look at the number 008507219, we note that there is no difference between
number 8507219 and 008507219.
Thus, it can be said the two zeros (00) preceding 8507219 have no meaning.
The other way of expressing the zero fact is that the two digits `00’ before 8507219
have no significant, i.e. these are not significant digits for two numbers. Similarly, in
092168308, the digit `0’ preceding 92168308 is not significant and in 0007095862, the
digits `000’ preceding 7095862 are not significant.
In general, it can be said that all the digits 0 written before a number (be a natural
number) are not significant.
It shall not be mistaken that all the digits 0 used in a number are not significant. In
fact, in any number, all 0s (except those written before a number) are significant.
Thus, in 921128009300, all the four `0s’ are significant. Thus, in a number:
•
•
•
Digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 are always significant.
Digits 0 occurring between two significant digits (1 to 9) and also on the right of the
number are all significant.
Digits 0 on the left of the number are not significant.
What can thought of about significant digits in a decimal number?
As regards the digits 1 to 9 is concerned, there is no change in the situation. However,
in the case of digit 0, the situation is a little bit reverse. Note that in decimals, 2.73,
2.730 and 2.7300 are considered as the same decimal.
Thus, 0 used at the end of the decimal have no meaning, i.e. they are not significant.
Thus, digits 0 used on the right of a decimal number are not significant, while the
digits 0 used on the left of a number (natural number) are not significant.
124
Thus, for a decimal number, we have the following rules for significant digits:
•
Digits 1 to 9 are always significant.
•
Digits 0 between two significant digits are always significant. For example, in
decimal number 2.31008095, all `0s’ are significant.
•
Digits 0s to the right of the decimal part and digits 0s to the left of the whole number
part of a decimal are not significant.
•
Digits 0s used only for spacing the decimal point (i.e. as place holders) is not
significant. For example, in the decimal 0.02507, one 0 to the left of the decimal point
is not significant. However, the remaining five digits 0, 2, 5, 0 and 7 after the decimal
point are all significant in this decimal.
If we consider the decimal 1.02507, then all the six digits 1, 0, 2, 5, 0 and 7 are
significant.
Example 85: Write all the significant digits in each of the following decimals:
(i)
345.098070
(ii)
450.108007
(iii)
0.001503
Solution:
(i)
The significant digits are: 3, 4, 5, 0, 9, 8, 0, 7
(ii)
The significant digits are: 4, 5, 0, 1, 0, 8, 0, 0, 7
(iii)
The significant digits are: 9, 0, 0, 1, 5, 0, 3
(iv)
The significant digits are: 1, 5, 0, 3
125
09.001503
(iv)
STUDENT’S WORKSHEET – 1
IMPORTANCE OF NUMBERS
WARM UP W1
Name of the student _____________________
1.
Date ______
What are natural numbers. How did our ancestors use them ?
____________________________________________________________________________
____________________________________________________________________________
2.
What are whole numbers? How are whole numbers different from natural numbers.
Give some examples of whole numbers?
____________________________________________________________________________
____________________________________________________________________________
3.
What are directed numbers? What was the need to introduce directed numbers. Write
a few lines to explain .
____________________________________________________________________________
____________________________________________________________________________
4.
What are the part of the whole numbers known as? Give some situations in life where
you use such numbers?
____________________________________________________________________________
____________________________________________________________________________
5.
What do the numbers like 32.9, $4.50, 162.5cm mean ? What are they known as?
____________________________________________________________________________
____________________________________________________________________________
126
6.
Divide the figure into equal number of parts (number of parts mentioned alongside)
a.
8 parts
b.
c.
3 parts
2 parts
d.
7 parts
SELF ASSESSMENT RUBRIC – WARM UP (W1)
Parameter
Recall different type of numbers
they have come across or they
have used.
Divide a quantity in equal parts
127
e.
9 part
STUDENT WORKSHEET 2
TANGRAMS
WARM UP W2
Name of the student ______________________
Date ______
The Tangram puzzle set given above ( literally "seven boards of skill") is a dissection puzzle
consisting of seven flat shapes, called tans, which are put together to form shapes. The
objective of the puzzle is to form a specific shape using all seven pieces, which may not
overlap.
Observe and answer the following questions:
1.
Total number of pieces
= ___________
2.
Number of 3-sided pieces
= ___________
3.
Number of 4-sided pieces
= ___________
4.
There are ___________ number of 3-sided pieces out of total ________pieces.
5.
There are ___________ number of 4-sided pieces out of total ________pieces.
128
Thus we can say that ________ out of total 7 pieces are 3-sided pieces.
In the same manner ________ out of total 7 pieces are 4-sided pieces.
We can say that
are 4-sided.
Recall how the number ½ , ¾ , ¼ etc are used by you and your family in day to day life.
Write five such incidences and discuss it with the class.
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
Recall how the number 0.5, 6.25, $1.5 etc are used by you and your family in day to day life.
Write five such incidences and discuss it with the class.
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
129
SELF ASSESSMENT RUBRIC – WARM UP (W2)
Parameter
Compare
quantities
using
concept of division
Represent a quantity as a part of
a whole
130
STUDENT’S WORKSHEET – 3
PRECONTENT P1
A QUANTITY AS A PART OF A WHOLE
Name of the student ______________________
Date ______
A student is given a chocolate to share it equally among two friends
1.
If the chocolate is to be divided into two friends equally, then how much does each
friend get?
2.
Represent the number of chocolates each friend gets as a part of the total number of
pieces in the given box.
3.
Now complete the statement: Each friend gets ________ of the chocolate.
4.
Now repeat the same activity by dividing the chocolate equally among three friends
131
5.
Now complete the statement: Each friend gets ________ of the chocolate.
6.
Can you find the part of chocolate that each friend gets, if there were 5 friends?
SELF ASSESSMENT RUBRIC
PRE CONTENT (P1)
Parameter
Comprehend given problem
situation
Represent quantities using
concept of division
Represent a quantity as a part of
a whole
132
STUD
DENT’S WORKS
SHEET – 3
W
WHAT
IS
S IT ANY
YWAYS
PRECO
ONTENT
T P2
Name of the studentt ________________________
133
Date ______
Do as directed (complete and discuss)
A)
Instructions for students:
(i)
Cut out figure 1
(ii)
Total number of triangles = _______________
(iii)
Color the figure as given below:
(vi)
How many green triangles are there in your figure? ______________
(v)
Green portion is _____________ of the total triangles.
Conclusion:
The green portion is half of the total triangles.
B)
Instructions:
Now cut out figure 2.
C)
(i)
Color the triangles in figure 2 in green.
(ii)
How many parts of the whole have you colored? _____________
(iii)
Remove one green triangle.
(iv)
What part of the figure has been removed? ____________
(v)
Numerator is __________ and Denominator is __________.
Instruction:
(i)
Cut out and color figure 3.
(ii)
Remove one green triangle.
(iii)
What part has been removed? __________
(iv)
State the numerator and denominator. _________________.
134
D)
E)
F)
Instruction:
(i)
Color figure 5 with Blue triangles.
(ii)
Remove one triangle.
(iii)
What part of the blue triangles are left? _______________
(iv)
Specify the numerator and the denominator. _________________
Instruction:
(i)
Color figure 6 with blue blocks.
(ii)
Remove one blue block.
(iii)
What part of the figure is covered by blue? ___________
Instruction:
(i)
Color half of the figure 7 with red blocks.
Extension:
(i)
Color 2
(ii)
Color 3
3
5
of figure 2 with blue triangles.
of figure 5 with brown triangles.
SELF ASSESSMENT RUBRIC – PRE CONTENT (P2)
135
Parameter
Is
able
to
follow
multiple
instructions for a question
Represent
quantities
using
concept of division
Represent a quantity as a part of
a whole
STUDENT’S WORKSHEET 5
CONTENT WORKSHEET C1
UNDERSTANDING OF REPRESENTING A FRACTION AND ITS
COMPONENTS
Name of the student ____________________
1.
Date ______
Complete the following table(for shaded parts):
Fraction
Numerator
136
Denominator
2.
Circle the denominator and box the numerator in each of the following:
4
4
a)
1
4
1
5
1
2
1
8
1
7
Do you observe something special in all these fractions? Is there any exception?
_______________________________________________________________
b)
Which of the above numbers is odd one out? Why?
_______________________________________________________________
3. Read the clues and give an example for each of the following:
(a) A fraction whose denominator is three times the numerator.______
(b) A fraction whose numerator is four less than the denominator.________
(c) A fraction which has no common factor in the numerator and denominator.
(d) A fraction whose denominator is 2 times the numerator.
4.
A pizza is sliced into 8 parts. One part was given to the child and 2 parts was taken by
the father. Draw a figure to show how much of the pizza was left.
5.
Approximately, what fraction of chocolate has been eaten?
6.
Observe the fractions and answer the
following questions-
1/4 ,
10/25,
5/7 ,
7/2 ,
3/11,
½,
4/7 ,
5/3,
8/14,
137
4 ,
6 ,
2/3 ,
3/9,
7/14,
12/34,
8/9,
Write
(a)
Unit fractions
__________________________________________________________________________
(b)
Proper fraction
__________________________________________________________________________
(c)
Improper fraction
__________________________________________________________________________
(d)
Write like fractions
__________________________________________________________________________
7. I bought ten chocolates. Each chocolate is divided into eight equal parts. After
distributing ½ of the chocolate to each of the 16 friends, find out:
(a) How many chocolates am I left with?
___________________________________________________________________________
(b) Write the number of chocolates left as a fraction of the total number of the
chocolates?
__________________________________________________________________________
8. An egg crate contains 12 egg in all. Two eggs have been removed.
(a) Write the number of eggs left as a fraction of the total number of eggs.
(b) Explain why the fraction is written as 1/6 in the figure and not 2/12.
(c)
9. If an apple
represents the number 1 then,
138
represents_________________
Now draw & show how can you represent 4 ½ apples?
10. Observe the figure & write the fraction for the shaded part.
139
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET 1
Parameter
Has knowledge of various types
of numbers.
Is able to relate the given
number to different kind of
numbers
Can
speak
creatively
about
numbers and their need and
usage
140
STUD
DENT’S WORKS
SHEET – 6
LOCA
ATING FRACTIO
F
ONS ON
N A NUM
MBER LIN
NE
CON
NTENT WORKS
W
SHEET C2
C
Name of th
he student ________________________
Date ______
_
Activity 1 – Locating fractions on
o a numbeer line
Look at
a your ruleer and take a guess to the
t location
n of the fracction ½ , 1 ½ , 2 ½ and
d so
on……
………………
…
Write and expreess what thought
t
prrocess/ straategy will you apply
y to locatee these
numbeers?
_________________
_____________________________________________________________________
_________________
_____________________________________________________________________
Answer th
he followin
ng question
ns on the baasis of videeo clip 4.
1
What did
d you lea
arn from vid
deo clip?
________________
______________________________________________________________________
________________
________________________________________________________________
2
For rep
presenting the
t fraction
n 4
a)
10
on th
he number line
l
Th
he portion between
b
0 and
a 1 is div
vided into _______
_
equ
ual parts.
b) Eaach part of the
t numberr representss _______________ fraction.
c)
3
Drraw a numb
ber line and
d mark the position off 4
10
on it.
Now lo
ook at the number
n
linee given belo
ow and ansswer the folllowing queestions.
141
If the whole line is divided into 10 equal parts, then explain:
1. Why does ‘A’ represent 1
10
(the first division out of 10)?
________________________________________________________________________
2. Why does the point ‘B’ represent 5
10
?
_______________________________________________________________________
3. What fractions do C, D and E represent?
__________________________________________________________________________
4.
Represent 4 7 on the following number line.
5. Write another fraction of your choice.
(a)
____________
To represent it on number line the portion between 0 and 1 is divided into
_______ parts.
(b)
(c)
Draw a number line and mark the fraction on it.
Explain the procedure in detail. (Refer to the previous question for some
ideas.)
______________________________________________________________
______________________________________________________________
______________________________________________________________
(d)
Now represent the following fractions on the number line:
2
5
, 3
8
, 1
7
, 1
9
, 5
9
142
143
ACTIVITY 2 –‘GEM POWER’FRACTIONS ARE NUMBERS
1
Do as directed
You will need a pack of gems to answer the questions. But before
you eat them!!
1. Count the number of gems of each colour in your pile .
Red: ___
Blue: ____
Orange:___ Yellow: ____Green: ____
Any other ____
2. Write down each color gems as a fraction of the total number of gems you had.
Red: ___
Blue: ____
Orange:___ Yellow: ____Green: ____
Any other ____
3. Draw a number line in the space provided:
4. Using a ruler, divide your number line into as many equal parts as the total number
of gems you have.
5. Starting from left to right, mark the first point as 0 and the last point as 1.
6. Explain why is the last point was marked as 1? Is there any other way you could
mark it as?
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
144
Mark the fractions representing different color gems on the number line using that
colour pen/pencil.
ACTIVITY 3 – LOCATING MORE
FRACTIONS ON A NUMBER LINE
1. Observe 11/4 on number line.
What do we observe? 11 = 2 ¾ (remember! Mixed fraction)
4
11
4
= 4
4
+ 4
4
+¾=1+1+¾
The numbers on the number line can be regrouped in groups of 4 every time.
Portion between each pair of consecutive integer on number line is divided into four
equal parts.
•
We see that 11
•
Portion between 3 and 4 is divided into 4 equal parts and 3rd part is 11 .
4
lies between 2 and 3.
4
2. Now use number line to mark following fractions:
15 ,
7
24 ,
5
16 ,
9
145
20 .
3
SELF ASSESSMENT RUBRIC –
CONTENT WORKSHEET C2
Parameter
146
Knowledge of parts of fraction.
Divide a unit distance into as
many
equal
parts
as
the
denominator.
Represent fractions on a number
line.
STUDENT’S WORKSHEET – 7
INTRODUCING DECIMALS
CONTENT WORKSHEET C3
Name of the student ______________________
Date ______
1.
Due to over use, the markings are faded and I can see markings only till 10.
2.
How much length is visible here?
3.
What portion of the visible length is the arrowed length?
_______________
147
_______________
a)
Represent it as a fraction of visible length.
_______________
b)
Is there any other way in which this information can be represented? __
c)
What can you say about this form of representation?
Note: A decimal is a special case of a fraction with denominator in powers of 10.
Examples:
•
The fraction 3
•
The fraction 54
•
The fraction 456
10
means 3 out of 10 which is also represented as 0.3
100
means 54 out of (10 x 10) or 5.4 out of 10 or 0.54
1000
means 456 out of (10 x 10 x 10) or 45.6 out of (10 x 10) or 4.56 out
of 10 or 0.456.
4.
Now, observe this image and answer the following;
The measuring
tape shown above is a meter
long.
a)
Write down the length of the measuring tape in centimeters.
b)
A toy was measured by the tape and found to be 32 cm long.
Express the length of the toy as a fraction of the total length of the tape._____
c)
5.
Express your answer to part (b) as a decimal. ______
Refer to online math dictionary and write what is meant by a decimal, decimal
notation and its component.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
6.
Complete the following table:
148
Fraction
1
10
0.1
5
0.5
10
7
100
9
4
1000
10
3
100
39
100
435
7.
Decimal
1000
If each shaded box is worth 1, then represent the shaded part using decimal.
Decimals
8.
Observe the figure of the decimal blocks below and complete the table given below
149
150
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C3
Parameter
Exhibit knowledge of fraction
and parts of fraction.
Represent
fraction
with
denominator in powers of 10
using concept of decimal.
Represent
quantities
using
decimal.
STUDENT’S WORKSHEET – 8
151
PLACING DECIMALS ON A NUMBER LINE
CONTENT WORKSHEET C4
Name of the student ______________________
Date ______
Activity 1 –‘Gem power’ - Only 10 gems
1.
Based on video clip 5, describe what did you understand?
____________________________________________________________________________
____________________________________________________________________________
2.
Now the gems activity (take any 10 gems for this activity)
I.
Count gems of different colors and write your answer here:
Red_______, green ______, yellow______, pink______,
other colors_____.
II.
Represent all color gem fractions and decimals.
Red_______, green ______, yellow______, pink______,
other colors_____.
III.
Now draw a number line and represent all your different color gems as
fractions on the number line.
IV.
Also write their corresponding decimal representation below the fraction
number
now on a number line. What is 3
in other words known as?
3.
Represent 3
4.
Draw a number line and represent 0.7 on the number line.
5.
If the line below represents a unit length, which is divided into 10 equal parts. The
10
10
line is marked with points labeled A - G. Write the decimal numbers for each of these
points.
A=_____,
B=______,
G=_____.
152
C=_____,
D=______,
E=_____,
F=______,
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C4
Parameter
Has knowledge of various types
of numbers.
Is able to relate the given
number to different kind of
numbers
Can
speak
creatively
about
numbers and their need and
usage
153
STUDENT’S WORKSHEET – 9
EQUIVALENT FRACTIONS
CONTENT WORKSHEET C 5
Name of the student ______________________
1.
The fractions 2
3
and 4
6
are represented diagrammatically.
2
What do you observe? Is 2
2.
Check for
1
Date ______
3
4
3
=4
6
6
? _______________________________
3
5 and 15
They are also called______________________________________.
3.
Write a general procedure for converting fractions to equivalent fractions.
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
4.
Find the missing terms in each of the following:
(a)
4/7 = __/21
(d)
2/9 = 18/__
(b)
40/13=80/__
(e)
24/18= 3/__
(c)
25/30= 5/__
154
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C5
Parameter
Represent fractions
diagrammatically
Understand and find equivalent
fractions to a given fraction
155
STUDENT’S WORKSHEET 10
CONVERTING FRACTIONS TO DECIMALS
CONTENT WORKSHEET C6
Name of the student ______________________
Date ______
1.
Represent the shaded portion as a fraction:
2.
Now divide the other two circles in equal portions and shade to represent equivalent
fractions
3.
Divide each bar into equal portions and shade in the bars to represent three equivalent
fractions:
If the rectangle is cut into 10 equal parts instead, shade and represent an equivalent
fraction for the above fractions.
156
4.
Why do you think we want to cut the shape into ten equal parts?
____________________________________________________________________________
____________________________________________________________________
5.
Is there another way to represent this fraction?
(Hint: 3
6.
5
= 10 = 100)
Discuss the procedure for converting a fraction to decimal and summarize with your
partner
Hint:
7.
Based on the video clip 8, write the procedure for the conversion of a fraction to a
decimal by an algorithm and summarize:
Hint:
157
8.
Now Try to convert following fractions to decimals:
a)
6
b)
7
25
10
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C6
Parameter
Represent
fractions
diagrammatically
Understand equivalent fractions
Convert fractions to decimals
158
STUDENT’S WORKSHEET 11
SIMPLIFICATION OF FRACTIONS
CONTENT WORKSHEET C7
Name of the student ______________________
Date ______
Observe the pattern and explain what you see/?
24
28
,
18
21
,
12
14
,
6
7
____________________________________________________________________________
____________________________________________________________________________
Now watch video clips 9 and then answer the following questions:
1.
Observe the picture and write down the simplification process. Also mark how many
slices were joined or merged together and show the arrows and the division process as
in the example
a) ______slices merged together
b) ______slices merged together
c) ______slices merged together
4. Find the HCF of the numerator and the denominator of each of the original fractions
given above. Explain how the knowledge of HCF would help in simplification.
159
________________________________________________________________________________
________________________________________________________________________________
_______________________________________________________________________________
5. Some of the following fractions are such that you CANNOT simplify them. Cross them
out. Why is that? Discuss. Simplify the ones you can
4 , 15
, 15 , 13 , 13 , 12 , 12
18
22
51
52
21
17
7
6. Write each of the following decimal as a fraction (denominator as a power of 10)
Decimal
Fraction
0.2
2
10
0.3
0.4
0.33
5.7
7. Write each decimal in the fraction form (simplify wherever possible)
0.27
0.01
0.55
0.95
0.82
0.17
0.48
0.25
0.02
1.05
160
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C7
Parameter
Represent fractions
diagrammatically
Knowledge of equivalent
fractions
Understanding of simplification
of fractions
161
STUDENT’S WORKSHEET – 12
INDEPENDENT PRACTICE
CONTENT WORKSHEET C8
Name of the student ______________________
Date ______
Activity 1: Independent Practice
1. a) Put a tick in the appropriate column and complete the table:
Proper fraction
1
4
Mixed fraction
3
3
11
7
2 1
6
Improper fraction
7
11
b) Explain what do you understand by fractions 4
3
, 1 1 and 2 1 ?
7
7
c) Represent each one of them diagrammatically
2. From the number line given below, identify n and express your answer as a fraction.
3.
Now, Place the following improper fractions on the number line (guided practice)
4. Observe the pattern in part a) and b) and choose the appropriate answers for all questions
in part c).
a.
162
b.
c. Circle the correct improper fraction for the parts that are shaded.
(i)
5
,
6
5
2 ,
6
1
3
(ii)
13
,
14
1
1 ,
6
1
2
(iii)
9
,
4
5
1 ,
4
2
1
1 ,
3
(iv)
4
,
6
1
4
4
3
5. Complete the following table:
Improper fraction to Mixed number
Mixed
fraction
12
16
11
10
7
2 1
7
13
9
5
7
2½
5
32
3
3
4¼
4
163
number
to
Improper
6.
Fill in the missing numbers to make equivalent fractions:
1 2
=
3
2 4
=
9
3 9
=
7
1 10
=
10
2 8 ? ?
= = =
7 ? 69 ?
10
=
2
5
3 3000
=
8
? ? 13
39
= =
=
? 8 104
?
7. Write each of the following fractions as an equivalent fraction with:
Denominator 24
1
2
1
3
3
4
5
12
3
4
6
7
4
5
Numerator 12
1
6
8. Some of the following equivalent fractions are correct but two of them are wrong.
Find the incorrect ones and explain why they are incorrect.
2 6
=
5 15
9.
2 4
=
3 9
3 6
=
7 14
4 12
=
9 27
7
77
=
10 100
Observe and complete the pattern
½, 2
2
4
3
5
, 3
4
, 4
, 8
6
10
6
,_____,_______,_____,______
, 6
, 12
9
, _____,______,______,_____
15
,_____,______,_______,______
Discuss with your partner how you would decide on the pattern.
10.
Draw a rectangle and cut it into 4 equal parts,
a)
Represent 3
4
by shading on the rectangle.
164
9 18
=
13 26
Now divide the rectangle into 16 equal parts.
b)
Represent by shading a fraction equivalent to the previous fraction. Place the two
rectangles one below the other.
c)
Explain, how you know by observation, that the two fractions are equal.
d)
How will you be able to show and explain fractions smaller and bigger than this
fraction?
11.
Convert 0.8 into a fraction. Represent 4
5
and 0.8 on the same number line. Explain
what you observe.
12.
Convert ( if required ) and then reduce the given fractions to lowest terms in the space
provided.( try to work with prime factors)
1)
0.5
2)
0.6
3)
32
4)
99
5)
96
6)
600
7)
525
8)
1.25
40
126
1125
132
800
165
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C8
Parameter
Has
knowledge
of
various
types of fraction.
Is able to relate the given
number to different kind of
fractions.
Can speak creatively about
numbers and their need and
usage
166
STUDENT’S WORKSHEET 13
PLACE VALUE CHART OF DECIMALS
CONTENT WORKSHEET C9
Name of the student ______________________
1.
2.
Date ______
Use the above place value table and write the following numbers in the expanded form.
a)
6402307
b)
703297
c)
109990
Based on the Place value chart above and the knowledge of expanded form, write the
expanded form of following (teacher to explain if required)
a)
54.2
b)
231.42
167
c)
5910.016
9.
Fill in the missing words and numbers on this chart
10.
a)
Place the digits of the number mentioned in the place value chart appropriately.
b) Write its number name
c)
Write each number in the expanded form.
Is there a special name for the last number in the series?
(Hint: You researched on this in unit1)
168
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C9
Parameter
169
Understands the place value
chart for decimals
Read and write decimals in
expanded form
STUDENT’S WORKSHEET 14
COMPARISON OF FRACTIONS / DECIMALS
CONTENT WORKSHEET C10
Activity1: Cut out one fractional part of each type
Students to cut out fractional parts 1, 1/2 , ¾ , 1/6 , 2/5 , 1/10 , 1/12
Now arrange them on your desk from the smallest to greatest to the smallest
170
Activity 1- Like and unlike fractions
171
Image1
Image 2
1. Observe the above images carefully and discuss:
In image 1: The heights of the two people were measured and we came to the conclusion
that the man is taller than the women. What do you think?
In image 2 it was stated that Tower T 4 is smaller than T 7 and Tower T 2 is taller than T
5. What can you say about this analysis?
Give an argument to justify your answer.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
2. Convert all the fractions given above with the same denominator.
(Hint - 5, 4, 8 all have common multiple 40. Remember!)
_______________________________________________________________________
_______________________________________________________________________
Verify
2/5=16/40
¼=10/40
3/8=15/40
3. Convert the following fractions to like fractions and arrange them in the ascending order.
1/5, 1/2, 4/5, 3/2, 12/10
4. Which of the following pairs of fraction are like fractions and which are unlike
fractions.
(2/5, 3/5),
(12/15, 3/5),
(½, 6/14),
ACTIVIT 3 - DECIMALS
Observe the following examples carefully:
a)
Write 1.4 as a fraction in two parts?
If 1.4 can be shown as
172
(4/5, ¾)
OR
1.4 = 1 and 0.4 = 10
10
+ 4
10
= 14
10
true! What kind of fraction is this?
b) 2.4 is made of 2 and .4 = 1 and 1 and .4
= 10
10
= 24
1
c)
+ 10
10
10
100
10 +
10
10
10
1
23.4 = 20 and 3 and .4 =
+ 4
100
4
10 +
30
3
10
10 +
4
= 0.4
10 =
234
10 (Why?)
.4
1.
Using the algebra tiles express diagrammatically the decimals 14.3, 21.7, 20.8.
2.
The decimal number 23.4 is read as twenty three point four.
Now complete the following sentences
1.3 is read as one point three four and so on……
1.4 is read as_______________________________________
109.09 is read as_________________________________________
173
25.002 is read as _______________________________________
3. Put the correct sign in the blank: (<,>or=)
a)
9.16________9.166
b)
2.034__________2.03
c)
2.16__________54/25
d)
12.61_________12.16
e)
2.54________2.45
f)
1.045_________1.405
g)
0.402__________0.420
h)
1/2_________0.5
i)
63.24_________163.24
j)
16.78_______1.678
k)
41.56__________14.56
l)
0.0045_______0.0405
m)
$4.85________$417/50
n)
$6and 5 cents_________$6 and 50 cents
o)
0.22__________ 12/48
p)
18.05_________1.805
q)
5.95___________5.59
r)
13/51__________0.3
s)
2.54_________2.6
t)
5/7__________7/10
u)
4/10__________0.34
Parameter
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C10
174
Is able to understand fraction
strip and use it creatively
Is able to understand like and
unlike fractions
Is able to compare decimals
STUDENT’S WORKSHEET 15
ADDITION AND SUBTRACTION OF FRACTIONS
CONTENT WORKSHEET C11
Name of the student ______________________
175
Date ______
ACTIVIT
TY 1 – DO
OMINO ACTIVITY
A
Y
Begin with
h the 'Start' domino an
nd then add
d dominoes so that the matching expressions
e
s are
equivalentt to form a continuous
c
loop.
(a)
1.
ACT
TIVITY 2 – SOME MO
ORE PROB
BLEMS
Add
d and then verify yourr answer ussing fraction
n strips.
(a)
½+½
(b)
(c)
(d)
(e)
1/6 + 1/
/6 +1/6
1/3 + ¼
1/3 + 1/
/5
1/5 + ¼.
¼
2.
Now
w try ½
+½
3.
ve the follow
wing :
Solv
using fraction strips and then
n without using
u
fractio
on strips.
a)
1/4 + 2/4
4
c)
4/15 + 6//15 - 3/15
b
b)
d)
½+
1/2
/2
1/9 + 2/3 +2/9
176
e)
4/5 + 7/10 + - ½
f)
7/15 + 1/3 – 1/5
4.
If Ali spends 3/5 of his allowance to buy candies, then spends 1/5 of the money for a
sandwich , how much portion of his allowance is spent in total?
5.
During Easter, my aunt put leftover cakes into the fridge. She noticed that the fridge
had 5/11 of a cake filled with cherries, 8/11 of a cake filled with blueberries, and 1/11
of a cake filled with peaches. How many left over cake did my aunt have in all?
Simplify your answer and write it as a proper fraction or as a whole or mixed number.
( assuming that all the cakes were of same size )
6.
Do as directed :-
Add the
fractions.
Add the whole
numbers.
2
a)
e)
7.
4
1
5
2
b)
3
1
8
f)2
1
2
2
7
8
c)1
5
6
3
2
5
g)5
2
5
2
3
4
Fill in the missing number.
1
6
______
4
7
________
2
3
1/5
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C11
177
d) 3
2
5
h) 5
1
1
4
7
10
3
1
6
STUDENT’S WORKSHEET 16
ADDITION AND SUBTRACTION OF DECIMALS
CONTENT WORKSHEET C12
Name of the student ______________________
Date ______
ACTIVITY1 : STAR BLOCK
Solve the given problems and use the given table for coloring
Parameter
Has
knowledge
of
various
types of fractions
Is able to
add and subtract
fractions
Can solve related problems on
fractions
178
If the answer lies between
0 and 10
11 and 99
100 and 999
Color the shape
red
Blue
yellow
ACTIVITY -2 : Do as directed.
1.
Louis has $4.50 and Ronald $3 ¼. How much money do they have together?
179
2.
What is the combined thickness of these five strips of ribbon 0.008, 0.125, 0.15,0.185,
0.005 cm?
3.
The ice cream parlor has been charging $3.70 for its jumbo banana split
4.
Because of inflation the prices were increased by 75 percent.
5.
a)
Write the increased price as decimal
b)
What is the new price.
Jhon is making a telescope for his science project. He joined two metal tubing, one
30.45 inches long
and other 12.75 inches long. How long would the telescope be
when the two tubes are joined?
6. For winter’s snowstorm, Mrs. Brown bought two coats for her children. One coat cost
$55.75 and the other cost $9.30.What was the difference in the cost of the two coats?
7. In the school café the cost of items are as followed.
Ryan ordered one of the each item
a)
How much money does Ryan have to pay
b)
He paid with a $10 note. How much money will he get back?
8. A piece of webbing is 7.6 m long. If two pieces each 2.3 m and 1.5 m long respectively
cut off, how much is left?
Pizza (small)
$5.50
Sandwich
$1.20
Juice
75 cents
Veg burger
$1 .60
180
SELF
ASSESSMENT RUBRIC
CONTENT WORKSHEET C12
Parameter
Has knowledge of decimal
Add and subtract decimals
Can solve related problems on
decimals
181
STUDENT’S WORKSHEET 17
EXTEND MULTIPLICATION AND DIVISION AS REPEATED
ADDITION AND SUBTRACTION.
CONTENT WORKSHEET C13
Name of the student ______________________
Date ______
Activity 1: Repeated addition and subtraction
1. Do as directed : Solve
______
a)
b)
Rewrite this addition problem as a multiplication problem.__________
Verify your answer by using fraction strips.
_____________
c)
Now frame this information as a multiplication fact and solve :
1/12 + 1/12 +1/12 +1/12+1/12+ 1/12 =1/12 X _____=________
2. What can you say about the multiplication of a decimal by whole number :
(a)
What is 0.4 + 0.4=_________?
(b)
Write this as a multiplication fact : _______
(c)
Now complete the statement- 0.4 X ______=______
(d) Now model the solution using decimal blocks and paste the tiles/draw a
diagram if possible
3.
Do as directed:
(a)
Explain what do you understand by 1 divided by
(b)
Represent this diagrammatically.
(c)
What does ½ divided into two mean?
(d)
Confirm this by superimposing the result on the ¼ fraction strip. Paste your
result.
(e)
Write the steps of the process you have just followed.
4. Now try to write the algorithm for the fraction division.
182
5. Can you do the same for division of decimal with a whole number. Explore with
decimal blocks and draw a diagram to show your findings.
Take help from this graphic:
(a)
1 ÷ 2 , in a pile of 10 slabs each represents 0.1 so
(b)
4 slabs divided into two groups = 2 slabs each. So 0.4 ÷ 2 = 0.2
Try to demonstrate the same for the following cases (you may use a
manipulative if you need)
½x5
¾ x4
½÷3
¼ ÷3
0.9 ÷ 3
0.9 x 3
2.4 ÷ 6
2.4 ÷ 4
Activity 2: Fraction of a whole
Fraction of a quantity
Give a scenario: A student is given the chocolate to share it equally among two friends
1. Count the number of pieces the chocolate has? (1 chocolate =_____ pieces)
2. If the chocolate is to divided into two friends equally, how much does each friend get?
183
3. Represent the number of chocolates each friend gets as a fraction of the total number of
pieces.
4. Simplify your answer.
_______________________________________________________
____________________________________________________
5.
Now complete the statement: Each friend gets __________________ of the chocolate
______________________________________________________________________
Now repeat the same activity by dividing the chocolate
equally among three friends
1
3
of 24 = _____________
Now let us find out 2/3 of a chocolate
Write and explain how you would solve such a problem
________________________________________________________________________
________________________________________________________________________
11.
If I have a stack of 60 coins arranged like this plan a strategy and verify that
1
3
of 60 = 20
Now Try these: 2
5
of 100
1
184
3
of 90
4
7
of 49
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C13
Parameter
Understand
multiplication as
repeated addition
Understand
division
as
repeated subtraction
185
STUDENT’S WORKSHEET 18
WORD PROBLEMS INVOLVING FRACTIONS AND DECIMALS
CONTENT WORKSHEET C14
Activity 1 – Try them all!!!
1. Annabelle cut a piece of rope and labeled it 1 4 5 . Ryan cut another piece of ropeand
labels it as 1.8. They measured the rope lengths and found them to be the same.
Explain how are they equal?
2.
Ravi and Karan were both given some math questions to do. Ravi completed the
whole work in 1 2 5 hours and karan the same task in 1.5 hours. Who completed the
work late?
3. The thickness of four sheets of metal is 35/1000 inches, 13/100 inches, 97/1000 inches
and 1/10 inches. Arrange these sheets in ascending order of thickness.
4. Four swimmers entered into a competition, the timings of the four of the swimmers
were 9.8o seconds, 9.75 seconds, 9.79 seconds, 9.801 seconds. In what minimum time
the fifth swimmer should swim in order to win the competition?
5. To make a miniature ice-cream truck we need tyres with a diameter between 1.465 cm
and 1.42 cm. Will a tyre with 1.469 cm in diameter work? Explain why or why not?
6. ”I SCREAM ICE-CREAM”
All right you bums, I got your at bats and hit right here, and if any of you have less
than 0.250 average you don’t get any ice cream.
186
Check for all the players getting ice-cream
Players
/
Mickey:
bats averages
15/30
Roger:
10/25
Joe:
19/35
Yogi:
9/42
Boby:
6/15
Activity 2 - The Match Maker
In this card game, there are two clues one on the front side and one at the back side of the
card. This is just played as a game in the class.
Students will record their observations on the worksheet. Students get 10 minutes to
complete the whole chain of clues.
For e.g.
In the following cards Card 1 follows card 12 and card 7 follows card 1 and so on.
Card No.
12
Front
Back
6
I have 6. Who has 2 .5 less?
1
3 .5
7
3
I have 3 1 . Who has its double?
2
I have 3
8
8
. Who has 1
8
more?
Now try it out yourself.
Card No.
Front
Back
1
3 .5
I have 3 1 . Who has its double?
2
½
I have ½. Who has 1/4 more?
3
33
4
2
8
2
I have 3 3 . Who has 2 less?
8
I have 2. Who has 7
187
8
more?
5
43
6
11
7
3
8
67
9
3
4
4
I have 4 3
8
less?
4
I have 1.25. Who has 3 .5 more?
I have 3
8
. Who has 3
4
. Who has 1
8
I have 6 7
more?
8
. Who has 6 1 less?
12
8
I have 3. Who has 3 more?
10
13
11
4
I have 4. Who has 5/8 less?
12
6
I have 6. Who has 2 .5 less?
13
7
I have 7. Who has 1/8 less?
14
1
15
27
16
0.75
I have 1 3 . Who has 5
8
8
I have 1
8
8
8
I have 2 7
8
. Who has 1 1 more?
8
8
. Who has 1
I have 0.75. Who has 5
188
5/8 more?
8
8
more?
less?
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C14
Parameter
Has knowledge of fractions
and decimals
Can solve related problems
on decimals and fractions
189
STUDENT’S WORKSHEET – 19
ROUNDING OFF AND ESTIMATION
CONTENT WORKSHEET C15
Name of the student ______________________
Date ______
Activity 1 –
1.
Estimate your waist meaurement and write it down. __________________________
Now pair up and measure up each others waist and record_____________________
Write what do you observe?
______________________________________________________________________________
______________________________________________________________________________
Discuss the possible reasons for these variations
______________________________________________________________________________
______________________________________________________________________________
What specifications do you need to give to the shop keeper, if you wish to buy a dress
for yourself.
______________________________________________________________________________
______________________________________________________________________________
If a person’s waist size is measured as 22.6 inches, what waist size would the shop owner
give? 23 or 22
______________________________________________________________________________
Why does this happen?
______________________________________________________________________________
What are the consequences of buying a size 23?
______________________________________________________________________________
What are the consequences of buying a size 22?
190
After watching video clip no 15: to understand the process of rounding off. Recall:
What did we learn?
__________________________________________________________________________________
__________________________________________________________________________________
What are the rules of rounding off?
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
What are the exceptions?
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
How do we take care of these?
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
Based on what you have learnt:
Test yourself:
1. Complete the following sentences and round the number as directed:
(a)
1.248 is between 1.24 and 1.25 and when rounded to (nearest hundredth) is 1.25
(b)
4. 534is between ______and _____ and when rounded to (nearest tenths) is ______.
(c)
47.203 is between _____and _____and when rounded to ( nearest ones) is _____.
(d) 125.598 is between _____and _____ and when rounded to (nearest tenths) is ______
2. Round each to the nearest tenths
6.25
4.23
17.65
122. 456
3. Round the following to the nearest 100:
101+ 34
295+ 67
345- 32
798
191
54.37
90.099
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C15
Parameter
Understand
the
need
of
estimation in real life
Can analyse the possible reasons
for the variations in the two
answers
Understand
the
process
of
rounding off
192
STUDENT’S WORKSHEET – 20
ESTIMATION AND SIGNIFICANCE
CONTENT WORKSHEET C16
Name of the student ______________________
1.
Date ______
Measure and write your height to the nearest 10 mm, to the nearest cm, correct to 2
decimal places.
2.
In the number 1.2304
Write which is the most significant figure. ______________________
Write the second most significant digit___________________
Write the third most significant digit__________________________
4.
Observe the examples and answer the questions that follow:
(a)
170.6, 1(the non zero figure) is the first, 7 is the second and 0 is the third
significant figure.
(b)
In 0.02509, 2 is the first (the non zero figure), 5 the second and 0 is third significant
figure.
For figure 0.001503, write down
(a)
the first significant figure( 1 the first non zero figure)
(b)
The third significant figure (0 )
(c)
Watch video clip and write down the rules for significant digits:
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
5.
Rewrite the answer in 2 significant figures:
1) 7803
________________
2)
1009 ________________
3) 918.010_______________
4)
0.0001 _______________
5) 0.00390_______________
6)
8120 ________________
193
7) 7.991
_______________
8)
729 ________________
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C16
Parameter
Understand the rules for the
significant figures.
Solving questions on significant
figures.
194
STUDENT’S WORKSHEET – 21
RATIO AND PROPORTION
CONTENT WORKSHEET C 17
Name of the student ______________________
Date ______
Activity 1: Ratio
1.
Discuss and write where in your daily life situations do you use ratio and proportion.
______________________________________________________________________________
______________________________________________________________________________
2.
Based on the discussions in class:
Write how would you define a ratio?
______________________________________________________________________________
______________________________________________________________________________
3.
Now try the following questions: Look at the pattern below and answer the following questions:
(a)
Ratio of white beads to total beads ? ________________
(b)
Ratio of white beads to total beads ? _______________________
(c)
Ratio of black to white ?________________________
(d)
Ratio of white to black ? ___________________
(Remember to simplify a ratio, wherever possible)
4.
If length of the string is now increased
(a)
Express the number of black beads of the total
number of beads as a fraction. ________________
(b)
Observe and describe what is the change in the fraction of black beads to the total
beads in the both the cases.
___________________________________________________________________
195
5.
Write the following ratio as instructed
6.
Colour 4 of these tins red and 3 tins blue.
What is the ratio of red to blue?
Activity 2: Proportion
Complete the following information, as the discussion takes place in class.
1.
To paint a picture I need to make color , mixing 2 scoops of white and 8 scoops of
blue.How much color do I need to mix to get the same shade but for three similar
paintings?________________________________________________________________
________________________________________________________________________
2.
In a group there are 15 males and 12 females.
(a)
What is the ratio of males to females? Give your answer in its simplest
form.____________________________________________________________________
(b)
What is the ratio
of males to the total number of people in the group.
_________________________________________________________________________
(c)
What does it mean to you? ______________________________________________
(d)
I observe that for every 27 people, there are 15 males. If the number of people is
trebled, what would the number of males be?
196
3.
What is the difference between a ratio and a proportion:
______________________________________________________________________________
______________________________________________________________________________
4.
Which of the following are in proportion?
35 7
=
20 4
3 12
=
8 16
54 3
=
81 4
64 8
=
40 5
Activity3: Its Maggi, maggi maggi
Read the following instructions carefully
For each packet of maggi add two cups of water, and allow it to boil.
Once the water comes to boil add one maggi cake, and a packet of the spices given along
with maggi.
Simmer for 2 minutes and presto, your maggi is ready.
Now tabulate the ingredients used by you:
No.
of
cakes Cups
Maggi
Water
1
2
of
water
used Ratio
½
2
3
No.
of
cakes Time taken to prepare Ratio
Maggi
maggi
1
2 min
½
2
3
197
SELF ASSESSMENT RUBRIC
CONTENT WORKSHEET C17
Parameter
Is able to relate the given
problem to ratio and solve them
Is able to relate the given
problem to proportion and solve
them
198
STUDENT’S WORKSHEET – 22
POST CONTENT WORKSHEET PC 1
Name of the student ______________________
Date ______
Activity : Test yourself
1. Observe the colored stircase and answer the following;
How can you compare the length of the orange staircase to the red one?
What is the ratio of the length of the
orange to the red staircase?
What is the ratio of the length of the
green to the blue staircase?
Is there any other pair you could choose
with same result ?
2. In a bag there are 14 apples and 16 oranges. Write this as a ratio of
Oranges: apples
Apples: Oranges
Apples: total fruit
Oranges: total fruit
3. Find the missing number n if
48 n
=
72 6
35 7
=
60 n
4. Find the missing number n if
2, 5, 32 and n respectively are in proportion? ( 2
4, 7, n and 84 respectively are in proportion? ( 4
199
5
7
= 32 )
n
= n
84
)
5. The ratio of girls to boys in a class is 2: 3. If there are 12 boys, how many girls are there?
6. The ratio of the marks received by Ali and Amir are in the ratio 6:7. If Ali received 42 out
of 50 in the test how many marks did Amir get out of 50?
7. The length and the breadth of a rectangle are in the ratio 1:3. If the breadth of the
rectangle is 21cm what is the length of this rectangle?
8. An investment of $5200 is divided in the ratio 6:7between two brothers Ram and
Lakshman.
(a)
How much money does each brother get?
(b)
Represent the money received by Ram as a fraction of the total money and
simplify it.
(c)
Represent the money received by Lakshman as a fraction of the total money and
simplify it.
(d) Is there an easier method of writing these fractions (observe the ratio carefully)
STUDENT’S WORKSHEET – 23
POST CONTENT WORKSHEET PC 2
Name of the student ______________________
Date ______
1. Choose the correct answer:
(a)
Which is the biggest?
4.013
(b)
4.0014
Calculate 2.15 + 0.37?
2.87
(c)
Write 16
1
(d)
4.004
3
2.62
20
2.52
in its simplest form.
2
6
4
5
Three friends compare their weights. Who is the heaviest?
200
Ann is 41.58kg
(e)
Christine is 41.62kg
Sabah calculates that she spends 2
3
Drew is 41.47kg
of her pocket money on video games, and 1
5
on magazines. What fraction of her pocket money does Sarah spend on games and
magazines?
5
11
15
6
(f)
What is 42 minutes as a fraction of 1 hour?
3
(g)
13
15
2
5
7
3
10
What is the missing number from this ratio?
11:7 = [ ]:35
55
(h)
10
7
A map has a scale of 1:25000. How much does 1cm on the map represent in real
life?
250cm
I.
2m
25000cm
Bruce, Alice and Sally enter a 'guess the weight of the cake' competition. The
actual weight of the cake is 800g. Who is closest to the correct answer?
Bruce guesses 788g
II.
Sally guesses 784g
Round 57.53m to the nearest ten meters
54
III.
Alice guesses 856g
60
50
Round 3.643 to 2 decimal places.
3.6
3.64
3.7
2. Evaluate the following;
6
2
3
+ 11
3
4
5
7
3
−1
12
4
1 + 1 − 1
4
3
2
1 +2 − 1 +2
2
3
6
9
1
7
10
2
−1
11 7
201
10
+41
5
1 − 1
2
8
6 3
4
- 4
3
3. Arrange the following in ascending order:
7
10
, 13
20
,2
3
0.875, 0.833, 13
16
4.
Jane used ½ a piece of ribbon and her sister used 2
3
of the piece. Who used more
ribbon and by how much?
5.
Write the natural numbers from 102 to 113. What fraction of them are prime numbers?
6.
Mrs. Bell made 40 cookies. Her son ate 1
7.
Harry was given $15 allowance each week. He spent 3
5
of them .How many cookies did he eat?
5
of it. What fraction did he
save? How much did he save in dollars?
8.
At a sale some shirts are sold at ½ of their original price. If the original price of these
shirts is $30, what is the sale price?
9.
Represent the value 4.56 on the number line.
10.
Read the value indicated by the pointer;
202
STUD
DENT’S
S WORK
KSHEET
T – 24
Posst CONT
TENT Wo
orksheett PC.3
1. These notices
n
werre seen on two
t
markett stalls. At which
w
stalll was the prrice of one orange
cheaperr and by ho
ow much?
H father gave
g
him a choice
c
of getting
g
it on
n a weekly or on a
2. Rob waanted an allowance. His
daily basis. He sa
aid he woulld either paay him $1.225 a week or pay him
m in the folllowing
mannerr for a weeek: On Monday
M
he would giv
ve him $0.01; On Tu
uesday $0.02; On
Wednesday $0.04 and so on through Sunday. Wh
hat would you
y tell Ro
ob to do so he can
get morre allowancce?
3. A Drug
g Store park
king lot hass space for 1000 cars. 2
5
of the spaces aree for compaact cars.
On Tueesday, theree were 200 compact caars and som
me standard
d size cars in the park
king lot.
The parrking lot was 3
4
full.. How man
ny standard
d size cars were
w
in the parking
p
lott?
4. Shane the
t Snail sttarted at thee dot. Whaat side will he be on when
w
he haas crawled 13
20
of
the disttance aroun
nd the regular pentago
on of equal sides?
5. Jenny bought
b
7 t--shirts, onee for each of
o her seveen brothers,, for $9.95 each. The cashier
charged
d her an ad
dditional $13.07 in salees tax. She left
l the storre with a measely
m
$7.28. How
much money
m
did Jenny
J
start with?
203
6. On an average day, Canadians spend $958904.00 buying video games. Of this total
$767123.00 is spent on Nintendo games. In one week how much do Canadians spend on
Nintendo Games. How much do they spend on other video games in a week?
STUDENT’S WORKSHEET – 25
POST CONTENT WORKSHEET PC 4
Name of the student ______________________
Date ______
Activity 3 - Test yourself
1. Solve the following:
a) 4 + 1
5
2
b) 1 + 2
3
5
c) 2 − 1
3
4
d) 3 − 3
7
14
e) 1 + 2 + 2
9
3
9
f) 4 + 7 − 1
5
10
2
g) 1 + 2 + 1
6
3
4
h) 7
2. From what number should 4
15
+ 1 − 1
3
5
be subtracted to get 1? Give your answer in the simplest
7
form wherever possible.
3. Of the students in the musical, 1
12
play the flute and another 1 dance. What fraction of
3
the students in the band play either the flute or dance?
Simplify your answer and write it as a proper fraction or as a whole or mixed number.
4. In the morning, Aman drove to school and used 7
he drove to the movie theater and used 1
3
15
of a gallon of gas. In the afternoon,
of a gallon of gas. How much gas did Aman
use in all? Simplify your answer and write it as a proper fraction or as a whole or mixed
number.
5. Louisa initially filled a measuring cup with 3
she poured 1
3
5
of a cup of syrup from a large jug. Then
of a cup back into the jug. How much syrup remains in the measuring
cup? Simplify your answer.
204
6. Mr. Higgins cut two pieces of wood for making a toy. One piece of wood measured 5/8
of a meter and the other piece of measured 1
4
of a meter. How much more is the first
piece of wood more than the second piece? Simplify your answer.
7. Ben watched a cockroach and an ant on the floor. The cockroach walked 5
6
of a yard
and the ant moved ½ of a yard in the same time. How much farther did the cockroach
crawl than the ant?
8. What is the combined thickness of these five strips of ribbon 0.008, 0.125, 0.15, 0.185, and
0.005 cm?
9. John is making a telescope for his science project. He joined two metal tubing, one 30.45
inches long and the other 12.75 inches long. How long would the telescope be when the
two tubes are joined?
10. For winter’s snowstorm, Mrs Brown bought two coats for her children. One coat cost
$55.75 and the other cost $9.30.What was the difference in the costs of the two coats?
11. Two boys were riding their bicycles to school. One boy was traveling at the speed of
15 1
2
km per hour and the other was traveling at a speed of 11.2 km per hour. How much
faster is the first boy traveling than the other boy?
12. I took my car for servicing .The usual price for a service is $30.50. However, I got a
coupon in a lucky draw in which I got a $15.75 off on the car servicing. How much do I
still have to I pay for the service?
13. The planet Mars takes 1 4
5
of our year to go around sun. The planet Jupiter which is
farther away from Mars takes 11.9 of our years to go around the sun. How much longer
does Jupiter take then Mars to go around the sun?
14. George has three pieces of wood cut from a 22ft long log. One is 10 1 ft long, another is
8
6 1 ft long and the third is 5 7 ft long.
4
8
How far do they stretch if they are laid end to end? How much wood is left over after
the three pieces are cut?
15. In the school café the cost of the items are as follows:
205
Ryan ordered one of each item
a)
How much money does Ryan have to pay?
b)
He paid with a $10 note. How much money will he get back?
16. A piece of webbing is 7.6m long. If two pieces each 2.3 m and 1.5 m long respectively are
cut off, how much is left?
17.
Answer the following:
(a)
Which row shows the highest prices?
(b)
Which row shows the lowest prices?
(c)
Which is the cheapest kind of fuel you can buy?
The difference between the highest and the lowest rates is called the range. The higher the
range the less consistent the fuel is.
(d)
Which is the most consistent fuel price in the table given above?
(e)
Which is the least consistent fuel price in the table given above?
206
ACKNOWLEDGMENTS:
www.aaamath.com/fra.html
www.helpingwithmath.com› BySubject › Fractions
www.homeschoolmath.net/worksheets/fraction-decimal.php
www.mathsisfun.com/worksheets/decimals.php
www.teach-nology.comMathWorksheets
www.teachingideas.co.uk/.../contents08fracdecperratprop.htm
ww.math-drills.com/decimal.shtml
www.aplusmath.com/Worksheets/index.html
www.theteachersguide.com/mathlessonplans1.html
ww.superteacherworksheets.com/fraction-cont.html
www.newtonanddescartes.com/protected/content/ipe_na/.../g6_03_05.pdf
http://www.jamit.com.au/worksheets/
http://www.mystfx.ca/special/mathproblems/grade6.html
http://www.lessonplanspage.com/Math45.htm
http://staff.argyll.epsb.ca/jreed/math7/strand1/1201.htm
http://www.dr-mikes-math-games-for-kids.com/fractions-worksheets.html
http://www.learningwave.com/lwonline/workingfront/decimalsfront.html
TILTLE/LINK
Video clip 1
Introduction to fractions
http://www.youtube.com/watch?v=cy-8lPVKLIo
http://www.youtube.com/watch?v=gJgusNWTIkA&feature=related
Video clip 2
Introduction to decimals
http://www.youtube.com/watch?v=M_xskCkyAmo&feature=relatedi
mproper
207
Video clip 3
placing fractions on a number line
http://www.youtube.com/watch?v=7X3sn2Bj-AM&feature=related
Video clip 5
placing decimals on a number line
http://www.youtube.com/watch?v=uCAXYAu1y7g&feature=related
Video Clip 6
fraction converted to mixed fraction
http://il.youtube.com/watch?v=cLYa05dOy8E&feature=channel
Video clip7
introduces concept of equivalent fractions
Video clip 8
Converting fraction to a decimal
http://www.youtube.com/watch?v=SGiSW2fvKdw&feature=relatedc
onvertin
Video clip 9
Process of simplifying fractions
http://www.youtube.com/watch?v=7P7zT-iuH80&feature=related
Video clip 11
Comparing decimals
http://il.youtube.com/watch?v=HCC96awA-FM&feature=related
Video clip 12
Adding unlike fractions
http://www.youtube.com/watch?v=MEo0rSZ4O3w&NR=1
Video clip 13
how to add and subtract mixed fractions
http://www.youtube.com/watch?v=KW4XN0fs1K8&feature=related
Video clip 15
rounding of decimals
http://www.youtube.com/watch?v=SfdFD1bZatU&feature=channel
Video clip 16
Significant figures
http://www.youtube.com/watch?v=QXKMj-kaXcE
Web link 1
review of four operations on fractions ppt
http://www.slideshare.net/kamstrak/fractions-power-point-1261979
Web link 2
check fraction and related words
http://www.amathsdictionaryforkids.com/
208
Web link 3
Ordering of decimals
http://www.mathsisfun.com/numbers/ordering-game.php?m=DecTricky
Web link 4
Presentation on fractions and important concepts ( To be referred
continuously)
http://www.kidsolr.com/math/fractions.html
Web link 5
Game on Simplification of fractions
http://www.funbrain.com/cgi-bin/fob.cgi?A1=s&A2=0
Web link 6
more drill on fraction and decimals
http://home.avvanta.com/~math/FDU1.HTM
Web link 8
more work for remedial / practice
http://www.math-drills.com/fractions.shtml
Web link 10
more work for remedial / practice
http://www.ixl.com/math/grade/sixth/
209