Mathematics Textbook for Class VI 1st Edition Shveta Uppal

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Mathematics Textbook for Class VI 1st Edition Shveta
Uppal Digital Instant Download
Author(s): Shveta Uppal; Indien National Council of Educational Research and
Training
ISBN(s): 9788174504821, 8174504826
Edition: 1
File Details: PDF, 10.74 MB
Year: 2006
Language: english

Mathematics
TEXTBOOK FOR CLASS VI
2020-21

ISBN 81-7450-482-6
ALL RIGHTS RESERVED
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or otherwise without the prior permission of the publisher.
q Thisbookissoldsubjecttotheconditionthatitshallnot,bywayoftrade, belent, re-
sold, hired out or otherwise disposed of without the publisher’s consent, in any form
of binding or cover other than that in which it is published.
q The correct price of this publication is the price printed on this page, Any revised
price indicated by a rubber stamp or by a sticker or by any other means is incorrect
andshouldbeunacceptable.
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First Edition
February 2006 Phalguna 1927
Reprinted
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December 2007 Agrahayana 1929
January 2009 Pausa 1930
November 2009 Agrahayana 1931
November 2010 Kartika 1932
January 2012 Magha 1933
December 2012 Pausa 1934
December 2015 Agrahayana 1937
November 2017 Agrahayana 1939
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© National Council of Educational
Research and Training, 2006
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Publications Ltd., Plot No. 2/4, Dr. Rajkumar
Road, Rajaji Nagar, Bengaluru - 560 010
2020-21

Foreword
The National Curriculum Framework (NCF), 2005, recommends that children’s life at
school must be linked to their life outside the school. This principle marks a departure
from the legacy of bookish learning which continues to shape our system and causes a
gap between the school, home and community. The syllabi and textbooks developed on
the basis of NCF signify an attempt to implement this basic idea. They also attempt to
discourage rote learning and the maintenance of sharp boundaries between different
subject areas. We hope these measures will take us significantly further in the direction
of a child-centred system of education outlined in the National Policy on Education (1986).
The success of this effort depends on the steps that school principals and teachers
will take to encourage children to reflect on their own learning and to pursue imaginative
activities and questions. We must recognise that, given space, time and freedom, children
generate new knowledge by engaging with the information passed on to them by adults.
Treating the prescribed textbook as the sole basis of examination is one of the key reasons
why other resources and sites of learning are ignored. Inculcating creativity and initiative
is possible if we perceive and treat children as participants in learning, not as receivers of
a fixed body of knowledge.
These aims imply considerable change in school routines and mode of functioning.
Flexibility in the daily time-table is as necessary as rigour in implementing the annual
calendar so that the required number of teaching days are actually devoted to teaching.
The methods used for teaching and evaluation will also determine how effective this textbook
proves for making children’s life at school a happy experience, rather than a source of
stress or boredom. Syllabus designers have tried to address the problem of curricular
burden by restructuring and reorienting knowledge at different stages with greater
consideration for child psychology and the time available for teaching. The textbook
attempts to enhance this endeavour by giving higher priority and space to opportunities
for contemplation and wondering, discussion in small groups, and activities requiring
hands-on experience.
The National Council of Educational Research and Training (NCERT) appreciates the
hard work done by the Textbook Development Committee responsible for this textbook.
We wish to thank the Chairperson of the advisory group in Science and Mathematics,
Professor J.V. Narlikar and the Chief Advisor for this textbook, Dr. H.K. Dewan for guiding
the work of this committee. Several teachers contributed to the development of this textbook;
we are grateful to their principals for making this possible. We are indebted to the
institutions and organisations which have generously permitted us to draw upon their
resources, material and personnel. We are especially grateful to the members of the National
Monitoring Committee, appointed by the Department of Secondary and Higher Education,
Ministry of Human Resource Development under the Chairpersonship of Professor Mrinal
Miri and Professor G.P. Deshpande, for their valuable time and contribution. As an
organisation committed to the systemic reform and continuous improvement in the quality
of its products, NCERT welcomes comments and suggestions which will enable us to
undertake further revision and refinement.
Director
New Delhi National Council of Educational
20 November 2006 Research and Training
2020-21

Textbook Development Committee
CHAIRPERSON, ADVISORY GROUP IN SCIENCE AND MATHEMATICS
J.V. Narlikar, Emeritus Professor, Inter University Centre for Astronomy & Astrophysics
(IUCCA), Ganeshkhind, Pune University, Pune
CHIEF ADVISOR
Dr. H.K. Dewan, Vidya Bhawan Society, Udaipur, Rajasthan
CHIEF COORDINATOR
Hukum Singh, Professor, DESM, NCERT, New Delhi
MEMBERS
Anjali Gupte, Teacher, Vidya Bhawan Public School, Udaipur, Rajasthan
Avantika Dam, TGT, CIE Experimental Basic School, Department of Education, Delhi
Dharam Prakash, Reader, CIET, NCERT, New Delhi
H.C. Pradhan, Professor, Homi Bhabha Centre for Science Education, TIFR, Mumbai,
Maharashtra
Harsha J. Patadia, Senior Reader, Centre of Advance Study in Education,
M.S. University of Baroda, Vadodara, Gujarat
Jabashree Ghosh, TGT, DM School, RIE, NCERT, Bhubaneswar, Orissa
Mahendra Shankar, Lecturer (S.G.) (Retd.), NCERT, New Delhi
Meena Shrimali, Teacher, Vidya Bhawan Senior Secondary School, Udaipur, Rajasthan
R. Athmaraman, Mathematics Education Consultant, TI Matric Higher Secondary School
and AMTI, Chennai, Tamil Nadu
S. Pattanayak, Professor, Institute of Mathematics and Application, Bhubaneswar, Orissa
S.K.S. Gautam, Professor, DESM, NCERT, New Delhi
Shraddha Agarwal, PGT, Sir Padampat Singhania Education Centre, Kanpur, (U.P.)
Srijata Das, Sr. Lecturer (Mathematics), SCERT, New Delhi
U.B. Tewari, Professor, Department of Mathematics, IIT, Kanpur, (U.P.)
Uaday Singh, Lecturer, DESM, NCERT, New Delhi
MEMBER-COORDINATORS
Ashutosh K. Wazalwar, Professor, DESM, NCERT, New Delhi
Praveen K. Chaurasia, Lecturer, DESM, NCERT, New Delhi
2020-21

CONSTITUTION OF INDIA
Fundamental Duties
Fundamental Duties – It shall be the duty of every citizen of India —
(a) to abide by the Constitution and respect its ideals and institutions, the National
Flag and the National Anthem;
(b) to cherish and follow the noble ideals which inspired our national struggle for
freedom;
(c) to uphold and protect the sovereignty, unity and integrity of India;
(d) to defend the country and render national service when called upon to do so;
(e) to promote harmony and the spirit of common brotherhood amongst all the people
of India transcending religious, linguistic and regional or sectional diversities; to
renounce practices derogatory to the dignity of women;
(f) to value and preserve the rich heritage of our composite culture;
(g) to protect and improve the natural environment including forests, lakes, rivers,
wildlife and to have compassion for living creatures;
(h) to develop the scientific temper, humanism and the spirit of inquiry and reform;
(i) to safeguard public property and to abjure violence;
(j) to strive towards excellence in all spheres of individual and collective activity so
that the nation constantly rises to higher levels of endeavour and achievement;
(k) who is a parent or guardian, to provide opportunities for education to his child or,
as the case may be, ward between the age of six and fourteen years.
Part IV A (Article 51 A)
2020-21

Acknowledgements
The Council acknowledges the valuable comments of the following participants of the workshop
towards the finalisation of the book — K.K. Gupta, Reader, U.N.P.G. College, Padrauna, Uttar
Pradesh; Deepak Mantri, Teacher, Vidya Bhawan Basic School, Udaipur, Rajasthan; Shagufta
Anjum, Teacher, Vidya Bhawan Senior Secondary School, Udaipur, Rajasthan; Ranjana Sharma,
Teacher, Vidya Bhawan Secondary School, Udaipur, Rajasthan. The Council acknowledges the
suggestions given by Utpal Chakraborty, Lecturer, SCERT, Raipur, Chattisgarh.
The Council gratefully acknowledges the valuable contributions of the following
participants of the Textbook Review Workshop : K. Balaji, TGT, Kendriya Vidyalaya, Donimalai,
Karnataka; Shiv Kumar Nimesh, TGT, Rajkiya Sarvodaya Bal Vidyalaya, Delhi; Ajay Singh,
TGT, Ramjas Senior Secondary School No. 3, Delhi; Rajkumar Dhawan, PGT, Geeta Senior
Secondary School No. 2, Delhi; Shuchi Goyal, PGT, The Airforce School, Delhi; Manjit Singh,
TGT, Government High School, Gurgaon, Haryana; Pratap Singh Rawat, Lecturer, SCERT,
Gurgaon, Haryana; Ritu Tiwari, TGT, Rajkiya Pratibha Vikas Vidyalaya, Delhi.
The Council acknowledges the support and facilities provided by Vidya Bhawan Society
and its staff, Udaipur for conducting the third workshop of the development committee at
Udaipur, and to the Director, Centre for Science Education and Communication (CSEC), Delhi
University for providing library help.
The Council acknowledges the academic and administrative support of Professor
Hukum Singh, Head, DESM, NCERT.
The Council also acknowledges the efforts of Uttam Kumar (NCERT) and Rajesh Sen
(Vidya Bhawan Society, Udaipur), DTP Operators; Monika Saxena, Copy Editor; and Abhimanu
Mohanty, Proof Reader; APC office and the administrative staff DESM, NCERT and the
Publication Department of the NCERT.
2020-21

CONSTITUTION OF INDIA
Part III (Articles 12 – 35)
(Subject to certain conditions, some exceptions
and reasonable restrictions)
guarantees these
Fundamental Rights
Right to Equality
• before law and equal protection of laws;
• irrespective of religion, race, caste, sex or place of birth;
• of opportunity in public employment;
• by abolition of untouchability and titles.
Right to Freedom
• of expression, assembly, association, movement, residence and profession;
• of certain protections in respect of conviction for offences;
• of protection of life and personal liberty;
• offreeandcompulsoryeducationforchildrenbetweentheageofsix and fourteenyears;
• of protection against arrest and detention in certain cases.
Right against Exploitation
• for prohibition of traffic in human beings and forced labour;
• for prohibition of employment of children in hazardous jobs.
Right to Freedom of Religion
• freedom of conscience and free profession, practice and propagation of religion;
• freedom to manage religious affairs;
• freedom as to payment of taxes for promotion of any particular religion;
• freedom as to attendance at religious instruction or religious worship in educational
institutions wholly maintained by the State.
Cultural and Educational Rights
• for protection of interests of minorities to conserve their language, script and culture;
• for minorities to establish and administer educational institutions of their choice.
Right to Constitutional Remedies
• by issuance of directions or orders or writs by the Supreme Court and High
Courts for enforcement of these Fundamental Rights.
2020-21

A Note for the Teachers
athematics has an important role in our life, it not only helps in day-to-day situations
but also develops logical reasoning, abstract thinking and imagination. It enriches life
and provides new dimensions to thinking. The struggle to learn abstract principles develops
the power to formulate and understand arguments and the capacity to see interrelations
among concepts. The enriched understanding helps us deal with abstract ideas in other subjects
as well. It also helps us understand and make better patterns, maps, appreciate area and
volume and see similarities between shapes and sizes. The scope of Mathematics includes
many aspects of our life and our environment. This relationship needs to be brought out at all
possible places.
Learning Mathematics is not about remembering solutions or methods but knowing
how to solve problems. We hope that you will give your students a lot of opportunities to
create and formulate problems themselves. We believe it would be a good idea to ask them
to formulate as many new problems as they can. This would help children in developing an
understanding of the concepts and principles of Mathematics. The nature of the problems
set up by them becomes varied and more complex as they become confident with the ideas
they are dealing in.
The Mathematics classroom should be alive and interactive in which the children should
be articulating their own understanding of concepts, evolving models and developing
definitions. Language and learning Mathematics have a very close relationship and there
should be a lot of opportunity for children to talk about ideas in Mathematics and bring in
their experiences in conjunction with whatever is being discussed in the classroom. There
should be no obvious restriction on them using their own words and language and the shift
to formal language should be gradual. There should be space for children to discuss ideas
amongst themselves and make presentations as a group regarding what they have
understood from the textbooks and present examples from the contexts of their own
experiences. They should be encouraged to read the book in groups and formulate and
express what they understand from it.
Mathematics requires abstractions. It is a discipline in which the learners learn to generalise,
formulate and prove statements based on logic. In learning to abstract, children would need
concrete material, experience and known context as scaffolds to help them. Please provide
them with those but also ensure that they do not get over dependent on them. We may point
out that the book tries to emphasise the difference between verification and proof. These two
ideas are often confused and we would hope that you would take care to avoid mixing up
verification with proof.
There are many situations provided in the book where children will be verifying
principles or patterns and would also be trying to find out exceptions to these. So, while on
the one hand children would be expected to observe patterns and make generalisations,
they would also be required to identify and find exceptions to the generalisations, extend
patterns to new situations and check their validity. This is an essential part of the ideas of
Mathematics learning and therefore, if you can find other places where such exercises can
be created for students, it would be useful. They must have many opportunities to solve
problems themselves and reflect on the solutions obtained. It is hoped that you would give
children the opportunity to provide logical arguments for different ideas and expect them
to follow logical arguments and find loopholes in the arguments presented. This is necessary
for them to develop the ability to understand what it means to prove something and also
become confident about the underlying concepts.
M
2020-21

There is expectation that in your class, Mathematics will emerge as a subject of exploration
and creation rather than an exercise of finding old answers to old and complicated problems.
The Mathematics classroom should not expect a blind application of ununderstood algorithm
and should encourage children to find many different ways to solve problems. They need
to appreciate that there are many alternative algorithms and many strategies that can be
adopted to find solutions to problems. If you can include some problems that have the
scope for many different correct solutions, it would help them appreciate the meaning of
Mathematics better.
We have tried to link chapters with each other and to use the concepts learnt in the initial
chapters to the ideas in the subsequent chapters. We hope that you will use this as an
opportunity to revise these concepts in a spiraling way so that children are helped to appreciate
the entire conceptual structure of Mathematics. Please give more time to ideas of negative
number, fractions, variables and other ideas that are new for children. Many of these are the
basis for further learning of Mathematics.
We hope that the book will help ensure that children learn to enjoy Mathematics and
explore formulating patterns and problems that they will enjoy doing themselves. They should
learn to be confident, not feel afraid of Mathematics and learn to help each other through
discussions. We also hope that you would find time to listen carefully and identify the ideas
that need to be emphasised with children and the places where the children can be given
space to articulate their ideas and verbalise their thoughts. We look forward to your comments
and suggestions regarding the book and hope that you will send us interesting exercises that
you develop in the course of teaching so that they can be included in the next edition.
2020-21

Contents
FOREWORD iii
A NOTE FOR THE TEACHERS ix
CHAPTER 1 KNOWING OUR NUMBERS 1
CHAPTER 2 WHOLE NUMBERS 28
CHAPTER 3 PLAYING WITH NUMBERS 46
CHAPTER 4 BASIC GEOMETRICAL IDEAS 69
CHAPTER 5 UNDERSTANDING ELEMENTARY SHAPES 86
CHAPTER 6 INTEGERS 113
CHAPTER 7 FRACTIONS 133
CHAPTER 8 DECIMALS 164
CHAPTER 9 DATA HANDLING 184
CHAPTER 10 MENSURATION 205
CHAPTER 11 ALGEBRA 221
CHAPTER 12 RATIO AND PROPORTION 244
CHAPTER 13 SYMMETRY 261
CHAPTER 14 PRACTICAL GEOMETRY 274
ANSWERS 293
BRAIN-TEASERS 315
2020-21

Counting things is easy for us now. We can count objects in large numbers,
for example, the number of students in the school, and represent them
through numerals. We can also communicate large numbers using suitable
number names.
It is not as if we always knew how to convey large quantities in conversation
or through symbols. Many thousands years ago, people knew only small
numbers. Gradually, they learnt how to handle larger numbers.They also learnt
how to express large numbers in symbols. All this came through collective
efforts of human beings. Their path was not easy, they struggled all along the
way. In fact, the development of whole of Mathematics can be understood
this way.As human beings progressed, there was greater need for development
of Mathematics and as a result Mathematics grew further and faster.
We use numbers and know many things about them. Numbers help us
count concrete objects. They help us to say which collection of objects
is bigger and arrange them in order e.g., first, second, etc. Numbers are
used in many different contexts and in many ways. Think about various
situations where we use numbers. List five distinct situations in which
numbers are used.
We enjoyed working with numbers in our previous classes. We have added,
subtracted, multiplied and divided them.We also looked for patterns in number
sequences and done many other interesting things with numbers. In this chapter,
we shall move forward on such interesting things with a bit of review and
revision as well.
1.1 Introduction
Chapter
1
Chapter
1
Chapter
1
Chapter
1
Chapter
1
Knowing our
Knowing our
Knowing our
Knowing our
Knowing our
Numbers
Numbers
Numbers
Numbers
Numbers
2020-21

MATHEMATICS
2
1.2 Comparing Numbers
As we have done quite a lot of this earlier, let us see if we remember which is
the greatest among these :
(i) 92, 392, 4456, 89742
(ii) 1902, 1920, 9201, 9021, 9210
So, we know the answers.
Discuss with your friends, how you find the number that is the greatest.
Can you instantly find the greatest and the smallest numbers in each row?
1. 382, 4972, 18, 59785, 750. Ans. 59785 is the greatest and
18 is the smallest.
2. 1473, 89423, 100, 5000, 310. Ans. ____________________
3. 1834, 75284, 111, 2333, 450 . Ans. ____________________
4. 2853, 7691, 9999, 12002, 124. Ans. ____________________
Was that easy? Why was it easy?
We just looked at the number of digits and found the answer.
The greatest number has the most thousands and the smallest is
only in hundreds or in tens.
Make five more problems of this kind and give to your friends
to solve.
Now, how do we compare 4875 and 3542?
This is also not very difficult.These two numbers have the
same number of digits. They are both in thousands. But the digit
at the thousands place in 4875 is greater than that in 3542.
Therefore, 4875 is greater than 3542.
Next tell which is greater, 4875 or
4542? Here too the numbers have the
same number of digits. Further, the digits
at the thousands place are same in both.
What do we do then? We move to the
next digit, that is to the digit at the
hundreds place.The digit at the hundreds
place is greater in 4875 than in 4542.
Therefore, 4875 is greater than 4542.
Find the greatest and the smallest
numbers.
(a) 4536, 4892, 4370, 4452.
(b) 15623, 15073, 15189, 15800.
(c) 25286, 25245, 25270, 25210.
(d) 6895, 23787, 24569, 24659.
2020-21

KNOWING OUR NUMBERS
3
9 8 6 7
1 0 2 7
4
4
9
9
1
If the digits at hundreds place are also same in the two numbers, then what
do we do?
Compare 4875 and 4889 ; Also compare 4875 and 4879.
1.2.1 How many numbers can you make?
Suppose, we have four digits 7, 8, 3, 5. Using these digits we want to make
different 4-digit numbers in such a way that no digit is repeated in them. Thus,
7835 is allowed, but 7735 is not. Make as many 4-digit numbers as you can.
Which is the greatest number you can get? Which is the smallest number?
The greatest number is 8753 and the smallest is 3578.
Think about the arrangement of the digits in both. Can you say how the largest
number is formed? Write down your procedure.
1. Usethegivendigitswithoutrepetitionandmakethegreatestandsmallest4-digit
numbers.
(a) 2, 8, 7, 4 (b) 9, 7, 4, 1 (c) 4, 7, 5, 0
(d) 1, 7, 6, 2 (e) 5, 4, 0, 3
(Hint : 0754 is a 3-digit number.)
2. Now make the greatest and the smallest 4-digit numbers by using any one
digit twice.
(a) 3, 8, 7 (b) 9, 0, 5 (c) 0, 4, 9 (d) 8, 5, 1
(Hint : Think in each case which digit will you use twice.)
3. Make the greatest and the smallest 4-digit numbers using any four different
digits with conditions as given.
(a) Digit 7 is always at Greatest
ones place
Smallest
(Note, the number cannot begin with the digit 0. Why?)
(b) Digit4isalways Greatest
at tens place
Smallest
(c) Digit 9 is always at Greatest
hundreds place
Smallest
(d) Digit 1 is always at Greatest
thousandsplace
Smallest 1
2020-21

MATHEMATICS
4
Ramhari
(160 cm)
Dolly
(154 cm)
Mohan
(158 cm)
Shashi
(159 cm)
` 2635 ` 1897 ` 2854 ` 1788 ` 3975
4. Take two digits, say 2 and 3. Make 4-digit numbers using both the digits equal
numberoftimes.
Whichisthegreatestnumber?
Whichisthesmallestnumber?
Howmanydifferentnumberscanyoumakeinall?
Stand in proper order
1. Who is the tallest?
2. Who is the shortest?
(a) Can you arrange them in the increasing order of their heights?
(b) Can you arrange them in the decreasing order of their heights?
Which to buy?
SohanandRitawent
to buy an almirah.
There were many
almirahs available
withtheirpricetags.
(a) Can you arrange their prices in increasing
order?
(b) Can you arrange their prices in
decreasing order?
Ascending order Ascending order means arrangement from the smallest to the
greatest.
Descending order Descending order means arrangement from the greatest to
the smallest.
Think of five more situations
where you compare three or
more quantities.
2020-21

KNOWING OUR NUMBERS
5
1. Arrange the following numbers in ascending order :
(a) 847, 9754, 8320, 571 (b) 9801, 25751, 36501, 38802
2. Arrange the following numbers in descending order :
(a) 5000, 7500, 85400, 7861 (b) 1971, 45321, 88715, 92547
Make ten such examples of ascending/descending order and solve them.
1.2.2 Shifting digits
Have you thought what fun it would be if the digits in a number could shift
(move) from one place to the other?
Think about what would happen to 182. It could become as large as 821
and as small as 128. Try this with 391 as well.
Now think about this. Take any 3-digit number and exchange the digit at
the hundreds place with the digit at the ones place.
(a) Is the new number greater than the former one?
(b) Is the new number smaller than the former number?
Write the numbers formed in both ascending and descending order.
Before 7 9 5
Exchanging the 1st and the 3rd tiles.
After 5 9 7
If you exchange the 1st and the 3rd tiles (i.e. digits), in which case does the
number become greater? In which case does it become smaller?
Try this with a 4-digit number.
1.2.3 Introducing 10,000
We know that beyond 99 there is no 2-digit number. 99 is the greatest 2-digit
number. Similarly, the greatest 3-digit number is 999 and the greatest 4-digit
number is 9999. What shall we get if we add 1 to 9999?
Look at the pattern : 9 + 1 = 10 = 10 × 1
99 + 1 = 100 = 10 × 10
999 + 1 = 1000 = 10 × 100
We observe that
Greatest single digit number + 1 = smallest 2-digit number
Greatest 2-digit number + 1 = smallest 3-digit number
Greatest 3-digit number + 1 = smallest 4-digit number
2020-21

MATHEMATICS
6
We should then expect that on adding 1 to the greatest 4-digit number, we
would get the smallest 5-digit number, that is 9999 + 1 = 10000.
The new number which comes next to 9999 is 10000. It is called
ten thousand. Further, 10000 = 10 × 1000.
1.2.4 Revisiting place value
You have done this quite earlier, and you will certainly remember the expansion
of a 2-digit number like 78 as
78 = 70 + 8 = 7 × 10 + 8
Similarly, you will remember the expansion of a 3-digit number like 278 as
278 = 200 + 70 + 8 = 2 × 100 + 7 × 10 + 8
We say, here, 8 is at ones place, 7 is at tens place and 2 at hundreds place.
Later on we extended this idea to 4-digit numbers.
For example, the expansion of 5278 is
5278 = 5000 + 200 + 70 + 8
= 5 × 1000 + 2 × 100 + 7 × 10 + 8
Here, 8 is at ones place, 7 is at tens place, 2 is at hundreds place and 5 is at
thousands place.
With the number 10000 known to us, we may extend the idea further. We
may write 5-digit numbers like
45278 = 4 × 10000 + 5 × 1000 + 2 × 100 + 7 × 10 + 8
We say that here 8 is at ones place, 7 at tens place, 2 at hundreds place,
5 at thousands place and 4 at ten thousands place. The number is read as forty
five thousand, two hundred seventy eight. Can you now write the smallest
and the greatest 5-digit numbers?
Read and expand the numbers wherever there are blanks.
Number Number Name Expansion
20000 twenty thousand 2 × 10000
26000 twenty six thousand 2 × 10000 + 6 × 1000
38400 thirty eight thousand 3 × 10000 + 8 × 1000
four hundred + 4 × 100
65740 sixty five thousand 6 × 10000 + 5 × 1000
seven hundred forty + 7 × 100 + 4 × 10
2020-21

KNOWING OUR NUMBERS
7
1.2.5 Introducing 1,00,000
Which is the greatest 5-digit number?
Adding 1 to the greatest 5-digit number, should give the smallest
6-digit number : 99,999 + 1 = 1,00,000
This number is named one lakh. One lakh comes next to 99,999.
10 × 10,000 = 1,00,000
We may now write 6-digit numbers in the expanded form as
2,46,853 = 2 × 1,00,000 + 4 × 10,000 + 6 × 1,000 +
8 × 100 + 5 × 10 +3 × 1
This number has 3 at ones place, 5 at tens place, 8 at hundreds place, 6 at
thousands place, 4 at ten thousands place and 2 at lakh place. Its number
name is two lakh forty six thousand eight hundred fifty three.
Read and expand the numbers wherever there are blanks.
Number Number Name Expansion
3,00,000 three lakh 3 × 1,00,000
3,50,000 three lakh fifty thousand 3 × 1,00,000 + 5 × 10,000
3,53,500 three lakh fifty three 3 × 1,00,000 + 5 × 10,000
thousand five hundred + 3 × 1000 + 5 × 100
4,57,928 _______________ _______________
4,07,928 _______________ _______________
4,00,829 _______________ _______________
4,00,029 _______________ _______________
89324 eighty nine thousand 8 × 10000 + 9 × 1000
three hundred twenty four + 3 × 100 + 2 × 10 + 4 × 1
50000 _______________ _______________
41000 _______________ _______________
47300 _______________ _______________
57630 _______________ _______________
29485 _______________ _______________
29085 _______________ _______________
20085 _______________ _______________
20005 _______________ _______________
Write five more 5-digit numbers, read them and expand them.
2020-21

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Most a Mary otthonáról beszéltek és ő egyszerre utálni kezdte a
férje mellett eltöltött nyugalmas öt esztendőt.
Az Oszkárral együtt töltött napokra gondolt és egész este
nyugtalan, ideges, türelmetlen volt.
Érezte, hogy megöregedett és elszélesedett, restelte a nagy krigli
sörét és a vadászkalapját. Szerette volna, ha ma is olyan vékony
törékeny a csuklója, mint régen volt, mikor még a Halensee partján
ebédelt együtt Oszkárral.
Vacsora után moziba akart menni Judit, Mary is velük tartott, nem
mert elválni tőlük, félt attól a pillanattól, ha majd egyedül marad.
Pedig únta az előadást, egyáltalán nem szerette az ostobán
humoros és szentimentális történeteket a fehér vásznon. Olyan jól is
tudott máskor a szerinte együgyű élvezettel gúnyolódni, de ma este
nem nevetett a fiatalasszony mulatságán, irigyelte tőle azt is,
irigyelte a gyerekes örömét és minden nevetését.
A gép berregett, a levegő rossz volt, a zenekar hamisan játszott.
Mary egyedül, elfogultan ült az asszony mellett, a ki az ura kezét
fogta, hozzásimult és úgy mulatott.
Mary is szeretett volna fiatal, boldog asszony lenni, mindent
elfelejteni, a mi volt és csak ma egyetlen egyszer a Judit helyében
lenni.
Együtt indultak el hazafelé, egy szép, új hotel előtt Oszkár
megállt. – Na, mi itthon vagyunk – mondta.
– Kísérjük el Maryt, – ajánlotta az asszony, aztán felmutatott a
negyedik emeletre – ott lakunk a sarkon, gyönyörű szobánk van,
egyáltalában én még sohasem láttam ilyen szép hotelt, egészen új
és hihetetlenül izléses. A mi szobánkban fehér a butor, vörös selyem
huzatokkal, az egész padló szürke plüssel van bevonva és még az
ágyban a pokróczok mintája is egész különösen szép.
Mary egy penzióban lakott, ott, a hol régen leánykorában is, ha
bennmaradt a városban. Oszkárék hazáig kísérték. A ház előtt egy
nagy kulcsot vett ki a táskájából és kizárta a kaput. Már kezében

fogta a kilincset, mikor hirtelen érezte, hogy most így nem válhat el
tőlük, mit fog különben gondolni róla az az asszony.
– Holnap mit fognak csinálni? – kérdezte.
– Még nem tudom, nincs semmi tervünk – mondta Oszkár. – Este
az operába megyünk, csak az bizonyos még, a Pillangókisasszonyt
nézzük meg. Maga meddig marad még itt, Mary?
– Holnap este megyek vissza Feldaffingba.
Csend volt. Judit érezte, hogy most valami kedveset kell
mondania.
– Vacsorázzék velünk holnap is. Várjon, de hol? Tudja mit, jőjjön
el a szállodánk éttermébe, hétkor eszünk és aztán megyünk az
operába.
– Jó, majd meglátom, talán – felelte Mary és ez a barátságtalan
válasz jól esett neki.
Aztán elváltak, Mary bezárta a nagy kaput, felment a sötét
lépcsőn, fölnyitotta az előszobát is, meggyujtotta a számára
odakészített gyertyát és végre eljutott a szobájába.
Az ágy szélére ült le és gondolkodott a mai estén. Valami
bosszantó megszégyenítő érzés fogta el, bántotta az asszony
minden féltékenységtől ment kedvessége, a melylyel vele bánt és
bosszantotta, hogy ő meg annyira elveszítette a biztonságát, a
nyugalmát.
A gyertya pislogva égett, a villanyt már féltizenegykor lezárta a
háziasszony. Mary a félhomályban ülve nézte az egyenes, barnalábú
íróasztalt, a téglavörös mintás tapétát, az ízléstelen, olcsó függönyt.
Ő itt lakott, azok pedig most már hazaértek a fehér szobába, a
vörös selyemhuzatos butorok közé. A szürke bársonyszőnyegen jár
az asszony, rakosgat, vetkezik és Oszkár behúzza a szép, nagy
függönyöket.

Az asszony bizonyosan csodálkozik, hogy ő, Mary, olyan, a
milyen, másnak képzelte. Más is azon a képen, a melyet
Frohnauban ajándékozott Oszkárnak. Akkor még hüvelyk- és
mutatóujjával tudta átfogni a csuklóját, most pedig egyáltalán nem
érte át. Ha ő olyan lenne, mint akkor volt, ha holnap megjelenhetne
megint a régi rozsdaszínű ruhájában, talán még féltékeny is lenne
Judit. – Mary olyan, mint egy rózsaszínű orchidea! – Erre csak úgy
emlékezett már, mint öregasszonyok az ifjúkori hódításaikra.
És Oszkár, Oszkár vajjon mit gondol most róla? Semmit, Oszkár
nem törődik már, csak a feleségével, most, míg ő itt ül az ágya
szélén, talán épen megöleli, úgy mint régen őt. Nem, nem úgy,
másként, nagy szerelemmel és boldogok, olyan boldogok, a milyen ő
nem volt és nem lehet sohasem.
Marynek tovább mozgott a képzelete, látta a gyönyörű mintás
pokróczokat, látta az asszony barna fejét a kis fehér párnán, hallotta
kaczagni és egyszerre ott állt előtte, úgy a mint ma este látta, a mint
tudta, hogy most már mindig látni fogja őt, a sötétkék ruhájában,
vörös szegélylyel a fekete áttört selyemzsinórok alatt, a kemény kis
alakjával, a mint profilba fordul, ránéz az urára és mosolyog.
Mary haragudott rá, féltékenynyé szerette volna tenni és nagy,
nagy szomorúságot okozni neki. Irigyelte a boldogságát, de már nem
érezte azt a fájón édes, egész testét megrázó vágyakozást, a melyet
régen érzett, ha Oszkárra gondolt. Az a vágyakozás már meg volt
ölve egy együttélésben, a mely fáradttá és unottá tette csak,
boldoggá soha. Már, már elhitette magával, hogy a nő számára a
férfi csak gyermeket jelent, az ifjúságban kívánt, remélt, sejtett
gyönyört soha.
Most látta Juditot, a mint mosolygó biztonsággal nézett az ura
szemébe, látta hozzásimulni, hallotta a szobájukról beszélni és
csónakázásaikról a Starnbergi-tavon.
Mary felugrott az ágy széléről, az ablakhoz ment. Vidám diákok
énekelve vonultak el a szemközti utczasoron. Lenn a vendéglőben
még sokan ültek és a lámpák széles fénysugarakat vetettek a
járdára. Póstások robogtak hazafelé autoczikliken, egy fiatal pár

nevetve jött át az utczán és nevetésük felhallatszott Maryhez. Ezek
is boldogok! Talán Oszkár és Judit most róla beszélnek, Oszkár
elmesél a feleségének a megismerkedésüktől az elválásukig
mindent, az egész tovatűnt berlini barátságukat.
Az asszony meg bizonyosan kitalálja a Mary nagy, reménytelen
szerelmét és sajnálja őt. Ebben a pillanatban sajnálja, a fehér
szobában, a férje karjaiban, mikor ő itt áll a nyitott ablakban
elhagyatva, egyedül.
Bizonyosan azt hiszi, hogy ő még mindig szereti Oszkárt s hogy
boldogtalan nagyon a mai találkozás után. Ezt nem akarta Mary,
együtt kell lennie velük okvetetlenül, meg fogja mutatni, hogy boldog
ő is, el fogja felejtetni velük a mai estét.

XI.
Egy vonat robogott az éjszakában Münchenből Feldaffing felé,
hideg, ködös, októberi éjben fázva simult Mary a kupé sarkába.
Egyedül és nagyon szomorúan.
Ismét találkozott Oszkárékkal. Már egész nap kereste őket, a
képtárakban, a sétatéren, a legszebb utczákon, a nagy árúházban,
szeretett volna túlesni a rá nézve olyan nehéz együttléten, de nem
találta meg őket sehol.
Végre eljött az este és együtt ültek ismét, látta az asszonyt
estélyi ruhában is, világos szürkén, selymesen, nagy tollas, kis
fekete kalapban, Oszkárt meg frakkban úgy, mint régen az Eriksen
Lolo hangversenyén. Most még szebb volt, férfiasabb és még
derültebben, fölényesebben nyugodt, mint akkor.
Arról az elmúlt estéről kezdett beszélni Mary, a mosolygó
visszaemlékezés derüs hangján, mintha egy együtt leélt gyermekkor
eseményeiről beszélne csak, olyan vidáman.
Aztán a kis fiáról mesélt sokat és nevelési kérdések felett vitába
elegyedett Judittal. Most végre fölényben érezte magát. – Istenem –
mondta végül – mit is tudja azt az, a ki csak pedagogus, a
legegyszerűbb anya is jobban ért a gyermekekhez. Maguk még nem
is tudják, milyen boldogság a gyerek, minden nő elsősorban anya, a
többi mind mellékes. Én csak azóta élek igazán, mióta megvan a
fiam. Még a hiúságról is lemondtam, a szép ruhák sem érdekelnek
többé, pedig az nálam nagy szó.
– Igen, leánykorában nagyon szerette a különös és szép színű
ruhákat. Nekem nagyon tetszettek is, én csak a jól öltözött nőket

szeretem, – mondta Oszkár.
Mary Juditot nézte, a gyönyörű ruháját vizsgálta. – Nincs pedig
annak semmi értelme – jegyezte meg aztán.
Így mentette ki a vadászkalapját, füző nélküli reformruhás
megjelenését, a ragyogóan fiatal Judit mellett.
Oszkárék gyorsan vacsoráztak, siettek az operába. Bucsúzóul az
asszony virágokat vett ki egy csomagból és átadta Marynek.
Orchideák voltak, rózsaszinű orchideák.
– Fogadja el tőlem – mondta – Oszkár úgy emlékezett, hogy
ezek a kedvencz virágai.
Mary megköszönte és végtelen szomorú lett egy pillanatra a
szeme, a mint ránézett Oszkárra, a ki már nem is gondol arra, hogy
egyszer féltékenyen őrizte az ő orchideáját a gomblyukában.
Aztán Mary kinn állt az omnibusz megállóhelyen, egyik kezében
az esernyője és az orchideák, másikban a kis sárga táskája és sok
apró csomag, az egyikben czipő volt a kis Ervin számára, a
csomagoló papiros kiszakadt és kilátszott egy keskeny, fekete sarok.
Judit várni akart, míg jön a Mary omnibusza, de Oszkár nem
engedte, félt, hogy elkésnek és elmulasztják a nyitányt.
Elmentek hát. Judit még egyszer visszafordult és mosolyogva
intett Marynek, a ki utánuk nézett. Oszkár senkivel sem törődve,
átfogta a felesége derekát, úgy tartotta össze a nagy köpenyegét, az
asszony a széllel küzdve előre szegte a fejét és nagy strucztollai
libegtek a kalapján. Még egyszer előbújt szürke selyem harisnyája a
lakkczipője felett, aztán eltüntek a tömegben. Csak néhány könnyű,
rózsaszínű virág maradt belőlük. Még most is ott voltak a Mary
ölében, míg a vonat vitte őt hazafelé. Ölében voltak az orchideák és
össze-vissza zürzavarosan jártak fejében a gondolatok.
Nagyon levert volt, ez a találkozás kizaklatta őt a megszokott,
csendes életéből. Ő szótfogadott a férjének és az elmult öt év alatt
keveset gondolkozott az életen, keveset magán, szeszélyes volt

néha és levert, de magyarázatot nem keresett. Hozzászokott ahhoz,
hogy ne várjon a napoktól mást, csakhogy elteljenek csendesen,
izgalom nélkül. Mióta a fia is meg volt, meg az ezer kedves, apró
gond, azóta épen nem törődött mással, mint az étellel, itallal, a
lakásával és a bútorokkal. Magáról elfelejtkezett egészen, úgy
mintha minden napra viseljük már az ünneplő ruhánkat és elfelejtünk
vigyázni arra, a mit valamikor annyi gonddal őriztünk.
A férje boldog volt vele, neki kényelmesebb volt a lecsiszolt,
eltompult Mary, ő szépnek látta és szerette, az ő szemében örökre
élt a régi varázsa. Csendes, nagy érzelem volt, a melyről sohasem
beszélt és óvatos, kicsit megszorított járása jellemezte egész
életüket. Maryt mindig bosszantotta ez a járás és most ott a kupé
sarkában egyszerre gyűlölte, épen úgy gyűlölte, mint a kifaragott,
élesen, biztosan körülhatárolva előadott véleményeit, szigorú nagy
szavait és az Ervin egész nyugodt, meggondolt szerelmét.
Oszkárra gondolt és arra, milyen lehet ha szeret, ha
szenvedélyesen, fiatalon öleli magához az asszonyát.
A halenseei sétájuk jutott eszébe, a csókjaik, visszagondolt a
búcsújukra az ő szobájában, mikor fejét a vállára hajtva ülhetett
mellette. Akkor Oszkár is szerette egy kicsit. Most már más ül
mellette, mást csókol, más az övé egészen.
Mary sohasem lehet az övé, soha azé a férfié, a kit szeret és
sohasem ölel senkit úgy, a hogy átvirrasztott fiatal éjszakáin az
álmait ölelte.
Ostobán hinni próbált Róth Ervinnek azon a rég elmúlt tavaszi
napon és a vágyak elcsitulását várta nála, de az ő vágyai nem
elcsitultak egy szerelemben, hanem lassan haltak meg, múltak el az
évekkel.
Most üresen áll és nincs semmije és senkije.
Hazagondolt, a kékvirágos levesestál jutott eszébe, a nagy kanál,
aztán hirtelen gyöngédséget érzett, a kis fiát látta, a mint feléje
nyujtja a tányérját. De mindjárt a férje is megjelent, szemben ült a
fiával, most óvatosan dugja nyakába a szalvétáját és a kis fia, a

Mary baloldalán épen úgy a trikója kivágásába. Épen úgy, mindent
épen úgy, mint az apja, olyan óvatosan, tisztán, rendesen.
A szemei az apja jóságával, gyengédségével vannak tele,
minden szava az ő türelmével és kis kezei fehérek, keskenyek, mint
az Ervin kezei.
Egy nagy Ervin és egy kis Ervin, ez az ő élete. Oszkár és Judit
pedig most az operában ülnek, egymáshoz simulva zenét
hallgatnak, az asszony az ura kezét fogja, úgy mint tegnap, és
Oszkár ránéz meleg pillantással. Mosolyog is hozzá, de a szeme
komoly marad, csak a sűrű szőke bajusza mozdul meg. Milyen jól
ismerte Mary ezt a mosolyt.
Aztán hazamennek majd a Puccini zenéjének emlékével, a
legszerelmesebb zene emlékével és kicsit ünnepélyes lesz a
csókjuk, mintha a színpadon lennének ők is.
Egyszer Juditnak is lesz bizonyosan kis fia, de az Oszkárhoz fog
majd hasonlítani. Megszagolta az orchideáit és nézte őket soká. Ha
ő akkor régen mégis ott maradt volna Berlinben, talán most neki is
volna egy olyan kis gyermeke. Bolond gondolat, de elképzelni jó.
Minek is ment ő feleségül Ervinhez? Hátha találkozott volna még
valakivel, a kit tudott volna szeretni is. Vagy maradt volna inkább
egyedül, akkor nem pusztult volna el a művészete és nem halt volna
meg a vágyakozása, hanem végtelen, gyönyörű bánattá finomodik
eddig.
Apró kezekről álmodozott akkor, kis gyermekkarokat kívánt a
nyaka köré, minden szenvedést megváltó gyermekfejeket látott
maga előtt, a melyekről Frau Petersen beszélt azon a régi esős
délutánon, a Bettina néni szalonjában, a sok asszonynak.
Most megvolt a kis fia, de az apjához hasonlított, az apjához
egészen és Mary nem tudott benne gyönyörködni, mert nem szerette
a férfit, a ki adta és lealacsonyító volt számára a szenvedély, a
melyből támadt.

Csak most gondolta végig mindezt Mary, most kapott
magyarázatot sok nehéz órája, most megértette a gyermeke
születése után, nagy, ösztönszerű elégedetlenségéből fakadt sirását.
Megérkezett a vonat Feldaffingba, hideg volt, köd ülte meg a
tájat, mintha láthatatlan kezek lengetnének fehér fátyolokat. Igazi
ősz volt, szeles, hideg október és a fákról hullottak a levelek.
Mary gyalog ment haza, nem félt az éjszakától, egy vonat füttye
hallatszott messziről és a templom-óra ütni kezdett. Csak itt-ott égett
pislogva egy-egy lámpa az út mellett és Marynek vigyáznia kellett
minden lépésére. A küzdelem az úttal, a sötéttel új életkedvet öntött
belé, önkéntelenül valami kibúvót keresett nagy keserűségéből.
Zizegett a sok sárga levél a lába alatt, reménytelensége
szomorúság lett lassan, mély igazságokkal telt meg, függetlennek
érezte magát mindattól, a mi vele történt, külön vált a saját sorsától
és elhatározta, hogy könyvet ír róla.
Nagy tanulsággal teli könyvet az asszonyoknak, a leányoknak,
hogy sohase fogják meg férfi kezét nagy igérettel, ha nem szeretnek,
mint ő régen, régen egyszer tavaszszal.
Igen így is fogja kezdeni. – Tavasz volt, Mária az arczához
szorította az Egon kezét és… –
Gyorsan ment, sietett haza, még ma este el akarta kezdeni. Úgy
érezte, most már elmenekült zürzavaros, sötét helyzetéből és valami
nagyon világosnak és tisztának kell jönnie. Már a finom fejléczeket is
látta, a melyeket a könyvéhez rajzol, felébredt régen elaludt
munkakedve és majdnem vidám lett.
Otthon nem várták, csendesen zárta ki a kaput, felosont a
szobájába, ledobta a kalapját, kabátját, leült az íróasztalához,
meggyujtotta a lámpát, állát gondolkodva támasztotta az
összekulcsolt kezére és maga elé nézett. A fény ráesett a
Bartholomé domborművének képére, a mely a régi helyén lógott,
szemben vele.

Mary rajta felejtette a pillantását. Egy fiatal férfi és egy nő aludt
egymás mellett holtan, keresztülfektetve rajtuk kis gyermekük. Ők
emelt fővel mennek be az elmúlás kapuján, míg körülöttük
roskadozva gyülekeznek öregek és fiatalok, más férfiak és nők, a
kiknek nem telt be a sorsuk úgy, mint nekik.
Mary nézte, nézte a képet, a lámpa bizonytalan fényében a férfi
lassan Oszkár lett, a nő olyan mint Judit és a gyermek az ő
gyermekük.
Piros rózsák nyíltak most is körülöttük a Mary képzeletében, mint
régen kinn a párisi temetőben… mi ehhez képest minden szó és
minden írás?
Mary lehajtotta a fejét az asztalra és sírt.

*** END OF THE PROJECT GUTENBERG EBOOK MARY ***
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