Teacher’sManual

MathsQuest

6

S Purkayastha

(An imprint of New Saraswati House (India) Pvt. Ltd.)New Delhi-110002 (INDIA)

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

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Second Floor, MGM Tower, 19 Ansari Road, Daryaganj, New Delhi-110002 (India) Phone : +91-11-43556600Fax : +91-11-43556688E-mail : delhi@saraswatihouse.comWebsite : www.saraswatihouse.comCIN : U22110DL2013PTC262320Import-Export Licence No. 0513086293

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First published 2016

ISBN: 978-93-5199-701-6

Published by: New Saraswati House (India) Pvt. Ltd.19 Ansari Road, Daryaganj, New Delhi-110002 (India)

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This book is meant for educational and learning purposes. The author(s) of the book has/have taken all reasonable care to ensure that the contents of the book do not violate any copyright or other intellectual property rights of any person in any manner whatsoever. In the event the author(s) has/have been unable to track any source and if any copyright has been inadvertently infringed, please notify the publisher in writing for any corrective action.

PrefaceThe 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.

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

ContentsContents

5

Learning Objectives

Students will be able to ➢ recapitulate the concept of natural numbers,whole numbers ➢ understand Indian and International systems of numeration ➢ � nd 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 ➢ � nd how many numbers are there between two given numbers ➢ use large numbers in real life ➢ apply operations in large numbers ➢ round o� numbers and estimate the sum, di� erence, 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 di� erence 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 di� erence 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 a� er the 5th place, the places in both the systems are called by di� erent names. Emphasise on the use of commas in both the systems at di� erent 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.

1 Knowing Our Numbers

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Successor and Predecessor; Comparison of numbers; Ordering of numbers• Rules of comparing numbers and successor and predecessor need to be applied to bigger

numbers as well.• A� er 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 di� erent colours, and one

big chart paper to the class.• Cut out 10 cm × 10 cm pieces of thick chart paper/cardboard of di� erent colours (one for

each student).• Divide the class into � ve groups and give di� erent 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 speci� ed range

on the piece of chart paper in � gures and in words.• On a big chart paper, draw 10 cm × 10 cm squares (as many squares as the number of

students) in � ve 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.• � is will continue till all squares are � lled 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 � nd 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 � nd the numbers using a simpler method as mentioned below.

• First � nd the di� erence between the numbers, then: (a) Add 1 to the di� erence, if both the numbers are included. (b) Subtract 1 from the di� erence, 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, di� erence, product and quotient; Roman numbers• Explain the method of rounding o� numbers to their nearest tens, hundreds and

thousands and explain to students that we can also estimate the sum, di� erence, products and quotients by � rst rounding o� the numbers involved and then rounding them o� .

• Students have learnt enough about the 7 symbols of the Roman system and the di� erent 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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1. Complete the following statements with suitable words or � gures.

(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) � ere 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 di� erence (47029 – 39385) to the nearest thousand.

Worksheet 1

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1. From the numeral 40256801, pick out the digits in each of the following place. (a) � ousands = ________ (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 di� erence between the greatest and the smallest numbers that are formed using the digits 6, 5, 2, 0, 9, 7 only once.

___________________________________________________________________

4. � e population of a city is 5236845. If the number of males is 2469850, � nd the number of females in the city.

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5. Find the estimated sum of 5486 and 62548 by estimating the numbers to their nearest (a) hundreds and (b) thousands. Also, � nd the di� erence between the estimated sum and the actual sum.

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6. Estimate the following by rounding o� 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) � e smallest 6-digit number using the digits 0 and 1 with repetition of digits is

110000. (c) � e place value and face value of the digit 1 in the number 12548 is same. (d) � e Roman numeral CCLIV in Hindu-Arabic numeral is 254. (e) Number of ten thousand in 100 millions is ten thousand.

Worksheet 2

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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 di� erent 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. � e number 1 is

the smallest natural number and there doesn’t exist largest natural number.

� e number zero; Whole numbers• Explain to the students how a whole number di� ers 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.

2 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. � erefore, 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. � erefore, 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 veri� es 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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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 a� er 10999.

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3. Which is the smallest whole number?

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4. How may whole numbers are their between 35 and 58?

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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

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10. � e 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 1

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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) � e predecessor of a 2-digit number is never a 1-digit number. ________ (h) � e natural number 1 has no predecessor. ________ (i) � e whole number 1 has no predecessor. ________ (j) � e whole number 0 has its predecessor. ________ (k) � e successor of a 2-digit number is always a 2-digit number. ________

2. Study the given pattern and � ll 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?

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Worksheet 2

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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) ➢ � nd the factors and multiples of the given numbers ➢ identify prime and composite numbers from 1 to 100 ➢ de� ne 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) ➢ � nd HCF by prime factorisation method and division method ➢ know about common multiples and least common multiples (LCM) ➢ � nd 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.

Simpli� cation of numerical expressions; Order of operations; Simpli� cation 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 di� erent answers. To get the correct answer while simplifying, we have to follow the speci� ed 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.

3 Playing With Numbers

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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 � nding 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 di� erent examples

from the textbook.• How does test of divisibility help students to � nd whether the given number is divisible

by a speci� c number.• Explain the general properties of divisibility, i.e., – If a number is divisible by other number, then the � rst 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 di� erence. • 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 � nding 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 � nd the HCF of three numbers, we � rst � nd the HCF of two numbers. � en, we � nd the HCF of the third number and HCF of the � rst 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 � nding 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.

• De� ne the properties of HCF and LCM of the given numbers as follows: (i) � e HCF of the given numnbers cannot be greater than any one of the numbers. (ii) � e LCM of the given numbers cannot be less than any one of the numbers.

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(iii) � e LCM of two co-prime numbers is equal to their product. (iv) � e HCF of two co-prime numbers is equal to 1. (v) � e 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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1. State whether the following statements are true or false:

(a) � e sum of three odd numbers is even.

(b) � e sum of two even numbers and one odd number is even.

(c) � e product of three even numbers is even.

(d) � e 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) � e product of two even numbers is always even.

2. � e 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.

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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

Worksheet 1

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1. Using divisibility tests, determine which of the following numbers are divisible by the given numbers as directed.

Number Divisibility testBy By By By By By By By By2 3 4 5 6 8 9 10 11

128 Yes No Yes No No Yes No No No9901586275668663921042971428563060406839

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. � e HCF of two numbers is 25. If the numbers are 75 and 100, � nd their LCM.

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

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Worksheet 2

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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 ➢ � nd 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. � us, 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. � is di� culty 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.� e opposite integers whose sum is zero are

called additive inverse.

4 Integers

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• Every integer has its successor or predecessor. � e smallest positive integer is 1, but the smallest negative integer is not known. � e 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.• De� ne di� erent 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.• De� ne di� erent 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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1. Represent the following numbers as integers with appropriate signs. (a) An aeroplane is � ying at a height of two thousand and � ve hundred metres above

the sea level. (b) A submarine is moving at a depth of seven hundred � � y metres below the sea

level. (c) A deposit of rupees � ve hundred. (d) A withdrawal of rupees six hundred.

2. Following is the list of temperatures of � ve 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 pro� t of `1000 on the sale of a refrigerator and loses `300 on the sale of a camera. Find what is Amar’s actual pro� t or loss?

Worksheet 1

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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. � e sum of two integers is 50. If one of them is 100, � nd the other.

___________________________________________________________________

8. If P – (–7) = –1, � nd the value of P.

___________________________________________________________________

Worksheet 2

24

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,

110

, 15

, 150

,...,etc.

• � e 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.

5 Fractions

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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.• De� ne two or more fractions representing the same part of a whole as equivalent

fractions.• Use the cut-outs of circles having di� erent 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. � e 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. – � nd the HCF of numerator and denominator. � en, 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• De� ne 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 di� erence of the numeratorsCommon 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, � rst

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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1. Draw number lines and represent the following fractions on them.

(a) 12

, 14

, 34

, 44

(b) 18

, 38

, 58

, 78

(c) 25

, 35

, 85

, 115

2. Express the following as improper fractions.

(a) 734

(b) 578

(c) 267

(d) 1023

(e) 937

(f) 859

3. Find the equivalent fraction of 35

having

(a) denominator 20 (b) numerator 9 (c) denominator 30 (d) numerator 27 (e) denominator 45

4. Find the equivalent fraction of 3648

with

(a) numerator 9 (b) denominator 4

5. Reduce the following fractions into simplest form.

(a) 4560

(b) 15090

(c) 8498

(d) 1696

(e) 1442

6. Match the equivalent fractions.

(a) 250400

(b) 180200

(c) 660990

(d) 180360

(e) 220550

(i) 910

(ii) 25

(iii) 12

(iv) 23

(v) 58

7. Compare which is bigger, 45

or 56

?

___________________________________________________________________

8. Simplify.

(a) 123

+ 3 12

(b) 930

+ 724

+ 518

(c) 314

+ 2 12

– 113

Worksheet 1

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1. Write Yes or No for each of the following.

(a) Is 59

equal to 45

? (c) Is 916

equal to 59

?

(d) Is 45

equal to 1620

? (d) Is 115

equal to 430

?

2. Solve.

(a) 128

+ 128

(b) 77

+ 57

(c) 122

+ 2122

(d) 815

+ 815

(e) 58

+ 38

(f) 1 – 13

(g) 17

+ 07

(h) 4 – 135

3. Fill in the missing fractions.

(a) 710

– 310

= 310

(b) 36

= 36

– 310

(c) 310

– 321

= 521

(d) 310

+ 527

= 1227

4. Solve.

(a) 23

+ 34

+ 12

(b) 113

+ 323

(c) 423

– 314

(d) 34

– 13

5. Shreya’s house is 910

km from her school. She walked some distance and then took

a bus for 12

km to reach the school. How far did she walk?

___________________________________________________________________

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

___________________________________________________________________

Worksheet 2

29

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. � ey can read and write decimals. • Discuss with them the meaning of decimal, use of decimal point. � e 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.

6 Decimals

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Decimals as fractions; Representation of decimals on a number line• 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• De� ne 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 � rst 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.• De� ne 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 a� er 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 le� ) the decimal point.

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• De� ne 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 a� er 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, � rst 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. � en 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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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-� ve (c) Fourteen point seven (d) One hundred point zero three (e) Five hundred point nine

3. Write the following as decimals. (a) � ree hundred six and seven-hundredths (b) Nine and twenty-� ve 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?

___________________________________________________________________

Worksheet 1

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1. Convert the following into decimals.

(a) 5100

(b) 810

(c) 45210

(d) 961000

(e) 4581000

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

Worksheet 2

34

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.

7 Introduction to Algebra

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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 di� erent 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 coe� cients; 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, coe� cients, 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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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. � e 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

Worksheet 1

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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. � e 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 n2

represent?

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) � e product of m and 23 (d) s times 10 (e) � e 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

Worksheet 2

38

Learning Objectives

Students will be able to ➢ recapitulate the concept of algebraic expressions and di� erent operations on algebraic

terms ➢ de� ne the meaning of mathematical statement and equation ➢ understand the concept of an algebraic equation especially a linear equation ➢ di� erentiate 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 di� erent

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.• � e 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 le� hand

side (LHS) and the right hand side (RHS). (b) solving an equation means to � nd 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.

8 Algebraic Equations

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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 � nd 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 di� culty.

• 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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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) 62 = 3 (d) (7 × 5) – 21 = 14

(e) 4 × 6 – 18 = 2n (f) 3x2 < 7

(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 (c) 5.5 x = 16.5 (d) 4y7 = 28

5. Madhav’s son is two times as old as his daughter. After 10 years, the son will be 32 times as old as Madhav’s daughter. Find the present age of Madhav’s son and daughter.

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Worksheet 1

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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) x2 = 8 (15, 16, 17, 18)

(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 � nd 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) 2m3 + 13 = 4 (b)

y + 12 +

y – 12 = 2

(c) 23x3 –

45

3x2 = 17 (d)

2x – 1 3 –

13 5 –

x – 1 2 = 10

4. � e sum of three consecutive even natural numbers is 48. Find the numbers.

___________________________________________________________________

5. � e breadth of a rectangle is 5 cm less than its length. If the perimeter of the rectangle is 54 cm, � nd the length and breadth of the rectangle.

___________________________________________________________________

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

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

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

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Worksheet 2

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Learning Objectives

Students will be able to ➢ de� ne the meaning of ratio ➢ know about the terms of ratio ➢ express a ratio in its simplest form ➢ � nd 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 � nding an equivalent ratio(s) of a given ratio either

by multiplying or dividing the terms of a ratio by any non-zero integer.

9 Ratio, Proportion and Unitary Method

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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 ⇒ ab = cd ⇒ ad = bc

• Explain in proportion, the � rst 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, � nd the value for one quantity. Next, from the value of the one quantity � nd 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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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) 1518 = 156 =

1018 =

1518 (b)

2127 =

159 =

718 =

1572

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. � e 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 a� er 10 years to the age of the son a� er 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.

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8. Find the mean proportional between 5 and 45.

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Worksheet 1

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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?

___________________________________________________________________

Worksheet 2

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MathematicsModel Test Paper 1

Time: 2½ hoursClass 6

Total Marks: 70

General Instructions: (i) All questions are compulsory. (ii) � e 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) � ere 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 pro� t of `250.

2. Represent 238 in the form of a mixed fraction. 3. If 5y = 35, then � nd 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 + 710 + 8

100

10. Convert DCCCLXXV into Hindu-Arabic numeral.

11. Find the equivalent fraction of 2535 having the denominator 7.

12. Solve: 67 – 35

13. Write all integers between –7 and 7. 14. Express in litres using decimals: 4 L 50 mL

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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 56 of an hour, while Suresh works for 89 of an hour. Find who works

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.

46,

311,

65, 1

37,

58

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, � nd 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 o� ce. Find out how far is his o� ce 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.

48

Learning Objectives

Students will be able to ➢ recapitulate the concepts of point, line, line segment, angle, etc. ➢ know what is geometry and its purpose ➢ de� ne a point, plane, line, curve, angle, ray, line segment ➢ identify di� erent types of lines such as intersecting lines, parallel lines, concurrent

lines, etc. ➢ de� ne and identify collinear and non-collinear points ➢ measure and compare the line segments ➢ de� ne angle and measure an angle ➢ learn about the di� erent 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 de� ne 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• Con� rm that each student has a pencil, a ruler, a divider, an eraser, etc.• Read the related sections from the textbook.

10 Basic Geometrical Concepts

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• Demonstrate how to compare two line segments using a divider and a ruler.• De� ne 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 di� erent 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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1. Draw a rough � gure 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)

Worksheet 1

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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, re� ex, 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 � gure name

(a) three line segments

AE B F

DC

(b) three rays

4. State which of the following statements are true or false.

(a) � e 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 re� ex angle.

(g) A line is a set of all points which have length only, i.e., no breadth, no height.

Worksheet 2

52

Learning Objectives

Students will be able to ➢ de� ne 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 ➢ � nd the perimeter of triangle ➢ know about quadrilaterals and types of quadrilaterals ➢ de� ne 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 � gures such as triangle,

square, rectangle, circle, etc.• Recapitulate these concepts with the help of some examples.

Polygons• De� ne 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.

11 Understanding Elementary Shapes

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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 stud