6

Maths
Quest
Teacher’s
Manual
S Purkayastha

(An imprint of New Saraswati House (India) Pvt. Ltd.)

New Delhi-110002 (INDIA)
 R

(An imprint of New Saraswati House (India) Pvt. Ltd.)

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First published 2016

ISBN: 978-93-5199-701-6

Published by: New Saraswati House (India) Pvt. Ltd.

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 Preface
The Math Quest Teacher’s Resource Pack is based on guidelines and aids to support
and supplement classroom teaching. The aim of this pack is to empower teachers so
that the process of teaching and learning becomes interesting and interactive. The tools
and techniques provided will ensure a seamless flow of knowledge so that the students
take an inherent interest in the subject. The main purpose of the pack is to allay the fear
of Maths from the minds of the students such that they develop an inherent liking for
the subject and become curious to know more. A wide array of resources are included
in the Teacher’s Resource Pack to provide maximum support to teachers.
The main components of the Teacher’s Resource Pack are as follows.

Teacher’s Manual
Teacher’s Manual has been developed to provide teaching guidelines to teachers so that
they are prepared to teach a topic in the best possible manner. The manual comprises
detailed lesson plans, which are supported by ample practice material in the form of
Worksheets and Model Test Papers and their answers. There is a Teacher’s CD as a
digital support so that students are familiarised with the modern ways of teaching.

Lesson plans
Each lesson plan explains each topic in detail. Its components are as follows.
• Learning objectives list out the measurable aims of each chapter, which should
be achieved after teaching the chapter.
• Concept explanation gives a detailed method of explaining the important
concepts of the chapter using various teaching aids.
• Reinforce puts emphasis on important points that should not be missed while
teaching.

Practice material
Worksheets and Model Test Papers along with their answers form the part of the
practice material. These ensure that the students learn to solve the questions based on
the concepts taught. This will help students have a good base right from the beginning
on tackling tricky questions.

Teacher’s CD
Teacher’s CD comprises flip book, animated concepts, interactive activities, lesson
plans, along with worksheets and Model Test Papers and their answers.

Web Support
The web support consists of worksheets, Model Test Papers, and answers to worksheets
and Model Test Papers. These would help teachers in assessing students on the concepts
taught in the class.
Contents
1. Knowing Our Numbers 5
2. Whole Numbers 10
3. Playing With Numbers 14
4. Integers 20
5. Fractions 24
6. Decimals 29
7. Introduction to Algebra 34
8. Algebraic Equations 38
9. Ratio, Proportion and Unitary Method 42
Model Test Paper 1 46
10. Basic Geometrical Concepts 48
11. Understanding Elementary Shapes 52
12. Understanding Three-Dimensional Shapes 56
13. Practical Geometry 60
14. Symmetry 65
15. Mensuration 69
16. Data Handling 73
Model Test Paper 2 77
Answer Key 81
 1 Knowing Our Numbers

Learning Objectives
Students will be able to
➢ recapitulate the concept of natural numbers,whole numbers
➢ understand Indian and International systems of numeration
➢ find place value and face value, successor and predecessor
➢ understand how to compare and order the given numbers
➢ form numbers, greatest and the smallest numbers using the given digits
➢ find how many numbers are there between two given numbers
➢ use large numbers in real life
➢ apply operations in large numbers
➢ round off numbers and estimate the sum, difference, product and quotient
➢ recognise Roman numerals and rules to form Roman numerals

Concept Explanation
• Students are already familiar with the large numbers.
• Read the Introduction section to recapitulate these concepts.
• Explain to students the difference between natural numbers and whole numbers and let
them understand that all natural numbers are whole numbers but all whole numbers are
not natural numbers, thus focusing on exception of zero.
Indian System of numeration (Place value, face value, expanded form); International
System of numeration (Place value, face value, expanded form)
• Make the students understand the difference between the terms notation and numeration.
• Have a quick recap of the Indian and International systems of numeration and the
common place values in both, that is, up to 5 places. Make the students understand
that after the 5th place, the places in both the systems are called by different names.
Emphasise on the use of commas in both the systems at different places in both the
systems and on how to read the numbers in both the systems.
• Students have already learnt place value and face value up to 8-digit numbers in their
previous class. So just do a quick recall and then extend the concept towards bigger
number.
• To reinforce, ask them to do Check Point 1.1 from the textbook.

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Successor and Predecessor; Comparison of numbers; Ordering of numbers
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• Rules of comparing numbers and successor and predecessor need to be applied to bigger
numbers as well.
• After making them understand the comparison of numbers, conduct an activity with
students wherein they will reinforce ascending and descending orders.
• Bring 10 cm × 10 cm pieces of thick chart paper/cardboard of different colours, and one
big chart paper to the class.
• Cut out 10 cm × 10 cm pieces of thick chart paper/cardboard of different colours (one for
each student).
• Divide the class into five groups and give different coloured chart papers/cardboard
pieces to each group. For example, group A gets green, group B gets red and so on.
• Give a certain range of numbers to each group, for example, group A gets 100-1000,
group B gets 1000-10000 and so on.
• Each member writes any number of his/her choice that falls within the specified range
on the piece of chart paper in figures and in words.
• On a big chart paper, draw 10 cm × 10 cm squares (as many squares as the number of
students) in five columns. Students of group A will then go and paste their slips (with the
written numbers) in their column in ascending order.
• Students of other groups will also follow the steps taken by group A.
• This will continue till all squares are filled with multicoloured pieces.
• To reinforce, ask them to do Questions 1, 6, 7 and 8 of Check Point 1.2 from the textbook.
Forming numbers using the given digits; Writing the greatest and smallest numbers the
using given digits; Finding how many numbers are there between two numbers
• Read the related sections from the textbook.
• Explain to students that the largest number using the given number of digits is formed
by arranging the given digits in descending order and the smallest number using the
given digits is formed by arranging the digits in ascending order, for example, if they
have to use the digits 9,8,5,1,2,6,7,4,3,0 to make the largest and the smallest number
using all digits, then the largest number will be 987653210 and the smallest number will
be 1023456789 (without repeating any digit).
• Now let students find out how many numbers are there between two given numbers, for
example, 20-30 by counting orally or by writing and then tell the simple way to find the
numbers using a simpler method as mentioned below.
• First find the difference between the numbers, then:
(a) Add 1 to the difference, if both the numbers are included.
(b) Subtract 1 from the difference, if both the numbers are excluded.

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+% • To reinforce ask the students to do Questions 2, 3, 4, 5 and 10 to 16 of Check Point 1.2
from the textbook.
Use of large numbers in daily life
• Read the related sections from the textbook.
• Students have enough understanding of bigger numbers now. So make them understand
the application of bigger numbers in real life.
• Make students attempt some more real life applications of bigger numbers, for example,
conversion of units of length, weight and capacity, etc. and the word problems based on
the same. Make them practise the four operations involving bigger numbers.
• To reinforce, ask them to do Maths in Real Life and Values sections from the textbook.
• To reinforce, ask the students to do Check Point 1.3 from the textbook.
Estimation of numbers; Estimating sum, difference, product and quotient; Roman
numbers
• Explain the method of rounding off numbers to their nearest tens, hundreds and
thousands and explain to students that we can also estimate the sum, difference, products
and quotients by first rounding off the numbers involved and then rounding them off.
• Students have learnt enough about the 7 symbols of the Roman system and the different
rules to form the same. Have a revision of their understanding so far by giving them
some numbers and asking them to convert them into Roman numerals and vice versa.
Now make them understand the meaning and usage of the bar used in Roman numbers
by illustrating a few examples on the board.
• For better understanding of the concept of Roman numerals, do the Maths Lab Activity
section from the textbook.
• Ask the students to do Check Point 1.4 and Check Point 1.5 from the textbook.
To revise the concepts learnt in the chapter, students will do Test Yourself and Brain Workout
sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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Worksheet 1
1. Complete the following statements with suitable words or figures.
(a) 1 crore = ________ lakh
(b) 1 million = ________ thousand
(c) 1 lakh = ________ thousand
(d) 1 crore = ________ ten lakh
(e) 1 billion = ________ million
(f) 2 crore = ________ million
(g) 5 km = ________ m
(h) 2 mm = ________ cm
(i) 5 g = ________ kg
(j) 486 L = ________mL
(k) 89 cm = ________m
(l) There are only seven symbols in ________ numerals. (Roman/Hindu-Arabic)
(m) MMM = 3 × ________
(n) 400 is represented by ________ in Roman system of numeration.
2. Do as directed.
(a) Write:
(i) 1 lakh in thousands (ii) 1 million in lakhs (iii) 100 million in crores
(b) Write the smallest whole number. Is it a natural number also?
(c) Write the smallest natural number. Is it a whole number?
(d) Write place-value and face value of 2 in 7852146.
(e) Write the number 90 lakhs 9 thousands 9 ones in International system of
numeration.
3. (a) Estimate the sum (395 + 170) to the nearest hundred.
(b) Estimate the difference (47029 – 39385) to the nearest thousand.

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+ % Worksheet 2
1. From the numeral 40256801, pick out the digits in each of the following place.
(a) Thousands = ________ (b) Ten lakhs = ________
(c) Hundreds = ________ (d) Crores = ________
2. Write the greatest and the smallest 5-digit numbers using the digits
(a) 0, 2, 6, 7, 8 only once (b) 1, 9, 3, 7, 4 only once
3. Find the difference between the greatest and the smallest numbers that are formed
using the digits 6, 5, 2, 0, 9, 7 only once.

___________________________________________________________________
4. The population of a city is 5236845. If the number of males is 2469850, find the
number of females in the city.

___________________________________________________________________
5. Find the estimated sum of 5486 and 62548 by estimating the numbers to their
nearest (a) hundreds and (b) thousands. Also, find the difference between the
estimated sum and the actual sum.

___________________________________________________________________
6. Estimate the following by rounding off each number to its greatest place.
(a) 256 + 458 + 8457 (b) 1265 + 8754 – 3564
7. Convert the following into Hindu-Arabic numerals:
(a) DCLVIII (b) DXV (c) MCDXXV (d) CDXLV
8. State whether the following statements are True or False.
(a) One million is equal to ten lakh.
(b) The smallest 6-digit number using the digits 0 and 1 with repetition of digits is
110000.
(c) The place value and face value of the digit 1 in the number 12548 is same.
(d) The Roman numeral CCLIV in Hindu-Arabic numeral is 254.
(e) Number of ten thousand in 100 millions is ten thousand.
9
 2 Whole Numbers

Learning Objectives
Students will be able to
➢ understand whole numbers, successor and predecessor of whole numbers
➢ compare, add, subtract, multiply and divide the whole numbers
➢ learn the properties of addition, subtraction, multiplication and division of whole
numbers
➢ identify different patterns in whole numbers

Concept Explanation
• Students are already familiar with the natural numbers and can apply the basic operations
on these.
• Read the Introduction section to recapitulate these concepts.
• Explain to students the collection of natural numbers i denoted by N. The number 1 is
the smallest natural number and there doesn’t exist largest natural number.
The number zero; Whole numbers
• Explain to the students how a whole number differs from a natural number. Let the
students understand that all natural numbers are whole numbers but all whole number
are not natural number. Hence, focus on zero (0).
• Explain to the students the meaning of 0 which means absence of item or no item.
• Tell students why the whole number 0 does not have any predecessor and there is no end
of successor. Make the student understand that adding 1 to any number we get the next
number called successor and on subtracting 1 from any number (except 0) we get the
previous number called predecessor.
• Use related examples to make the students understand the concept.
• Make them understand how we can represent whole numbers on a number line.
• Ask the students to do Check Point 2.1 from the textbook.
Properties of addition; Properties of subtraction; Properties of multiplication;
Properties of division
• Use the related examples to explain to the students the properties of whole numbers.
• Explain why the closure property of subtraction and division does not hold good for a
whole number. Also, explain why commutative law and associative law do not hold in
subtraction and division of whole numbers.
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+% • Students have enough understanding of commutative law of multiplication of whole
numbers.
• Now, make them understand the application of commutative law in real life by making
them do an activity.
• For this form two groups. Ask one group to arrange the chairs in the class in 6 rows with
8 chairs in each row. Therefore, the number of chairs in the arrangement is 6 × 8 = 48.
• Ask other group to arrange the chairs in the class in 8 rows with 6 chairs in each row.
Therefore, the number of chairs in the arrangement is 8 × 6 = 48.
• Ask both the groups to observe in both the arrangements that the total number of chairs
are same, that is, 6 × 8 = 8 × 6 = 48, which verifies that multiplication of whole numbers
is commutative.
• Make students attempt some more activity by arranging the chairs to verify commutative,
associative property of addition, distributive property of multiplication over addition,
etc.
• To reinforce, ask the students to do the Maths Lab Activity, Values and Maths in Real Life
sections from the textbook.
• Ask the students to do Check Point 2.2, Check Point 2.3, Check Point 2.4 and Check
Point 2.5 from the textbook.
Patterns in whole numbers
• Read the related section from the textbook.
• Tell students how to represent whole numbers using dots in the form of triangular,
rectangular, square patterns.
• Make students understand the use of patterns like addition of 9, 99, 999, …, etc.,
subtraction of 9, 9, 99, 999, …, Multiplying by 9, 99, 999 … and division by 5, 25, 125, …
etc., of a whole number for quick and smart calculations.
• Instruct the students to do Check Point 2.6 from the textbook.
To revise the concepts learnt in the chapter, students will do Test Yourself and Brain Workout
sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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Worksheet 1
1. Fill in the blanks.
(a) (200 + 5) × (100 – 5) = ______ × ______
(b) 666 + 555 + 444 = ______ × 120
(c) 225 × 55 = (200 + ______) × (50 + ______)
(d) 76 × (100 – 3) = 76 × ______
(e) (10 + 6) (10 – 6) = ______ – 36
2. Write the next three natural numbers after 10999.
___________________________________________________________________
3. Which is the smallest whole number?
___________________________________________________________________
4. How may whole numbers are their between 35 and 58?
___________________________________________________________________
5. Write the successor of:
(a) 428301 (b) 199001 (c) 45667236
6. Write the predecessor of:
(a) 63 (b) 100 (c) 9612054
7. Find the sum by suitable arrangement.
(a) 837 + 208 + 363 (b) 1692 + 345 + 1358 + 745
8. Find the product by suitable arrangement.
(a) 16 × 625 × 297 (b) 4 × 166 × 25 (c) 8 × 125 × 431
(d) 2 × 1695 × 50 (e) 125 × 40 × 8 × 25
9. Simply: 125 × 55 + 125 × 45
___________________________________________________________________
10. The school canteen charges `20 for lunch and `4 for milk for each day. How much
money would you spend in 5 days on these things?
___________________________________________________________________
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+ % Worksheet 2
1. Which of the following statements are true (T) and which are false (F)?
(a) Zero is the smallest whole number. ________
(b) 600 is the predecessor of 599. ________
(c) Zero is the smallest natural number. ________
(d) 599 is the successor of 600. ________
(e) All the natural numbers are whole numbers. ________
(f) All whole numbers are natural number. ________
(g) The predecessor of a 2-digit number is never a 1-digit number. ________
(h) The natural number 1 has no predecessor. ________
(i) The whole number 1 has no predecessor. ________
(j) The whole number 0 has its predecessor. ________
(k) The successor of a 2-digit number is always a 2-digit number. ________
2. Study the given pattern and fill in the blanks.
(a) 1 × 8 + 1 = 9
(b) 12 × 8 + 2 = 98
(c) 123 × 8 + 3 = 987
(d) 1234 × 8 + 4 = 9876
(e) 12345 × 8 + 5 = 98765
(f) ______ × 8 + 6 = ______
(g) _______ × 8 + 7 = _______
3. Give two examples of each of the following properties of whole numbers.
(a) Closure property of addition and multiplication.
(b) Commutative property of addition and multiplication.
(c) Associative property of addition and multiplication.
(d) Distributive property of multiplication over addition and subtraction.
4. 15 laddoos can be packed in box. How many boxes are required to pack 200
laddoos?

___________________________________________________________________
13
 3 Playing With Numbers

Learning Objectives
Students will be able to
➢ simplify the numerical expressions, perform mathematical operations in the correct
order (the rule of DMAS)
➢ simplify the numerical expressions having brackets in the correct order (the rule of
BODMAS)
➢ find the factors and multiples of the given numbers
➢ identify prime and composite numbers from 1 to 100
➢ define twin primes, prime triplets, perfect number, co-primes and Goldbach’s
conjecture
➢ learn when a number is divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11
➢ learn the general properties of divisibility
➢ examine if a given number is prime or composite
➢ know the concept of prime factorisation
➢ know about common factors and highest common factors (HCF)
➢ find HCF by prime factorisation method and division method
➢ know about common multiples and least common multiples (LCM)
➢ find LCM by prime factorisation method and division method
➢ properties and relation between the HCF and LCM of given numbers

Concept Explanation
• Students are already familiar with the whole numbers, properties of addition, subtraction,
multiplication and division and can apply the basic operations with whole numbers.
• Read the Introduction section to recapitulate these concepts.
Simplification of numerical expressions; Order of operations; Simplification of brackets
• Explain to the students how and why the DMAS and BODMAS rules should be followed.
If we do not follow the rules, then we will get different answers. To get the correct answer
while simplifying, we have to follow the specified rules.
• Make the students understand the meaning of DMAS, i.e., simplifying numerical
expressions in the order Division, Multiplication, Addition and Subtraction. Also,
BODMAS stands for Bracket Of division, multiplication, addition and subtraction.

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+% • Use the examples given in the textbook to better understand these concepts to the
students.
• To reinforce, ask the students to do Check Point 3.1 from the textbook.
Factors and multiples; Prime and composite numbers
• Let students understand the meaning of factors and multiples of any number, so that
they can distinguish between factors and multiples.
• Tell the students about the twin primes, prime triplet, perfect number, co-prime number.
Now students have learnt enough to understand that the prime number are always
co-prime but the co-prime numbers may or may not be prime. For example, 1, 5 and
7 are prime as well as co-prime, but 20, 21 are co-primes but both the numbers are
composite.
• Explain the procedure of finding the prime numbers and composite numbers between 1
and 100 by the method known as the Sieve of Eratosthenes.
• Draw their attention to the fact that two prime numbers are always co-primes but two
co-primes need not be both prime numbers.
• Instruct the students to do Check Point 3.2 from the textbook.
Tests of divisibility; To examine if a given number is prime or not; Prime factorisation;
• Explain to students why the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
59, 61, 67, 71, 73, 79, 83, 89, 97 are prime numbers by test of divisibility.
• Also, explain that the other numbers except 1, are composite numbers by the test of divisibility.
• Make students understand clearly the test of divisibility using related different examples
from the textbook.
• How does test of divisibility help students to find whether the given number is divisible
by a specific number.
• Explain the general properties of divisibility, i.e.,
– If a number is divisible by other number, then the first number is divisible by every
factor of the second number.
– If a number is divisible by two or more co-prime numbers, then the number must be
divisible by their product.
– If a number is the factor of two or more numbers, then the number is the factor of
their sum.
– If a number is the factor of two given numbers, then the number is also the factor of
their difference.
• To examine the test the divisibility of the given number by the prime numbers unless
we get a quotient which is less than the next prime divisor, then the number is prime,
otherwise composite.
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• Explain to express a given number as a product of prime factors is called prime
%+
factorisation or complete factorisation of the given number.
• Use the illustrative examples given in the related topic to make them better understand
the concept.
• Instruct students to do Check Point 3.3 and Check Point 3.4 from the textbook.
Common factors; Highest common factors; Finding HCF by prime factorisation method
and division method
• Read the related sections from the textbook.
• Use the related illustrative examples to make the students understand these concepts.
• Explain the HCF or GCD is the greatest number which divides two or more numbers
without leaving any remainder.
• Make a note for the point that if two numbers have no common factor except 1 (co-
prime numbers), then their HCF is 1.
• Discuss the step by step procedure of Euclid’s Algorithm of finding HCF of two or more
numbers. Give examples so that students can understand the method and will be able to
solve independently.
• Ask the students to focus their attention to the point, i.e., to find the HCF of three
numbers, we first find the HCF of two numbers. Then, we find the HCF of the third
number and HCF of the first two numbers already found.
• To reinforce, ask students to do the Maths Lab Activity, Values and Maths in Real Life
sections from the textbook.
• Instruct the students to do Check Point 3.5 from the textbook.
Common multiples; Least common multiples; Finding LCM by prime factorisation
method and long division method; Properties of HCF and LCM of given numbers and
relation between them
• Read the related sections from the textbook.
• Use the related illustrative examples to make the students understand these concepts.
• Explain the least common multiple (LCM) of two or more numbers is the smallest
number which is divisible by each of the given numbers.
• Discuss the step by step procedure while finding LCM of two or more numbers by prime
factorisation method and division method. Use the related examples so that students
can understand the method and will be able to solve independently.
• Define the properties of HCF and LCM of the given numbers as follows:
(i) The HCF of the given numnbers cannot be greater than any one of the numbers.
(ii) The LCM of the given numbers cannot be less than any one of the numbers.

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+% (iii) The LCM of two co-prime numbers is equal to their product.
(iv) The HCF of two co-prime numbers is equal to 1.
(v) The HCF of a group of numbers is always a factor of their LCM.
(vi) If ‘a’ and ‘b’ are the given numbers such that ‘b’ is a multiple of ‘a’, then, their HCF
is ‘a’ and their LCM is ‘b’.
• Write, product of two numbers = Product of their HCF and LCM.
• Strengthen the concept with the help of related examples given in the textbook.
• Instruct the students to do Check Point 3.6 from the textbook.
To revise the concepts learnt in the chapter students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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Worksheet 1
1. State whether the following statements are true or false:
(a) The sum of three odd numbers is even.
(b) The sum of two even numbers and one odd number is even.
(c) The product of three even numbers is even.
(d) The product of two odd numbers and one even number is odd.
(e) When an odd number is divided by 2, the quotient is always even.
(f) All prime numbers are odd.
(g) Prime numbers do not have any factors.
(h) Sum of two prime numbers is always even.
(i) All even numbers are composite numbers.
(j) Number which is divisible by 2 is also divisible by 4.
(k) Number which is divisible by 9 is also divisible by 3.
(l) Number which is divisible by 2 and 3 is also divisible by 6.
(m) Number which is divisible by 15 is also divisible by 3 and 5.
(n) The product of two even numbers is always even.
2. The numbers 13 and 31 are prime numbers. Both these numbers have same digits
1 and 3. Find such pair of prime numbers up to 100.

___________________________________________________________________
3. Find the HCF and LCM of the following numbers by the prime factorisation
method.
(a) 8, 12 (b) 20, 35 (c) 33, 36, 39 (d) 15, 20, 25
4. Find the HCF and LCM of the following numbers by the division method.
(a) 10, 70 (b) 25, 50, 75 (c) 14, 42, 98 (d) 30, 90, 120, 150

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+ % Worksheet 2
1. Using divisibility tests, determine which of the following numbers are divisible by
the given numbers as directed.
Number Divisibility test
By By By By By By By By By
2 3 4 5 6 8 9 10 11
128 Yes No Yes No No Yes No No No
990
1586
275
6686
639210
429714
2856
3060
406839
2. Find the HCF and LCM of the following numbers by the prime factorisation
method.
(a) 30, 45 (b) 169, 52 (c) 100, 75, 40 (d) 150, 200, 225
3. Find the HCF and LCM of the following numbers by the division method.
(a) 12, 60 (b) 30, 60, 90 (c) 18, 36, 50, 68 (d) 7, 28, 42, 35, 63
4. The HCF of two numbers is 25. If the numbers are 75 and 100, find their LCM.

___________________________________________________________________
5. Find the greatest number of 5 digits which is divisible by 5 and 10 both.

___________________________________________________________________

19
 4 Integers

Learning Objectives
Students will be able to
➢ recapitulate the concept of natural numbers and whole numbers
➢ learn about integers, negative integers and positive integers
➢ know how integers are related in our daily life
➢ represent integers on a number line
➢ learn about opposite of integers, comparing and ordering integers
➢ find successor and predecessor of integers, and absolute value of an integer
➢ learn the concept of addition of integers on a number line, rules for addition of
integers and properties of addition of integers
➢ learn the concept of subtraction of integers on a number line, rules for subtraction of
integers and properties of subtraction of integers

Concept Explanation
• Students are already familiar with the natural numbers, whole numbers, properties of
addition, subtraction, multiplication and division and can apply the basic operations
with whole numbers.
• Read the Introduction section to recapitulate these concepts.
Integers; Representation of integers on the number line; Absolute value of an integer
• Read the related sections from the textbook.
• Explain that the collection of integers is denoted by Z or I.
Thus, Z = ..., – 4, – 3, – 2, – 1, 0, +1, +2, +3, +4, ... .
• Use blackboard to explain why Z = {Z–} {0} {Z+}.
• Explain to students what is integer and what is the need of integers. We have seen that
subtraction on whole numbers does not hold good for closure property, also subtraction
on whole numbers does not hold for commutative, associative properties. This difficulty
is overcome by the introduction of integers.
• Demonstrate on the blackboard how an integer can be represented on a number line.
• Draw their attention to the fact,i.e., the number zero (0) is neither positive nor negative.
It is called a non-negative integer.
• Every integer has its opposite except zero.The opposite integers whose sum is zero are
called additive inverse.
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+% • Every integer has its successor or predecessor. The smallest positive integer is 1, but
the smallest negative integer is not known. The greatest negative integer is —1, but the
greatest positive integer is not known.
• Explain that the absolute value of an integer is the numerical value of the integer.
• Use illustrative examples given in the textbook to better understand the concepts.
• To reinforce, ask the students to do Check Point 4.1 from the textbook.
Addition of integers; Properties of addition of integers
• Read the related sections from the textbook.
• Use the Maths Lab Activity section to make the students understand the concept.
• Explain the rules associated with the addition of integers
• With the help of related illustrative examples make them better understand the concept.
• Define different properties of addition of integers with the help of examples given in this
section.
• For more practice ask them to do Check Point 4.2 from the textbook.
Subtraction of integers; Properties of subtraction of integers
• Read the related sections from the textbook.
• Conduct an activity similar to addition of integers to make the students understand the
concept of subtraction of integers.
• Explain the rules associated with the subtraction of integers
• With the help of related illustrative examples make them better understand the concept.
• Define different properties of subtraction of integers with the help of examples given in
this section.
• To reinforce, ask the students to do the Values and Maths in Real Life sections from the
textbook.
• For more practice, ask them to do Check Point 4.3 from the textbook.
To revise the concepts learnt in the chapter, students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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Worksheet 1
1. Represent the following numbers as integers with appropriate signs.
(a) An aeroplane is flying at a height of two thousand and five hundred metres above
the sea level.
(b) A submarine is moving at a depth of seven hundred fifty metres below the sea
level.
(c) A deposit of rupees five hundred.
(d) A withdrawal of rupees six hundred.
2. Following is the list of temperatures of five places in India on a particular day of
the year. Write these temperatures with appropriate sign in the blanks given.
Place Temperature
a. Shimla 2°C below 0°C _________
b. Agartala 30°C above 0°C _________
c. Delhi 20°C above 0°C _________
d. Shrinagar 7°C below 0°C _________
e. Mumbai 25°C above 0°C _________
3. Write all the integers between the given pairs of integers.
(a) 0 and –7 (b) –4 and 4 (c) –16 and –8 (d) 51 and 39
(e) –31 and –25
4. Write the absolute values of the following integers.
(a) 10 (b) –21 (c) –111 (d) 275
5. Write the opposite of the following integers.
(a) +25 (b) +54 (c) –87 (d) –123
6. Simplify.
(a) (–74) + (–34) (b) 500 + (–55) – (–100)
7. Evaluate.
(a) |– 41 – 9| – |16 – (–5)| (b) |– 78 – (– 11)| + |– 68 + 29|
8. Subtract the sum of – 35 and 12 from the sum of – 29 and – 90.
9. On a day Amar earns a profit of `1000 on the sale of a refrigerator and loses
`300 on the sale of a camera. Find what is Amar’s actual profit or loss?
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1. Write the solution of the following using a number line.
(a) (+7) + (–11) (b) (–13) + (+10) (c) (–7) + (–9) (d) (+7) + (–10)
2. Find the sum of:
(a) 137 and – 314 (b) – 52 and 52
(c) – 312, 39 and 192 (d) 37 + (– 2) + (– 65) + (– 8)
3. Subtract.
(a) 6 from 9 (b) – 14 from 25 (c) – 23 from – 36 (d) 31 from – 31
4. Fill in the blanks.
(a) – 8 + ___ = 0 (b) 13 + ___ = 0
(c) 15 + (–15) = ___ (d) (–4) + ___ = 15
(e) ___ – 16 = – 10 (f) 26 + ___ = –26
5. Fill in the blanks with >, < or = sign.
(a) (–4) + (–6) ___ (–4) – (–6)
(b) (–21) + (–10) ___ (–31) + (–8)
(c) 45 – (–11) ___ 57 + (–4)
(d) (–52) – (–24) ___ (–24) – (–52)
6. Simplify.
(– 4) + 6 + (– 4) + 33 + (– 23) + 24 + (– 26)
7. The sum of two integers is 50. If one of them is 100, find the other.

___________________________________________________________________
8. If P – (–7) = –1, find the value of P.

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23
 5 Fractions

Learning Objectives
Students will be able to
➢ recapitulate the meaning of fractions both as a part of a whole and a part of a
collection, and numerator and denominator of a fraction
➢ represent fractions on a number line
➢ learn more about proper, improper and mixed fractions
➢ convert a mixed fraction into an improper fraction and vice versa
➢ understand the concept of equivalent fractions
➢ reduce a fraction into its lowest term
➢ Identify like and unlike fractions
➢ compare like fractions and unlike fractions
➢ add and subtract like fractions and unlike fractions

Concept Explanation
• Students are already familiar with the fractions as a part of a whole and a part of a
collection, and numerator and denominator of a fraction.
• Read the related sections to recapitulate these concepts.
• Use examples given in these sections to strengthen the concepts.
• To reinforce, ask the students to do Check Point 5.1 from the textbook.
Integers; Representation of integers on the number line; Absolute value of an integer
• Use blackboard to explain the process of representation of fractions on a number line.
• Call out some fractions and ask students to represent these on the number line.
• Clear the concept of proper, improper and mixed fractions.
• In a proper fraction, if the numerator is 1, it is called a unit fraction. For example,
1 , 1, 1 ,...,etc.
10 5 50
• The value of a proper fraction is less than 1 while the value of an improper fraction is
always greater than 1.
• Also, let them understand the process of conversion of mixed fractions into improper
fractions and improper fractions into mixed fractions.
• Use the illustrative examples given in the textbook to strengthen these concepts in the
students.
24
 0 ÷
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+% • Instruct the students to do Check Point 5.2 and Check Point 5.3 from the textbook.
Equivalent fractions; Reducing a fraction into to its lowest term
• Read the related sections from the textbook.
• Define two or more fractions representing the same part of a whole as equivalent
fractions.
• Use the cut-outs of circles having different equal parts but each shaded with equal parts
to demonstrate the concept of equivalent fractions.
• Let them understand the process of converting the fraction to its equivalent fraction by
reducing it into lowest term. The process can be used to
– express the numerator and denominator as the product of prime factors and cancel
the common factors to obtain the fraction in its lowest term.
– find the HCF of numerator and denominator. Then, divide the numerator and
denominator by their HCF.
• Go through the illustrative examples and ask them to do Check Point 5.4 and Check
Point 5.5 from the textbook.
Like and unlike fractions; Comparing like and unlike fractions
• Define the like and unlike fractions. Give some fractions and let them distinguish
between like and unlike fractions.
• Discuss the process of converting the unlike fractions into like fractions.
• Discuss all the process of comparison of two fractions as follows.
– Comparing the fractions with same denominator.
– Comparing the fractions with same numerator.
– Comparing unlike fractions by LCM method and cross-multiplication method.
• Use illustrative examples of the related sections to make them understand these concepts.
• To reinforce, ask the students to do Check Point 5.6 from the textbook.
Operations on fractions
• Read the related section from the textbook.
• Discuss the process of operation of fractions as
Addition or subtraction of like fractions = Sum or difference of the numerators
Common denominator
Addition and subtraction of unlike fractions by changing the given unlike fractions into
equivalent like fractions, or by taking LCM of denominators.
• Tell students that to add or subtract mixed fractions having the same denominator, first
convert them into improper fractions and then add or subtract as we add or subtract like
fractions.
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• Use illustrative examples to make the students understand these concepts thoroughly.
%+
• To reinforce, ask the students to do the Maths Lab Activity, Values and Maths in Real Life
sections from the textbook.
• Instruct the students to do Check Point 5.7 and Check Point 5.8 from the textbook.
To revise the concepts learnt in the chapter, students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths section to do a quiz contest in the
class.
Use the At a Glance section to revise the key points of the concepts.

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 0 ÷
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+ % Worksheet 1
1. Draw number lines and represent the following fractions on them.
(a) 1, 1, 3, 4 (b) 1, 3, 5, 7 (c) 2, 3, 8, 11
2 4 4 4 8 8 8 8 5 5 5 5
2. Express the following as improper fractions.
(a) 73 (b) 57 (c) 26 (d) 102 (e) 93 (f) 85
4 8 7 3 7 9
3. Find the equivalent fraction of 3 having
5
(a) denominator 20 (b) numerator 9 (c) denominator 30
(d) numerator 27 (e) denominator 45

4. Find the equivalent fraction of 36 with

48
(a) numerator 9 (b) denominator 4
5. Reduce the following fractions into simplest form.
(a) 45 (b) 150 (c) 84 (d) 16 (e) 14
60 90 98 96 42
6. Match the equivalent fractions.
(a) 250 (b) 180 (c) 660 (d) 180 (e) 220
400 200 990 360 550

(i) 9 (ii) 2 (iii) 1 (iv) 2 (v) 5

10 5 2 3 8
7. Compare which is bigger, 4 or 5?
5 6

___________________________________________________________________

8. Simplify.
(a) 12 + 3 1 (b) 9 + 7 + 5 (c) 31 + 2 1 – 11
3 2 30 24 18 4 2 3

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Worksheet 2
1. Write Yes or No for each of the following.
(a) Is 5 equal to 4? (c) Is 9 equal to 5?
9 5 16 9
(d) Is 4 equal to 16? (d) Is 1 equal to 4 ?
5 20 15 30
2. Solve.
(a) 1 + 1 (b) 7 + 5 (c) 1 + 21 (d) 8 + 8
28 28 7 7 22 22 15 15
(e) 5 + 3 (f) 1 – 1 (g) 1 + 0 (h) 4 – 13
8 8 3 7 7 5
3. Fill in the missing fractions.
(a) 7 – 3 = 3 (b) 3 = 3 – 3
10 10 10 6 6 10
(c) 3 – 3 = 5 (d) 3 + 5 = 12
10 21 21 10 27 27
4. Solve.
(a) 2 + 3 + 1 (b) 11 + 32 (c) 42 – 31 (d) 3 – 1
3 4 2 3 3 3 4 4 3
5. Shreya’s house is 9 km from her school. She walked some distance and then took
10
a bus for 1 km to reach the school. How far did she walk?
2

___________________________________________________________________
6. In class A of 25 students, 20 passed in first class; in class B of 30 students, 24
students passed in first class. In which class did a greater fraction of students got
first class?

___________________________________________________________________

28
 6 Decimals

Learning Objectives
Students will be able to
➢ recapitulate reading and writing a decimal
➢ represent fractions in the form of decimals
➢ learn decimal as an extension of place value table
➢ represent decimals as fractions
➢ represent decimals on the number line
➢ identify like and unlike decimals
➢ convert unlike decimals into like decimals
➢ compare decimals
➢ learn conversion of a given fraction into a decimal and vice versa
➢ add and subtract the decimals
➢ learn the uses of decimal notation in real life

Concept Explanation
• Students are already familiar with the decimals. They can read and write decimals.
• Discuss with them the meaning of decimal, use of decimal point. The decimal point
divides the number into two part, i.e., integral part and decimal part.
• Read the related sections to recapitulate these concepts.
Decimals as an extension of place value table
• Make a decimal place value table on the blackboard.
• Ask the students to observe the tenths, hundredths and thousandths places to the right
of the decimal point in the table.
• Instruct the students to draw the same table in their notebooks.
• Call out a decimal number and make the students understand how to represent it in the
place value table.
• Use illustrative examples to strengthen the concept.
• Ask them to do more practice for writing and reading a decimal in words.
• To reinforce, ask the students to do Check Point 6.1 from the textbook.

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Decimals as fractions; Representation of decimals on a number line
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• Now read the Decimals as fractions section from the textbook.
• Use rectangular strips, 10 × 10 square grids, and blocks to make them understand the
concepts of tenths, hundredths and thousandths.
• Discuss the place value system in the extension of decimal. Give them some problems to
solve by themselves.
• Let the students understand the decimal fractions as 10, 100 and 1000 as denominator.
Also, demonstrate how to represent the decimals on a number line.
• Explain the process of representing the decimals on a number line.
• Draw their attention to the fact that we can reduce a decimal fraction into its lowest term
by dividing the numerator and denominator by their HCF.
• Use illustrative examples of related sections to make them better understand these
concepts.
• To reinforce, ask the students to do Maths Lab Activity section from the textbook.
• Ask them to do Check Point 6.2 from the textbook.
Like and unlike decimals; Conversion of unlike decimals into like decimals; Comparing
decimals
• Define the like and unlike decimals in the class.
• Let the students understand the process of converting the unlike decimals into like
decimals by adding the required number of zeroes to the extreme right of the decimal
part. Give some examples and solve.
• Discuss the process of comparing decimals so that they can understand the concept and
can then arrange the decimals in ascending order and descending order.
• Ask the students to first go through the illustrative examples given in the related section
and then do Check Point 6.3 for more practice.
Conversion of a given fraction into a decimal; Conversion of a given decimal into a
fraction
• Read the related sections from the textbook.
• Define the procedure for the conversion of a given fraction into a decimal as follows,
(a) Write the given fraction.
(b) See the number of zeroes in the denominator.
(c) Insert a decimal point after as many digits from the extreme right as the number of
zeroes in its denominator.
• Draw their attention to fact, that in the decimal number, if there is no whole number
part, then we always put zero before (or left) the decimal point.

30
 0 ÷
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+% • Define the procedure for the conversion of a given decimal into a fraction as follows,
(a) Write the given decimal.
(b) Remove the decimal point from the numerator and at the same time write in the
denominator as many zeroes after 1 as there are digits in the decimal part.
(c) Simplify to the lowest term (if possible).
• Use illustrative examples to strengthen the concepts.
• Instruct the students to do Check Point 6.4 from the textbook.
Operations on decimals
• Read the related sections from the textbook.
• Clarify that to add or subtract, first we convert unlike decimals into like decimals, then
we arrange them vertically in such a way that the decimal point comes in the same
column. Then we subtract as usual.
• Use the illustrative examples to make them understand the concept.
• To reinforce, ask the students to do the Values and Maths in Real Life sections from the
textbook.
• Ask the students to do Check Point 6.5 from the textbook.
Uses of decimal notation in real life
• Ask students to solve some real life problems such as
– Conversion of rupees into paise and vice versa
– Conversion of mm into cm, cm into metre, metre into cm, km into metre and m
into km, etc.
• Read the related section from the textbook.
• Make them understand the uses of decimal notation in money, length, mass and capacity.
• Instruct the students to do Check Point 6.6 and Check Point 6.7 from the textbook.
To revise the concepts learnt in the chapter, students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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Worksheet 1
1. Write the following decimals in the place value table.
(a) 19.4 (b) 0.5 (c) 11.7 (d) 205.8
(e) 0.7 (f) 2.8 (g) 1.0 (h) 3.5
(i) 13.8 (j) 21.2
2. Write each of the following as a decimal.
(a) Seven point one (b) Two hundred point twenty-five
(c) Fourteen point seven (d) One hundred point zero three
(e) Five hundred point nine
3. Write the following as decimals.
(a) Three hundred six and seven-hundredths
(b) Nine and twenty-five thousandths
4. Write each of the following decimals as a word.
(a) 0.03 (b) 1.20 (c) 108.56 (d) 10.07 (e) 0.032
5. Which is greater?
(a) 0.3 or 0.4 (b) 0.099 or 0.19 (c) 1.5 or 1.50
(d) 1.421 or 1.439 (e) 3.3 or 3.300 (f) 0.07 or 0.008
6. Add the following.
(a) 0.25, 3.26, 1.258 (b) 5.25, 2.35, 2.058 (c) 23.245, 125.250, 100.024
7. Subtract.
(a) 4.25 from 5 (b) 8 from 9.025 (c) 117.45 from 223.5
8. Subtract.
(a) 100 – 45.36 (b) 999.99 – 9.9
9. What should be subtracted from 500 to get 258.75?
10. Manish purchased a bat for `1245.50 and a ball for `145.50. He gave two 1000-rupee
notes to the shopkeeper. What amount did he get back?

___________________________________________________________________
32
 0 ÷
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+ % Worksheet 2
1. Convert the following into decimals.
(a) 5 (b) 8 (c) 452 (d) 96 (e) 458
100 10 10 1000 1000
2. Convert the following into fractions.
(a) 2.5 (b) 0.8 (c) 0.36 (d) 41.02 (e) 3.025
3. Express in rupees using decimals.
(a) 8 paise (b) 70 paise (c) 25 paise (d) 725 paise
(e) 50 rupees 90 paise
4. Express in metres using decimals.
(a) 18 cm (b) 5 cm
(c) 3 m 45 cm (d) 491 cm
5. Express in cm using decimals.
(a) 5 mm (b) 55 mm
(c) 175 mm (d) 9 cm 75 mm
6. Express in km using decimals.
(a) 8 m (b) 70 m
(c) 7777 m (d) 70 km 8 m
7. Add the following.
(a) 15 + 0.632 + 13.8
(b) 25.65 + 9.007 + 8.7
(c) 0.72 + 11.425 + 2
(d) 280.69 + 25.2 + 3
(e) 27.076 + 0.55 + 0.004
8. Find the value of
(a) 9.756 – 6.28 (b) 41.06 – 32.28 (c) 18.5 – 7.97
(d) 11.6 – 7.974 (e) 88.009 – 9.088 (f) 16.815 – 15.681

33
 7 Introduction to Algebra

Learning Objectives
Students will be able to
➢ recapitulate the concept of counting numbers, operations on numbers, and arithmetic
expressions
➢ understand how letters of alphabets (English, Greek, etc.) are used to represent
numbers
➢ make matchstick pattern in algebra as generalisation
➢ apply operations on literal numbers, such as
– addition of literals and properties of addition
– subtraction of literals and properties of subtraction
– multiplication of literal and properties of multiplication
– division of literals
– power of literal numbers
➢ recognise the like and unlike terms

Concept Explanation
• Students are already familiar with the natural numbers, whole numbers and can apply
operations on numbers and can simplify arithmetic expressions.
• Read the related sections to recapitulate these concepts.
Use of literals to denote numbers; Matchstick pattern in algebra as generalisation
• Make the students understand how do we use literals (a, b, c … x, y, z, etc.) to represent
(denote) numbers.
• Explain to them that a symbol which can have any value is called a variable or a literal.
• Discuss with students giving some suitable examples, how matchstick patterns help to
develop generalisation formula in algebra.
• Provide some ice cream sticks to each student.
• Instruct the students to perform some activity in group as well as individually and make
a pattern using these sticks and write a generalisation formula for the pattern in their
notebooks.
• To reinforce, ask the students to do the Maths Lab Activity and Values sections from the
textbook.
• Ask the students to do Check Point 7.1 from the textbook.
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+% Operations on literal numbers; Power of literal numbers
• Go through the concept of operations (addition, subtraction, multiplication and division)
on numerals and literals.
• Explain how to express a phrase using numbers, literals and different basic operations.
• Use the related examples of the section to strengthen these concepts to them.
• Explain 0 is the additive identity and 1 is the multiplicative identity.
• Make the students understand about the power of literals.
• Explain that power of a literal indicates the number of times the literal has been multiplied
by itself.
• To reinforce, ask the students to do Check Point 7.2 from the textbook.
Algebraic Expressions; Factors and coefficients; Like and unlike terms
• Read the related section from the textbook.
• Discuss what is an algebraic expression and its types.
• Explain and exemplify about monomial, binomial, trinomial and polynomial, so that
they can very easily identity them as mono, di, tri, and polynomials.
• Also, tell them about factors, coefficients, like terms and unlike terms.
• Use illustrative examples for more practice and to make them understand the concepts.
• Instruct the students to do Check Point 7.3 from the textbook.
To revise the concepts learnt in the chapter, students should do the Maths in Real Life, Test
Yourself and Brain Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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Worksheet 1
1. Find the rule which gives the number of matchsticks required to make the following
matchstick patterns. Use a variable to write the rule.
(a) A pattern of letter T. (b) A pattern of letter Z.
(c) A pattern of letter U. (d) A pattern of letter V.
(e) A pattern of letter E . (f) A pattern of letter S.
(g) A pattern of letter A . (h) A pattern of letter R.
2. If there are 50 mangoes in a box, how will you write the total number of mangoes
in terms of the number of boxes? (use y for the number of boxes)

___________________________________________________________________
3. The side of a regular hexagon is denoted by ‘m’. Express the perimeter of the
hexagon using ‘m’.

___________________________________________________________________
4. Which of the following are expressions with numbers only?
(a) x + 5 (b) (9 × 15) – 7z (c) 10 (d) 5x
(e) 6 – 3y (f) 5(25 – 8) + 11 × 3 (g) (5 × 18) – (6 × 10) – 48 + y
5. Write expressions for the following cases.
(a) 6 added to x (b) 5 subtracted from x (c) x multiplied by 7
(d) x divided by 7 (e) 8 subtracted from –y (f) –y multiplied by 6
(g) –x divided by –5 (h) –y multiplied by –7
6. Write each of the following in the product form.
(a) m3n2 (b) r2s3 (c) 144x4 (d) 25m7n8p9
7. Write down each of the following in the exponential form.
(a) 3m2 × 5m2n × 4n2 (b) 12a3b3 × 6a3b4 × 3a2b

36
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+ % Worksheet 2
1. Change the following statements using expressions into statements in ordinary
language.
(a) A notebook cost `P. A book cost `3P.
(b) Our class has n students. The school has 20n students.
(c) Jaggu is z years old. His uncle is 4z years old and his aunt is (4z – 3) years old.
2. Given, Suman’s age is x years.
(a) Can you guess what (x – 2) may show?
(b) Can you guess what (x + 4) may show?
(c) Can you guess what (5x + 4) may show?
3. Given, n students in a class like football.
(a) What may 2n represent? (b) What may n represent?
2
4. Form expressions using t and 4. Use not more than one number operation. Every
expression must have t in it.

___________________________________________________________________
5. Write each of the following phrases using numbers, literals, and the basic
operations.
(a) x divided by 7 (b) Divide y by 11
(c) The product of m and 23 (d) s times 10
(e) The sum of y and z (f) subtracting n from 50
6. Give expressions for the following cases.
(a) Increase –7 by x.
(b) m less then product of –8 and n.
(c) 7 subtracted from the product of p and q.
7. Write each of the following in the expanded form.
(a) a5b3 (b) 4x3 (c) 12r3s4t5 (d) 500a5b5c5

37
 8 Algebraic Equations

Learning Objectives
Students will be able to
➢ recapitulate the concept of algebraic expressions and different operations on algebraic
terms
➢ define the meaning of mathematical statement and equation
➢ understand the concept of an algebraic equation especially a linear equation
➢ differentiate between an equation and an identity
➢ Solve linear equations by various methods as follows:
– by trial and error method
– by systematic method
– by transposition method
➢ know the applications of linear equations

Concept Explanation
• Students are already familiar with the concept of algebraic expressions and different
operations on algebraic terms.
• Read the related sections to recapitulate these concepts.
Algebraic equation; Solving an algebraic equation by trial and error method
• Tell students that a statement that involves mathematical symbols like ‘+’, ‘–’, ‘×’. ‘÷’ and
numbers is called mathematical statement.
• The equality of two mathematical statements is called an equation.
• Read the related section from the textbook. Use the related examples given in the
textbook to explain these concepts.
• Tell them when the degree or power of a variable (x, y, z,.., etc.) is one (1), then the
equation is called the linear equation in one variable.
• Point out the facts, i.e.,
(a) the sign of equality in an equation divides it into two sides, namely, the left hand
side (LHS) and the right hand side (RHS).
(b) solving an equation means to find the value of the variable.
• Explain with examples the process of solving linear equation by Trial and Error method.
Now the students are capable enough to solve the problems independently. Assess the
students by giving some problems from the textbook and some from your side.
38
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+% • To reinforce, ask the students to do Maths Lab Activity from the textbook.
• Ask the students to do Check Point 8.1 from the textbook.
Solving an algebraic equation by systematic method
• Read the related section from the textbook.
• Explain the rules associated with the concept of solving linear equations by systematic
method.
• Use illustrative examples of the related section for better understanding of the process of
solving linear equation by systematic method.
• Assess the students with the help of problems from the textbook and from your side.
• To reinforce, ask the students to do Check Point 8.2 from the textbook.
Solving an algebraic equation by transposition method
• Read the related section from the textbook.
• Explain the rules associated with the concept of solving linear equations by transposition
method.
• Use illustrative examples of the related section for better understanding of the process of
solving linear equation by transposition method .
• Assess the students with the help of problems from the textbook and from your side.
• To reinforce, ask the students to do Check Point 8.3 from the textbook.
Applications of linear equations
• Explain the rules associated with the concept of solving linear equations by transposition
method.
• Discuss with the students step by step procedure of an equation taking unknown quantity
formulate (to be find out) as x, y, z, etc. It is observed that generally students face a lot
of problem in the formation linear equation. Sometimes they try to formulate equation
before formulating mathematical statements and face a lot of difficulty.
• Ask them to do illustrative examples of the related section for better understanding of
the concept.
• To reinforce, ask the students to do the Values and Maths in Real Life sections from the
textbook.
• Ask the students to do Check Point 8.4 from the textbook.
To revise the concepts learnt in the chapter, students should do the Maths in Real Life, Test
Yourself and Brain Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.
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Worksheet 1
1. Read the given phrase and answer the questions that follow.
‘Sarita’s present age is x years.’
(a) What will be her age 5 years from now?
(b) What was her age 4 years back?
(c) Sarita’s grandfather is 6 times her age. What is the age of her grandfather?
(d) Her grandmother is 2 years younger than grandfather. What is the age of
grandmother?
(e) Sarita’s father’s age is 6 years more a than 2 and a half times Sarita’s age. What is
her father’s age?
2. State which of the following are equations (with one variable). Give reason for
your answer. Identify the variable from the equations with a variable.
(a) 16 = x + 9 (b) x – 7 > 8
(c) 6 = 3 (d) (7 × 5) – 21 = 14
2
(e) 4 × 6 – 18 = 2n (f) 3x < 7
2
(g) 7 = (11 × 2) + p (h) 20 = 4x
(i) 23 – (10 – 7) = 4 × 5 (j) x + 12 > 25
3. Solve each of the following equations by the trial and error method.
(a) a + 5 = 17 (b) b + 3 = 20 (c) 6x = 42 (d) 19y = 57
4. Solve each of the following equations.
(a) m + 9 = 63 (b) 84 – n = 100 (d) 4y = 28
(c) 5.5 x = 16.5
7
5. Madhav’s son is two times as old as his daughter. After 10 years, the son will be
3 times as old as Madhav’s daughter. Find the present age of Madhav’s son and
2
daughter.

_________________________________________________________________

40
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1. Pick out the solution from the values given in the bracket next to each equation.
Show that the other values do not satisfy the equation.
(a) 5x = 75 (5, 10, 15, 20) (b) x + 12 = 25 (9, 10, 11, 13)
(c) x – 5 = 5 (0, 5, 10, 20) (d) x = 8 (15, 16, 17, 18)
2
(e) 6n – 2 = 10 (0, 1, 2, 3) (f) x + 4 = 2 (–2, 0, 2, 4)
2. Complete the table and by inspection of the table find the solution to the equation
x + 10 = 16

x 1 2 3 4 5 6 7 8 9 10 ...
x + 10
3. Solve the following equations.
(a) 2m + 1 = 4 (b) y + 1 + y – 1 = 2
3 3 2 2
(c) 2 x – 4 3x = 17 (d) 2x – 1 – 1 5 – x – 1 = 10
3 3 5 2 3 3 2
4. The sum of three consecutive even natural numbers is 48. Find the numbers.

___________________________________________________________________
5. The breadth of a rectangle is 5 cm less than its length. If the perimeter of the
rectangle is 54 cm, find the length and breadth of the rectangle.

___________________________________________________________________
6. Divide 60 into two parts such that one part is two times the other part.

___________________________________________________________________
7. 8 added to 9 times of a number gives 107. Find the number.

___________________________________________________________________
8. Find three consecutive multiples of 5 whose sum is 165.

___________________________________________________________________
41
 9 Ratio, Proportion and Unitary Method

Learning Objectives
Students will be able to
➢ define the meaning of ratio
➢ know about the terms of ratio
➢ express a ratio in its simplest form
➢ find equivalent ratios of a given ratio
➢ compare the given ratios
➢ learn about proportion and continued proportion
➢ understand and apply the concept of unitary method

Concept Explanation
• Students are already familiar with the concept of fractions and can apply basic operations
in real life problems.
• Recapitulate these concepts with some examples.
Ratio; Terms of ratio; Ratio in the simplest form
• Explain to the students that ratio is a fraction which shows how many times a quantity
is of another quantity of the same kind and same unit. Give some suitable examples to
justify the above statement.
• Also, let them understand a : b ≠ b : a. Also, ‘ : ‘ is the symbol of ratio and read as ‘is to’.
• Discuss the process of converting the ratio in simplest form by dividing the terms
(antecedent and consequent) by their HCF.
• Tell them that a ratio a : b is said to be in the simplest form or lowest term, if its antecedent
(a) and consequent (b) have no common factor other than 1.
• Draw their attention to the facts given in the Tidbits section of the related concept.
• Use the examples given in the textbook to strengthen the concepts in the students.
• To reinforce, ask the students to do the Maths Lab Activity given in the textbook.
• Ask the students to do Check Point 9.1 from the textbook.
Equivalent ratios; Comparison of ratios
• Let then understand the process of finding an equivalent ratio(s) of a given ratio either
by multiplying or dividing the terms of a ratio by any non-zero integer.

42
 0 ÷
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+% • Two or more ratios can be compared by converting them into equivalent ratios with
common denominator.
• Ask the students to go through the procedure given in the related section while comparing
the two ratios.
• Use the examples given in the related section to better grasp the concept.
• To reinforce, ask the students to do Maths in Real Life section from the textbook.
• Instruct the students to do Check Point 9.2 from the textbook.
Proportion; Continued proportion
• Tell students that the equality of two ratios is called proportion.
So, if a, b, c, d are in proportion, then
a : b : : c : d ⇒ a = c ⇒ ad = bc
b d
• Explain in proportion, the first term ‘a’ and fourth term ‘d’ are called the extreme terms
or extremes whereas the second term ‘b’ and the third term ‘c’ are called the mean terms
or means.
• Write the formula, Product of extremes = Product of means on the board.
• Clarify, if ad ≠ bc, then a, b, c and d are not in proportion.
• Also explain the continued proportion, third proportional and mean proportional.
• Use the illustrative examples to strengthen these concepts.
• To reinforce, ask the students to do the Values section from the textbook.
• Instruct the students to do Check Point 9.3 from the textbook.
Unitary method
• Let the students understand the meaning of unitary method. From the value of given
quantity, find the value for one quantity. Next, from the value of the one quantity find the
value of the required quantity.
• Use illustrative examples to make them understand the concept.
• To reinforce, ask them to do Check Point 9.4 from the textbook.
To revise the concepts learnt in the chapter, students should do the Maths in Real Life, Test
Yourself and Brain Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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Worksheet 1
1. Out of 30 students in a class, 6 like football, 12 like cricket and the remaining like
tennis.
Find the ratio of the:
(a) number of students liking football to the number of students liking tennis.
(b) number of students liking cricket to the total number of students.
2. Fill in the following blank boxes.

(a) 15 = 15 = 10 = 15 (b) 21 = 15 = 7 = 15
18 6 18 18 27 9 18 72
Are these equivalent ratios?
3. Find the ratio of the following.
(a) 81 to 108 (b) 98 to 63 (c) 33 km to 121 km
(d) 30 minutes to 45 minutes (e) 25 kg to 400 kg (f) `35 to `115
(g) 100 mL to 250 mL (h) 4.5 to 18
4. Compare the following ratios.
(a) 2 : 3 and 6 : 5 (b) 1 : 3 and 1 : 4 (c) 4 : 5 and 5 : 4
(d) 12 : 14 and 5 : 6 (e) 3 : 7 and 2 : 8 (f) 4 : 20 and 8 : 24
5. Divide 20 pens between Sheela and Sangeeta in the ratio of 3 : 2.
6. The present age of a father is 42 years and that of his son is 14 years. Find the ratio of the
(a) present age of father to the age of son.
(b) age of the father to the age of son, when the son was 12 years old.
(c) age of the father after 10 years to the age of the son after 10 years.
(d) age of the father to the age of the son when the father was 30 years old.
7. Find the third proportional to 6, 12.

___________________________________________________________________
8. Find the mean proportional between 5 and 45.

___________________________________________________________________

44
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+ % Worksheet 2
1. Determine if the following are in proportion.
(a) (a) 15, 45, 40, 120 (b) 33, 121, 9, 96 (c) 24, 28, 36, 48
(d) 4, 6, 8, 12 (e) 33, 44, 75, 100 (f) 32, 48, 70, 210
2. Write True (T) or False (F) against each of the following statements:
(a) 12 : 18 : : 28 : 12 ______ (b) 6 : 21 : : 10 : 35 ______
(c) 16 : 20 : : 24 : 30 ______ (d) 8 : 24 : : 9 : 27 ______
(e) 5.2 : 3.9 : : 3 : 4 ______ (f) 0.9 : 0.36 : : 10 : 4 ______
3. Find the value of x in each of the following proportions.
(a) 3 : 5 :: 48 : x (b) x : 3 :: 4 : 6
4. Find the value of x so that the given four numbers are in proportion.
(a) 8, x, 24 and 15 (b) 9, 5, 63 and x
5. Find the mean proportion between:
(a) 25 and 36 (b) 0.16 and 0.49
6. Find the fourth proportional to:
(a) 5, 15, 25 (b) 3.5, 1.4, 7
7. If the cost of 6 cans of juice is `210, what will be the cost of 4 such cans of juice?

___________________________________________________________________
8. If the cost f a dozen soaps is `153.60, what will be the cost of 15 such soaps?

___________________________________________________________________
9. Anish made 42 runs in 6 overs and Anup made 63 runs in 7 overs. Who made more
runs per over?

___________________________________________________________________

45
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Mathematics
Model Test Paper 1
Class 6
Time: 2½ hours Total Marks: 70
General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 29 questions divided into four sections A, B,C and D.
Section A consists of 8 questions of 1 mark each.
Section B consists of 6 questions of 2 marks each.
Section C consists of 10 questions of 3 marks each.
Section D consists of 5 questions of 4 marks each.
(iii) There is no overall choice. However, an internal choice has been provided in some questions.
Attempt only one options in such questions.

SECTION - A
1. Write the opposite of profit of `250.
2. Represent 23 in the form of a mixed fraction.
8
3. If 5y = 35, then find the value of y.
4. Write 4.65 in words.
5. Find the ratio of 40 to 160.
6. Using the divisibility test, check whether 526537 is divisible by 11 or not.
7. Write 360,256,201 in words.
8. Represent 5 + 2 on a number line.

SECTION - B
9. Write the following as decimal.
6+ 7 + 8
10 100
10. Convert DCCCLXXV into Hindu-Arabic numeral.
11. Find the equivalent fraction of 25 having the denominator 7.
35
12. Solve: 6 – 3
7 5
13. Write all integers between –7 and 7.
14. Express in litres using decimals: 4 L 50 mL

46
 0 ÷
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+% SECTION - C
15. Find the greatest number of 5 digits exactly divisible by 8, 12 and 50.
16. Find the sum: 5 + 7.205 + 25.25
17. Find all possible 3-digit numbers using the digits 2, 0 and 5.
18. Subtract the sum of 2.36 and 4.01 from the sum of 6.32 and 2.39.
19. Check whether 12, 10, 6 and 5 are in proportion or not?
20. Mahesh works for 5 of an hour, while Suresh works for 8 of an hour. Find who works
6 9
for a longer time?
21. Find the HCF of 24, 36, 54 and 60 using the division method.
22. Use the distributive property to simplify the following.
(a) 1458 × 1458 + 1458 × 542 (b) 98 × 59
23. Write the following fractions in ascending and descending orders.
4, 3 , 6, 13, 5
6 11 5 7 8
24. Write expressions for the following statements.
(a) 13 more than m.
(b) 7 subtracted from the product of a and 4.
(c) 5 added to the product of 5 and b.

SECTION - D
25. A student multiplied 2678 by 95 instead of multiplying it by 59. By how much was
his/her answer greater than the correct answer?
26. A bike consumed 15 litres of petrol in going from town A to town B and another
12 litres of petrol in going from town B to town C. If the cost of petrol is `65 per litre,
find the total amount spent by him on petrol.
27. Determine the longest tape which can be used to measure exactly the lengths
5 m 50 cm, 3 m 85 cm and 10 m 45 cm.
28. Kamal travelled 15 km 250 m by train, 5 km 500 m by bus and 400 m on foot to reach
his home from his office. Find out how far is his office from his house.
29. State whether the following statements are true (T) or false (F).
(a) 8 : 120 : : 11 : 165 (b) HCF × LCM = Product of the given numbers
(c) –3 is smaller than –8 (d) 0 is the smallest positive integer.

47
 10 Basic Geometrical Concepts

Learning Objectives
Students will be able to
➢ recapitulate the concepts of point, line, line segment, angle, etc.
➢ know what is geometry and its purpose
➢ define a point, plane, line, curve, angle, ray, line segment
➢ identify different types of lines such as intersecting lines, parallel lines, concurrent
lines, etc.
➢ define and identify collinear and non-collinear points
➢ measure and compare the line segments
➢ define angle and measure an angle
➢ learn about the different types of angles

Concept Explanation
• Students are already familiar with the concept of point, line, line segment, angle, etc.
• Explain the meaning of geometry, ‘geo-metron’ ‘measurement of earth’ and the use of
geometry to define point, plane, line, etc. Give example of each of the term.
• Read the Introduction, Point and Plane sections from the textbook.
• Recapitulate these concepts with the help of some examples.
Line; Collinear and non-collinear points
• Go through the section ‘Line’ in the textbook.
• Provide a sheet of paper and a string to each group.
• Ask students to follow the instructions given in Experiment 1 and Experiment 2 of this
section and observe the straight lines formed.
• Use strings or sheets to demonstrate intersecting lines, parallel lines and concurrent lines.
• Read the Collinear and Non-collinear points section from the textbook.
• Use some practical examples and blackboard to make them better understand these
concepts.
• To reinforce, ask the students to do Check Point 10.1 from the textbook..
Line segments; Comparison of line segments; Ray; Curves
• Confirm that each student has a pencil, a ruler, a divider, an eraser, etc.
• Read the related sections from the textbook.
48
 0 ÷
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+% • Demonstrate how to compare two line segments using a divider and a ruler.
• Define a ray and a curve.
• Explain that a curve which does not cut itself is called an open curve and a curve which
cuts itself is called a closed curve.
• Use examples given in the textbook to make them understand the concept.
• Instruct them to do Check Point 10.2 from the textbook.
Angles; Measuring an angle which is less than 180°; Types of angles
• Read the related sections from the textbook.
• Draw some points in the interior and exterior of the angle. Ask them to identify which
points are in the interior and exterior of the angle.
• Bring a cardboard to the class. Fix a nail in it. Tie a string to it.
• Draw a horizontal line passing through the nail. Move the string around it (both in
clockwise and anti clockwise directions) to demonstrate the different types of angles.
• Draw an angle on the blackboard. Use a protractor and demonstrate how to measure an
angle less than or more than 180°.
• Use some examples to make them understand the concept that sum of all the angles
around a point is 360°.
• Also, let the students learn the relation between degree, minute and seconds such that
1° = 60ʹ (60 minutes) and 1ʹ = 60ʹʹ (60 seconds).
• To reinforce, ask the students to do the Maths Lab Activity section from the textbook.
• Instruct the students to do Check Point 10.3 from the textbook.
To revise the concepts learnt in the chapter, students should do the Maths in Real Life, Test
Yourself and Brain Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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Worksheet 1
1. Draw a rough figure and label suitably in each of the following cases.
(a) Point P lies on AB.
(b) XY and PQ intersect at M.
(c) Line l contains E and F but not D.
(d) OP and OQ meet at O.
2. Draw the rough diagrams to illustrate the following
(a) Open curve (b) Closed curve
3. Draw a polygon and shade its interior.
4. Illustrate, if possible, each of the following with a rough diagram.
(a) A closed curve that is not a polygon.
(b) A polygon with three sides.
5. Fill in the blanks.
(a) If there is a point common to two lines drawn, we say that the two lines
_____________ at the common point.
(b) When three or more lines in a plane are passing through a point, then lines
are called _____________.
(c) Three or more points in a plane are said to be _____________, if they all lie
on the same line.
(d) Two lines in a plane either intersect at exactly one point or are _____________.
6. From your surroundings, give the examples of:
(a) concurrent lines (b) intersecting lines
(c) parallel lines (d) an obtuse angle
7. Classify the following curves as open or closed.
(a) (b) (c)

(d) (e)

50
 0 ÷
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+ % Worksheet 2
1. Draw rough diagrams of two angles such that they have
(a) one point in common.
(b) two points in common.
(c) three points in common.
(d) four points in common.
(e) one ray in common.
2. Classify the following angles as acute, straight, right obtuse, reflex, zero and
complete angles.
(a) 103° (b) 68° (c) 175° (d) 210°
(e) 360° (f) 90° (g) 0° (h) 91°
(i) 180° (j) 71° (k) 358° (l) 180.5°
3. In the given figure name
(a) three line segments
E B A F

C D

(b) three rays

4. State which of the following statements are true or false.
(a) The sum of angles around a point is 180°.
(b) An acute angle is always less than 90° and greater than 0°.
(c) An obtuse angle is always less than 90° and greater than 180°.
(d) An angle whose measure is equal to 180° is called a straight angle.
(e) An angle whose measure is equal to 0° is called a complete angle.
(f) An angle greater than 180° and less than 360° is called a reflex angle.
(g) A line is a set of all points which have length only, i.e., no breadth, no height.

51
 11 Understanding Elementary Shapes

Learning Objectives
Students will be able to
➢ define polygon and types of polygons
➢ identify adjacent sides, opposite sides, adjacent angles and opposite angles of a
polygon
➢ identify and locate vertices and diagonals of a polygon
➢ learn about regular and irregular polygons
➢ know about triangle and its properties
➢ learn about types of triangle as scalene triangle, isosceles triangle, equilateral triangle,
acute-angled triangle, right-angled triangle, obtuse-angled triangle and isosceles
right- angled triangle
➢ learn and verify the angle sum property of a triangle
➢ find the perimeter of triangle
➢ know about quadrilaterals and types of quadrilaterals
➢ define circle and its various terms such as centre, radius, diameter, chord, circular
region, semi-circle, arc of a circle, secant, tangent to a circle, sector of a circle and
segment of a circle

Concept Explanation
• Students are already familiar with the concept of angles, plane figures such as triangle,
square, rectangle, circle, etc.
• Recapitulate these concepts with the help of some examples.
Polygons
• Define and describe polygons and their various terms.
• Use blackboard and draw a polygon. Label adjacent sides, vertices and diagonals of the polygon.
• Provide some cut-outs of regular and irregular polygons and ask the students if these are
regular or irregular.
• Help them to identify regular and irregular polygons.
• Use the table given in the related section to interpret the attributes of these polygons.
• Give some examples of concave and convex polygons on the board and ask them to
identify if this is concave or convex.
• To reinforce, write some problems on the board and ask the students to answer these.
52
 0 ÷
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+% Triangle and its properties
• Provide a chart to each student. Instruct each student to draw a triangle of their choice
on their charts.
• Now the teacher will draw a triangle on the board and label its various parts. Ask the
students to do the same.
• Discuss the six elements of the triangle with class. Ask the students to show interior and
exterior parts of their triangles on theirs charts.
• Define interior, exterior and triangular regions of a triangle.
• Use cut-outs of different triangles and define one by one these triangles on the basis of
their sides and angles.
• Do an activity to verify the angle sum property of a triangle. For this the teacher can ask
the students to find the measure of three angles of the triangle the students have drawn
on their charts and find the sum of these angles.
• To reinforce, ask the students to do Check Point 11.1 from the textbook.
Quadrilaterals
• Draw a quadrilateral on the board and label its various parts. Ask the students to do the
same on their chart paper (the student can draw any kind of quadrilateral).
• Instruct students to show interior and exterior of the quadrilaterals they have drawn.
• Define interior, exterior and quadrilateral regions of a quadrilateral.
• Use cut-outs of different types of quadrilaterals and define their attributes and their properties.
• To reinforce, ask the students to do the Maths Lab Activity and Maths in Real Life
sections from the textbook.
• Ask the students to do Check Point 11.2 from the textbook.
Circle
• Read the related section from the textbook.
• Show some real life examples of a circle.
• Define circle and all the terms related to circle such as centre, radius, diameter,
circumference, arc, chord, semi-circle, secant, tangent, quadrant, sector, segment, etc.
• Write a relation, diameter of a circle is twice its diameter, i.e.,
Diameter = 2 × Radius on the board.
• Discuss some examples of concentric circles in the class.
• To reinforce, ask the students to do Check Point 11.3 from the textbook.
To revise the concepts learnt in the chapter, students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.
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Worksheet 1
1. Observe the given figure and A

(a) identify three triangles in the figure.

(b) write the names of any seven angles.
(c) write the names of six line segments.
B
D C
(d) find which of the two angles have D as common.
2. Draw a rough sketch of a quadrilateral KLMN. State
(a) two pairs of opposite sides.
(b) two pairs of opposite angles.
(c) two pairs of adjacent sides.
(d) two pairs of adjacent angles.

3. Draw rough sketch of a quadrilateral PQRS. Draw its diagonals. Name them. Is
the meeting pint of the diagonals in the interior or exterior of the quadrilateral.

4. State whether the following statements are true or false.

(a) A triangle having all three sides equal is called a scalene triangle.
(b) The sum of interior angles of a triangle is 180°.
(c) The line segments joining the opposite vertices of a quadrilateral are called its
tangents.
(d) If a line segment joining any two points in the interior of a quadrilateral does not
lie completely within it, then it is a convex quadrilateral, otherwise it is a concave
quadrilateral.
(e) A parallelogram whose all sides are equal is called a rhombus.
(f) A quadrilateral in which the pair of opposite sides are parallel, is called a
trapezium.
(g) The perimeter of a circle is known as the diameter.
(h) Two or more circles are said to be concentric, if they have the same centre.

54
 0 ÷
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+ % Worksheet 2
1. From the figure, identify
M
(a) centre of the circle
B
(b) three radii A E C
(c) a diameter
(d) a chord O

(e) two points in the interior F

(f) a point in the exterior D

(g) a sector
N
(h) a segment

2. Write yes or no to answer the following questions.

(a) Is every diameter of a circle also a chord?
(b) Is every chord of a circle also a diameter?

3. State whether the following statements are true or false.

(a) Two diameters of a circle will necessarily intersect.
(b) The centre of a circle is always in its interior.

4. Draw a circle and mark

(a) its centre (b) a radius (c) a diameter
(d) a minor sector (e) a major segment (f) an arc
(g) a point in its interior (h) a point in its exterior

5. Fill in the blanks.

(a) If the two sides of a polygon have a common end point, then these sides are
called _______ sides of the polygon.
(b) A polygon is said to be ________ if sides are not equal and angles are not equal.
(c) The point of intersection of three altitudes of a triangle is called ________.
(d) The point of intersection of the three medians is G, which is known as ________.
(e) A triangle having any two sides equal is called an ________ triangle.

55
 12 Understanding Three-Dimensional Shapes

Learning Objectives
Students will be able to
➢ recapitulate the concepts of plane figures
➢ identify and define three-dimensional shapes
➢ relate three-dimensional shapes (as cuboids, cubes, cylinder, cone, sphere, prism and
pyramid) from our surroundings
➢ learn about the types of prism and pyramid
➢ find the number of edges, faces and vertices of a given solid
➢ know about the nets of solids

Concept Explanation
• Students are already familiar with the concept of plane figures.
• Recapitulate these concepts with the help of some examples.
Cuboid; Cube; Right circular cylinder; Righ circular cone; Sphere; Prism and types of
prism; Pyramid and types of pyramid
• Introduce and explain with examples the basics of three-dimensional shapes.
• Ask the students to identify the edges, faces and vertices of a solid.
• Instruct them to collect some objects with three-dimensional shapes from their
surroundings.
• From the collected objects, separate and classify the different objects.
• Group the objects with similar shapes, and then introduce these as cuboid, cube, cylinder,
cone, sphere, prism and pyramid.
• Ask the students to find the number of vertices, faces and edges of the solids of different
shapes.
• Introduce students with different types of prisms and pyramids based on the shape of
base.
• Define types of prisms and pyramids on the basis of their properties.
• To reinforce, ask the students to do the Maths in Real Life section from the textbook.
• Ask the students to do Check Point 12.1 from the textbook.
Nets of solids
• Take any box having a shape of a cube or a cuboid and made up of same hard paper.
56
 0 ÷
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+% • Cut it along the edges as required and open the box.
• Ask the students to observe the flat surface so obtained.
• Introduce in this way the net of the solid so obtained.
• Also show them that by folding different segments of the flat surface, we can obtain the
solid again.
• In this way, students will be able to draw the net of any solid.
• Thus, by cutting and posting the required solid can be obtained.
• To reinforce, ask the students to do the Maths Lab Activity section from the textbook.
To revise the concepts learnt in the chapters students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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Worksheet 1
1. A cuboid looks like a rectangular box. Draw a cuboid and label it.
Also, name its each face, vertex and edge.
2. A cube is a cuboid whose edges are all of equal length. Now fill in the blanks.
(a) It has __________ faces.
(b) Each face has __________ edges.
(c) Each face has __________ vertices.
(d) It has __________ edges.
(e) It has __________ vertices.

3. A triangular pyramid has a triangle as its base. It is also known as tetrahedron.

Now answer the following questions.
(a) How many faces does a tetrahedron have?
(b) How many edges does a tetrahedron have?
(c) How many vertices does a tetrahedron have?

4. A square pyramid has a square as its base. Now answer the following questions.
(a) How many faces does a square pyramid have?
(b) How many edges does a square pyramid have?
(c) How many vertices does a square pyramid have?

5. A triangular prism looks like the shape of a kaleidoscope. It has a triangle as its
base. Now answer the following questions.
(a) How many faces does a kaleidoscope have?
(b) How many edges does a kaleidoscope have?
(c) How many vertices does a kaleidoscope have?

6. Study the given polygon PQRS and name its:

S R
(a) sides (b) vertices
(c) diagonals (d) adjacent sides

P Q

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1. Match the following.

(a) (i)

(b) (ii)

(c) (iii)

(d) (iv)

(e) (v)

(f) (vi)

2. Draw the nets of a cube and a cuboid in the space given below.

59
 13 Practical Geometry

Learning Objectives
Students will be able to
➢ recapitulate the use of a ruler, a protractor, compasses for drawing some geometrical
shapes
➢ construct a line segment using a ruler and a pair of compasses
➢ construct a line segment equal to the sum and difference of the measures of two given
line segments
➢ construct a line segment perpendicular to a given line segment at a point on it by
– using a ruler and set squares
– using a ruler and a pair of compasses
➢ construct a perpendicular to a line segment from a point outside it by using
– a ruler and set squares
– a ruler and a pair of compasses
➢ construct a perpendicular bisector of a given line segment
➢ construct angles
– by using set squares
– by using a protractor
– by using a ruler and a pair of compasses
➢ construct the bisector of an angle
➢ construct a circle by using
– any circular object
– a ruler and a pair of compasses

Concept Explanation
• Students are already familiar with basic geometrical tools.
• Read the Basic Geometrical Tools section to recapitulate these concepts.
• Introduce students to the basic geometrical tools such as a ruler, a divider, compasses, a
protractor, set squares and their functions.

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perpendicular to a given line segment
• Discuss the process of measuring the line segments by using a ruler as well as a divider.
• Ask students to draw a line segment of any measure. Then ask them to measure its length
by using a ruler and a divider.
• Explain the process of constructing a line segment by using a ruler and a pair of compasses.
Now, construct two line segments PQ and RS of any length. Then, draw a ray MN longer
than the total length of PQ and RS. Using compasses, measure and cut off MC = PQ and
CT = RS along MN, i.e., MT = MC + CT = PQ + RS.
• Similarly, ask them to draw a line segment of measure MG = PQ - RS
• Explain the process of constructing a perpendicular on a given line segment at a point
on it by using a ruler and set squares as well as by using a ruler and compasses.
• Demonstrate how to construct a perpendicular on a given line segment from a point
outside it by using a ruler and set squares as well as by using a ruler and compasses.
• Also, demonstrate how to draw a perpendicular bisector of a line segment.
• To reinforce, ask the students to do the Maths Lab Activity section from the textbook.
• Instruct the students to do Check Point 13.1 from the textbook.
Construction of angles using set squares
• Explain the process of construction of ‘angles’ by using set squares, by using a protractor
and by using a ruler and a pair of compasses.
• Also, discuss the process of drawing the angle bisector of a given angle.
For example, first draw an angle of 50°, then using a ruler and compasses draw the
bisector of the angle.
• Use the examples given in the textbook to explain the process and to reinforce the concepts.
• Instruct the students to do Check Point 13.2 from the textbook.
Construction of circle
• Bring some circular objects to the class.
• Demonstrate how to draw a circle using these objects.
• Now provide each with student a circular object or ask them to find any circular object
from their surroundings in the class and draw a circle on a sheet of paper.
• Now, demonstrate how to draw a circle using a ruler and compasses.

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• To reinforce, ask the students to do the Maths in Real Life section from the textbook.
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• For more practice, instruct the students to do Check Point 13.3 from the textbook.
To revise the concepts learnt in the chapter, students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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1. Draw a line segment of length 8.5 by using a ruler.
2. Construct a line segment AB of length 7.5 cm. From this, cut off length 5.3 cm.
Measure the length of BC.

___________________________________________________________________
3. Given AB of length 7.3 cm and CD of length 4.3 cm. Construct a line segment XY
such that the length of XY is equal to the difference of the lengths of AB and CD.
Verify the length of XY by measuring it using a ruler.

___________________________________________________________________
4. Draw any line segment PQ. Take any point R on it. Through R, draw a perpendicular
to PQ. (Use a ruler and set squares.)
5. Draw the perpendicular bisector of XY whose length is 10.3 cm.
(a) Take any point P on the bisector drawn. Examine whether PX = PY.
(b) If M is the mid-point XY, what can you say about the lengths of MX and XY?
6. Draw a line segment of length 13.8 cm. Using compasses, divide it into four equal
parts. Verify the length of each by actual measurement.
7. Construct angles of the following measures using a ruler and a protractor.
(a) 40° (b) 57° (c) 123° (d) 149°
8. Construct angles of the following measures using a ruler and compasses.
(a) 60° (b) 22.5° (c) 150° (d) 180°
9. Draw an angle of measure 80° and bisect it. Also, find the measure of each angle.
10. Construct the circles with the following radii.
(a) 4 cm (b) 7 cm (c) 4.5 cm (d) 6.2 cm

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Worksheet 2
1. Draw an angle of measure 143° by using a protractor. Also, construct its bisector.
2. Using set squares, construct the following angles.
(a) 30° (b) 60° (c) 45° (d) 150°
3. Construct angles of the following measures by using a ruler and compasses.
(a) 60° (b) 30° (c) 90° (d) 75°
(e) 120° (f) 150° (g) 135° (h) 45°
4. Draw an angle of measure 45° and bisect it.
5. Draw an angle of measure 70°. Make a copy of it using only a straight edge and
compasses.
6. Draw a circle of radius 3.5 cm.
7. Draw a circle and any two of its diameters.
If we join the end points of these diameters, what is the figure so obtained?
Which figure is obtained if the diameters are perpendiculars to each other?
How do we check our answer?
8. Given XY of length 5 cm and MN of length 2.5 cm. Construct a line segment
PQ such that the length of PQ is equal to the sum of the lengths of XY and MN.
Verify the length of PQ by measuring it using a ruler.

___________________________________________________________________
9. An angle ABC is given, whose measure is not known. Construct another angle
PQR such that its measure is twice that of angle ABC.

___________________________________________________________________

64
 14 Symmetry

Learning Objectives
Students will be able to
➢ recapitulate the concept of symmetrical and asymmetrical objects
➢ identify and draw symmetrical figures with
– one line of symmetry
– two lines of symmetry
– more than two lines of symmetry
➢ recognise symmetry in different geometrical figures
➢ learn about the reflection and reflection line

Concept Explanation
• Students are already familiar with symmetrical and asymmetrical objects.
• Bring some pictures of symmetrical objects to the class.
• Show these pictures one by one to the class and ask if these are symmetrical or not.
• Read the Introduction section to recapitulate these concepts.
One line of symmetry
• Read the related section from the textbook.
• Tell students that a figure which is identical on both the sides of a line, is said to be
symmetrical about the line.
• When a symmetrical object is folded along the line of symmetry, the two parts
superimpose, i.e., fall on one another exactly.
• Draw an isosceles triangle on the board. Demonstrate how to draw the line of symmetry
of this triangle.
• Ask them to find some other examples of figures having exactly one line of symmetry.
For this the teacher can give examples of some letters of English alphabet.
• To reinforce, ask the students to do Check Point 14.1 from the textbook.
Symmetrical figures with two lines of symmetry; Symmetrical figures with more than
two lines of symmetry; Symmetry of different geometrical figures
• Read the related sections from the textbook.
• Draw a rectangle, an equilateral triangle, a circle, etc. on the blackboard.
• Demonstrate how to draw the lines of symmetry for these shapes.
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• Ask them to identify the shapes having exactly two or more than two lines of symmetry.
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• Use the examples given in the textbook and ask the students to observe the symmetry of
different geometrical figures.
• To reinforce, ask the students to do the Maths Lab Activity section from the textbook.
• Ask the students to do Check Point 14.2 from the textbook.
Reflection
• Bring a mirror, a chart to the class.
• Make the mirror stand vertically on the chart with the help of a student.
• Place an object in front of the mirror. Ask the students to observe the reflection (or
image) of the object formed behind the mirror.
• Tell to them that the image of the object and the object both are equidistant from the
mirror.
• Also, explain to them that here mirror is playing the role of a mirror line.
• Verify the concept by using the picture of a symmetrical object. For this, first place it on
the chart paper and draw its line of symmetry. Now place the mirror along the line of
symmetry.
• Ask the students to observe the reflection formed behind the mirror.
• Repeat the same with the pictures of different symmetrical objects.
• To reinforce, ask the students to do the Maths in Real Life section from the textbook.
• Ask the students to do Check Point 14.3 from the textbook.
To revise the concepts learnt in the chapter, students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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1. List any four symmetrical objects that you see at your surroundings.
2. Draw a triangle which has
(a) exactly one line of symmetry.
(b) exactly three lines of symmetry.
(c) no line of symmetry.
3. On a square–shaped paper, sketch the following.
(a) A triangle with a horizontal line of symmetry but no vertical line of symmetry
(b) A quadrilateral with both horizontal and vertical lines of symmetry.
(c) A quadrilateral with a horizontal line of symmetry but no vertical line of
symmetry.
(d) A hexagon with exactly two lines of symmetry.
(e) A hexagon with six lines of symmetry.
4. Draw all possible lines of symmetry in each of the following symmetrical
shapes.

(a) (b) (c)

(d) (e) (f)

(g) (h)

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Worksheet 2
1. Draw the line of symmetry in each of the following letters.

A D M H
S Z E V
Also, name the letters which do not have any line of symmetry.
2. State whether the following statements are True or False.
(a) A square has two lines of symmetry. _______
(b) A rectangle has two lines of symmetry. _______
(c) An isosceles trapezium has one line of symmetry. _______
(d) An equilateral triangle has no lines of symmetry. _______
(e) Every diameter of a circle is a line of symmetry for the circle. _______
(f) A rhombus has two lines of symmetry. _______
(g) A line segment has two lines of symmetry, its perpendicular
bisector and itself. _______
(h) An angle has one line of symmetry, which is its bisector. _______
3. Draw all possible lines of symmetry in each of the following shapes.

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

(k) (l)

68
 15 Mensuration

Learning Objectives
Students will be able to
➢ recapitulate the concept of plane figures and their perimeters
➢ identify closed curve, simple curve, simple closed curve, interior and exterior regions
of simple closed curves
➢ find the perimeter of some irregular and regular figures
➢ understand the concept of area and its units
➢ find the area of a given figure
– by using a square-shaped paper
– by using formulae
➢ apply the concept of perimeter and area in our day-to-day activities

Concept Explanation
• Students are already familiar with the concept of plane figures, perimeter and area.
• Give some examples and recapitulate these concepts.
Simple closed curve
• Read the related section from the textbook.
• Explain the concept of curve and its different shapes such as open curve, closed curve,
simple curve, simple closed curve, etc.
• Draw some of these curves on the blackboard.
• Now ask the students to identify the types of curves drawn on the blackboard.
• For more practice, ask the students to do Check Point 15.1 from the textbook.
Perimeter
• Bring a string, a ruler, a meter scale, tape, etc. to the class.
• Demonstrate how to find the perimeter of an object.
• Tell them that the perimeter of a plane figure is the measure of the length of its boundary.
• Develop the formulae for calculating the perimeter of the figures like rectangle, square,
triangle and equilateral triangle.
• Draw their attention to the fact that to find the perimeter of a plane figure (with/without
formulae), first convert all the sides of the plane figure into the same unit.

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• Read the Tidbits section of the related topic to make them understand the relation
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between different units of length.
• Use the examples given in the textbook to make them understand the concept.
• To reinforce, ask the students to do Check Point 15.2 from the textbook..
Area and its measurement
• Go through the related section in the textbook.
• Explain the meaning of area by using some real life examples.
• Tell them an area is the amount of surface enclosed within the boundary of a plane
figure.
• Make students familiar with the different units of area such as cm², m², dm², etc.
• Bring a square-shaped sheet to the class and give a sheet to each student.
• Ask the students to draw any plane figure on their sheets.
• Explain the rules associated with the concept of finding the area of plane figures (regular
or irregular) using the square-shaped sheet paper.
• Use the related examples given in the textbook to make them understand the concept.
• Develop the formulae for calculating the area of the figures like a rectangle and a square.
• Write the formulae for calculating the area of a square and a rectangle.
• Ask the students to observe the interrelationship between the different units of area.
• Use examples given in the textbook to make them understand the concept.
• To reinforce, ask the students to do the Values, Maths in Real Life and Maths Lab Activity
sections from the textbook.
• To reinforce, ask the students to do Check Point 15.3 from the textbook.
To revise the concepts learnt in the chapter, students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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1. Measure and write the lengths of the four sides of a page of your notebook.
The sum of the lengths of the four sides
= AB + BC + CD + DA
= ____ cm + ____ cm + ____ cm + ____ cm = ____ cm
What is the perimeter of the page?
2. Find the perimeter of the following figures:
(a) Perimeter = AB + BC + CD + DA = ____ + ____ + ____ + ____ = ____
A 40 cm B

10 cm 10 cm

D C
40 cm

(b) Perimeter = AB + BC + CD + DA = ____ + ____ + ____ + ____ = ____

D 6 cm C

6 cm 6 cm

A 6 cm B

3. A piece of wire is 84 cm long. What will be the length of each side if the wire is used
to form
(a) a square (b) an equilateral triangle (c) a regular hexagon
4. Find the perimeters of the rectangles whose lengths and breadths are given below.
(a) 6 cm, 4 cm (b) 9.1 cm, 7.2 cm (c) 15 m, 6.5 m
5. Find the perimeters of the squares whose sides are given below.
(a) 5 cm (b) 4.5 cm (c) 50.5 m
6. Each side of a square field is 20 m. Find the
(a) perimeter of the field.
(b) cost of fencing the field at the rate of `20 per metre.

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Worksheet 2
1. Find the perimeter of the following figures. H 1 cm G

3 cm
(a) Perimeter = AB + BC + CD + DE + EF + FG + GH 3 cm
+ HI + IJ + JK + KL + LA 3 cm 3 cm E
J
1 cm I F
1 cm
= ____ + ____ + ____ + ____ + ____ L C
K 3 cm 3 cm D
+ ____ + ____ + ____ + ____ + ____
+ ____+ ____ = ____ 3 cm 3 cm

(b) Perimeter = AB + BC + CD + DE + EF + FA A B
1 cm
= ____ + ____ + ____ + ____ + ____ + ____ = ____

A 100 m B
45 m
120 m
F 65 m E
80 m C
D 90 m

2. Find the perimeter of a rectangle whose length and breadth are 150 cm and 1 m,
respectively.
3. Find the perimeter of a regular pentagon with each side measuring 3 cm.
4. Pinky runs around a square field of side 75 m. Bob runs around a rectangular field
of length 160 m and breadth 105 m. Who covers more distance and by how much?
5. Find the area of a square plot of side 8 m.
6. The area of a rectangular piece of cardboard is 36 cm2 and its length is 9 cm. What
is the width of the cardboard?
7. Sameer wants to cover the floor of a room 3 m wide and 4 m long with square tiles.
If the length of each tile is 0.5 m, then find the number of tiles required to cover
the floor of the room.
8. Find the area of each shaded region. Given, each square is 1 cm² .
(a) (b) (c) (d)

72
 16 Data Handling

Learning Objectives
Students will be able to
➢ recapitulate the concept of data and representation of data
➢ know the concept of frequency distribution and construction of frequency distribution
table
➢ know how to interpret and draw a pictograph
➢ know how to interpret and draw a bar graph

Concept Explanation
• Students are already familiar with the concept of data and representation of data.
• Give some examples and recapitulate these concepts.
Data; Frequency distribution table
• Discuss the need and application of data handling in real life.
• Tell students that the information in numerical facts is called data.
• Discuss the process of collection and representation of data.
• Explain that the data can be arranged in a serial order, ascending order and descending
order.
• Ask students to collect data of age, qualification of his/her family member and to
represent in ascending or descending order.
• Explain to them the process of constructing the frequency distribution table.
• Introduce tally marks, frequency, etc.
• Use the illustrative examples given in the textbook to make students understand the
concept.
• Instruct them to do Check Point 16.1 from the textbook.
Interpretation of a pictograph; Drawing a pictograph
• Bring a pictograph in the class.
• Discuss in class how to read and interpret the graph.
• Now ask students some questions based on the graph.
• Now provide a chart to each student and ask him/her to make same cut-outs on their
own choices related to the pictograph.

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• Collect information on the favourite subjects of the students and write it on the board.
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• Demonstrate how to draw a pictograph for the information recorded on the board.
• Use the examples given in the textbook to make them understand these concepts.
• For more practice, ask the students to do Check Point 16.2 and Check Point 16.3 from
the textbook.
Interpretation of a bar graph; Drawing a bar graph
• Tell students that a bar graph is another way of representation of data. We can read
(interpret) the bar graph to get information and also based on the information (data), we
can construct the bar graph.
• Demonstrate how to draw a bar graph for the pictograph they have already made.
• Clarify to them that a bar graph can be horizontal or vertical.
• Draw their attention to the fact that they should not forget to label both the axes of the
graph and write the title of the bar graph.
• Use examples given in the textbook to make the students understand the concept of
interpreting and drawing a bar graph.
• To reinforce, ask the students to do the Maths in Real Life and Maths Lab Activity
sections from the textbook.
• Instruct the students to do Check Point 16.4 from the textbook.
To revise the concepts learnt in the chapter, students should do the Test Yourself and Brain
Workout sections from the textbook.
Use the Multiple Choice Questions and Mental Maths sections to conduct a quiz contest in
the class.
Use the At a Glance section to revise the key points of the concepts.

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1. In a mathematics test, the following marks were obtained by 40 students. Arrange
these marks in a table by using tally marks.
8 1 3 7 6 5 5 4 4 2 4 9
5 3 7 1 6 5 2 7 7 5 8 4
2 8 9 5 8 6 7 4 5 6 9 6
4 4 6 6
(a) Find the how many students scored marks equal to or more than 7.
(b) How many students scored marks less than 4?
2. Following are the favourite sweets of 30 students of Class VI.
Rasgulla, Gulab jamun, Sandesh, Laddoo, Laddoo, Gulab jamun, Sandesh, Jalebi,
Laddoo, Sandesh, Rasgulla, Ragulla, Laddoo, Sandesh, Gulab jamun, Rasgulla,
Laddoo, Sandesh, Jalebi, Jalebi, Gulab jamun, Laddoo, Gulab jamun, Rasgulla, Jalebi,
Jalebi, Laddoo, Rasgulla, Gulab jamun, Rasgulla
(a) Arrange the names of the sweets in a table using tally marks.
(b) Which sweet is liked by the maximum number of students?
3. The pictograph given below shows the number of milk bottles sold each day in a
dairy. Use the information from the graph and answer the questions that follow.
Milk bottles sold
Day Number of milk bottles
Monday

Tuesday

Wednesday

Thursday

Friday
Key = 500 milk bottles

(a) On which of the two days did the dairy sell minimum number of milk bottles?
(b) How many more bottles of milk were sold on Friday than Thursday?

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Worksheet 2
1. The total number of animals in five villages is as follows.
Village A Village B Village C Village D Village E
80 120 90 40 60
Prepare a pictograph of these animals using the symbol (∆) to represent 10 animals
and answer the following questions.
(a) How many symbols represent the animals of village E?
(b) Which village has the maximum number of animals?
(c) Which village has more animals: Village A or Village C?
2. The total number of students of a school in different years is shown in the following
table.
Years 1996 1998 2000 2002 2004
Number of students 400 550 450 600 623
(a) Prepare a pictograph of students using a symbol () to represents 100 students
and answer the following questions.
(i) How many symbols represent the total number of students in the year 2002?
(ii) How many symbols represent the total number of students in the year 1998?
(b) Prepare another pictograph of students using any other symbol representing
50 students. Which pictograph do you find more informative?
3. A survey of 120 students was carried out to find out which activity they preferred
to do in their free time.
Preferred activity Number of students
Playing 45
Reading storybooks 30
Watching TV 30
Listening to music 10
Painting 15
(a) Draw a bar graph to illustrate the above data taking the scale of
1 unit length = 5 students.
(b) Which activity is preferred by the maximum number of students?
(c) Which two activities are preferred by the same number of students?
(d) Which activity is preferred by the least number of the students?
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Model Test Paper 2
Class 6
Time: 2½ hours Total Marks: 70
General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 28 questions divided into four sections A, B,C and D.
Section A consists of 8 questions of 1 mark each.
Section B consists of 6 questions of 2 marks each.
Section C consists of 10 questions of 3 marks each.
Section D consists of 4 questions of 5 marks each.
(iii) There is no overall choice. However, an internal choice has been provided in some questions.
Attempt only one options in such questions.

SECTION - A
1. What is the shape of a Globe.
2. Define a regular polygon.
3. What kind of angle 280° is?
4. Write the name of the plane shape which has countless number of lines of symmetry.
5. What is the sum of the interior angles of a triangle?
6. Perimeter of a square = 4 × ______
7. Length and breadth of a rectangular plot are ‘l’ and ‘b’, respectively. Use these
dimensions to find the perimeter of the plot.
8. Define tetrahedron.

SECTION - B
9. Find the number of sides, vertices and faces in a pentagonal prism.
10. With the help of a ruler and compasses, construct an angle of measure 30°.
11. Complete the following table.
Shape Number of line(s) of symmetry
Isosceles triangle
Kite
12. Draw the net of a cube.

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13. Find the perimeter of the following figure.
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7m

5m
4m

3m 1m

14. Following are the number of hits out of 10 chances of 20 students in a game of hitting
a target.
5 7 6 2 4
9 1 3 0 3
5 8 5 10 6
2 4 4 3 7
Prepare a frequency distribution table for the given data.

SECTION - C
15. Find the number of lines of symmetry for the following shapes.
(a) (b)

16. In the following figure, ABCDEF is a regular hexagon.

(a) What kind of triangle is ABC according to angles?
(b) What kind of triangle is ADC according to sides?
(c) What kind of quadrilateral is ADEF?

E D

F C

A B

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18. Find the area of the following figures.
(a) (b)

19. The number of monkeys in five cities of India is depicted by the given pictograph.

City Number of monkeys

Delhi

Bengaluru

Faizabad

Jaipur

Pune

Key: = 500 monkeys

Look at the pictograph and answer the following questions.

(a) Which city has the minimum number of monkeys?
(b) Is the number of monkeys in Faizabad less than the number of monkeys in Jaipur?
(c) How many monkeys are there in all?
20. Use a ruler and compasses to draw the following angles.
(a) 90° (b) 135°
21. The area of a rectangular garden is 240 m2. If the length of the garden is 24 m, find the
breadth of the garden.
or
Find the area of a square field of side 400 m2.
22. Draw a line segment MN = 7 cm. Mark a point P on it. Through point P, draw a
perpendicular to MN by using a ruler and compasses only.

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23. Find the number of right angles turned through by the hour hand of the clock when
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it goes from
(a) 3 to 9 (b) 2 to 11
24. Write the special names of the following.
(a) Square prism (b) Rectangular prism (c) Circular pyramid

SECTION - D
25. Draw a circle of radius 5.8 cm and centre O. Draw a chord PQ of length 8 cm. Shade
the major segment of the circle.
26. The bar graph given below shows the eye colours of a group of students.
Students’ eye colour

20
18
16
14
Number of students

12
10
8
6
4
2

Blue Black brown green

Eye colour

Now observe the graph and answer the following questions.

(a) How many students have blue eyes?
(b) Which eye colour do most students have?
(c) Which eye colour do least students have?
(d) How many students were there in the group?
27. Arun wants to cover the floor of a room 6 m wide and 9 m long by squared tiles. If
each square tile is of side 1.5 m, then find the number of squared tiles required to
cover the floor of the room.
D'
28. With reference to the adjacent figure
E' C'
(a) Write the name of the solid.
(b) How many vertices does it have in all? A' B'
(c) How many faces does the solid have? D
E C
(d) Name alll the vertical edges.
(e) Name all the horizontal edges. A B
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+% Answer Key
Chapter-1 Worksheet-2
Worksheet-1 1. (a) True (b) False (c) False (d) False
1. (a) 100 (b) 1000 (c) 100 (d) 10 (e) True (f) False (g) False (h) True
(i) False (j) False (k) False
(e) 1000 (f) 20 (g) 5000 (h) 0.2 cm
2. (f) 123456 × 8 + 6 = 987654
(i) 0.005 kg (j) 486000 mL (k) 0.89 m
(g) 1234567 × 8 + 7 = 9876543
(l) Roman (m) 1000 (n) CD
3. (a) 3 + 5 = 8, 4 + 1 = 5; 3 × 5 = 15, 4 × 1 = 4
2. (a) (i) 100 thousands (ii) 10 lakhs
(b) 2 + 3 = 3 + 2, 2 × 3 = 3 × 2; 4 + 7 = 7 + 4,
(iii) 10 crores 4×7=7×4
(b) 0; No (c) 1; Yes (c) (1 + 2) + 3 = 1 + (2 + 3), (4 + 6) + 11
(d) PV = 2000, FV = 2 (e) 9,009,009 = 4 + (6 + 11); (1 × 2) × 3 = 1 × (2 × 3);
3. (a) 600 (b) 8000 (4 × 6) × 11 = 4 × (6 × 11)
Worksheet-2 (d) 5 × (3 + 4) = 5 × 3 + 5 × 4,
1. (a) 6 (b) 0 (c) 8 (d) 4 5 × (4 – 3) = 5 × 4 – 5 × 3;
2. (a) 87620, 20678 (b) 97431; 13479 7 × (9 + 2) = 7 × 9 + 7 × 2,
3. 770841 4. 2766995 7 × (9 – 2) = 7 × 9 – 7 × 2
5. (a) 68000 4. 195 laddoos will be packed in 13 boxes such as
15 laddoos in each box. Also, there will remain 5
(b) 68000; In both cases the difference is 34. laddoos, which will be packed in another box.
6. (a) 8800 (b) 6000 So, total number of boxes required
7. (a) 658 (b) 515 (c) 1425 (d) 445 = 13 + 1 = 14 boxes
8. (a) True (b) False (c) False (d) True
(e) True Chapter-3
Worksheet-1
Chapter-2 1. (a) False (b) False (c) True (d) False
Worksheet-1 (e) False (f) False (g) False (h) True
1. (a) 205 × 95 (b) 111 × 120 (i) False (j) False (k) True (l) True
(c) (200 + 25) × (50 + 5) (m) True (n) True
(d) 76 × 97 (e) 100 – 36 2. (a) 17 and 71, 37 and 73, 79 and 97
2. 11,000; 11,001; 11,002 3. (a) HCF = 4, LCM = 24
3. 0 4. 22 (b) HCF = 5, LCM = 140
5. (a) 428302 (b) 199002 (c) 45667237 (c) HCF = 3, LCM = 5148
6. (a) 62 (b) 99 (c) 9612053 (d) HCF = 5, LCM = 300
7. (a) 1408 (b) 4140 4. (a) HCF = 10, LCM = 70
8. (a) 2970000 (b) 16600 (c) 431000 (b) HCF = 25, LCM = 150
(d) 169500 (e) 1000000 (c) HCF = 14, LCM = 294
9. (a) 12500 (b) `120 (d) HCF = 30, LCM = 1800

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Worksheet-2 3. (a) 3 (b) 39 (c) –13 (d) –62
%+
1. Number Divisibility test 4. (a) –8 + 8 = 0 (b) 13 + (–13) =0
By By By By By By By By By
2 3 4 5 6 8 9 10 11
(c) 15 + (–15) = 0 (d) (–4) + 19 = 15
128 Yes No Yes No No Yes No No No (e) 6 + –16 = –10 (f) 26 + (–52) = –26
990 Yes Yes No Yes Yes No Yes Yes Yes
5. (a) < (b) > (c) > (d) <
1586 Yes No No No No No No No No
275 No No No Yes No No No No Yes 6. 6 7. –50 8. –8
6686 Yes No No No No No No No No
639210 Yes Yes No Yes Yes No No Yes Yes Chapter-5
429714 Yes Yes No No Yes No Yes No No Worksheet-1
2856 Yes Yes Yes No Yes No No No No
1. (a) 0 1 2 3 4
3060 Yes Yes Yes Yes Yes No Yes Yes No 4 4 4 4
406839 No Yes No No No No No No No
(b) 0 1 2 3 4 5 6 7 1
2. (a) HCF = 15, LCM = 90 8 8 8 8 8 8 8

(b) HCF = 13, LCM = 676

(c) HCF = 5, LCM = 600 (c) 0 2 3 1 8 2 11 3
5 5 5 5
(d) HCF = 25, LCM = 1800
3. (a) HCF = 12, LCM = 60 2. (a) 31 (b) 47 (c) 20 (d) 32
4 8 7 3
(b) HCF = 30, LCM = 180 66 77
(e) (b)
(c) HCF = 2, LCM = 15300 7 9
(d) HCF = 7, LCM = 1260 3. (a) 12 (b) 9 (c) 18 (d) 27
20 15 30 45
4. LCM = 300 5. 99990
(e) 27
45
Chapter-4
4. (a) 9 (b) 3
Worksheet-1 12 4
1. (a) + 2500 m (b) – 750m 5. (a) 3 (b) 5 (c) 6 (d) 1 (e) 1
4 3 7 6 3
(c) + `500 (d) – `600
6. (a) v (b) i (c) iv (d) iii (e) ii
2. Shimla – 2°C Agartala + 30°C
7. 5 8. (a) 5 1 (b) 313 (c) 4 5
Delhi + 20°C Shrinagar – 7°C 6 6 360 12
Mumbai + 25°C Worksheet-2
3. (a) –1, –2, –3, –4, –5, –6 1. (a) No (b) No (c) Yes (d) No
(b) –3, –2, –1, –0, 1, 2, 3 2. (a) 1 (b) 12 (c) 22 = 1 (d) 16 = 1 1
14 7 22 15 15
(c) –15, –14, –13, –12, –11, –10, –9
8
(e) = 1 (f) 2 (g) 1 (h) 7
(d) 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40 8 3 7 5
4. (a) 10 (b) 21 (c) 111 (d) 275 3. (a) 4 (b) 0 (c) 8 (d) 7
10 6 21 27
5. (a) –25 (b) –54 (c) 87 (d) 123
6. (a) –108 (b) 545 7. (a) 29 (b) 106 4. (a) 111 (b) 5 (c) 1 5 (d) 5
12 12 12
8. (a) –96 9. `700
Worksheet-2 5. 14 km = 1 2 km
10 5
1. (a) –4 (b) –3 (c) –16 (d) –3 6. In both the classes same fractions of students got
2. (a) –177 (b) 0 (c) –81 (d) –38 the first class.
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+% Chapter-6 5. (a) 0.5 cm (b) 5.5 cm (c) 17.5 cm
Worksheet-1 (d) 16.5 cm
1. 6. (a) 0.008 km (b) 0.070 km (c) 7.777 km
(d) 70.008 km
Hundreds Tens Ones . tenths hundredths
7. (a) 29.432 (b) 43.357 (c) 14.145
(a) 1 9 . 4
(d) 308.89 (e) 27.63
(b) 0 . 5
8. (a) 3.476 (b) 8.78 (c) 10.53
(c) 1 1 . 7
(d) 3.626 (e) 78.921 (f) 1.134
(d) 2 0 5 . 8
(e) 0 . 7 Chapter-7
(f) 2 . 8 Worksheet-1
(g) 1 . 0 1. (a) 2n (b) 3n (c) 3n (d) 2n
(h) 3 . 5 (e) 4n (f) 5n (g) 3n (h) 6n
(i) 1 3 . 8 2. (a) 50y 3. 6m 4. (c), (f)
(j) 2 1 . 2 5. (a) x + 6 (b) x – 5 (c) 7x
2. (a) 7.1 (b) 200.25 (c) 14.7 (d) 100.03 (d) x ÷ 7 (e) –y – 8 (f) (–y)6
(e) 500.9 (g) (–x) ÷ (–5) (h) (–y) × (–7)
3. (a) 306.07 (b) 9.25 6. (a) m × m × m × n × n (b) r × r × s × s × s
4. (a) Three-hundredths (c) 2 × 2 × 2 × 2 × 3 × 3 × x × x × x × x
(b) One point two zero (d) 5 × 5 × m × m × m × m × m × m × m × n ×
(c) One hundred right point five six n×n×n×n×n×n×n×p×p×p×p×p
×p×p×p×p
(d) Ten point zero seven
7. (a) 22 × 3 × 5 × m4 × n3 (b) 23 × 33 × a8 × b8
(e) Zero point zero three two
Worksheet-2
5. (a) 0.4 (b) 0.19 (c) both are equal
1. (a) The cost of a book is three times the cost of a
(d) 1.439 (e) both are equal (f) 0.07 notebook.
6. (a) 4.768 (b) 9.658 (c) 248.519 (b) The strength of the school is 20 times the
7. (a) 0.75 (b) 1.025 (c) 106.05 strength of our class.
8. (a) 54.64 (b) 990.09 (c) Jaggu’s uncle is four times as old as he and his
aunt is 3 years less than four times as old as he.
9. 241.25 10. `609
2. (a) Age of Suman 2 years ago.
Worksheet-2
(b) Age of Suman 4 years after.
1. (a) 0.05 (b) 0.8 (c) 45.2 (d) 0.096
(c) 4 more thaan 5 times the age of Suman.
(e) 0.458
3. (a) Two times the number of students like football
2. (a) 25 (b) 8 (c) 36 (d) 4102 in a class.
10 10 100 100
3025 (b) Half the number of students like football in a
(e)
1000 class.
3. (a) `0.08 (b) `0.70 (c) `0.25 (d) `7.25 4. (a) t + 4, t – 4, 4t, t ÷ 4, 4 – t, 4 ÷ t
(e) `50.90 5. (a) x ÷ 7 (b) y ÷ 11 (c) 23 × m
4. (a) 0.18 m (b) 0.05 m (c) 3.45 m (d) s × 10 (e) y + z (f) 50 – n
(d) 4.91 m 6. (a) –7 + x (b) –8 × n – m (c) p × q – 7
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7. (a) a × a × a × a × a × b × b × b 5. Sheela = 12 pens, Sangeeta = 8 pens
%+
(b) 4 × x × x × x 6. (a) 3:1 (b) 10:3 (c) 13:6 (d) 15:1
(c) 2 × 2 × 3 × r × r × r × s × s × s × s × t × t × t × 7. 24 8. x = 15
t×t
Worksheet-2
(d) 2 × 2 × 5 × 5 × 5 × a × a × a × a × a × b × b ×
b×b×b×c×c×c×c×c 1. (a) Yes (b) No (c) No (d) Yes
(e) Yes (f) No
Chapter-8 1. (a) False (b) True (c) True (d) True
Worksheet-1 (e) False (f) True
1. (a) (x + 5) years (b) (x – 4) years 3. (a) x = 80 (b) x = 2
(c) 6x years (d) (6x – 2) years 4. (a) x = 5 (b) x = 35
(e) 5x + 6 years 5. (a) 30 (b) 0.28 6. (a) 75 (b) 2.8
2
2. (a) x (e) n (g) p (h) x 7. `140 8. `192 9. Anup
3. (a) a = 12 (b) b = 17 (c) x = 7
(d) y = 3 Model Test Paper 1
4. (a) m = 54 (b) n = –16 (c) x = 3 Section A
(d) y = 1 1. loss of `250 2. 27 3. 7
8
5. (a) Son = 20 years, Daughter = 10 years 4. Four point six five 5. 1:4
Worksheet-2 6. Yes, 526537 is divisibile by 11.
1. (a) 15 (b) 13 (c) 10 (d) 16 (e) 2 (f) –2 7. Three hundred sixty million two hundred fifty-
six thousand two hundred one
2. x 1 2 3 4 5 6 7 8 9 10 ...
8. 5+2=7
x + 10 11 12 13 14 15 16 17 18 19 20

Thus, x = 6 0 1 2 3 4 5 6 7 8 9

3. (a) m = 11 (b) y = 2 (c) x = –765 Section B

2 44
(d) x = 73 9. 6.78 10. 875 11. 5
5 7
4. 14, 16, 18 12. 9 13. –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6
35
5. length = 16 cm, breadth = 11 cm 14. 4.050 L
6. (a) 20, 40 7. 11 8. 50, 55, 60 Section C
15. 99600 16. 37.455 17. 205, 250, 502, 520
Chapter-9 18. 2.34 19. Yes
Worksheet-1 20. Suresh works for a longer time.
1. (a) 6:12 or 1:2 (b) 12:30 or 2:5 21. 6 22. 2916000
2. (a) 15 = 5 = 10 = 15; yes
6 12 18 23. Ascending order: 3 > 5 > 4 > 6 > 13
18 11 8 6 5 7
Descending order: 13 < 6 < 4 < 5 < 3
(b) 21 = 7 = 7 = 56; yes 7 5 6 8 11
27 9 9 72
24. (a) m + 13 (b) 4a – 7 (c) 5b + 5
3. (a) 3:4 (b) 14:9 (c) 3:11 (d) 2:3
Section D
(e) 1:16 (f) 7:23 (g) 2:5 (h) 1:4
25. 96408 26. `1755 27. 55cm
4. (a) 2:3 < 6:5 (b) 1:3 > 1:4 (c) 4:5 < 5:4
28. 21 km 150 m
(d) 12:14 > 5:6 (e) 3:7 > 2:8 (f) 4:20 < 8:24
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Worksheet-1
Chapter-10
1. (a) ∆ABD, ∆ADC, ∆ABC
Worksheet-1
P (b) ∠ABD, ∠ADB, ∠BAD, ∠ADC, ∠ACD,
1. (a) A P B (b) X

D ∠DAC, ∠BAC
l M
(c) (c) AB, AC, BC, AD, BD, DC
E F
P Y Q (d) ∠ADB and ∠ADC
(d) O 2. (a) KL, NM and KN, LM
Q
(b) ∠NKM, ∠LMN and ∠KLM, ∠KNM
2. (a) (b) (c) KL, ML and KN, MN
(d) ∠LKN, ∠MNK and ∠KLM, ∠NML
3. 4. (a) (b) 3. P Q The meeting point of the
diagonals of the quadrilateral
M
5. (a) intersect (b) intersecting lines PQRS, that is, M here, lies in its
(c) collinear (d) parallel R interior.
S

6. (a) Ashoka chakra

4. (a) False (b) True (c) False (d) False
(b) Two roads crossing each other
(e) True (f) False (g) False (h) True
(c) Railway track (d) Golf stick
Worksheet-2
7. (a) open (b) closed (c) closed
1. (a) O (b) OD, OC, ON (c) DC
(d) closed (e) open
(d) AB (e) O, E (f) F
Worksheet-2
A (g) minor or major sector ODN
1. (a) A C
(b) (h) segment AMB
D
2. (a) Yes (b) No
O
B D B C 3. (a) True (b) True
F
B P
S 4. (a) O (b) OP A
B

(c) D
(d) (c) PQ (d) Sector OPER P
O Q

E M N
(e) segment APERQB (f) arc AFB E L
R
B C Q R (g) O (h) L
M
5. (a) adjacent (b) irregular (c) orthocentre
(e)
(d) centroid (e) isosceles
Q

N P Chapter-12
1. (a) obtuse (b) acute (c) obtuse Worksheet-1
(d) reflex (e) complete (f) right 1. Faces: ABCD, PQRS, A B
D
(g) zero (h) obtuse (i) straight ADSP, BCRQ,
C
P Q
(j) acute (k) reflex (l) reflex ABQP, DCRS R
S
3. (a) AB, BC, CA, (b) BE, AF, AD Vertices: A, B, C, D, P, Q, R, S
4. (a) False (b) True (c) False (d) True Edges: AB, BC, CD, DA, PQ, QR, RS, SR, AP, DS,
(e) False (f) True (g) True BQ, CR
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2. (a) 6 (b) 4 (c) 4 (d) 12 (e) 8 (g) (h)
%+
3. (a) 4 (b) 6 (c) 4
4. (a) 5 (b) 8 (c) 5
Worksheet-2
5. (a) 5 (b) 9 (c) 6
6. (a) PQ, QR, RS, SP (b) P, Q, R, S (c) SQ, RP 1.
(d) SP, PQ; PQ, QR; QR, RS; RS, SP Letters S and Z do not have any lines of symmetry
Worksheet-2 2. (a) False (b) True (c) True (d) False
1. (a) → (iii) (b) → (iv) (c) → (i) (e) True (f) True (g) True (h) True
(d) → (vi) (e) → (ii) (f) → (v)
2. (a) (b) 3. (a) (b) (c)

Cube Cuboid
Chapter-14 (d) (e) (f)
Worksheet-1
1. a leaf, a butterfly, face of a human, flowers

2. (a) (b) (g) (h) (i)

(Equalateral
(isosceles triangle) triangle)
(c)
(j) (k) (l)

(scalene triangle)
Chapter-15
3. (a) (b) Worksheet-1
2. (a) 100 cm (b) 24 cm
3. (a) 21 cm (b) 28 cm (c) 14 cm
(c) (d) 4. (a) 20 cm (b) 32.6 cm (c) 43 cm
5. (a) 20 cm (b) 18 cm (c) 202 cm
(e) 6. (a) 80 m (b) `1600
Worksheet-2
1. (a) 28 cm (b) 500 m
4. (a) (b) (c) 2. 500 cm 3. 15 cm
4. Bob, 230 m 5. 64 m2
6. 4 cm 7. 48 tiles
(d) (e) (f)
8. (a) 45 cm2 (b) 42 cm2
(c) 10 cm2 (d) 49 cm2

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Worksheet-1 9. sides = 15, vertices = 10, faces = 7
1. Marks Number of students 11. Number of line(s) of
Shape symmetry
1
Isosceles triangle 1
2
Kite 1
3
4 12. 13. 33 m
5
6
7 14. Hits Tally Marks Frequency
8
0 1
9
1 1
(a) 12 (b) 7 2 2
2. (a) Sweet Number of students 3 3
Rasgulla 4 3
Gulab jamun 5 3
Sandesh 6 2
Laddoo 7 2
Jalebi 8 1
9 1
(b) Rasgulla and laddoo 10 1
3. (a) Tuesday and Thursday (b) 1500
Section C
Worksheet-2
15. (a) 5 (b) 5
1. (a) 6 (b) Village B (c) Village C
16. (a) Isosceles (b) Right-angled triangle
2. (a) (i) 6 (ii) Five and half symbols
(c) Isosceles trapezium
(b) Graph with a symbol representing 50 students.
18. (a) 5 unit2 (b) 5 unit2
3. (b) Playing
19. (a) Delhi (b) No (c) 12500
(c) Reading story books and watching TV
21. 10 m
(d) Listening music
23. (a) 2 right angles (b) 3 right angles
24. (a) cube (b) cuboid (c) cone
Model Test Paper 2
Section D
Section A
26. (a) 4 (b) black (c) green (d) 34
1. Sphere
27. 24 tiles
2. A polygon having all sides and interior angles
equal. 28. (a) pentagonal prism (b) 10 (c) 7
3. Reflex 4. Circle 5. 180° 6. Side (d) AA', BB', CC', DD', EE',
7. Perimeter of the plot = 2(l + b) (e) AB, BC, CD, DE, EA, A'B', B'C', C'D', D'E',
E'A'
8. A tetrahedron is a pyramind whose base is a
triangle.
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