202110719-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G06-FY_Optimized

practice workbook Mathematics Grade 6 Name: Roll No: Section: School Name:

by classklap IMAX is India’s only customised content and educational assessment m. 610+ Published Books Team of IITians & PhDs IMAX Program has authored about 610+ Content development and automation are publications which are used by more than led by a team of IITians, data scientists and 4,50,000+ students. education experts Workshops Lakh Assessments IMAX Program has conducted IMAX Program has conducted more than teacher training workshops for more 85,00,000+ assessments across 17 states in the last 10 years. than 15,000+ teachers. Copyright © 2020 BGM Policy Innovations Pvt Ltd) All rights reserved. No part of this publication, including but not limited to, the content, the presentation layout, session plans, themes, key type of sums, mind maps and illustrations, may be reproduced, stored in a retrieval system or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior, written permission of the copyright owner of this book. 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. Some of the images used in the books have been taken from the following sources www.freepik.com, www.vecteezy.com, www.clipartpanda.com Creative Commons Attribution This book is part of a package of books and is not meant to be sold separately. For MRP, please refer to the package price.

This practice book is designed to support you in your journey of learning Mathematics for class 6. The contents and topics of this book are entirely in alignment with the NCERT syllabus. For each chapter, a concept map, expected objectives and practice sheets are made available. Questions in practice sheets address different skill buckets and different question types, practicing these sheets will help you gain mastery over the lesson. The practice sheets can be solved with the teacher’s assistance. There is a self-evaluation sheet at the end of every lesson, this will help you in assessing your learning gap.



TABLE OF CONTENT • Assessment Pattern: 40 Marks • Assessment Pattern: 80 Marks • Syllabus & Timeline for Assessment Page 1: 1. Knowing Our Numbers Page 9: 2. Whole Numbers Page 16: 3. Playing with Numbers Page 23: 4. Basic Geometrical Ideas Page 32: 5. Understanding Elementary Shapes Page 43: 6. Integers Page 49: 7. Fractions Page 58: 8. Decimals Page 67: 9. Data Handling Page 77: 10. Mensuration Page 85: 11. Algebra Page 93: 12. Ratio and Proportion Page 100: 13. Symmetry Page 110: 14. Practical Geometry







AASSSSEESSSSMMEENNTTPPAATTTTEERRNN Marks: 40 Grade 6/Mathematics Marks: 40 Grade 6 / Maths Max Internal PAPER: BEGINNER PAPER: PROFICIENT Mark Option Q.No Skill Level Difficulty Level Skill Level Difficulty Level Easy Medium Difficult Easy Medium Difficult Section A (Question Type: MCQ) · · · ·· 11 Remembering · · Remembering ·· ·· Applying ·· · 21 Applying · · · ·· · Understanding · · 31 Understanding · ·· Applying · · ·· · Analysing · 41 Applying · · · 51 Analysing Section B (Question Type: VSA) 61 Remembering Remembering Understanding 71 Understanding Remembering ··· Understanding 81 Remembering Understanding ·9 1 Understanding 10 1 Understanding Section C (Question Type: SA) ·11 2 Remembering Remembering Understanding 12 2 · ·Understanding ·13 2 Applying Applying Analysing Analysing 14 3 Understanding ·15 3 Understanding Remembering 16 3 Remembering Section D (Question Type: LA) · 17 3 Remembering Remembering Remembering ·18 4 Remembering Understanding Understanding Remembering 19 4 20 4 Remembering Beginner Paper: (Easy: 40%, Medium: 50%, Di icult:10%) Proficient Paper: (Easy: 25%, Medium: 50%, Di icult: 25%) Easy Question (E): Direct reference to concept fact, definition, theories and laws (mostly from worked examples in the book or end of chapter exercise). Medium Di iculty Question (M): Combination of concepts, definition, theories and laws, solving through direct or indirect methods, numerical with direct substitution, drawing diagram and direct labelling, uses and applications, balance equation (mostly modified concepts). Di icult Question (D): Complex numerical, justification, interpret some info and draw diagram, working of appliance, functioning (on-the-fly thinking of solutions based on understanding of concepts).

AASSSSEESSSSMMEENNTTPPAATTTTEERRNN MMarakrsk:s8: 080 Grade 6/Mathematics Grade 6 / Maths Max Internal PAPER: BEGINNER PAPER: PROFICIENT Mark Option Q.No Skill Level Difficulty Level Skill Level Difficulty Level Easy Medium Difficult Easy Medium Difficult Section A (Question Type: MCQ) 11 Remembering • Remembering • 21 Remembering • Remembering • 31 Applying • Applying • 41 Applying • Applying • 51 Understanding • Understanding • 61 Understanding • Understanding • 71 Remembering • Remembering • 81 Understanding • Understanding • 91 Applying • Applying • 10 1 Applying • Applying • Section B (Question Type: VSA) 11 1 Remembering • Remembering • 12 1 • Understanding • Understanding • 13 1 Analysing • Analysing • 14 1 Remembering • Remembering • 15 1 Analysing • Analysing • 16 1 • Applying • Applying • 17 1 • Applying • Applying • 18 1 • Understanding • Understanding • 19 1 Analysing • Analysing • 20 1 Understanding • Understanding • Section C (Question Type: SA) 21 2 • Applying • Applying • 22 2 • Understanding • Understanding • 23 2 Applying • Applying • 24 2 Remembering • Remembering • 25 2 Analysing • Analysing • 26 2 Remembering • Remembering • Section D (Question Type: SA) 27 3 • Remembering • Remembering • 28 3 Remembering • Remembering • 29 3 • Understanding • Understanding • 30 3 • Applying • Applying • 31 3 Remembering • Remembering • 32 3 Remembering • Remembering • 33 3 • Remembering • Remembering • 34 3 Analysing • Analysing • Section E (Question Type: LA) 35 4 Understanding • Understanding • 36 4 • Remembering • Remembering • 37 4 Remembering • Remembering • 38 4 Understanding • Understanding • 39 4 • Understanding • Understanding • 40 4 Understanding • Understanding •

SYLLABUS FOR ASSESSMENT Grade 6/Mathematics CHAPTERS PT-1 TE-1 PT-2 TE-2 Chapter 1: Knowing Our Numbers ✓ ✓ Chapter 2: Whole Numbers ✓ ✓ ✓ ✓ Chapter 3: Playing with Numbers ✓ ✓ ✓ Chapter 4: Basic Geometrical Ideas ✓ ✓ ✓ ✓ Chapter 5: Understanding Elementary Shapes ✓ ✓ Chapter 6: Integers ✓ ✓ Chapter 7: Fractions ✓ sChapter 8: Decimals ✓ Chapter 9: Data Handling ✓ Chapter 10: Mensuration ✓ Chapter 11: Algebra ✓ Chapter 12: Ratio and Proportion Chapter 13: Symmetry Chapter 14: Practical Geometry Assessment Timeline Periodic Test-1 22nd July to 12th August Term 1 Exam 23rd September to 21st October Periodic Test-2 16th December to 13th January Term 2 Exam 1st March to 30th March



PRLAECSTSIAOCNENDSPHLEAENTS (This section contains lesson plans for all lessons listed in annual plan. This section also has a set of practice questions grouped into different sheets based on different skill buckets. These worksheets are designed for usage within the allocated periods. Answer keys for these questions are also provided. By solving these questions in the class, students will strengthen and enhance their knowledge in all aspects related to Mathematics. The same has to be administered in the allocated period. Peer check should be instilled at the end)



1. Knowing Our Numbers Learning Outcome By the end of this chapter, students will be able to: • Describe Indian and International system of • Read and expand large numbers numeration • Carry out comparison of numbers • Appreciate usage of brackets • Write and explain Roman numerals Concept Map Addition, Knowing our numbers Multiplication, Subtraction, Dividing of numbers Comparing numbers Introducing 5-digit System of Estimation of Shifting digits and 6-digit numbers Numeration numbers Large numbers Roman numbers Aid in reading and writing large numbers Use of commas Key Points If the given numbers have same number of digits then, comparison of value of digits at different • Numbers help us to count concrete objects. place must be made to decide the largest and Numbers help us in ordering of objects (largest to smallest number. smallest & vice versa) Ex: To find the largest of the given numbers 6567, Comparing numbers: 6578 we have to first count number of digits in the To compare the numbers, the number of digits in given numbers. Here, number of digits is same for both the numbers. So we have to compare the digit the number is counted. value starting from the thousands place which Ex: To find the largest and smallest number in the is also same and equal to 6. Now compare the hundreds place digits which is also equal to 5. Then given set of numbers 35, 564, 4567, 56789 we have comparing the tens place digits i.e. 6 & 7 we can say to first count the number of digits in the given that number 6578 is larger than 6567. number. The number with highest number of digits will be the largest number i.e.56789 and number Making of numbers: with lowest number of digits i.e. 35 will be the The numbers can be made by using the digits (with lowest number. or without repetition) in appropriate places (thou- 1

1. Knowing Our Numbers sands place, hundreds place etc.) hundred seventy eight. Ex: From the digits 3, 2 & 9 three digits numbers • Larger Numbers If we add one to the greatest 6-digit number we that can be formed are 239, 293, 392, 329, 932, 923. get the smallest 7-digit number, called 10 lakh, i.e., Ascending and descending order of numbers: 999999 + 1 = 1000000 (Ten Lakh) Ascending order means arrangement from the Similarly, if one is added to the greatest 7-digit number we get the smallest 8-digit number called smallest to the greatest. Descending order means One Crore, i.e., 9999999 + 1 = 10000000 (One Crore) arrangement from the greatest to the smallest. • Remember • Shifting Digits 1 Hundred = 10 Tens If a digit in a number is shifted from one position to 1 Thousand = 10 Hundreds = 100 Tens another, the value of the number also changes. 1 Lakh = 100 Thousands = 1000 Hundreds Example: 1 Crore = 100 Lakhs = 10,000 Thousands 978 • Aid in Reading and Writing Large Numbers By exchanging the 1st and 2nd digits we get the Indicators: number These are very useful in writing the expansion of 798 numbers. By comparing the two number, 978, 798 we can say O stands for One’s place, T for Ten’s place, H for that 978 is greater than 798. Hundred’s place. • Introducing 10,000 (5-digit number) Example: 93625578 We know that 99 is the greatest 2-digit number, 999 is the greatest 3-digit number and 9999 is the Number Cr TLakh Lakh TTh Th H T O Number greatest 4-digit number. name We observe that: – Greatest 1-digit number + 1 = Smallest 2-digit Nine Crore Thirty Six number, i.e., 9 + 1 = 10 Lakh – Greatest 2-digit number + 1 = Smallest 3-digit Twenty Five 93625578 9 3 6 2 5 5 7 8 Thousand number, i.e., 99 + 1 = 100 – Greatest 3-digit number + 1 = Smallest 4-digit Five Hundred Seventy number, i.e., 999 + 1 = 1000 Eight – Greatest 4-digit number + 1 = Smallest 5-digit Expansion: number, i.e., 9999 + 1 = 10000 93625578=9x10000000+3x1000000+6x100000+ • Revisiting Place Value Expansion of a 5-digit number: 2 x 10000 + 5 x 1000 + 5 x 100 + 7 x 10 + 8 x 1 Example: Expand the number 78536 • Use of Commas 78536 = 7 x 10000 + 8 x 1000 + 5 x 100 + 3 x 10 + 6 Commas help us in reading and writing large x1 numbers. In Indian system of numeration, we have Here 6 is at one’s place (six), 3 is at ten’s place (thirty), one ones, tens, hundreds, thousands, lakhs and crores. 5 is at hundred’s place (five hundred), 8 is at Commas are used to mark thousands, lakhs and thousand’s place(eight thousand) and 7 is at ten crores. thousand’s place (seventy thousand). The number First comma comes after hundreds place, i.e., is seventy eight thousand five hundred thirty six. 3-digits from the right and marks thousands place. • Introducing 1,00,000 (6-digit number) Second comma comes two digits later i.e., 5-digits By adding 1 to the greatest 5-digit number we get from the right and marks lakh. the smallest 6-digit number, Third comma comes after another two digits and 99999 + 1 = 100000, marks crore. The number 100000 is named one Lakh. To write a 6-digit number in expanded form: Example: Expand the number 513278 513278 = 5 x 100000 + 1 x 10000 + 3 x 1000 + 2 x 100 + 7 x 10 + 8 x 1 Here 5 is at lakh’s place, 1 is at ten thousand’s place, 3 is at thousand’s place, 2 is at hundred’s place, 7 is at ten’s place and 8 is at one’s place. The number is five lakh, thirteen thousand two 2

1. Knowing Our Numbers Example: 93625578 = 9,36,25,578 reading a particular magazine across the country, • International System of Numeration: etc. Ones, tens, hundreds, thousands, millions are used Estimation involves approximating a quantity to an accuracy required, i.e., estimating to the nearest in International system of numeration. tens, hundreds, thousands by rounding off. Commas are used to mark thousands and millions. Example: – Number 28 can be rounded off to 30 as 28 is Commas are placed every three digits from right. First comma marks thousands and second comma nearer to 30 – Number 53 can be rounded off to 50 as 53 is marks millions. Example: 78,258,391 is read in International system near to 50 than 60. –  441 can be rounded off to 400 as seventy eight million, two hundred fifty eight –  981 can be rounded off to 1000 thousand three tundred ninety one. –  78536 can be rounded off to 79000 Here 1 Million = 10 Lakh • Estimating Outcomes of numbers 10 Million = 1 Crore Conventional method of addition, subtraction, 1 Billion = 1000 Million = 100 Crore • Large Numbers in Practice multiplication of numbers may be time consuming Numbers used in measuring distances in many cases. So estimating outcomes of numbers 10 millimeter = 1 centimetre by approximation can be used to calculate quickly. 100 centimeter = 1 metre = 1000 millimetre There are no rigid rules to estimate the outcomes 1000 meter = 1 kilometre of numbers. The procedure depends on the degree Numbers used in measuring weight of accuracy required and how quickly the estimate 1000 milligram = 1 gram is needed. Most important is how sensible is the 1000 gram = 1 kilogram guessed number: Numbers used in measuring quantity of liquid Example: 1 litre = 1000 millilitre –  Estimate 8275 + 7896 Note: kilo is 1000 times greater than milli 8275 can be rounded off to 8000 and centi is 100 times smaller than 1 metre. 7896 can be rounded off to 8000 and • Estimation of numbers: Estimated sum will be 8000 + 8000 = 16000. There are many situations in which the exact –  Estimate 53 x 890 quantity is not needed and a reasonable 53 can be rounded off to 50 and 890 to 900 approximation of number can be estimated. 50 x 900 = 45000 is the estimated product. Example: Number of spectators in a stadium • Using Brackets watching a cricket match, number of people Brackets can be used when we need to carry out more than one operation. This will avoid confusion and wrong calculations in the problem. 3

1. Knowing Our Numbers When brackets “( )” are used, first turn everything • Rules inside the brackets into a single number and then 1) If symbol is repeated, its value is added as many the operation outside is carried out. times as it occurs. 2) A symbol is not repeated more than three times. Example: (12 + 3 – 2) x 10 = (15 – 2) x 10 = (13) x 10 Symbols V, L and D are never repeated. = 130 3) If a symbol of smaller value is written to the right of a symbol of a greater value, its value • Roman Numerals gets added to the value of the greater number. We use Hindu-Arabic numeral system. The other Example: XV = 10 + 5 = 15 VII = 5 + 2 = 7 system of writing numerals is the system of Roman 4) If a symbol of smaller value is written to left of a numerals. This is used in clocks, copy right date on symbol of greater value, its value is subtracted films, television programs, etc. from the value of greater symbol. Example XL = 50 – 10 = 40 1 2 3 4 5 6 7 8 9 10 IV = 5 – 1 = 4 I II III IV V VI VII VIII IX X 5) Symbols V, L and D are never written to the left 1 5 10 50 100 500 1000 of a symbol of greater value, i.e., V, L and D are I V X L CDM never subtracted. 6) Symbol ‘I’ can be subtracted from V and X only. Work Plan 7) Symbol X can be subtracted from L and C only. CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Addition, Subtraction, Multiplication and PS – 1 Division of numbers Knowing our numbers • Comparing numbers • Shifting digits in numbers • 5-digit and 6-digit numbers, expansion of PS – 2 same, use of placement boxes • Large numbers and aid in reading & writing large numbers • System of numeration • Estimation of numbers • Use of brackets • Roman numerals Worksheet for “Knowing Our Numbers” PS - 3 Evaluation with Self Check or ---- Self Evaluation Sheet Peer Check* 4

PRACTICE SHEET - 1 (PS-1) 1. Girish has 57 chocolates in his bag. His friends ate 22 chocolates from Girish’s bag. How many chocolates are now there in Girish’s bag? 2. Add the following numbers: 578 and 362. 3. A shopkeeper sells 3kg of brinjal for Rs. 25 per kg, 1 kg of tomato for Rs. 12 and 5 kg of onion for Rs. 16 per kg. Calculate the total sale value. 4. Find the value of the products. i) 123 × 56  ii) 998 × 223 5. In a school there are 10 sections. 2800 books are needed to be divided among these sections equally. Find the number of books each section gets. 5

PRACTICE SHEET - 2 (PS-2) 1. Find the greatest and the smallest numbers: i) 582, 2876, 7827, 827 ii) 9827, 9926, 9956, 9986, 988 2. Using the following given digits, make greatest and smallest 4-digit numbers: 0, 3, 7, 9. a) Without repetition of digits b) By using any one digit thrice. 3. Arrange the following numbers in ascending order: 5986, 5672, 5708, 5996, 5921. 4. Arrange the following numbers in descending order: 08276, 7920, 9008, 810, 9100. 5. Expand and give number names for the following numbers. i) 76287  ii) 525387  iii) 1259876  iv) 31927654 6. Write these numbers in placement boxes indicating their expanded form and read the numbers. i) 38496  ii) 105827 7. The number of sales of a newspaper in the month of January was 525262. In the month of February the sales increased by 28572. Find the total number of sales of newspapers in the month of February. Show the various methods of calculation. 8. A motorcycle company manufactured 9,48,282 motorcycles in a year 2010-2011. Next year is 2011-2012 and did the company produced 10,53,814 motorcycles. In which year the company produced more motorcycles and by how much? 9. Cost of one pencil, one sharpener and one eraser is Rs. 8, Rs. 12 and Rs. 6 respectively. Calculate the cost of 50 pencils, 50 erasers and 50 sharpeners. 10. The distance between Rajesh’s house and his office is 5 km 250 m. Every day he travels by car both ways. Find the total distance covered by him in 5 days. 11. Find the sum of the following by rounding off each number to its nearest hundreds: i) 538 + 890 + 151  ii) 391 × 122 12. Write the roman numerals of the following numbers. i) 79  ii) 512 6

PRACTICE SHEET - 3 (PS-3) I. Choose the correct option. 1. What is the Roman numeral of 67? (A) LXVII (B) LXV (C) XLV (D) LXVI 2. 23716 when rounded to the nearest hundreds gives: (A) 23700 (B) 23800 (C) 24000 (D) 24800 3. The number 25378103 in Indian system is: (A) 253,78,103 (B) 25,37,81,03 (C) 2,53,78,103 (D) 2,53,78,10,3 4. Estimate of 358 + 282 is: (A) 600 (B) 700 (C) 800 (D) 1000 5. Compare the numbers and choose the correct sign that can be filled in the blank 137248 ______ 137428 is: (A) > (B) = (C) < (D) Δ 6. Compare and find the largest number among the given numbers: (A) 72864 (B) 75864 (C) 70846 (D) 75684 7. A scooter manufacturing company makes 158 scooters each day. Find the total number of scooters manufactured for the month of September? (A) 4898 (B) 4582 (C) 4740 (D) 4424 8. The weight of 6 cartons of apples of same size is 9 kg 412 g. What is the weight of each carton? (A) 9 kg 412 g – 6 (B) 9 kg 412 g ÷ 6 (C) 9 kg 412 g + 6 (D) 9 kg 412 g × 6 9. The expression for the following using brackets is: Forty-two multiplied by the difference of twenty and twelve (A) 40 × 20 × 12 (B) 42 × (20 – 12) (C) 42 – (20 × 12) (D) 40 × (20 + 12) 10. Arrange the given numbers in descending order: 2891, 4102, 5903, 4123 (A) 5903, 4123, 4102 and 2891 (B) 5903, 4123, 2891 and 4102 (C) 4123, 5903, 4102 and 2891 (D) 5903, 4102, 4123 and 2891 II. Short Answer Questions. 1. Estimate the following products using general rule: (i) 354 × 27 2. Form four-digit numbers with the digits 6, 9, 3, 5, Compare them and find which is the greatest and the smallest among them? 3. How many 4-digit numbers are there in all? Explain. III. Long Answer Questions. 1. Write the numbers in figures: (i) Ninety thousand five hundred five: (ii) Seventy-six thousand one hundred eight four: (iii) Thirty-one thousand eight hundred two: (iv) Sixty thousand two hundred fifty: 2. Arrange the given numbers in ascending order: (i) 62910, 4502, 39025, 4545 (ii) 53103, 33726, 4812, 2190 7

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Insert commas suitably and write names according to the Indian and International system of (4 Marks) numeration. i) 731206  ii) 12345678 2. Find the difference between the greatest and least numbers that can be formed by using the digits 1, 8, 5, 9 only once. (2 Marks) 3. Make 4 different numbers from 1, 8 , 5, 9 and arrange them in ascending order. Use all digits and repetition of digits is not allowed. (1 Mark) 4. Due to leakage from a large ship on the sea, many sea animals died. Number of fish that died was 25387 and number of prawns that died was 18371. i) Calculate the actual number of sea animals that died. ii) Estimate the approximate total number of fish and prawns that died by rounding off to nearest thousand. iii) Estimate the approximate total number of animals that died by rounding off to nearest hundreds. Comment on the estimated answer by comparing it with actual answer. (4 Marks) 5. A container has 6 litres and 750 ml of milk. In how many glasses, each of 75 ml, can the milk be filled?  (2 Marks) 6. Fill in the blanks (2 Marks) i) 1 million = ________ lakh ii) 1000 km = _______ meters 8

2. Whole Numbers Learning Outcome • Describe various properties of whole numbers • Identify various patterns in whole numbers By the end of this chapter, student will be able to: • Understand the concept of natural numbers and whole numbers • Draw and perform basic mathematical calculations on number line Concept Map Whole Numbers Natural Number Properties of Patterns in Numbers Line Whole Numbers Whole Numbers  Whole numbers  Addition on line  Commutative  Predecessor number property  Successor  Subtraction on line  Associative number property  Multiplication on  Distributive line number property Identity for Addition and Multiplication Key Points o Draw a line and mark a point at its left end. o Name that point as 0. • Natural Numbers o Now mark a second point to the right of 0, and Natural numbers are those which are used for counting and ordering. Natural numbers are called label it as 1. as counting numbers. o Distance between points 0 and 1 is called Unit Example: 1, 2, 3, 4, ……… Distance. • Predecessor and Successor o On the straight line, points 2, 3, 4, 5, etc., can be A predecessor is a number obtained by subtracting 1 from a natural number. marked by the same method. Example: 90 – 1 = 89 o This line with markings will be the number line When 1 is added to a natural number, the obtained number is called as its successor. for whole numbers. Example: 91 + 1 = 91 1 has no predecessor in natural numbers. On a number line, out of any two whole numbers, the number on the right of the other number is • Whole Numbers greater. Similarly, number on the left of the other Natural numbers along with zero form the number is a smaller number. collection of whole numbers, i.e., 0, 1, 2, 3, 4,….. Eg: 9 > 7, i.e., 9 is to the right of 7    8 < 15 i.e., 8 is to the left of 15. • Number Line • Addition on the Number Line It is a straight line with markings of numbers placed at equal intervals along its length. • To construct a Number Line 9

2. Whole Numbers Consider addition of 4 and 2 not closed under subtraction. Start from 4 Example: 11 – 2 = 9, 9 is a whole number To add 2, make two jumps to the right, i.e., 4 to 5,      2-11= -9 which is not a whole number 5 to 6 iv) Division of two whole numbers is not a whole The tip of the last arrow in the second jump is at 6. number all the time. So the collection of whole ∴ 4 + 2 =6 numbers is not closed under division. • Subtraction on the Number Line Example: 15 ÷ 3 = 5, 5 is a whole number Let us find 9 – 3      15 ÷ 0 is not defined Start from 9       1 ÷ 2 = 0.5 is not a whole number To subtract 3 from 9, make 3 jumps to the left, i.e., Note: Division of a whole number with zero is not 9 to 8, 8 to 7, 7 to 6. defined. ∴9–3=6 • Commutative Property of Whole Numbers i) Addition Consider 2 + 6 and 6 + 2 • Multiplication on Number Line By changing the order of addition, we get the To find 2 × 3 same sum, i.e., sum is not different when order of Start from 0 addition is changed. Move 2 units at a time to the right So we can add two whole numbers in any order, Make 3 such moves, we reach 6 thus addition is commutative for whole numbers. ∴2×3=6 ii) Multiplication Multiplication of two whole numbers in any order • Properties of Whole Numbers gives the same result, i.e., we can multiply two Properties help us to: whole numbers in any order. o Understand the numbers better So multiplication is commutative for whole o Make calculations under certain operations very numbers. simple Example: 2 × 4 = 8 , also by the changing the order i) Sum of any two whole numbers is a whole i.e. 4 × 2 = 8 number, i.e., the collection of whole numbers is closed under addition. This property is known as iii) Subtraction the closure property for addition of whole numbers. Subtraction of two whole numbers is not Example: 1 + 8 = 9, 9 is a whole number commutative in nature.    12 + 35 = 47, 47 is a whole number Example: 7 – 4 = 3 but 4 – 7 = -3 ≠ 3 ii) Multiplication of two whole numbers is a whole iv) Division number. Thus the system of whole numbers is Division of two whole numbers is not commutative. closed under multiplication. Example: 6 ÷ 3 = 2 but 3 ÷ 6 = 0.5 ≠ 2 Example: 10 × 6 = 60, 60 is a whole number • Associative Property of Whole Numbers    1 × 2 = 2, 2 is a whole number i) Addition of Whole Numbers Whole numbers are closed under addition and multiplication. iii) Subtraction of two whole numbers is not always a whole number, i.e., the set of whole numbers is 10

2. Whole Numbers (7 + 2) + 4 = 9 + 4 = 13, we first add 7 and 2 to get Example: 35 × 1 = 35 9 and then add 4 to 9 to get 13 • Any whole number when multiplied with zero 7 + (2 + 4) = 7 + 6 = 13, we first add 2 and 4 to get becomes zero. 6 and then add 7 to 6 to get 13 Example: 5 × 0 = 0 Results in both case is same, ie., • Patterns in Whole Numbers (7 + 2) + 4 = 7 + (4 + 2) This is associative property of addition of whole Numbers are arranged in elementary shapes made numbers. up of dots. ii) Multiplication of whole numbers Elementary shapes allowed are line, rectangles, 5 × ( 4 × 10) = 5 × 40 = 200 square, triangle. No other shape is allowed. (5 × 4) × 10 = 20 × 10 = 200 Every number can be arranged as a line. ∴ 5 × ( 4 × 10) = (5 × 4) × 10 Example: Number 4 can be shown as • • • • This is associate property of multiplication of      Number 6 can be shown as • • • • • • whole numbers. Some numbers can be shown as rectangles. Subtraction and division of whole numbers do not follow associative property. Example: Number 6 can be shown as ∞∞∞ or ∞∞ iii) Subtraction of whole numbers ∞∞∞ ∞∞ Example: (5 – 3) – 1 = 2 – 1 = 1      5 – (3 – 1) = 5 – 2 = 3 ∞∞ We find that the result is two cases are not same. Division of whole numbers ∴Some Numbers can be shown as triangles. Example: (32 ÷ 8) ÷ 2 = 4 ÷ 2 = 2      32 ÷ (8 ÷ 2) = 32 ÷ 4 = 8 Example: Number 3 can be shown as Thus the results in the two cases are not same. • Distributivity of Multiplication over Addition • Patterns Observation 9 × ( 2 + 5) = 9 × 7 = 63 i) Patterns to aid in addition/subtraction of the (9 × 2) + (9 × 5) = 18 + 45 = 63 number by 9, 99, 999, …. ∴ 9 × (2 + 5) = (9 ×2) + (9 × 5) a) 135 + 9 = 135 +10 – 1 = 145 – 1 = 144 This is known as distributivity of multiplication b) 135 – 9 = 135 -10 + 1 = 125 + 1 = 126 over addition. c) 135 + 99 = 135 +100 – 1 = 235 – 1 = 235 • Identity for addition and Multiplication d) 135 – 99 = 135 -100 + 1 = 35 + 1 = 36 i) Identity for Addition or Additive Identity of ii) Patterns helpful in multiplying a number by 9, whole numbers is Zero, because when any whole 99, 999, …. number is added to zero (0) we get the same whole a) 58 × 9 = 58 × (10 – 1) = 580 – 58 = 522 number. b) 58 × 99 = 58 × (100 – 1) = 5800 – 58 = 5742 Example: 22 + 0 = 22 iii) Patterns useful to multiply a number by 5 or ii) Multiplicative Identity or Identity for 25 or 125 Multiplication of whole numbers is One (1), because when any whole number multiplied by a) 36 × 5 = 36 × 10 = 360 = 180 one we get the same whole number. 22 b) 36 × 25 = 36 × 100 = 3600 = 900 44 c)36 × 125 = = 36 × 1000 = 36000 = 4500 88 Patterns are very useful for verbal calculations. 11

2. Whole Numbers Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Whole Numbers • Introduction to whole numbers • Number line • Commutative property of whole PS – 1 numbers • Associative property of whole numbers • Distributive property of whole numbers • Identity for addition and multiplication • Patterns in whole numbers Worksheet for “Whole Numbers” PS - 2 Evaluation with Self Check ---- Self Evaluation Sheet or Peer Check* 12

PRACTICE SHEET - 1 (PS-1) 1. Define natural numbers and whole numbers. 2. Write whether true or false. i) Zero is the smallest whole number. ii) All natural numbers are whole numbers. 3. Describe the construction of a number line for whole numbers. 4. Perform addition and multiplication of 4 and 4 on a number line. 5. 1000 ÷ ______ is not defined. 6. Compute the following by suitable rearrangement. i) 63 + 210 + 37 ii) 4 × 40 × 125 iii) 99 × 22 – 99 × 12 7. With example prove that set of whole numbers are not closed under subtraction and division. 8. Match the following. i) 2 8 + 29 = 29 + 28 a) A ssociative property of ii) (328×10)×598 = multiplication 328×(10×598) b) Distributivity of iii) 99×(25+35) = multiplication over 99×25+99×35 addition. c) Commutative property. 9. Express number 8 in two different patterns. 10. Use distributivity and simplify 105 ×20 11. Use distributivity and simplify 1008 × 256 12. Justify the statement with reasons and examples. Zero is the Additive Identity of numbers. 13. Varun owns a restaurant. From Monday to Saturday, each day 21 kg of tea powder is used to prepare tea for customers. On Sunday, 15 kg of tea powder is used to make tea. The price of tea powder per kg is Rs. 1000 Calculate the total expense on tea powder per week. PRACTICE SHEET - 2 (PS-2) I. Choose the correct option. 1. Which of the following number is a whole number but not a natural number? (A) 0 (B) 1 (C) 100 (D) Does not exist 2. The whole number represented by ‘A’ is ______. (A) 5 (B) 6 (C) 8 (D) 10 3. The successor of 44 lies to the _____ of 48 on number line. (A) right (B) left (C) above (D) below 4. Which of the following is the predecessor of 0? (A) Any natural number (B) Any whole number (C) No predecessor (D) 1 5. _______ is called as the additive identity for whole numbers. (A) Zero (B) One (C) The number itself (D) The successor of a number 6. (9 + 6) + 5 = 9 + (6 + 5) is true under _______ property of addition. (A) Closure (B) Associative (C) Commutative (D) Distributive 13

PRACTICE SHEET - 2 (PS-2) 7. If a + 17 = 17 + 8, then by commutativity of addition in whole numbers, x = ____. (A) 8 (B) 17 (C) 7 (D) 18 8. 14 × 22 + 14 × 18 = _________. (A) 14 × (22 + 18) (B) 22 (14 × 18) (C) 14 × 36 × 18 (D) 32 × 36 9. Identify the statement that is true with respect to the given number line. (A) 11 is the predecessor of 10 (B) 5 is the successor of 6 (C) 9 is the successor and predecessor of 8 (D) 7 is the successor of 6 and predecessor of 8 10. Identify the statement that is true: (A) Every whole number has a predecessor. (B) Zero is the least whole number. (C) Zero is the least whole number that has both predecessor and successor. (D) Every whole number has a predecessor and successor. II. Short answer questions. 1. Find the product using suitable properties: 732 × 103 2. Solve: (i) 9 × 5 × 2 × 5 × 2 × 5 (ii) 45 × 8 using the distributive property 3. We know that 0 × 0 = 0: Can you replace 0 by any other whole number such that a × a = a? If yes, what number is that? III. Long answer questions. 1. (i) Add: 2 + 9 + 6 using number line. (ii) Find the product of 3 × 4 using number line. 2. Explain the commutativity of whole numbers with an example. 14

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins (1 Mark) 1. Write the predecessor and successor of 9999.  2. Show 12 – 5 on a number line. (1 Mark) 3. Write whether true or false. (1 Mark) i) The successor of a 3-digit number is always a 3-digit number.  4. State whether the statement is true or false and give reason. (2 Marks) 99999 is largest natural number.  5. Fill in the blanks (3 Marks) i) Any whole number when multiplied by _________ results in zero. (2 Marks) ii) On a number line, 1037 appears to the ______ of 1068. iii) If multiplication of 2 numbers is 1, then both the numbers should be _______.  6. Explain the associative property of multiplication of whole numbers with example. 7. Find the product of 840 × 95.  (2 Marks) 8. Mayur owns a car and a bike and uses them on alternate days to travel from his house to office. The distance between his house and office is 20 km. The car consumes 2 liters of petrol for one day travel and bike consumes 1 litre of petrol for every day of travel. Per litre petrol cost is Rs. 70. Calculate the total expense of Mayur in a month on buying petrol. (Note: There are 4 Sundays in a month, and Mayur does not use any of the vehicles on Sundays. Also, there are 30 days in a month). (5 Marks) 15

3. Playing with Numbers Learning Outcome By the end of this chapter, students will be able to: • Find common factors and common multiples of • Describe factors and multiples of numbers numbers • Define and identify prime numbers, composite • Carry out prime factorisation numbers, even & odd numbers • Find HCF & LCM of given numbers • Understand and carry out tests for divisibility of numbers Concept Map Playing with numbers Factors & Prime numbers & Test for Divisibility of n Common Factors & Multiples Composite numbers Common Multiples • Composite numbers Co-primes • Even numbers Prime • Odd numbers Factorization HCF& LCM Key Points than or equal to that number. o  Number of multiples of a given number is • Factors and Multiples. Let us consider number 8. infinite. The number 8 can be written as product of two o  Every number is a multiple of itself. • Perfect Number numbers in different ways. A number for which sum of all its factors except 8 = 1 × 8  8 = 2 × 4  8 = 4 × 2  8 = 1 × 8 i.e., 1, 2, 4 & 8 are exact divisors of 8. They are also the number itself is equal to the number is called a perfect number. called factors of 8. Example: Number 6 and Number 28 are perfect • A number is a multiple of each of its factors, i.e., the numbers • Prime Number number 15 can be written as The numbers whose only factors are 1 and the   15 = 5 × 3  15 = 3 × 5 number itself are called numbers. Here 5 and 3 are factors of 15. Example: 15 is a multiple of 3 and 5. The factors of 2 are 1 and 2 , so it is a prime Each of the factor of 15 are less than 15. number. • Important points The factors of 5 are 1 and 5, so it is a prime o  1 is a factor of every number. o  Every number is a factor of itself. o  Every factor is less than or equal to the given number. o  Number of factors of a given number is finite. o Every multiple of a given number is greater 16

3. Playing with Numbers number. o  Odd numbers end with 1, 3, 5, 7 or 9. 2, 3, 5, 7, 11, 13, 17, etc are prime numbers.    Example: 1, 3, 39, 43, 103, 1009, etc The number 1 has only one factor and it is not a o Note: Odd numbers are in between even prime number. numbers and vice versa • Composite Number • To decide the given number is odd/even, check Numbers having more than two factors are called the digit in ones place of the number. If the digit composite numbers. is 0/2/4/6/8 then the number is even. If the digit is Example: 4, 6, 9, 10, 12, ….. 1/3/5/7/9 then the number is odd. • 1 is neither a prime number nor a composite • 2 is the smallest prime number which is even. All prime numbers except 2 are odd. number. • Test for Divisibility of Numbers • Sieve of Eratosthenes o Divisibility by 2: A number is divisible by 2, if it A method of finding prime numbers from 1 to has any one of the digits 0, 2, 4, 6 or 8 in its ones 100 without checking the factors of a number place. All even numbers are divisible by 2. was explained by a Greek mathematician named o Divisibility by 3: If the sum of the digits of a Eratosthenes. number is a multiple of 3, then the number is divisible by 3. 1 2 3 4 5 6 7 8 9 10 o Divisibility by 4: A number is divisible by 4, if the last two digits of the number (i.e., ones and 11 12 13 14 15 16 17 18 19 20 tens place) is divisible by 4. o Divisibility by 5: A number which has either 0 or 21 22 23 24 25 26 27 28 29 30 5 in its ones place is divisible by 5. o Divisibility by 6: If a number is divisible by 2 31 32 33 34 35 36 37 38 39 40 and 3, then it is divisible by 6. o Divisibility by 8: A number with 4 or more digits 41 42 43 44 45 46 47 48 49 50 is divisible by 8, if the number formed by the last three digits is divisible by 8. 51 52 53 54 55 56 57 58 59 60 o Divisibility by 9: If the sum of the digits of a number is divisible by 9, then the number is 61 62 63 64 65 66 67 68 69 70 divisible by 9. o Divisibility by 10: If the number has zero in 71 72 73 74 75 76 77 78 79 80 ones place, then it is divisible by 10. o Divisibility by 11: To check divisibility by 11 81 82 83 84 85 86 87 88 89 90 • First find the difference between the sum of 91 92 93 94 95 96 97 98 99 100 the digits at odd places (from right) and sum Step 1: Cross out 1, as it is not a prime number. of digits at even places (from right) of the Step 2: Encircle 2, cross out all multiples of 2, ie., 4, given number. • If the difference is either 0 or divisible by 11, 6, 8, …100. then the number is divisible by 11. Step 3: Encircle 3, cross out all multiples of 3. Example: Check the divisibility of 9317 by 11. Step 4: Encircle 5, cross out all multiples of 5. Sum of digits at even places from right Step 5: Continue this process till all the numbers in = 3 + 7 = 10 Sum of digits at odd places from right the list are either encircled or crossed. = 9 + 1 = 10 All enclosed numbers are prime numbers 10 – 10 = 0, hence the number is divisible by 11. • Common factors and Common Multiples and all crossed out numbers other than 1 are Consider numbers 6 and 20. composite numbers. This method is called Sieve of Factors of 6 are 1, 2, 3 and 6. Eratosthenes. Factors of 20 are 1, 2, 4, 5, 10 and 20. • Even Numbers Factors 1 and 2 are factors to both 6 and 20, thus o Integers that can be divided exactly by 2 are even numbers. o  All even numbers are multiples of 2. o  Even numbers end with 0, 2, 4, 6 or 8.    Example: 2, 4, 6, 8, 10, 150, 888, etc. • Odd Numbers o Integers that cannot be divided exactly by 2 are called odd numbers. 17

3. Playing with Numbers they are named as common factors. Thus, HCF of 18 and 24 = 2 × 3 = 6 • LCM: Lowest Common Multiple • Co-primes The Lowest Common Multiple (LCM) of two or more If two numbers have only 1 as common factor, then numbers is the smallest or lowest or least of their those two numbers are called co-primes. common multiples. Example: 4 and 15 are co-primes. Example: Find LCM of 20 and 30. • Let us consider the two numbers 3 and 2. The prime factorisation of 20 and 30 are Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, ……. 20 = 2 × 5 × 2 Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, ….. 30 = 3 × 5 × 5 6, 12, 18, …. are multiples of both 3 and 2. So they Maximum number of times the prime factor 2 occurs is twice, this happens for 20. are called common multiples. Prime factor 5 occurs once for 20 and 30, prime factor occurs once for 30, • Some more divisibility rules LCM = 2 × 2 × 3 × 5 o If a number is divisible by another number 2 20 , 30 i) Divide by the least prime then it is divisible by each of the factors of that 2 10 , 15 number which divides any number. 3 5 , 15 one of the given number. o If a number is divisible by two co-prime numbers, 3 5 ,5 Here it is 2. then it is divisible by their product also. 5 5 ,5 o If two given numbers are divisible by a number ii) Again divide by 2, continue then their sum is also divisible by that number. 1 ,1 this till we have no o If two given numbers are divisible by a number multiples of 2. then their difference is also divisible by that number. iii) Divide by next prime number, i.e., 3 • Prime Factorization When a number is expressed as a product of its iv) Divide by next prime factors, we say that the number has been factorised. number i.e., 5 Let us consider the number 18. 18 can be written as LCM = = 2 × 2 × 3 × 5 18 = 2 × 9 18 = 6 × 3 =2×3×3 =2×3×3 In above all factorisations of 18, we arrived at only one factorisation, i.e., 2 × 3 × 3. In this factorization, the only factors are 2 and 3 which are prime numbers. Such a factorisation which contains only prime numbers as factors is called Prime Factorisation. • HCF: Highest Common Factor Highest Common Factor (HCF) of a given number is the highest or greatest of their common factors. It is also known as Greatest Common Divisor (GCD). HCFofnumberscanbefoundbyprimefactorization. Example: Find HCF of 18 and 24 2 18 2 24 39 3 12 33 24 22 1 1 18 = 2 × 3 × 3 24 = 2 × 3 × 2 × 2 The common factors of 18 and 24 are 2 and 3. 18

3. Playing with Numbers Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Playing with numbers • Factors & multiples • Prime numbers, composite numbers, even & odd PS – 1 numbers • Test for divisibility of numbers • Common factors and common multiples • Prime factorization • HCF and LCM Worksheet for “Playing with Numbers” PS - 2 Evaluation with Self Check or ---- Self Evaluation Sheet Peer Check* 19

PRACTICE SHEET - 1 (PS-1) 1. The factors of 16 are: a) 1, 2, 5, 16  b) 1, 2, 4, 6, 16  c) 1, 2, 4, 8, 16 2. Find the factors of 25. 3. Write the multiples of 12 between 50 and 150. 4. Define a perfect number. 5. Answer the following: i) Show that sum of an even number an and odd number is always odd. ii)  Express 26 as sum of two odd prime numbers. iii)  List the composite numbers between 20 and 30. 6. Decide whether the following numbers are even/ odd. i) 10789206  ii) 99987653 7. Using divisibility test, determine which of the following numbers are divisible by 4 and 8. i) 2382572  ii) 5824792 8. Check whether the number 1335 is divisible by 15 or not. 9. Define co-prime numbers. Give examples. 10. Find the common factors of 8, 26 and 50. 11. Write 3 common multiples of 12 and 15. 12. Find the least number to be added or subtracted to make the number 732541 divisible by 3. 13. Write any two different factor trees for 50. 14. Find the prime factorisation of 988. 15. Express the smallest and largest 3-digit numbers in terms of their prime factors. 16. Find the LCM and HCF of 48, 96 and 108. 17. 4 bells commence tolling together and tolls at intervals of 4, 6, 10 and 12 seconds respectively. In 20 minutes, how many times do they toll together? 18. A rectangular room base is 5.9 meters long and 3 meters wide is paved exactly with square tiles of same size. Find the largest size of the tile used. 19. Find the least number which should be added to 3128 so that it is divisible by 3, 4, 5 and 6. 20

PRACTICE SHEET - 2 (PS-2) I. Choose the correct option. 1. The number 31205 becomes divisible by 10 when it is multiplied by ____. (A) 5 (B) 4 (C) 2 (D) 1 2. The missing factor in the given factor tree is ____. (A) 12 (B) 14 (C) 10 (D) 16 3. The least common multiple of 3 and 5 is _____. (A) 3 (B) 15 (C) 5 (D) 0 4. Which of the following numbers is divisible by 2 but not by 6? (A) 26318 (B) 27318 (C) 26316 (D) 26454 5. A pair of twin primes between 20 and 40 is __________. (A) (23, 25) (B) (21, 25) (C) (27, 30) (D) (31, 33) 6. The HCF of two co-prime numbers is always ____. (A) 0 (B) One of the two numbers (C) 1 (D) Product of the two numbers 7. The prime factorisation of 108 is _____ . (A) 4 × 27 (B) 3 × 36 (C) 2 × 2 × 3 × 3 × 3 (D) 2 × 2 × 27 8. The LCM of 12 and 24 is ____. (A) 24 (B) 12 (C) 36 (D) 1 9. If 6P6 is a three-digit number and is divisible by 6 then, the least value of P would be _______. (A) 2 (B) 3 (C) 4 (D) 5 10. Which of the statements is true? (A) A number divisible by 2 is always divisible by 6. (B) A number divisible by 3 is always divisible by 9 (C) A number divisible by 2 and 3 is always divisible by 6. (D) A number divisible by 3 and 6 is always divisible by 9. II. Short answer questions. 1. Write all the factors of 192 as a factor tree. 2. Find the least number which when divided by 12, 15, 18 and 20 leaves a remainder 5 in each case. 3. Write the smallest digit and the greatest possible digit in the blank space of the following number so that the number formed is divisible by 11. __4531 III. Long answer questions. 1. Find whether the given numbers are divisible by 2, 3, 5 and 6. (i) 340521 (ii) 185267 2. (i) 24 is divisible by both 2 and 3. We get 2 × 3 = 6. 24 is also divisible by the product of 2 and 3 i.e., 6. Similarly can we say the number divisible by 4 and 6 is also divisible by their product (4 × 6 = 24)? If not, justify your answer with 2 examples. (ii) The product of three consecutive numbers is always divisible by 6. Prove the statement with the help of 2 examples. 21

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Is 30 a perfect number? Justify the answer. (2 Marks) 2. How many prime numbers are there between 1 and 100? Make a list of them. Name (3 Marks) the greatest and smallest prime numbers from the list. 3. If a number 182X7365 is divisible by 11, then find the value of X (Single digit)? (2 Marks) 4. Fill in the blanks (2 Marks) i) --------- is the only prime number which is even. ii) ----------- is the factor of every number.  5. State whether the statement is true or false i) 0 is the smallest even number. ii) If two given numbers are divisible by a number then their difference is also divisible by that number. (2 Marks) 6. Find prime factorisation of 6789. (1 Marks) 7. Vijay, Varun and Vimal start at the same time in the same direction to run around a circular stadium. Vijay completes the round in 2 minutes 30 seconds, Varun in 3 min 20 seconds and Vimal in 4 minutes and 10 seconds. All start at the same point. After what time (in seconds) will they meet again at the starting point? (3 Marks) 22

4. Basic Geometrical Ideas Learning Outcome • Understand formation of angle • Draw & explain triangles and quadrilaterals By the end of this chapter, students will be able to: • Appreciate basic concepts of circle • Describe concept of point, line & line segment, ray, intersecting & parallel lines • Explain curves and its types Concept Map Key Points Line Segment: Geometry is derived from Greek word “geometron”. • It is a straight line that is bounded by two distinct Geo means Earth and metron means measurement. It end points. is a branch of mathematics which deals with shapes, sizes, relative position of figures. Euclid is called as • Line segment shown in the figure is denoted by father of geometry. MN or NM . Geometry finds its application in art, measurements, architecture, engineering, cloth designing, etc. • Points M and N are called End points of segment. Points: • Line segment is the shortest path possible between • A point in geometry determines a location, i.e., two points M and N. position. • Eg: An edge of a book, sides of a triangle, etc. • A point has no size, i.e., no width, no length and no Line: • A line is defined as a line of points that extends depth. • A point is shown by a dot which is invisibly thin. infinitely in two directions. • Line is straight with no bends. Eg: Tip of a compass, sharpened end of a pencil, • It has no bends, no thickness. pointed end of a needle, etc. • A line has no beginning or end points. • Point is usually denoted by a capital letter. The points shown in the figure can be read as point X, point Y and point Z. • Points are helpful in labeling, like labeling the end points of a line segment. 23

4. Basic Geometrical Ideas • Line consists of countless number of points. • Point Y is a point on the path of the ray. • Consider a line segment MN. Imagine that the line • Ray is denoted as XY segment is extended beyond points M and N Curves: infinitely. A line through points M and N is written • Any drawing (may be straight / non straight) done as MN . • A line is denoted by a lower case letter like l without lifting the pencil may be called a curve. • To determine a line two points are enough. • Curve is a generalization of line and can be named Intersecting lines: as continuous line. In maths a curve can be a straight line or it can be a bent line. • If a curve does not cross itself then it is called simple curve. • A curve is called as a closed curve it the ends of the curve are joined. • If the ends are not joined then the curve is said to be open curve. • If two lines have one common point or if two lines • In a closed curve there are 3 parts: meet at a point, then they are called intersecting o Interior (Inside) of the curve i.e., point A. lines. o Boundary (On) of the curve, i.e., point B. o Exterior (Outside) of the curve, i.e., point C. • Consider two lines l and m. Both lines pass through point X. So we can say that the lines l and m are The interior of curve along with its boundary is called intersecting lines. Region. Parallel lines: Polygon: • Parallel lines are those which never touch each • Polygon is a simple closed curve made up of line other or meet each other or do not intersect each segments. other and are always the same distance apart. • Polygon is a 2 dimensional figure. • If two lines MN and PQ are parallel, we represent • A polygon will have three or more line segments. • Eg: Triangle, rectangle, pentagon, etc. this as MN � PQ • The line segments forming a polygon are called lines m1 and m2 (OR) sides of polygon. • If the are parallel, we write as m1 � • The meeting point of a pair of sides is called m2. Vertices. Ray: • Any two sides with a common end point are called • Ray is a portion of a line which has a starting point adjacent sides of polygon. and goes endlessly in a direction, i.e., a ray is a line • The end points of the same side of the polygon are with single end point or a line with only one point of origin that extends infinitely in one direction. called adjacent vertices. • On a ray, 2 points X and Y are shown. Point X is the starting point or origin point. 24

4. Basic Geometrical Ideas The diagram shows ∠ZXY where o Point A is in the interior of the angle o Point B is in the exterior of the angle o Point C is on the angle. Triangles: • A diagonal is a line segment joining two non- • A triangle is a three sided polygon. It is the polygon adjacent vertices of a polygon with the least number of sides. • Consider the figure. A polygon MNOP is shown in • We denote triangle XYZ and ∆XYZ. the figure. The sides of ∆XYZ are XY , XZ and YZ Sides are MN , NO , OP , PQ , QM . 3 points X, Y and Z are called vertices of the triangle. M, N, O, P and Q are vertices. Point A is in the interior of the triangle. MN and NO are one pair of adjacent sides Point B is in the exterior of the triangle. P and Q are one pair of adjacent vertices. Point C is on the triangle. PM , PN , QO , QN , MO are the diagonals. Quadrilaterals: Angles: • A four-sided polygon is called a quadrilateral. • Angles are formed when corners are formed. • A quadrilateral has 4 sides, 4 angles and 4 vertices. • An angle is made up of two rays starting from a • The vertices of a quadrilateral are named in a cyclic common point or a single point manner. • The two rays forming the angle are called the arms or sides of the angle. • The common point is called the vertex of the angle. • Here the angle is formed by the rays ON and OM . • From the figure, the four sides are AB , BC , CD and ON and OM are called the arms of the angle O is the vertex. DA To name the angle: Four angles are ∠A, ∠B, ∠C and ∠D. o We can simple say angle O (or) Sides AB and BC is an example of adjacent sides o More specifically, we identity two points on each side and vertex to name the angle, i.e., Angle AB & CD , AD & BC are pairs of opposite sides. MON denoted by ∠MON. ∠A & ∠C, ∠B & ∠D are pairs of opposite angles o Vertex is written as the middle letter. ∠A is adjacent to ∠B and ∠D. Similar relation exists for other three angles. • An angle divides a region in three parts. They are Circles: on the angle, interior of the angle and exterior of • A circle is the path of a point moving at same the angle. distance from a fixed point. • The fixed point is centre of circle, fixed distance is radius of circle and distance around the circle is called circumference of the circle. • Circle is a simple closed curve which is not a 25

4. Basic Geometrical Ideas polygon. points on the circle. In the figure, XY is a chord of • Every point on the circle is at equal distance from • Dthiaemciercteler.WY is the chord that passes through the the centre of the circle. center of the circle. • Diameter of a circle is double the size of the radius • Figure shows a circle with centre O. of the circle and passes through the center of the circle. • Chord of a circle is a line segment joining any two Line segments OX = OY = OZ = OW = Radius of the circle. Radius is a line segment that connects the centre of circle to a point on the circle. WY is called diameter of circle. • Two distinct points are taken on the circle represent arc of the circle. In the figure arc AB is denoted by • A region in the interior of a circle enclosed by an arc on one side and a pair of radii on other two sides is called sector. • A region in the interior of a circle enclosed by a chord and arc is called segment of the circle. • Semi-circle is half of the circle. A semi-circle will have end points of diameter as a part of its boundary. • Diameter of circle divides circle into 2 equal parts. 26

4. Basic Geometrical Ideas Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Basic geometrical ideas • Points PS – 1 • Lines, line segment & ray • Intersecting & parallel lines PS - 2 • Curve & its types Self Evaluation Sheet • Polygons • Angles • Triangles • Quadrilaterals • Circles Worksheet for “Basic Geometrical Ideas” Evaluation with Self Check or ---- Peer Check* 27

PRACTICE SHEET - 1 (PS-1) 1. Define the following: a) Point b) Intersecting lines 2. Fill in the blanks: i) _________ has only one end point or origin point. ii) ________ lines never touch each other or meet. 3. Identify all line segments and at least three rays in the fig. 9. Is the given figure quadrilateral MNOP? If not, then name the quadrilateral correctly. 4. Illustrate the following with a neat sketch and 10. Draw a quadrilateral EFGH and its diagonals. define the terms. i) Indicate the intersection point of the diagonals. i) Simple curve ii) Closed curve iii) Open curve. 5. Draw a polygon EFGHIJ. Name all sides, vertices ii) State two pairs of adjacent sides iii) All pairs of opposite sides. and at least 3 diagonals. 11. Which of the following are polygons. 6. Name the angles in the given figure. 7. In the given figure, list the points which 12. Define the following: i) are interior of ∠POS   ii) are exterior of ∠POQ i) Chord of a circle   ii) Sector of a circle iii) are interior of ∠POR   iv) lie on ∠POQ 13. In the given figure, X is the center of the circle. i) Shade any two sectors. ii) Name the chords which are not diameters iii) Shade all the segments iv) Name all the radii v) Name only the diameters. 8. In the given figure i) Name the various triangles formed. ii) Name the triangles having ∠C as common. 28

PRACTICE SHEET - 2 (PS-2) I. Choose the correct option. 1. Name the line in the following. (A) (B) (C) (D) 2. A part of the circumference of a circle is called _______. (D) a segment (A) Radius (B) An arc (C) a chord (D) 3. Which of the following is a simple curve? (v) (A) (B) (C) (D) ii, iii and iv (D) uncountable 4. Identify the intersecting lines in the given pairs of lines. (i) (ii) (iii) (iv) (A) ii and iv (B) I, ii and v (C) I, iii and v 5. How many chords can be drawn for a given circle? (A) 3 (B) 4 (C) 10 6. How many line segments are there in the given figure? (A) 5 (B) 6 (C) 8 (D) 10 7. How many lines can be drawn passing through two points? (A) one (B) two (C) four (D) countless 8. If you are asked to draw a figure by using 6 line-segments, then identify the possible figure you get among the given. (A) (B) (C) (D) 9. In the figure, the angle ∠ABC cannot be written as ______. (A) ∠B (B) ∠CBA (C) ∠CAB (D) ∠MBA 10. Which of the following in not a chord in the given figure? (A) AC (B) DC (C) CE (D) OE 29

PRACTICE SHEET - 2 (PS-2) II. Short Answer Questions. 1. Identify the different parts of the circle and name them, like chord, radius, diameter, sector and centre of the circle in the given figure. 2. Give four examples of line segments from your surroundings. 3. Classify the angles with two examples from your surroundings as: right angle, acute angle and obtuse angle. III. Long Answer Questions. 1. Mark any three points to the interior of the triangle, exterior of the triangle and on the triangle. 2. Explain with example, the different situations where we use a point. 30

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. From the given figure  (3 Marks) 3. Fill in the blanks (5 Marks) i) The interior of a curve together with boundary i) How many lines are drawn? is ________ of a curve. ii) Find the number of line segments. ii) The end points of same side of a polygon are iii) Identify and name the points. called ________. iii) While naming the angle, ________ is written as the middle letter. iv) ________ is the polygon with least number of sides. v) Distance around a circle is called __________. 4. State whether the following statements are true or false.  (4 Marks) i) A polygon can be drawn with two line segments. ii) An angle leads to three divisions of a region. iii) Line PQ and line QP both refer to the same  line. iv) A quadrilateral consists of 4 sides and 6 angles. v) Circle is a simple closed curve which is not a polygon. vi) Radius of a circle is double the size of diameter. vii) A region in the interior of circle enclosed by a chord and arc is called segment. 2. In the given figure, (3 Marks) viii) A line segment has no finite length. i) Identify the triangles ii) Name any four quadrilaterals iii) Identify the other than diameter. 31

5. Understanding Elementary Shapes Learning Outcome By the end of this chapter, students will be able to: • Describe classification of triangles on basis of • Carry out measurement of line segments using sides and angles ruler • Identify types of quadrilateral and explain their • Explain and perform measurement of various properties angles using protractor • Name various polygons and explore three • Understand the concept of perpendicular lines dimensional shapes Concept Map Understanding Elementary shapes Measuring line Angles Perpendicular Quadrilateral Segments lines and its shape Comparison using Right angle Three ruler and divider Straight angle dimensional Acute angle Obtuse angle shapes Measuring angles using protractor Key Points • We can observe various shapes around us, and most of them are formed using curves and lines. Various But this crude way of comparison may not shapes possess different sizes and measurements. Generally, we can organize these shapes into line be useful when the differences in lengths are segments, angles, triangles and circles. small. Example: To decide on lengths of AB and XY • Measuring Line Segments • A line segment is a fixed portion of a line. The just by looking at the figure is difficult. measure of each line segment is unique number called Length of the line segment. • Ways of comparing line segments / Finding relation between lengths of line segments: o Comparing by observation By observing normally or by just looking at the line segments, lengths can be compared. Example: By observation, we can say that MN is longer than OP 32

5. Understanding Elementary Shapes To measure the length XY, place the end point of one of its arms at X and the other end point at the second arm at Y. Now without disturbing the opening of the divider, lift the divider and place it on ruler. Make sure that one end point is at zero mark of ruler. The mark on which the other end point meets will give the length XY. So there is a need of better methods of comparing line segments. o Comparison by Tracing T o compare RS and TU , a tracing paper is taken and TU line segment is traced on it. Now the traced line segment TU is placed on RS for comparison. • Angles: (Right and Straight) • There are 4 main directions and they are North (N), South (S), East (E) and West (W). This method depends upon the accuracy of tracing the line segment. It is also difficult to trace more number of lines repeatedly. o Comparison using ruler and a divider Use of Ruler – Markings on Ruler A ruler has markings along one of its edge. Ruler is divided into 15 equal parts. Each part is of length 1 cm. Each centimeter is divided into 10 sub parts. Each subpart of the division of 1 cm is • Consider a man is standing facing north. If the 1 mm. man turns himself to east, then we say he has turned through a Right angle. Now if the man To measure the length of the line segment XY , turns two right angles clockwise, he will be place the zero mark of the ruler at X. Now read facing South and this is a straight angle. the mark against Y. This will give the measure of length XY . Example: If the length is 3.5 cm, we may write it as length XY = 3.5 cm or XY = 3.5 cm • Turning by two straight angles or four right angles in the same direction, makes a full turn and we reach original position. One complete turn is called one revolution. Errors in measurement may occur due to • Angle of one revolution is called a complete • Difficulty in reading off the marks on ruler angle. • When the hand of a clock moves from one because of thickness of ruler. position to another, it turns through an angle. • Due to wrong eye position while reading the marks on the ruler. The errors can be minimized by positioning the eye vertically above the markings while reading. Use of Divider The divider consists of two arms with sharp end points. 33

5. Understanding Elementary Shapes be measured. • Angles (Acute, Obtuse and Reflex) Adjust the protractor so that YZ is along the • Acute angle: An angle smaller than the right straight edge of the protractor. angle is called acute angle, i.e., less than 90⁰ There are 2 scales on the protractor: Read the • Obtuse angle: An angle larger than a right angle scale which has 0ᵒ mark coinciding with the and smaller than a straight angle is called obtuse straight edge (i.e., with ray YZ ) angle, i.e., greater 90ᵒ and less than 180ᵒ. Mark shown by YX on the curved edge gives the • Reflex angle: A reflex angle is greater than degree measure of the angle. 180ᵒand lesser than 360ᵒ. m∠XYZ = 45ᵒ  or  ∠XYZ = 45ᵒ • Measuring Angles • Perpendicular Lines Right angle (RA) tester can only be used to • When two lines intersect each other and the compare angles with a right angle. It is possible angle between them is a right angle, then the to classify angles as acute, obtuse or reflex by a lines are said to be perpendicular. RA tester. • If MN is a perpendicular to PQ, we denote MN ⊥ To measure angles, an instrument called PQ. protractor is used. • Let XY be a line segment and AB be a line per- Measurement is done in degrees. pendicular to XY. If XB = BY then AB divide XY into One revolution is divided into 360 equal parts two equal parts XB and XY. and each part is called a degree (1ᵒ) So 1 revolution = 360 degrees or 360ᵒ Thus 1 Revolution = 1800 , 1 Revolution = 90° • So, AB is called perpendicular bisector of XY • Classification of Triangles 24 • A Triangle is a polygon with 3 sides and 3 angles. • Naming of triangles based on sides • A protractor has a curved edge and a straight o Scalene Triangle: A triangle having all three edge. The curved edge has the graduations which divide it into 180 equal parts. Each part unequal sides. is equal to a degree. Markings start from 0ᵒ from o Isosceles Triangle: A triangle having two the right side and ends with 180ᵒ on the left side and vice versa. equal sides. o Equilateral Triangle: A triangle having three equal sides. • Procedure to measure Angle using Protractor Place the protractor so that the midpoint of its straight edge lies on the vertex (Y) of the angle to 34

5. Understanding Elementary Shapes • Naming of triangles based on angles o Acute angle triangle: A triangle in which each angle is less than 90ᵒ. o Right angle triangle: A triangle in which any one angle is equal to right angle. o Obtuse angle Triangle: A triangle in which any one angle is greater than 90ᵒ. Note: i) Sum of all angles of a triangle is 180ᵒ. ii) In an isosceles triangle, 2 sides and 2 angles are equal. iii) In an equilateral triangle, all 3 sides and 3 angles are equal. Each angle is equal to 60⁰. • Quadrilateral • It is a polygon with 4 sides and 4 angles. An instrument box has 2 set squares. One is 30ᵒ, 60ᵒ, 90ᵒ set square and the other is 45ᵒ, 45ᵒ, 90ᵒ set square. By using these set squares in various combinations different types of quadrilateral can be formed. Square, rectangle, trapezium, p arallelogram and rhombus are major types of quadrilateral that are are shown in the figure below.       o Table of important properties of quadrilaterals: Quadrilateral Properties 1) Trapezium − One pair of opposite sides are parallel. 2) Parallelogram − Two pairs of opposite sides are parallel. − Opposite sides are equal in length. 3) Rectangle − Opposite angles are equal in measure. − Diagonals are unequal in length. 4) Rhombus − Parallelogram with 4 right angles. − Opposite sides are equal in length and parallel 5) Square − Diagonals are equal in length and bisect each other − Parallelogram with 4 sides of equal length − Opposite angle are equal − Diagonals are unequal and bisect each other. − Rhombus with 4 right angles. − Diagonals are equal in length and bisect each other. 35

5. Understanding Elementary Shapes • Polygon • Polygon is a closed curve made up of line segments only. No. of sides 3 4 5 6 8 Hexagon Octagon Name Triangle Quadrilateral Pentagon Example • Three Dimensional Shapes • Terminologies of 3D Shapes o Face: A face is a flat surface of a 3D object. It is 2 dimensional in nature. o Edge: Two faces meet at a line segment called an edge. o  Vertex: Three edges meet at a point called vertex. • Prism is a solid object with 2 identical ends and flat sides. The shapes of the ends give the prism a name. Prism has identical bases and other faces are rectangles. Example: 3)  Triangular Pyramid − It is a pyramid with triangle • Some important 3 dimensional shapes 1) Cuboid: as base. − Looks like a rectangular − It is also known as box. − It has six faces, each face tetrahedron − It has four faces, each face has four edges and four corners or vertices. has three edges 2) Cube − The solid has 4 vertices. − It is a cuboid whose edges 4)  Square Pyramid are all of equal length. − It has a square as its base. − It has six faces; each face − It has 4 triangular faces has four edges and four vertices. and one square base − Triangular faces have 3 edges each and 3 corner each. − It has 5 vertices. 36