mathematics today 6 - kopykitab€¦ · unit review exercises help in evaluating the assimilation...
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S.K. GUPTAPrincipal (Retd.)
Birla Vidya Mandir, NainitalFormer Chairman
Indian Public Schools’ Conference
O.P. MALHOTRAM.A. (Gold Medalist)Former Head of the
Mathematics DepartmentThe Doon School, Dehradun
ANUBHUTI GANGALM.A. (Gold Medalist), M.Ed.
Formerly, Senior Faculty MemberThe Daly College, Indore
Birla Vidya Mandir, Nainital
Mathematics Today
Book
6
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First Published in 1979First ICSE Edition 2006Thoroughly Revised Edition 2012 Revised Edition 2014This CD Edition 2016
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Mathematics Today: 6
PREFACE TO THE REVISED EDITION
The authors wish to express their satisfaction and gratitude for the warmwelcome that has been accorded to this series all these years.
1. The text matter and answers have been thoroughly re-checked.
2. The special feature of this edition is the inclusion of Multiple ChoiceQuestions, Challengers [High Order Thinking Skills (HOTS)], Worksheets andChapter Tests.
3. The authors are grateful to all those teachers and students who have providedvaluable feedback for the improvement of the series.
AUTHORS
Disclaimer : While the authors of this book have made every effort to avoid any mistakes or omissions and have used their skill,expertise and knowledge to the best of their capacity to provide accurate and updated information, the authors and S. Chanddo not give any representation or warranty with respect to the accuracy or completeness of the contents of this publication and areselling this publication on the condition and understanding that they shall not be made liable in any manner whatsoever. S. Chandand the authors expressly disclaim all and any liability/responsibility to any person, whether a purchaser or reader of this publicationor not, in respect of anything and everything forming part of the contents of this publication. S. Chand shall not be responsible forany errors, omissions or damages arising out of the use of the information contained in this publication.Further, the appearance of the personal name, location, place and incidence, if any; in the illustrations used herein is purelycoincidental and work of imagination. Thus the same should in no manner be termed as defamatory to any individual.
The Most Popular Series which has been in use for more than 30 years
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A NOTE FOR THE TEACHERSDear friends,
We feel happy to be able to present for your perusal and consideration the new andthoroughly revised and updated editions of our Mathematics Today Series for Classes VI,VII and VIII. No doubt it has been possible as a result of the motivation and feedback receivedin the form of valuable comments, suggestions and criticism from the learned teachers. Westrongly feel that a textbook howsoever good it may be is only a tool to help teachers to teacheffectively. It is the teacher and only the teacher who is competent to decide his/her teachingstrategies in the classroom and is the best judge of how to use the textbook to meet the specialneeds of his/her class. It is earnestly hoped that this series will be able to supplement yourefforts effectively to create interest of your pupils in the subject and make the study ofmathematics interesting and enjoyable and gain mastery over the subject.
Howsoever best one performs or creates there is always scope for improvement. We wouldbe very happy rather grateful to receive your comments, appreciation/criticism andsuggestions for further improvement of the books.
With regardsYours sincerely
AUTHORS
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A NOTE FOR THE STUDENTSDear students,
Wishing you the best in life. You are the best judge to evaluate whether the book you arestudying fulfills your needs and satisfies your thirst for knowledge or not. Please do nothesitate to write individually through the publishers or directly to the authors at the followingaddress if you come across any discrepancies or if you have some suggestions to make.
AUTHORS
PREFACE
It gives us great satisfaction to be able to bring out this new version of our old MathematicsToday Series for Classes VI to VIII. The old series has been rehashed and redesigned incorporatingthe current global trends and International practices and the latest philosophy and policy of providingstress free education.
The salient features of this series are :
1. It follows strictly the new syllabus of the ICSE Council.2. All the mathematical concepts have been presented in a very simple and lucid form and loading
the course content with unnecessary and irrelevant details has been avoided. The approachand orientation is to lay a strong foundation for the students through adequate emphasis onthe fundamentals.
3. It aims at complete involvement of the pupils in the learning process. The emphasis throughoutthe text is on a student-centered performance and the activity approach is freely used relatingthe mathematical concepts to real life situations.
4. Every unit is introduced by a motivating paragraph or story.5. To facilitate easy and better understanding each unit is divided into a number of subunits
with short and separate practice exercises on each subunit.6. An attempt has been made to expose the children more fully to the ‘Why’ of various
operations and made abundant use of diagrams, illustrations, cartoons, tables and charts tostimulate the student’s interest in the subject and to clarify more difficult concepts.
7. Colour panels are used throughout as a teaching aid to emphasize important terms andrelationships and present useful tips.
8. The problems given in the books avoid tedious calculations and help in strengthening theunderstanding of basic principles honing the faculties of thinking and reasoning.
9. Each chapter contains a unit summary of key points at the end. It reviews the main pointscovered and helps the students in remembering them.
10. Unit review exercises help in evaluating the assimilation of the concepts learnt in the chapter.11. Mental maths exercises have been given to help the students acquire speed and sharpen their
intellect.12. A special feature is the inclusion of ‘Mixed Review Exercises’ which would keep the students
in constant touch with all the topics studied earlier.13. Historical Notes, Quizzes, Just For Fun, Puzzles and Enrichment Material offer further
intellectrual challenge to sharp students and help them not only to maintain their interest inthe subject and widen their horizon of knowledge but would also be of immense help inpreparing for such competitions such as Mathematics Olympiad at various levels.
It is hoped that this series of books will meet more than adequately, the needs of the studentsthey are meant for. Any suggestions for the improvement of the books would be most welcome andgratefully acknowledged.
AUTHORS
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SYLLABUSTeaching Points Teaching Notes
1. Sets
Idea of a set notation (a) Set as a well defined collection of distinct objects.
Notation (b) Roster - listing and set builder methods of representing sets.
Finite/Infinite sets (c) Denoting sets by capital letters and elements by small letters.
The empty set (d) geometric figures as sets of points.
Cardinal number of a set (e) Sets of numbers : N, W, I or Z.
(f) Symbols : e.g., { }, , or , ( ), ,n A∈ ∉ ∅ ∪ ∩
2. Numbers
Number systems Hindu -Arabic system of numeration. Face and place value.
Integers Operations, use of integers as directed numbers.
Fractions Proper, Improper, mixed, equivalent,. Operations.
Decimals Tenths, hundredths, thousandths only. Fundamental operations,Conversion of decimals to fractions and fractions to decimals
(terminating decimals only).
The number line Representing numbers on the number line (natural numbers, whole numbers, integers,fractions, decimals). Use in illustrating the fundamental operations and properties ofnumbers.
Factors and multiples Prime factorization, HCF and LCM
Powers and roots Exponential notations (positive exponents only). Squares, cubes etc. Square root andcube root of positive integers by factor method.
3. Arithmetical Problems
Ratio Simple and direct problems only.
Percentage (Pupils must be fully conversant with the measures of money, length, weightProfit and loss and time).Simple interest
4. Algebra
Fundamental concepts Pupils will be expected to be familiar with terms such as : term, like and unlike terms,monomial, bionomial, trinomial, constant, variable literal or numerical, coefficient,degree of a polynomial.
Fundamental operation Addition, subtraction of polynomials, multiplication of a polynomial and a monomial.Multiplication of two binomials. Use of brackets as grouping symbols. (Use ofBODMAS rule is not desired at this stage.)
Substitution Substitution in polynomials of degree 1 or 2 involving at most three unknowns.
5. Geometry
Fundamental concepts Pupils will be expected to be familiar with the idea of a point, line, ray, plane, space,line segment, triangle, rectangle, square, circle.
Lines Parallel and intersecting lines, perpendicular bisector of a line segment.
The following incidence properties are to be observed and subsequently assumed(axioms);
(a) One and only one line passes through two distinct points in a plane.
(b) Two different lines in a plane are either parallel or intersect in exactly one point.
Idea of collinearity of points and concurrency of lines.
Angles Concept of an angle. Vertex, arm or sides of an angle. Interior and exterior of anangle. Measurement of angle – degrees, minutes, seconds. Use of a protractor tomeasure an angle.
Types Acute, right, obtuse, striaght, and reflex angles. Adjacent angles vertically oppositeangles, complementary and supplementary angles. Alternate, corresponding, interior,exterior angles with reference to parallel lines.
Properties of angles (a) If two straight lines intersect, the adjacent angles are
and lines supplementary and vertically opposite angles are equal.
(b) If two parallel lines are cut by a transversal line -
(i) the alternate angles are equal.
(ii) the corresponding angles are equal.
(iii) the interior angles on the same side of the transversal are supplementary.
Constructions (a) Using ruler and compasses :
(i) an angle equal to a given angle.
(ii) bisection of angle.
(iii) angles of 30°, 60°, 90°, 45°, 120°, 135°
(iv) perpendicular bisector of a line segment.
(v) perpendicular to a line from a point not on the line.
(vi) perpendicular to a line at a point on the line.
(b) Using set-squares :
(i) a right angle.
(ii) angles of 30°, 60°, 90°, 75°, 105°
(iii) perpendicular to a line from a point outside the line.
(iv) perpendicular to a line at a point on the line.
Triangles Concepts; vertices, sides and angles of a triangle. Denoting angles of a triangle. Interiorand exterior angles of a triangle.
Types Scalene, Isosceles, equilateral, acute, obtuse and right triangles.
Property the angle sum property of a triangle.
Constructions Construction of traingles given –
(i) two sides and included angle.(ii) two angles and a side.
(iii) three sides.
The Circle Terms : centre, radius, diameter, circumference, chord, secant, tangent, arc, sector,segment. Interior and exterior of a circle.
Linear symmetry Symmetric and non-symmetric figures. Line or axis of symmetry.
(a) a point symmetric to a given point with respect to a given line of symmetry,
Constructions (b) the line of symmetry given two points which are symmetric with respect to theline of symmetry.
6. Mensuration
Recognition of solids Recognition of faces, edges, vertices (corners) of solids. Prism, pyramid, cube, cuboid.Perimeter and area Perimeter of square, rectangle, triangle. concept of area :measurement of area using squared paper. Area of rectangle and square only (Usingformulae for area).
Volume and surfaces Cubes and cuboids.
(Pupils will be expected to be familiar with abbreviations cm, m, km, cm2, m2,cm3, m3.)
7. Statistics Making column graphs, pie graphs and line graphs and drawing simple inferences.
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CONTENTS
UNIT I : SET CONCEPTS
1. SETS 1 – 15
UNIT II : PURE ARITHMETIC
2. NUMBER SYSTEM 16 – 29
3. THE INTEGERS 30 – 42
4. OPERATIONS ON INTEGERS 43 – 60
5. FACTORS AND MULTIPLES 61 – 77
6. FRACTIONS 78 – 100
7. DECIMALS 101 – 119
8. POWERS AND ROOTS 120 – 130
UNIT III : COMMERCIAL ARITHMETIC
9. RATIO AND PROPORTION 131 – 142
10. PERCENTAGE 143 – 152
11. PROFIT, LOSS AND DISCOUNT 153 – 161
12. SIMPLE INTEREST 162 – 166
UNIT IV : ALGEBRA
13. ALGEBRAIC EXPRESSIONS 167 – 177
14. OPERATIONS ON ALGEBRAIC EXPRESSIONS 178 – 191
15. LINEAR EQUATIONS IN ONE VARIABLE 192 – 201
UNIT V : GEOMETRY
16. BASIC GEOMETRICAL CONCEPTS 202 – 211
17. ANGLES 212 – 231
18. PARALLEL LINES AND TRANSVERSAL 232 – 243
19. PRACTICAL GEOMETRY (BASIC CONSTRUCTIONS) 244 – 254
20. CIRCLE 255 – 259
21. TRIANGLES 260 – 274
22. LINEAR SYMMETRY 275 – 281
UNIT VI : MENSURATION
23. PERIMETER AND AREA OF PLANE FIGURES 282 – 295
24. VOLUME AND SURFACE AREA OF SOLIDS 296 – 311
UNIT VII : STATISTICS
25. BAR GRAPHS, PIE GRAPHS AND LINE GRAPHS 312 – 329
PRACTICE TEST PAPERS 330 – 335
ANSWERS 336 – 366
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What is a Set ?All around us we see a collection of things. There may be a collection of books, a collection of photographs,
a collection of flowers, a collection of animals, etc. There may be a group of students or a group of teachers ora group of foreigners.
In mathematics, a collection or a group of things of any kind whatsoever is called a set. We may say ‘a setof pictures’, ‘a set of students’, ‘a set of animals’, etc.
The first picture shows a set of flowers.The second picture shows a set of children.The third picture shows a set of numbers.
Definition of a Set
A set is a well defined collection of objects.
What do we understand by “well defined”?By “well defined” we mean that it must be possible to tell beyond doubt whether or not a given object
belongs to the collection under consideration. Let us clearly understand the meaning of “well defined” with thehelp of the following examples.
1. If we consider the group of months whose names begin with M, then you know that March and May areincluded in this group, but June is not. This collection of months is well defined and so is a set.
2. The collection of numbers 1,3,5,7 and 9 is well defined and so is a set.3. The group of students of your class is well defined and so is a set.4. The collection of vegetables which taste good is not well defined as tastes differ from person to person.
Different people may include different vegetables in this collection, so it is not a set.5. Collection of all good movies is again not a set as good movies is a relative term. You may like a
particular movie but your friend may not, hence the collection will not be well defined.
Sets
UNIT I : SET CONCEPTS
1
1
EXERCISE 1 (A)
1. Which of the following collections are sets ? If not a set, give reasons.(a) Planets in our solar system. (b) Interesting books in the library.(c) Colours of a rainbow. (d) All difficult problems in your Maths book.(e) Top five wicket takers in Test Cricket. (f) Intelligent boys of your class.(g) Presidents of India. (h) Good football players.(i) Counting numbers between 30 and 40. (j) All good schools of India.(k) All vowels. (l) All yellow fruits.
2 Mathematics Today for Class VI [ICSE]
Membership of a SetThe objects which make up a set are called its members or elements.
Generally the elements of a set are written inside a pair of curly braces and are separated by commas. Thename of a set is always written in capital letters.
Here is a set of first four days of the week.A = {Monday, Tuesday, Wednesday, Thursday}Now instead of writing in full that Monday is an element of the set A we use the Greek letter “∈” (Epsilon)
and write Monday ∈ A. Just as the symbol “=” means “is equal to” in the same way, symbol ∈∈∈∈∈ means “is amember of ” or “belongs to”.For example :
a ∈ {a, b, c}, b ∈ {a, b, c}, c ∈{a, b, c}Not a member of
5 is not a member of the set {0, 2, 4, 6}. Thisstatement can be written as 5 ∉ {0, 2, 4, 6}. Thesymbol “∉∉∉∉∉” means “is not a member of.”
Examples :1. 7 ∈ the set of odd numbers. 2. 8 ∉ the set of odd numbers.
*3. –20 ∉ the set of whole numbers. 4. Apple ∉ the set of vegetables.5. d ∉ the set of vowels. 6. 1∉ {0, 2, 4, 6, 8}.
*7. 4 ∉ set of multiples of 3. 8. 9 ∈ set of square numbers.
Notation (Representation) of a SetA set can be represented with the help of the following methods :A. Description MethodB. Roster or Tabular MethodC. Rule or Set Builder Method
A. Description method : A set can be represented by describing it clearly and carefully in words.
* The set of whole numbers = W = { 0, 1, 2, 3, ....., }.The set of integers = Z = { ........ –3, –2, –1, 0, 1, 2, 3, .... }.The set of multiples of 3 = {3 × 1, 3 × 2, 3 × 3, .... }, i.e., {3, 6, 9, ...}.
EXERCISE 1 (B)1. State whether the following sentences are true or false :
(a) b ∉{a, e, i, o, u} (b) 6 ∈ the set of even numbers.(c) 13 ∉ the set of odd numbers. (d) Gold ∉{Silver, Platinum, Gold}.(e) Geetanjali ∈ set of books written by Tagore. (f) Rose ∉the set of all vegetables.(g) 1998 ∈ the set of leap years. (h) Moon ∈ the set of planets of the Solar System.
2. Let A = {odd numbers ≤≤≤≤≤ 19}. Insert the appropriate symbol ‘ ∈∈∈∈∈’ or ‘∉∉∉∉∉’ in blanks spaces :(a) 4 .............. A (b) 7 ............. A (c) 9 ............. A(d) 13 ............. A (e) 19 ............. A (f) 2 ............. A
3. Write these sentences in set language using the proper symbols ‘∈∈∈∈∈’ or ‘∉∉∉∉∉’ :(a) 2 is a member of the set of first five counting numbers. (b) d is a member of the set of consonants.(c) 32 is not a member of the set of multiples of 3 . (d) India is not a member of the set of continents.
�Symbol for
‘'is not a member'' of
Sets 3
For example :1. The set of students in your school who have their birthday in March.2. The set of letters in the English Alphabet.3. The set of counting numbers less than 20.
These examples can also be written with the help of curly brackets or braces { }. { } means 'the set of'.∴∴∴∴∴ Examples (1) — (3) can be written as :1. {all students in your school who have their birthday in March}2. {All letters in the English Alphabet}3. {All counting numbers less than 20}B. The Roster method : In this method, the elements of the set are separated by commas and listed within
braces.For example :
1. A = {1, 2, 3, 4, 5} or A = {5, 3, 1, 2, 4}.2. If P is the set of counting numbers less than 100, we may list the members of set P as {1, 2, 3, 4 ... 99}.
The three dots after 4 means that the members after 4 continue in the same manner until 99 is reached.3. Let N be the set of counting numbers. Then you can list N as N = {1, 2, 3, 4, …}.
Here the three dots mean that the numbers continue in the same manner without end.4. Let B be the set of letters in the word ‘floor’, then B is listed as B = {f, l, o, r}.From the above examples we arrive at the following important properties of a set :1. The set is represented by a capital letter.2. The elements of a set are written inside a pair of curly brackets separated by commas.3. If the elements of a set are alphabets then they are always written in small letters.4. The elements of a set can be written in any order.5. Repetition is not done while listing the elements, e.g., the set of letters in the word 'toffee' is
written as {t, o, f, e} and not as {t, o, f, f, e, e}.
C. The Rule method or Set Builder form : Here the set is described by stating a condition or propertywhich an object or element must satisfy if it belongs to that set.
More concisely, when the members of a set S possess a property P and x is only element of S, then we writethe set S as
S = {x | x has a property P} or S = {x : x has a property P} and read it as, S is the set of element x, such thatx has the property P. The vertical line or colon means 'such that'. This is called Set Builder form.For example :
1. The set of rivers in India can be written as{x | x is a river of India}.
EXERCISE 1 (C)1. Specify each of the following sets in the Roster form :
(a) The set of odd numbers less than 15. (b) The set of letters in the word ‘SATELLITE’.(c) The set of last three months of the year. (d) The set of letters in the word ‘PENCIL’.(e) The set of multiples of 7 less than 100. (f) The set of colours of a rainbow.(g) The set of vowels in the word ‘INDIA’. (h) The set of the first four multiples of 9.(i) The set of consonants in the word ‘AMERICA’.(j) The set of all days of the week beginning with letter S.
4 Mathematics Today for Class VI [ICSE]
EXERCISE 1 (D)
1. Write each given set in the Set Builder Form :(a) {4, 8, 12, 16, 20, 24, .........} (b) {1, 4, 9, 16, 25, 36, .........}(c) {May, March} (d) {Saturday, Sunday}(e) {January, March, May, July, August, October, December}.(f) Set of even numbers between 22 and 40. (g) Set of letters used in the word MUMBAI.(h) Set of colours of a rainbow. (i) Set of planets in our Solar system.
2. The set of all even numbers can be written as{x | x is an even number}.
3. A = {a, e, i, o, u, } can be written as A = {x | x is a vowel of the English Alphabet}.
TYPES OF SETS
Finite SetA set that contains countable number of different members is called a finite set. The members or
elements of this set can be counted with counting coming to an end.For example :
1. A = {1, 3, 5, 7} is a finite set.2. The set of oceans is a finite set.3. B = {2, 4, 6, 8, …, 100} is a finite set.4. C = {x | x is the capital of all states in India} is a finite set.
Infinite SetA set that contains unlimited number of different members is called an infinite set. In this type of
set, the process of counting of the members cannot come to an end.For example :
1. The set of natural numbers is an infinite set.2. Set of all points on a line segment is an infinite set.3. Set of stars in the sky is an infinite set.
Singleton SetA set containing only one element is called a singleton set.
For example : {7} is a singleton set, containing only one element, namely 7.
Null SetA set that contains no member is called the null set. It is also called the empty set.
For example :The set of months in a year that have less than 20 days has no members. Therefore, it is a null set.The null set is denoted by { } or φ. (φ is pronounced as phi).Consider the following examples :1. The set of trees 1 km high = φ.2. If A = the set of pupils more than 150 years
old in your class, then A = { } or A = φ, sincethe set A has no member.
{0} and {φ} are not empty sets becauseeach of these sets contains one element.
or φ{ }
Note
Mathematics Today For Class 6
Publisher : SChand Publications ISBN : 9789352530625Author : S K Gupta AndAnubhuti Gangal
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