Math-6

CSS Middle Standard “Mathematics” 1 CSS Middle Standard Mathematics For Class 6 1 st term Syllabus Teacher’s Guide Fully Solved Exercises

CSS Middle Standard “Mathematics” 2 Table of Contents Sr. No. Description Page No. 1. Division of Syllabus 2 2. Unit # 1 9 3. Unit # 2 15 4. Unit # 3 28 5. Model Paper # 1 53 6. Model Paper # 2 56 7. Model Paper # 3 58 8. Unit # 4 60 9. Unit # 5 72 10. Unit # 6 81 11. Unit # 7 89 12. Unit # 8 97 13. Unit # 9 104 14. Unit # 10 113 15. Model Paper # 1 129 16. Model Paper # 2 132 17. Model Paper # 3 135 18. Unit # 11 137 19. Unit # 12 146 20. Unit # 13 150 21. Model Paper # 1 153 22. Model Paper # 2 156 23. Model Paper # 3 158 Division of Syllabus 1 stTerm Week 1 Week 2 Unit # 1 Sets Set Types of Sets Week 3 Week 4 Week 5 Unit # 2 Whole Numbers Natural and Whole Numbers Addition and subtraction of whole numbers Multiplication and division of whole numbers Distributive laws of whole numbers Week 6 Week 7 Week 8 Unit # 3 Factors and Multiples Factors and Multiples Tests for divisibility Factorization HCF LCM Applications of HCF and LCM Week 9 Week 10 Revision of Units + Final Exam of 1st Term 2 ndTerm Week 1 Unit # 4 Integers Integers Ordering of integers Absolute or numerical value of an integer

CSS Middle Standard “Mathematics” 3 Week 2 Addition of integers Subtraction of integers Multiplication of integers Division of integers Week 3 Week 4 Unit # 5 Simplification BODMAS Rule Week 5 Unit # 6 Ratio and Proportion Ratio Proportion Week 6 Week 7 Unit # 7 Finanacial Arithematic Percentage Profit Loss and Discount Week 8 Week 9 Unit # 8 Introduction of Algebra Algebra Algebraic Expression Week 10 Unit # 9 Linear Equation Algebraic Equations Linear Equations Week 11 Week 12 Unit# 10 Geometry Line Segments Construction of Angles Construction of Triangles Week 13 Week 14 Revision of Units + Final Exam of 2nd Term 3 rdTerm Week 1 Week 2 Unit # 11 Area and Perimeter Perimeter and Area Week 3 Unit # 12 Three Dimensional Solid Volume and surface area Week 4 Week 5 Unit # 13 Infermation Handling Types of Data Bar graph Pie graph Week 6, 7, 8 Revision of all Units + Final Exam of 3rd Term ‘Detail Division of Syllabus’ First term Week 1 Unit– 1 Sets Day 1 Definition of Set, Ex (1a) Day 2 Tabular Form of Set, Ex (1b), Q#1 Day 3 Notation for membership of a set, Ex (1b), Q#2,3 Day 4 Type of Sets, Finite and Infinite sets, Ex (1c),Q#1 Day 5 Type of Sets, Equal and Equivalent sets, Ex (1c),Q#2 Day 6 Test of Exercises Week 2 Day 1 Define Subsets through examples Day 2 Formula for finding number of subsets, Ex (1d), Q#1

CSS Middle Standard “Mathematics” 4 Day 3 Types of subsets, Ex (1d), Q#2,3,4 Day 4 Review Exercise Q#1,2 Day 5 Review Exercise Q#3,4,5 Day 6 Test of Unit 1 Week 3 Unit– 2 Whole Numbers Day 1 Recognized natural numbers and give examples Day 2 Revise comparison symbols “<, >”, Ex (2a), Q#1 Day 3 Introduce 0 as whole number, Sum of whole numbers using number line, Ex (2a), Q#2 Day 4 Difference of whole numbers using number line, Ex (2a), Q#3 Day 5 Addition of more than two numbers using number line, Ex (2a), Q#4 Day 6 Ex (2a), Q#5,6 Week 4 Day 1 Revise whole and natural numbers, Ex (2a), Q#7,8 Day 2 Representation of numbers on number line, Ex (2a) Q#9 Day 3 Addition of whole numbers, Ex (2b), Q#1 Day 4 Subtraction of whole numbers, Ex (2b), Q#2,3,4,5 Day 5 Commutative and associative laws w.r.t addition, Ex (2b), Q#6 Day 6 Ex (2b), Q#7,8 Week 5 Day 1 Introduce multiplication and division of whole numbers through examples, Ex (2c), Q#2,3 Day 2 Distributive laws, Ex (2c), Q#1 Day 3 Commutative and associative laws under multiplication, Ex (2c), Q#4,5,6 Day 4 Ex (2c), Q#7,8,9 Day 5 Review Exercise 2, Q#1,2,3,4 Day 6 Review Exercise 2, Q#5 to Q#10 Week 6 Unit– 3 Factors and multiples Day 1 Revise prime and composite numbers, Ex (3a), Q#1 Day 2 Define factors and multiples, Ex (3a), Q#2,3

CSS Middle Standard “Mathematics” 5 Day 3 Define even and odd numbers, Ex (3a), Q#4,5,6 Day 4 Ex (3a), Q#7,8,9 Day 5 Introduce rules of divisibility by 2,3,4,5, Ex (3b), Q#1 Day 6 Introduce rules of divisibility by 6,8,9,10, Ex (3b), Q#2 Week 7 Day 1 Introduce rules of divisibility by 11,12,15,25 Ex (3b), Q#3 Day 2 Ex (3b), Q#4 Day 3 Introduce factorization and factor trees, Ex (3c), Q#1 Day 4 Factorization by division method, Ex (3c), Q#2 Day 5 Index notation, Ex (3c), Q#3,4 Day 6 H.C.F by prime factorization method, Ex (3d). Q#1 Week 8 Day 1 H.C.F by long division method, Ex (3d), Q#2,3 Day 2 L.C.M by prime factorization method, Ex (3e). Q#1 Day 3 L.C.M by division method, Ex (3e), Q#2 Day 4 Application of H.C.F and L.C.M, Ex (3f) Day 5 Review Exercise 3, Q#1 to Q#6 Day 6 Review Exercise 3, Q#7 to Q#13 Week 9 + Week 10 Revision of Units + Final Exam of 1st Term Second term Week 1 Unit– 4 Integers Day 1 Recognize integers, Ex (4a), Q#1 Day 2 Representaion of integers on number line, Ex (4a), Q#2 Day 3 Absolute value of integers, Ex (4a), Q#3 Day 4 Ordering of integers, Ex (4a), Q#4,5,6 Day 5 Addition and subtraction of integers by using number line, Ex (4b), Q#1,2,3,4 Day 6 Ex, (4b), Q#5,6,7 Week 2 Day 1 Subtraction of integers, Ex (4c) Day 2 Multiplication of integers, Ex (4d)

CSS Middle Standard “Mathematics” 6 Day 3 Division of integers, Ex (4e), Q#1,2 Day 4 Ex (4e), Q#3,4,5 Day 5 Review Exercise 4, Q#1 to Q#5 Day 6 Review Exercise 4, Q#6 to Q#9 Week 3 Unit – 5: Simplification Day 1 Kinds of Brackets, BODMAS Rule, Ex (5a), Q#1 (i –iv) Day 2 Solve, Ex (5a), Q#1 (v – vii) Day 3 Solve, Ex (5b), Q#1 (i – v) Day 4 Solve, Ex (5b), Q#1 (vi – ix) Day 5 Solve, Ex (5c), Q#1 (i – iv) Day 6 Solve, Ex (5c), Q#1 (v –vii) Week 4 Day 1 Real life problems involving fractions and decimals, Ex (5d), Q#1,2,3 Day 2 Real life problems involving fractions and decimals, Ex (5d), Q#4,5,6 Day 3 Review Exercise 5, Q#1 Day 4 Review Exercise 5, Q#2 Day 5 Review Exercise 5, Q#3 to Q#8 Day 6 Test of Unit 5 Week 5 Unit - 6: Ratio Day 1 Define Ratio, Solve Ex (6a), Q#1,2 Day 2 Solve Ex (6a), Q#3 to Q#8 Day 3 Define Proportion, Solve Ex (6b), Q#1,2,3,4 Day 4 Solve Ex (6b), Q#5 to Q#10 Day 5 Review Exercise 6, Q#1,2,3 Day 6 Review Exercise 6, Q#4 to Q#9 Week 6 Unit – 7: Financial Arithmatic Day 1 Introduce percentage through examples Day 2 Conversion of fraction into percentage and vice versa, Ex (7a), Q#1 Day 3 Conversion of fraction into percentage and vice versa, Ex (7a), Q#2

CSS Middle Standard “Mathematics” 7 Day 4 Convert percentage into decimals and vice versa, Ex (7b), Q#1 Day 5 Convert percentage into decimals and vice versa, Ex (7b), Q#2 Day 6 Real life problems involving percentage, Ex (7c), Q#1,2 Week 7 Day 1 Solve, Ex (7c), Q#3,4,5,6 Day 2 Solve, Ex (7c), Q#7 to Q#10 Day 3 Introduce selling and cost price, Ex (7d), Q#1,2,3 Day 4 Introduce profit and loss, Ex (7d), Q#4 to Q#7 Day 5 Review Exercise 7, Q#1,2,3 Day 6 Review Exercise 7, Q#4 to Q#7 Week 8 Unit – 8: Introduction of Algebra Day 1 Introduce Algebra, Ex (8a), Q#1 Day 2 Mathematical Statements, Ex (8a), Q#2,3 Day 3 Define Algebraic expressions and its parts, Ex (8b), Q#1,2 Day 4 Solve, Ex (8b), Q#3,4 Day 5 Addition of like and unlike, Ex (8c), Q#1,2 Day 6 Subraction of like and unlike, Ex (8c), Q#3,4 Week 9 Day 1 Evaluation of algebraic expressions, Ex (8d), Q#1 Day 2 Solve Ex (8d), Q#2 Day 3 Solve Ex (8d), Q#3,4 Day 4 Review Exercise 8, Q#1,2,3 Day 5 Review Exercise 8, Q#4,5,6 Day 6 Test of Unit 8 Week 10 Unit – 9: Linear Equation Day 1 Define algebraic equation and algebraic expression, Ex (9a) Day 2 Solution of linear equations, Ex (9b) Day 3 Solve Ex, (9c), Q#1,2 Day 4 Real life word problems, Ex, (9c), Q#3 to Q#7

CSS Middle Standard “Mathematics” 8 Day 5 Review Exercise 9, Q#1,2 Day 6 Review Exercise 9, Q#3 to Q#7 Week 11 Unit – 10: Geometry Day 1 Define Geometry, Line Segment, Addittion and Subtraction of Line Segments, Ex (10a), Q#1 Day 2 Draw line segment and its bisector, Ex (10a), Q#2,3,4 Day 3 Solve, Ex (10a), Q#5 to Q#9 Day 4 Define angle and its construction, Ex (10b), Q#1 Day 5 Draw an angle using compass, Ex (10b), Q#2 Day 6 Division of angles, Ex (10b), Q#3,4 Week 12 Day 1 Solve, Ex (10b), Q#5,6 Day 2 Define triangle and how to construct a triangle, Ex (10c), Q#1,2 Day 3 Define triangle and how to construct a triangle, Ex (10c), Q#3,4,5 Day 4 Review Exercise 10, Q#1 to Q#4 Day 5 Review Exercise 10, Q#5,6 Day 6 Review Exercise 10, Q#7 to Q#11 Week 13 + Week 14 Revision of Units + Final Exam of 2nd Term Third term Week 1 Unit – 11: Perimeter and Area Day 1 Define perimeter and area of square and rectangle, Ex (11a), Q#1 Day 2 Ex (11a), Q#2,3 Day 3 Area of square and rectangle, Ex (11b), Q#1 to Q#4 Day 4 Area of square and rectangle, Ex (11b), Q#5 to Q#8 Day 5 Discuss altitude and area of triangle, Ex (11c), Q#1 Day 6 Area of Parallelogram, Ex (11c), Q#2,3,4 Week 2 Day 1 Area of trapezium, Ex (11d), Q#1,2 Day 2 Area of trapezium, Ex (11d), Q#3 Day 3 Review Exercise 11, Q#1,2

CSS Middle Standard “Mathematics” 9 Day 4 Review Exercise 11, Q#3 Day 5 Review Exercise 11, Q#4,5 Day 6 Test of Unit – 11 Week 3 Unit– 12: Three dimensional solid Day 1 Define three dimensional figures and their parts, Day 2 Volume of cuboid, Ex (12a), Q#1 Day 3 Volume of cube, Ex (12a), Q#2 Day 4 Surface area of cube and cuboid, Ex (12a), Q#3 to 7 Day 5 Review Exercise 12, Q#1,2 Day 6 Review Exercise 12, Q#3 to 6 Week 4 Unit – 13: Information Handling Day 1 Define data and its types, Sources of data, Ex (13a), Q#1 Day 2 Ex (13a), Q#2 Day 3 Frequency distributiton, Ex (13a), Q#3 Day 4 Bar graphs and its types, Ex (13b), Q#1,2 Day 5 Bar graphs and its types, Ex (13b), Q#3 Day 6 Define pie graph and its construction Week 5 Day 1 Solve, Ex, (13b), Q#4 Day 2 Solve, Ex, (13b), Q#5 Day 3 Solve, Ex, (13b), Q#6,7 Day 4 Revision of Unit – 13 Day 5 Revision of Unit – 13 Day 6 Test of Unit – 13 Week 6, 7, 8 Revision of all Units + Final Exam of 3rd Term Unit No. 1 Sets Lesson # 1 Teaching Objectives: To introduce sets and their notations.

CSS Middle Standard “Mathematics” 10 To introduce tabular, descriptive and set builder notations. To introduce and explain different types of sets. Learning Outcomes: Students should be able to: Define sets, Recognize notation of a set and its objects/ elements. Describe tabular form of a set and demonstrate through examples. Define the following sets and demonstrate through examples: ● finite and infinite sets ● empty /void/null set ● singleton set ● equal and equivalent sets, ● subset and superset of a set, ● Proper and improper subsets of a set, Teaching Materials: CSS Middle Standard Mathematics Book 6 ● Writing Board ● Marker ● Eraser Procedure: This is the first time students are being taught sets. So, detailed explanation and easy simplification is necessary. Sets should be introduced with a 5-minute brainstorming quiz to create as many finite and infinite sets as possible. For example, planets of our galaxy, stars in the universe, prime numbers, multiples of 3 less than 20, etc. Students should be encouraged to suggest elements and types of sets. A group quiz can be created where flash cards of different types of sets can be named and each member of every group picks a card and gives an example of the type of set named on the card. To clarify the difference between subsets and proper subsets, pick two students. Give one of them three pencils and the other 5, of which 3 are identical to those of the first student. Place a set of different blocks or books in front of the students and tell them that it is a set because it is a collection of well defined and district objects. Give them examples of well defined objects e.g. item of clothes we wear, hand fingers, seven days in a week etc. Taking help from the book classify the definition of the set, set notation and elements of a set. Ask them to give examples of sets. Reinforce the fact that sets can be represented in three forms that is, tabular, descriptive and set builder form. Give simple examples of each before proceeding to the exercise. Simple examples: A = {a, b, c, d} and B = {1, 2, 3, 4} are both different sets. Their elements are a, b, c, d and 1, 2, 3, 4. These elements are distinguishable entities. Here they are written in tabular form. In descriptive form they will be written as, A = set of first four letters in the English alphabet B = set of first four natural numbers In set builder notation, sets A and B will be written as, A = { x x first four English alphabets} B = { x x N 1 ≤ x ≤ 4} Answer any queries the students may have regarding the three styles. Show them how they are important in different questions. Also, while explaining set builder notation, introduce the students with symbols used for specific sets such as N for natural numbers, W for whole numbers, Z for integers etc. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom.

CSS Middle Standard “Mathematics” 11 Fun Activity Catch Me Materials: Index cards with one set notation symbol on each o You will need one card per student so replicas will be necessary, however, try to keep the numbers of each symbol equal. Instructions: Write the six sets above on the board. Hand out one card to each student. Stand at the board and ask students to stand in a line against the opposite wall of the room. Call out instructions for students to move toward you based on their card identification. For example: o ''Move one step forward if your card means to create a new set out of all the numbers in two other sets.'' o ''Hop forward twice if your card would result in the set {4}.'' For clues that could have more than one response (like subset or intersection), ask students to explain why they have moved forward. The first student to reach you takes your place while the other students return to the starting place. Play as long as time allows swapping leaders each time a student reaches the leader. Exercise (1a) Q.1 Justify that the following statements form a set or not. i. The first 5 days of the week. Ans: First five days of the week are Monday, Tuesday, Wednesday, Thursday and Friday. All these days are well defined because Monday means Monday not Sunday and they are distinct because Friday has its own distinction from Tuesday, Thursday. Therefore first five days of a week form a set. ii. All the cups in the tea set. Ans: All the cups in the tea set are well defined items but not a set because all the cups are similar and there is not distinction of one cup from another cup. iii. The first 5 letters of the English alphabet. Ans: The first five letters of english alphabet {a, b, c, d, e} form a set because all the letters are well defined and are distinct from each other. iv. Students in a class-room. Ans: Students in a class form a set because each student is well defined and is distinct from another student. v. The first 4 odd numbers. Ans: The first 4 odd numbers {1, 3, 5, 7} form a set because each number is well defined and we know what 1 mean 1.We do not confuse it with 2. Secondly 1, 3, 5, 7 are distinct from each other because they are not similar. vi. Names of 4 students of class 6th of your school. Ans: Same as iv vii. All the glasses of a water set. Ans: Same as ii Exercise (1b) Q 1: Write the following sets in tabular form. i. The set of vowels in English alphabet. Ans: {a, e, i, o, u} ii. The set of names of lunar months in a year. Ans: {Muharram, Safar, Rabi’ al-awal, Rabi’ al-thani, Jumada al-awal, Jumada al-thani, Rajab, Sha’aban, Ramadan, Shawwal, Duh al-Q’idah, Duh al-Hijjah iii. The set of colours in our flag. Ans: {Green, White} iv. The set of names of your subjects in grade 6. Ans: {English, Urdu, Maths, Science, History, Geography, Islamiyat, Computer science} v. The set of your family members. Ans:Every student will have different family members vi. The set of names of your friends in grade 6. Ans: Every student will have different names Q 2: Count the number of elements in the following sets.

CSS Middle Standard “Mathematics” 12 i. {0} Ans: One ii. {5, 7} Ans: Two iii. {5, 7, 9, 11} Ans: Four iv. {1, 3, ... 9} Ans: Five v. {1, 2, 3, ... , 10} Ans: Ten vi. {4, 6, 8, 10, 12} Ans: Five vii. {2, 4, 6, … 10} Ans: Five Q 3: Write the following in tabular form of the set. i. 9 and 13 A. Ans: A={9,13} ii. Ali and Amna X. Ans: X={Ali, Amna} iii. 1,2,3,4,5,6 F. Ans: F={1, 2, 3, 4, 5, 6} iv. First 10 even numbers Z. Ans: {0, 2, 4, 6, 8, 10, 12, 14, 16, 18} v. Hammad and Amad W. Ans: W={Hammad, Amad} vi. 100,101,143 S. Ans: S={100, 101, 143} vii. The set natural numbers when even numbers Y. Ans: Y={1, 3, 5, ...} Lesson # 2 Procedure: Greet the students and ask them about the definition of the set. Ask them if sets are of differnet types or not. Start the lesson on types of set with finite and infinite sets. By now the students will have an idea on what a set is and what are its elements and their properties. Finite and infinite sets will be easier to handle at such a point. Furthermore, advanced topics in the chapter such as subsets require this solid foundation. Again, give examples from real life, so the students are able to understand the idea behind finite and infinite sets. The textbook is of great help in this regard. Moving on to empty set, teaching the concept of a null or void set can be a bit difficult. As the frequently asked question is that, “Why an empty set called a set when it is empty?” Such a question is tackled by relating to the conventions developed in mathematics and sciences for our convenience in building a theory. For example, positive and negative for charges of protons and electrons and choice of carbon-12 Avogadro’s number (6.022 1022) for mole. Teaching the students about singleton sets should not be a difficult task once they will have walked the tricky roads of sets and empty set. However, at least three examples are necessary. While dealing with equal and equivalent sets, make sure that every student grasps the difference between equal and equivalent sets. Follow the book and make some examples yourself. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (1c) Q 1: Identify finite and infinite set. i. The set of natural numbers between 3 and 10. Ans: finite

CSS Middle Standard “Mathematics” 13 ii. {1, 2, 3, …, 23} Ans: finite iii. The set of natural numbers between 0 and1. Ans: finite iv. Cities in the world. Ans: finite v. The set of natural number divisible by 5. Ans: infinite vi. Stars in the sky. Ans: infinite vii. {2, 4, 6, 8, … } Ans: infinite viii. Points on a line. Ans: infinite ix. {10, 9, – 9, – 14, 6} Ans: finite x. {1, 3, 5, ..., 51} Ans: finite xi. The set of natural numbers greater than 10. Ans: infinite xii. The set of natural numbers not divisible by 2. Ans: finite xiii. The set of factors of 10. Ans: finite Q 2: Pick out equal and equivalent signs for the following. i. {p, q, r}_____{P, Q, R} Ans: ii. {India, Srilanka, China}____{Pakistan, Iran, China} Ans: iii. {a, e, i, o, u}____vowels of the alphabet Ans: = iv. {10, 11, 12}_____{11, 12, 10} Ans: = v. {2, 4, 6, 8}____{1, 2, 3, 4} Ans: vi. {a, b, c, d, e}_____{e, b, c, a, d} Ans: = vii. {1,2,3,4}____{4,1,2,3} Ans: = viii. {Ali, Ahmed, Amna}_____{Ali, Amna, Haris} Ans: ix. {a, e, i, o, u}____{1,2,3,4,5} Ans: x. {tea, bread, egg}_____{shoes, socks, pants} Ans: xi. {Maths, English, Urdu}_____{Urdu, English, Maths} Ans: = xii. {Spring, Summer, Winter}_____{Autumn, Spring, Summer} Ans: Lesson # 3 Procedure: Write the word ‘sub’ on the board and ask studnts to tell what they know about it. Ask them to think about the word subset. At the level of grade 6, Subsets is perhaps the most difficult topic for students to understand. Emphasize on clarification of the definition of a subset. Start with simpler examples such as examples related to box containing packets of chocolates and candies. Each packet will be a subset of the box which is a superset of the packets. Then move on to the examples dealing with numbers. For this purpose, follow the book and improvise. Once the concept of subsets is clear, move on to the classification of subsets, that is, proper and improper subset. Every concept must be related to real life examples that are easier for the students to understand such as, cars, planes, fruits, toys etc. Invite them for book readind and understanding examples. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each questions further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom.

CSS Middle Standard “Mathematics” 14 Exercise (1d) Q 1: If a set has 5 number of elements. Find the number of subsets, proper subsets and improper subsets. Ans: Number of subsets = 25 = 32 Number of proper subsets = 31, Number of improper subsets = 1 Q 2: Find all the possible subsets of the following sets. i. A = {1,2} Ans: { }, {1}, {2}, {1, 2} ii. X = {a,b,c} Ans: { }, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} iii. Y = {,.,} Ans: { }, {}, {}, {}, {}, {, }, {, }, {, }, {, }, {, }, {, }, {, , }, {, , }, {, , }, {, , }, {, , , }, Q 3: Find the proper and improper subsets of the following. i. B = {x,y,z} Ans: proper subset = { }, {x}, {y}, {z}, {x, y}, {x, z}, {y, z} improper subset = {x, y, z} ii. C = {1,2,3} Ans: proper subset = { }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} improper subset = {1, 2, 3} Q 4: Which of the following sets are proper subset and which are improper subsets? Use the subset symbols in your answer. i. A = {7, 14, – 15, 23} B = {– 15, 23, 14, 7} Ans: A B ii. A = {e} B = {g, h, j, e} Ans: A ⸦ B iii. A = {book, pen, car, ruler} B = {book, pen, ruler} Ans: B ⸦ A iv. A = [1, 2, 3, 4} B = {4, 3, 2, 1} Ans: A B v. A = {2, 4, 6, 8, 10} B = {2, 4, 6, 8, 10, 12} Ans: A ⸦ B Review Exercise 1 Q 1: Choose the correct answer and fill the circle: i. If A= { 3,6,9,12} and B= { 9,12,3,6} then: A is subset of B they are equal set they are equivalent set all of them ii. All the even numbers in whole number are: empty set finite set infinite set void set iii. 14 A means: 14 is proper subset of A 14 is a member of A 14 does not belongs to A all of these iv. A is set of factors of 12. Which one of the following is not a member of A? 3 4 5 6 v. X is set of multiples of 3, Y is the set of multiples of 6, Z is the set of multiples of 9. Which one is true: X Y X Z Z Y Z X vi. S = {a,b,c,d,e}, how many proper subset does the set S have? 15 31 32 33 vii. A = {a, b, c, d}, how many subset does the set A have? 4 6 16 64 viii. If A = {3,4,6,7,8}, which of them is not a subset of A: B = {3,6,7} C = { 3,4} {3,4,5} {7,8} ix. If A has 18 elements. How many subset A has: 210 218 218–1 218–1 x. B = {3,4,6,7,8,9}, which one is superset of B: A = {1,3,4,6,7} C={3,4,6,7,8,9,10} D = {4,6,7,8} E = {1,3,4,7,8,10,11} xi. If Z has 16 elements. How many proper subset has:

CSS Middle Standard “Mathematics” 15 216 216-2 214 none of them xii. If X = {88,84,98} and Y = {88,101,10} then: X Y X Y X = Y X Y Q 2: Fill in the blanks with the symbol of subset. i. {a,b,c} {a,b,c,d} ii. { } {0,1,2} iii. {1,2} {1} iv. {1,2,3} {0,1,2...} Q 3: Name the following sets: i. {x} Ans: singleton set ii. A = {a,b,c }, B = {c,b,a} Ans: equal sets iii. { } Ans: empty set iv. X = {1,2,3...} Ans: infinite set v. A = {1,2,3}, B = {a,b,c} Ans: equivalent sets vi. X = set of all positive integers which is a multiple of 2: Ans: infinite set Q 4: Which of the following sets are subsets of other sets? Use the symbols. A = {0,1,2}, B = {1,2,–1} C = {1,2,0}, D = {1,0,–1} E = {e, f, g} F = {e,f} G = { } H = {g, f,e} Ans: A C, F E, F H, E H, G A, B, C, D, E, F Note: Empty set is subset of every set. Q 5: Find the possible subsets of the following sets. i. A = {a} Ans: { }, {a} ii. B = {a,b} Ans: { }, {a}, {b}, {a, b} iii. C = {1,2,3,4,5} Ans: { },{1},{2},{3},{4},{5},{1, 2},{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 3, 4, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}, {1, 2, 3, 4, 5} Unit No. 2 Whole Numbers and Number Line Lesson # 1 Teaching Objectives: To revise the difference between natural and whole numbers. To revise equalities and inequalities of whole numbers. To revise mathematical operations on whole numbers (also using number line). To revise commutative, associative and distributive properties of whole numbers under addition and multiplication. Learning outcomes: Students should be able to: Differentiate between natural and whole numbers. Identify natural and whole numbers, and their notations. Represent a given list of whole numbers, whole numbers < (or >) a given whole number, whole numbers > (or <) a given whole number, whole numbers > but < a given whole number, whole numbers > but < a given whole number, sum of two or more given whole numbers, on the number line. Add and subtract two given whole numbers. Verify commutative and associative law (under addition) of whole numbers. Recognize ‘0’ as additive identity. Multiply and divide two given whole numbers. Verify commutative and associative law (under multiplication) of whole numbers.

CSS Middle Standard “Mathematics” 16 Recognize ‘1’ as multiplication identity. Verify distributive law of multiplication over addition. Verify distributive law of multiplication over subtraction (with positive difference). Teaching Materials: ● CSS Middle Standard Mathematics Book 6. ● Writing Board. ● Marker. ● Eraser. Procedure: The large numbers and arithmetic operations have been handled in Mathematics Book 5. Set of whole numbers has been introduced in chapter 1. Handling whole numbers should not be difficult provided the students have understood their concepts in the previous years. Discuss the difference between natural and whole numbers clearly. Use a T-chart for this purpose. T-chart Natural Numbers Whole Numbers State clearly that natural numbers are ordinarily used counting numbers. Real-life Application and Activities: In Grade 6 we can discuss the importance of numbers and use clippings brought in from the business pages. Students can research, for example, the distance of the Earth from the Moon or the distance from Karachi to Chicago. A comparison of the population of Karachi and that of Faisalabad can also be done. Interestingly, they can do all the operations using population as the topic. 2v3 Natural Numbers and Whole Numbers Examples • How many times is the population of Karachi bigger than the population of Edinburgh? • They can Google the populations and then proceed. • What is the difference between the population of Karachi and that of Islamabad? • If a spaceship takes 3 days to circumnavigate a planet three times and covers a distance of 40,075 km, how long will it take to go around it ten times? Invite them for book reading and solivng quesitns. For comparisons and inequalities, go a step further and give numerical examples related to the formulae given in the book on page 16. For example, to explain x>a give an example that 5>2, here x = 5 and a = 2. Ask them what they know about a number line. Tell them that it is a pictorial representation of numbers. Give examples of other diagrammatical representations such as the world map, human body etc. to relate the concept. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (2a)

CSS Middle Standard “Mathematics” 17 Q 1: Fill in the blanks by using the symbol<or>. i. 21089346589 — 43586701 Ans: > ii. 415678910 — 483467890 Ans: < iii. 101023410 — 101022400 Ans: > iv. 1435861070 — 189345678 Ans: > v. 348964173 — 389814342 Ans: < vi. 4384642310 — 143210342 Ans: > vii. 1892341039 — 7189403479 Ans: < viii. 5678989610 — 437248910 Ans: > ix. 4132867180 — 63798543 Ans: > x. 4235115187 — 4237593480 Ans: < Q 2:Find the sum of the following whole numbers by using number line. i. 2 and 6 ii. 8 and 4 iii. 10 and 3 iv. 1 and 6 v. 24 and 6 vi. 40 and 20 vii. 15 and 25 Q 3: Find the difference of the following whole number using number line. i. 2, 6

CSS Middle Standard “Mathematics” 18 ii. 15, 20 iii. 9,15 iv. 10, 6 v. 5, 9 Q 4: Add more than two numbers on number line. i. 3, 6, 5 ii. 9, 12, 5 iii. 4, 8, 2 iv. 1, 4, 5 v. 3, 8, 12 vi. 4, 6, 10 +10 10 – 6 = 4 6 0 1 2 3 4 5 6 7 8 9 10 Ans: +9 9 – 5 = 4 0 1 2 3 4 5 6 7 8 9 10 Ans:

CSS Middle Standard “Mathematics” 19 Q 5: Find the difference on the number line and adjust the scale. i. 18 , 17 ii. 95 , 90 iii. 75 , 55 iv. 170 , 70 v. 110 , 50 vi. 250 , 200 vii. 350, 150 viii. 850, 50 ix. 475, 200

CSS Middle Standard “Mathematics” 20 x. 1700 , 300 Q 6: Which of the following statements are true or false. i. 14387234 > 14381123 Ans: true ii. 51432734 > 431411734 Ans: false iii. 411157832 > 21114321 Ans: true iv. 243789243 < 24534890 Ans: false v. 73429143 < 73429014 Ans: false vi. 73849672 < 83481432 Ans: true vii. 67892432 > 1143200 Ans: true viii. 805898724 > 984345670 Ans: false Q 7: Write all the natural numbers less than 15. Ans: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 Q 8: Write all the whole numbers less than 12 Ans: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 Q 9: Represent on number line. i. Whole numbers < 60. ii. Whole numbers > 6. iii. Whole numbers < 700. iv. Whole numbers ≥ 7. v. Whole numbers > 4 but <10. vi. Whole numbers ≥ 4 but < 10. vii. Whole numbers greater than 5. viii. Even whole numbers greater than or equal to 6 but less than 15. Lesson # 2 Procedure: Before moving on to addition and subtraction of very large numbers in the book, make the students recall the processes of these mathematical operations. A brainstorming session could be very helpful for this purpose. Draw two bubble maps on the board and ask students to tell you about addition and subtraction.

CSS Middle Standard “Mathematics” 21 Ask about the practical utilization of both of the concepts in the real life. Simple demonstrative examples are necessary to clarify the commutative and associative properties. For example, take identical tennis balls and add two of them by placing one by one on the table to in reverse orders to show the commutative property with respect to addition. Before jumping to additive identity, explain the mathematical meaning of an identity that is, “a number is an identity when it leaves another number unchanged under some operation.” Now relate this concept with zero in case of addition and tell them that zero is called “additive” identity because it leaves any number unchanged when it is added to it. Invite the students for book reading and solving questions. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (2b) Q 1: Fill in the blanks with correct statements. i. 100013601 + ______ = 100013601 Ans: 0 ii. 134134180 + 4281517 = ______ + 134134180 Ans: 4281517 iii. 710415 + (63 + 72) = (710415 + ______) + 72 Ans: 63 iv. ______ × 134712118123 = 0 Ans: 0 v. ______ + 0 = 0 + 1143 Ans: 1143 vi. ______ + 0 = 1110 Ans: 1110 vii. ______ ÷ 1101 = 0 Ans: 0 viii. 1231014 + ______ = 1231014 Ans: 0 Q 2: Draw first 100 even numbers on number line with the difference of 10. Q 3: Find the sum of smallest five digit number and largest four digit number. 10000 + 9999 19999 Q 4: Add the following. i. 32,963,508 ii. 284,492,334 + 59,300,456 +321,211,692 92,263,964 605,704,026 Addition Subraction

CSS Middle Standard “Mathematics” 22 iii. 256,321,413 +222,111,681 478,433,094 Q 5: Subtract the following. i. 4,690,882 ii. 284,591,621 iii. 4,580,198 –151,302 – 103,381,111 – 541,698 4,539,580 181,210,510 4,038,500 Q 6: Prove the commutative law and associative law w.r.t to addition in the following questions. i. 349, 7895 349+7895=7895+349 Ans: 8244=8244 ii. 10050, 35965 10050+35965=35965+10050 Ans: 46015=46015 iii. 285, 920, 1089 (285+920)+1089=285+(920+1089) 1205+1089=285+2009 Ans: 2294=2294 iv. 2132, 8931, 6754 (2132+8931)+6754=2134+(8931+6754) 11063+6754=2134+1585 Ans: 17819=17819 v. 7825, 8123, 9250 (7825+8123)+9250=7825+(8123+9250) 15948+9250=7825+17373 Ans: 25198=25198 Q 7: Find the total expenditures of a publishing company per annum. If the salaries of the labour is Rs. 7,950,438 per annum and the cost of maintance of the press is Rs. 380,540 per annum 7,950,438 +380,540 8,330,978 Q 8: Find is the increase in the population of Pakistan during 1998.census and 2017 census. The population of Pakistan in 1998 was 132352279 and in 2017 was 207774550. Ans: 207,774,550 – 132,352,279 75,422,271 Lesson # 3 Procedure: Addition, subtraction and division can be easily comprehended by senses. But multiplication is difficult. Addition and subtraction and their properties can easily be proved by identical objects such as tennis balls. Division is also easily exemplified by dividing candies among children but multiplication is a bit difficult to demonstrate with ordinary objects. This difficulty can be surpassed by the following Procedure: Ask one of the students to count the numbers of rows and columns in which they are seated (say 6 and 4). Ask him to write the numbers on the board. Now ask another student to count the total number of the students in the classroom. Finally write the equation “6 4 = 24”. Introduce a bubble map on the board and encourage the students to ttell what they know about multiplication. Invite the students for book reading. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should be: Multiplication

CSS Middle Standard “Mathematics” 23 Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (2c) Q 1: Fill in the blanks with suitable words. i. 14 × (5 × 60) = (14 × ______) × 60 Ans: 5 ii. ______ × (300 + 4) = ( _____ × 300) + (5 × 4) Ans: 5, 5 iii. 30 × (40 + 10) = (30 × 40) + ______ Ans: (30×10) iv. 115 × (7 + 21) = (115 × _____) + (115 × 21) Ans: 7 v. 13 × (50 + 30) = (_____) + (13 × 30) Ans: (13×50) Q.2 Multiply the following. i. 44 × 173 ii. 1842 × 22 1 7 3 1 8 4 2 × 4 4 × 2 2 6 9 2 3 6 8 4 6 9 2 0 3 6 8 4 0 7 6 1 2 4 0 5 2 4 iii. 17432× 100 1 7 4 3 2 × 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7 4 3 2 0 0 1 7 4 3 2 0 0 iv. 1872× 110×2 Step (i) 1 1 0 × 2 2 2 0 Step (ii) 1872×220 1 8 7 2 × 2 2 0 0 0 0 0 3 7 4 4 0 3 7 4 4 0 0 4 1 1 8 4 0 So 1870×110×2 = 411840 v. 3432 × 4440 × 110 Step (i)4440 × 110 4 4 4 0 × 1 1 0 0 0 0 0 4 4 4 0 0 4 4 4 0 0 0 4 8 8 4 0 0 vi. 1110043 × 4320 1 1 1 0 0 4 3 × 4 3 2 0 0 0 0 0 0 0 0 2 2 2 0 0 8 6 0 3 3 3 0 1 2 9 0 0 4 4 4 0 1 7 2 0 0 0 4 7 9 5 3 8 5 7 6 0 Step (ii) 3432 × 488400 4 8 8 4 0 0 × 3 4 3 2 9 7 6 8 0 0 1 4 6 5 2 0 0 0 1 9 5 3 6 0 0 0 0 1 4 6 5 2 0 0 0 0 0 1 6 7 6 1 8 8 8 0 0 So 3432 × 4440 × 110 = 1676188800

CSS Middle Standard “Mathematics” 24 vii. 8432 × 10 × 230 Step (i)10 × 230 2 3 0 × 1 0 0 0 0 2 3 0 0 2 3 0 0 Step (ii)8432 × 2300 8 4 3 2 × 2 3 0 0 0 0 0 0 0 0 0 0 0 2 5 2 9 6 0 0 1 6 8 6 4 0 0 0 1 9 3 9 3 6 0 0 So 8432 × 10 × 230 = 19393600 Q.3 Solve the following. i. 15,545 ÷ 5 3 1 0 9 5 1 5 5 4 5 –1 5 5 – 5 4 5 – 4 5 0 Remainder Hence, 15545 ÷ 5 = 3109 Quotient ii. 387616 ÷ 16 2 4 2 2 6 16 3 8 7 6 1 6 –3 2 6 7 – 6 4 3 6 – 3 2 4 1 – 3 2 9 6 – 9 6 0 Hence, 387616 ÷ 16 = 24226 Quotient iii. 695450 ÷ 350 1 9 8 7 350 6 9 5 4 5 0 –3 5 0 3 4 5 4 – 3 1 5 0 3 0 4 5 – 2 8 0 0 2 4 5 0 – 2 4 5 0 0 Hence, 695450 ÷ 350 = 1987 Quotient iv. 133427 ÷ 389 Ans: 3 4 3 389 1 3 3 4 2 7 –1 1 6 7 1 6 7 2 – 1 5 5 6 1 1 6 7 – 1 1 6 7 0 Hence, 133427 ÷ 389 = 343 Quotient Q.4 Prove the Commutative law under multiplication for the pair of whole number given below. i. a = 1777,b = 65 Ans: L.H.S. a × b = 1777 × 65 = 115505 R.H.S. b × a = 65 × 1777 = 115505 Hence, a × b = b × a ii. a = 6767, b = 12 Ans: L.H.S. a × b = 6767 × 12 = 80724 R.H.S. b × a = 12 × 6767 = 80724 Hence, a × b = b × a iii. a = 7777, b = 234 Ans: L.H.S. a × b = 7777 × 234 = 1819818 R.H.S. b × a = 234 × 7777 = 1819818 Hence, a × b = b × a iv. 2505,19423 Ans: a×b = b × a L.H.S. a × b = 2505 × 19423 = 48654615 R.H.S. b × a =19423 × 2505 = 48654615 Hence, 2505 × 19423 = 19423 × 2505 v. 8320,4302 Ans: a × b = b × a

CSS Middle Standard “Mathematics” 25 L.H.S. a × b = 8320 × 4302 = 35792640 R.H.S. b × a = 4302 × 8320 = 35792640 Hence, 8320 × 4302 = 4302 × 8320 Q.5 Verify associative law under multiplication in the following whole numbers. i. 110,160,300 Ans: a × (b× c) = (a × b) × c L.H.S. 110 × (160 × 300) = 5280000 R.H.S. (110 × 160) × 300 = 5280000 Hence, 110 × (160 × 300) = (110 × 160) × 300 ii. 141,213,10 Ans: a × (b× c) = (a × b) × c L.H.S. 141 × (213 × 10) = 300330 R.H.S. (141 × 213) × 10 = 300330 Hence, 141 × (213 × 10) = (141 × 213) × 10 iii. 21,2300, 404 Ans: a × (b× c) = (a × b) × c L.H.S. 21 × (2300 × 404) = 19513200 R.H.S. (21 × 2300) × 404 = 19513200 Hence, 21 × (2300 × 404) = (21 × 2300) × 404 iv. 191, 166, 511 Ans: a × (b× c) = (a × b) × c L.H.S. 191 × (166 × 511) = 16201766 R.H.S. (191 × 166) × 511 = 16201766 Hence, 191 × (166 × 511) = (191 × 166) × 511 v. 155,1341,40 Ans: a × (b× c) = (a × b) × c L.H.S. 155× (1341 × 40) = 8314200 R.H.S. (155 × 1341) × 40 = 8314200 Hence, 155× (1341 × 40) = (155 × 1341) × 40 vi. 2011, 800, 244 Ans: a × (b× c) = (a × b) × c L.H.S. 2011 × (800 × 244) = 392547200 R.H.S. (2011 × 800) × 244 = 392547200 Hence, 2011 × (800 × 244) = (2011 × 800) × 244 Q.6 Prove the distributive law of multiplication over addition in the following: i. 24, 17, 10 Ans: a × (b+ c) = (a × b) + (a × c) L.H.S. 24 (17 + 10) = 24 (27) = 648 R.H.S. 24 (17) + 24 (10) = 408 + 240 = 648 Hence, 24 (17 + 10) = 24 (17) + 24 (10) ii. 35, 42, 70 Ans: a × (b+ c) = (a × b) + (a × c) L.H.S. 35 (42 + 70) = 35 (112) = 3920 R.H.S. 35 (42) + 35 (70) = 1470 + 2450 = 3920 Hence, 35 (42 + 70) = 35 (42) + 35 (70) iii. 74, 100, 400 Ans: a × (b+ c) = (a × b) + (a × c) L.H.S. 74 (100 + 400) = 74 (500) = 37000 R.H.S. 74(100) + 74(400) = 7400+29600 = 37000 Hence, 74 (100 + 400) = 74 (100) + 74 (400) iv. 555, 621, 843 Ans: a × (b+ c) = (a × b) + (a × c) L.H.S. 555 (621 + 843) = 555 (1464) = 812520 R.H.S. 555 (621) + 555 (843) = 344655 + 467865 = 812520 Hence, 555 (621 + 843) = 555 (621) + 555 (843) v. 423, 700, 100 Ans: a × (b+ c) = (a × b) + (a × c) L.H.S. 423 (700 + 100) = 423 (800) = 338400 R.H.S. 423 (700) + 423 (100) = 296100 + 42300 = 338400 Hence, 423 (700 + 100) = 423 (700) + 423 (100) vi. 810, 7341, 100 Ans: a × (b+ c) = (a × b) + (a × c) L.H.S. 810 (7341 + 100) = 810 (7441) = 6027210 R.H.S. 810 (7341) + 810 (100) = 5946210 + 81000 = 6027210 Hence,810 (7341 + 100) = 810 (7341) + 810 (100) Q.7 Prove the distribution law of multiplication over subtraction in the following. i. 15, 13, 11 Ans: a × (b- c) = (a × b) - (a × c) L.H.S. 15 (13 – 11) = 15 (2) = 30 R.H.S. 15 (13) + 15 (11) = 195 – 165 = 30 Hence, L.H.S. = R.H.S. ii. 74, 44, 33 Ans: a × (b- c) = (a × b) - (a × c) L.H.S. 74 (44 – 33) = 74 (11) = 814 R.H.S. 74 (44) – 74 (33) = 3256 – 2442 = 814

CSS Middle Standard “Mathematics” 26 Hence, L.H.S. = R.H.S. iii. 100, 80, 17 Ans: a × (b- c) = (a × b) - (a × c) L.H.S. 100 (80 – 17) = 100 (63) = 6300 R.H.S. 100 (80) – 100 (17) = 8000 – 1700 = 6300 Hence, L.H.S. = R.H.S. iv. 378, 112, 100 Ans: a × (b- c) = (a × b) - (a × c) L.H.S. 378 (112 – 100) = 378 (12) = 4536 R.H.S. 378 (112) – 378 (100) = 42336 – 37800 = 4536 Hence, L.H.S. = R.H.S. v. 500, 675, 555 Ans: a × (b- c) = (a × b) - (a × c) L.H.S. 500 (675 – 555) = 500 (120) = 60000 R.H.S. 500 (675) – 500 (555) = 337500 – 277500 = 60000 Hence, L.H.S. = R.H.S. vi. 7341, 2340, 1132 Ans: a × (b- c) = (a × b) - (a × c) L.H.S. 7341 (2340 – 1132) = 7341 (1208) = 8867928 R.H.S. 7341 (2340) – 7341 (1132) = 17177940 – 8310012 = 8867928 Hence, L.H.S. = R.H.S. Q.8 The price of a mobile set is Rs. 92,500. What is the price of 312 such set of mobiles? Ans: Price of 1 mobile set = Rs. 92500/- Price of312 mobile sets = 312 × 92500 = Rs. 28860000/- 9 2 5 0 0 × 3 1 2 1 8 5 0 0 0 9 2 5 0 0 0 2 7 7 5 0 0 0 0 2 8 8 6 0 0 0 0 Q.9 The price of 380 bags of rices is Rs. 570,000. Find the price of one bag of rice. Ans: Price of 380 bags of rice = Rs. 570000/- Price of one bag of rice = 570000 380 = Rs. 1500/- 3 8 0 1 5 0 0 380 5 7 0 0 0 0 1 9 0 0 1 9 0 0 0 Review Exercise 2 Q 1: Choose the correct answer and fill the circle: i. If 20 ÷ 10 then the quotient is: zero 10 2 1 ii. “1” in whole numbers is called: additive identity inverse identity multiplicative identity subtraction iii. “0” in whole number is called: additive identity multiplicative identity inverse identity none of them iv. Name of property 4 + 3 = 3 + 4 is: commutative property of addition distributive property of addition associative property of addition none of these

CSS Middle Standard “Mathematics” 27 v. 311420 × 1 = ________: 311420 1 0 22 vi. 27070 + 0 = ________: 0 35 27070 1 vii. 10010 – 10000 = ________: 0 1 7 10 viii. “0” is a/an _______ number: natural numbers whole number odd numbers none of them ix. Natural numbers start with _________: 0 –1 1 000 Q 2: Complete the following. i. 6 × (5 – 2) = (___ × ___) – (6 × 2) Ans: 6, 5 ii. 27 + 330 = ____ Ans: 357 iii. 1 ×0 = _____ 100 Ans: 0 iv. 88 × 10 =____ Ans: 880 v. (15 × 31) × 4 = 15 × (___ × ___) Ans: 31, 4 vi. 100 + 0 = ____ Ans: 100 vii. 11 × (70 + 41) = (11 × ____) + (___ × ___) Ans: 70, 11, 41 viii. 44701 + (3424 + 2340) =____ Ans: 50465 ix. 150 × ____ = 150 Ans: 1 x. 150 = 150 Ans: 1 Q 3: Name and satisfy the following properties. i. (15 × 13) × 14 = 15 × ( 13 × 14 ) = Ans: Associative property of multiplication ii. 27 × 33 = 33 × 27 = Ans: Commutative property of multiplication iii. 11 + (70 + 41 ) = ( 11 + 70 ) + 41 = Ans: Associative property of addition Q 4: Write “T” for true and “F” for false in the following statements. i. 17 × 44 = 748 Ans: T ii. 14 + 81 + 31 = 125 Ans: F iii. 177 – 44 – 22 = 222 Ans: F iv. 4840 = 4840 1 Ans: T v. 44 – 0 = 44 Ans: T vi. 44,55,66,are all even numbers Ans: F Q 5: Do that following sums. i. 935, 695, 321 + 295, 481, 416 + 125, 328 Billion H-M T-M Million H-T T-T Thousands Hundreds Tens Ones 9 3 5 6 9 5 3 2 1 2 9 5 4 8 1 4 1 6 + 1 2 5 3 2 8 1 2 3 1 3 0 2 0 6 5 935, 695, 321 + 295, 481, 416 + 125, 328 = 1, 231, 302, 065 ii. 956, 442, 530 – 879, 532, 200 H-M T-M Million H-T T-T Thousands Hundreds Tens Ones 9 5 6 4 4 2 5 3 0 – 8 7 9 5 3 2 2 0 0 7 6 9 1 0 3 3 0 956, 442, 530 – 879, 532, 200 = 76, 910, 330

CSS Middle Standard “Mathematics” 28 Q.6 Solve the following. i. 3 8 2, 5 9 5 × 5 4 1 5 3 0 3 8 0 1 9 1 2 9 7 5 0 2 0 6 6 0 1 3 0 So, 382595 × 54 = 20660130 ii. 1 2 2, 5 0 0 × 1 2 5 6 1 2 5 0 0 2 4 5 0 0 0 0 1 2 2 5 0 0 0 0 1 5 3 1 2 5 0 0 So, 122500 × 125 = 15312500 Q.7 Solve the following. i. 36384 ÷ 96 ii. 1769768 ÷ 364 3 7 9 96 3 6 3 8 4 – 2 8 8 7 5 8 – 6 7 2 8 6 4 – 8 6 4 0 4 8 6 2 364 1 7 6 9 7 6 8 –1 4 5 6 3 1 3 7 – 2 9 1 2 2 2 5 6 – 2 1 8 4 7 2 8 – 7 2 8 0 Hence, 36384 ÷ 96 = 379 Hence, 1769768 ÷ 364 = 4862 Quotient Quotient Q.8 Public library has 73000 books in it .Due to some weather disaster 18340 books were damaged in the library and 2300 were shifted to other branch of the library. How many books left in public library? Ans: Number of public library books T = 73000 Number of damage books D = 18340 Number of books shifted = s = 2300 Number of books left in the library L = ? L = T – D – S = 73000 – 18340 – 2300 = 52360 Q.9 If the cost of a paint bucket is Rs. 700. For a house we need 300 bucket of paint. How much it will cost to paint the house? Ans: Cost of 1 bucket of paint = Rs. 700 = C Total cost of 300 buckets of paint = 300 × Rs. 700 = Rs. 21000/- Q.10 Anoral, Ibrahim and Alishaba got Rs. 18,000. They have to distribute it equally, what will be the share of Anoral? Ans: Total Amount = Rs. 18000 Anoral’s share = 18000 3 = Rs. 6000/- Unit No. 3 Factors and Multiples Lesson # 1 Teaching Objectives: To revise the concept of factor and its definition. To revise the concept of multiple and its definition. To distinguish between even and odd numbers. To distinguish between prime and composite numbers.

CSS Middle Standard “Mathematics” 29 To revise that 1 is neither prime nor composite but a factor of every number and 2 is the only even prime number. To revise the divisibility tests for factors from 2 to 25. To revise prime factorization method and index notation. To revise the concept of HCF and methods to find it. To revise the concept of LCM and methods to find it. To create examples from daily life that describe the applications of HCF and LCM. Learning Outcomes: Students should be able to: Define a factor as a number which divides the dividend completely leaving no reminder. Define a multiple as a dividend into which a factor can divide. Define even numbers as the numbers which are multiples of 2. Define odd numbers as the numbers which are not multiples of 2. Define prime numbers as numbers which have only two Factors (i.e., 1 and itself) Define composite numbers as numbers which have more than two factors. Know that 1 is neither prime nor composite as it has only one factor which is 1 itself. Know that 1 is a factor of every number. Know that 2 is the only even prime number where as all other prime number are odd. Tests by inspection whether the number 2, 3, 4, 5, 6,8, 9, 10, 11, 12, 15, and 25 candivide a given number. Define prime factorization as the process of factorizing a number into its prime factors. Recognize index notation. Factorize a given number and express its factor in the index notation. Define HCF as the greatest number which is a common factor of two or more numbers. Find HCF of two or more than two numbers by prime factorization long division method Define LCM as the smallest number which is a common multiple of two or more numbers. Find LCM of two or more numbers by: prime factorization division method Solve real life problems related to HCF and LCM. Teaching Materials: ● CSS Middle Standard Mathematics Book 6. ● Writing Board ●Marker ● Eraser. Procedure: Use divisibility rules when dividing given numbers. Differentiate between prime and composite numbers. Identify and find factors and multiples of a number. Find the HCF using short division, and long division. Find the LCM of 3 numbers by prime factorization and the short division method. Procedure: Factor and Multiple: In class 5 students were discussed the concept of factor, multiple, even number odd number prime and composite number, HCF and LCM, now in this unit we will discuss all above topic in detail. At the start of lesson revise all basic concept by conducting a quiz or have a class discussion or carry out some of the suggested activities. Factor is a number which divides the dividend completely having no remainder. For example the Factors of 18 are 1, 2, 3, 6, 9 and 18. Assign some number to students to find their factors. Multiple is the product of a number with the natural numbers 1, 2, 3, ….. For example the multiple of 8 are 8, 16, 24, 32,…..Bring some pencil and asked the student to make them the multiple of 15. Even and Odd Numbers The number which is multiple of 2 is called even number while the number which is not multiple of 2 is called odd number. List the numbers from 1 to 100 and ask the students to separate the even and odd numbers. Even Numbers: 2, 4, 6, 8, 10, … Odd Numbers: 1, 3, 5, 7, 9, … Check wheather the given number is even or odd by method explained on page#31. Prime and Composite Numbers Introduce Prime Numbers: 2, 3, 5, 7, 11, 13, and so on. A prime

CSS Middle Standard “Mathematics” 30 number is a natural number greater than 1 that can be divided only by the number itself and 1. All other numbers are composite numbers. 17 is a prime number because it has no factors other than 1 and itself. 17 ÷ 1 = 17 and 17 ÷ 17 = 1 Prime Numbers: 2, 3, 5, 7, 11, 13, 17 Composite Numbers: 4, 6, 8, 10, 12, 14, 16, 18 Here are the prime numbers below 100, which can be shown on the 1 to 100 number charts. 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. Prime and Composite Numbers: Discuss the meaning of prime and composite numbers and the importance of prime numbers. Explain the positioning of the prime and composite numbers in a 1 to 100 number square. Tell them that: No two prime numbers, other than 2 and 3, are consecutive. No prime number, other than 2, has an even number in its unit digit. No prime number, other than 5, has 5 or 0 in its unit digit. No prime number has the sum of its digits which is divisible by 3 or multiples of 3. No prime number has difference between the sums of alternate digits as 11, or a multiple of 11. The smallest prime number is 2. The smallest composite number is 4. Activity: Make the factors of the given numbers and tell whether they are prime or compsite number. Number Factor Composit or Prime number 10 11 12 13 14 15 16 17 18 19 Invite the class for the book reading. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should beL Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (3a) Q 1: Put all the prime numbers in a square and all the composite number in a circle. i. 11, 190, 37, 6, 9342, 8810, 420, 3301, 15123, 5, 6140, 49734, 173, 17, 107, 237, 250, 139, 197.

CSS Middle Standard “Mathematics” 31 Q 2: Write the factors of the following numbers. i. 4 Ans: 1, 2, 4 ii. 17 Ans: 1, 17 iii. 39 Ans: 1, 3, 13, 39 iv. 12 Ans: 1, 2, 3, 4, 6, 12 v. 15 Ans: 1, 3, 5, 15 vi. 18 Ans: 1, 2, 3, 6, 9, 18 vii. 24 Ans: 1, 2, 3, 4, 6, 8, 12, 24 viii. 48 Ans: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 ix. 100 Ans: 1, 2, 4, 5, 10, 20, 25, 50, 100 x. 120 Ans: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 xi. 146 Ans: 1, 2, 73, 146 xii. 250 Ans: 1, 2, 5, 10, 25, 50, 125, 250 xiii. 400 Ans: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400 xiv. 550 Ans: 1, 2, 5, 10, 11, 22, 25, 50, 55, 110, 275, 550 Q 3: Write the first five multiple for each number. i. 14 Ans: 14, 28, 42, 56, 70 ii. 22 Ans: 22, 44, 66, 88, 110 iii. 43 Ans: 43, 86, 129, 172, 215 iv. 82 Ans: 82, 164, 246, 328, 410 v. 10 Ans: 10, 20, 30, 40, 50 vi. 38 Ans: 38, 76, 114, 152, 190 vii. 93 Ans: 93, 186, 279, 372, 465 viii. 73 Ans: 73, 146, 219, 292, 365 Q 4: Write all the even and odd numbers separately. i. 13 Ans: odd ii. 76 Ans: even iii. 135 Ans: odd iv. 97 Ans: odd v. 400 Ans: even vi. 340 Ans: even vii. 700 Ans: even viii. 1057 Ans: odd ix. 3230 Ans: even x. 137843 Ans: odd Q 5: Write even numbers between 10 and 100. Ans: 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100. Q 6: Write odd numbers between 1400 and 1500. Ans: 1401, 1403, 1405, 1407, 1409, 1411, 1413, 1415, 1417, 1419, 1421, 1423, 1425, 1427, 1429, 1431, 1433, 1435, 1437, 1439, 1441, 1443, 1445, 1447, 1449, 1451, 1453, 1455, 1457, 1459, 1461, 1463, 1465, 1467, 1469, 1471, 1473, 1475, 1477, 1479, 1481, 1483, 1485, 1487, 1489, 1491, 1493, 1495, 1497, 1499. Q 7: Write prime numbers between 1 and 50. Ans: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 Q 8: Why 1 is neither prime nor composite? Ans: A natural number (i.e. 1, 2, 3, 4, 5, etc.) is called a prime number (or a prime) if it has exactly two positive factors, 1 and the number itself. Natural numbers that have more than two positive factors are called composite. 1 has only one positive factor i.e. 1 only. Hence, 1 is neither prime nor composite. Q 9: Write composite numbers between 1 and 75. Ans: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74. Lesson # 2 Procedure: Ask the students to describe the process of division and its utilization in life. Tell the students that there are some rules by which we can check that the given numbers are divisible by 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, and 25. These rules are called test for divisibility. Put the chart of natural numbers between 1 and 1000 in class and use the explanation as given about the method of test for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, and 25 on page# 39, 40, 41, 42, 43, 44, and 45 to check their divisibility. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5

CSS Middle Standard “Mathematics” 32 questions. Each question further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (3b) Q 1: Which of the following numbers are divisible by 2,3,4,5 using divisibility test. i. 786010 Ans: Number 786010 is divisible by 2 and 5 only. ii. 154262 Ans: Number 154262 is divisible by 2 only. iii. 104800 Ans: Number 104800 is divisible by 2, 4 and 5 only. iv. 960734 Ans: Number 960734 is divisible by 2 only. v. 777080 Ans: Number 777080 is divisible by 2, 4 and 5 only. vi. 542175 Ans: Number 542175 is divisible by 3 and 5 only. vii. 1296010 Ans: Number 1296010 is divisible by 2 and 5 only. viii. 477333 Ans: Number 477333 is divisible by 3 only. Q 2: Using the divisibility test, find out which of the following numbers are divisible by 6, 8, 9 and 10. i. 540 Ans: Number 540 is divisible by 6, 9 and 10 only. ii. 2340 Ans: Number 2340 is divisible by 6, 9 and 10 only. iii. 3040 Ans: Number 3040 is divisible by 8 and 10 only. iv. 7896 Ans: Number 7896 is divisible by 6 and 8 only. v. 17460 Ans: Number 17460 is divisible by 6, 9 and 10 only. vi. 15008 Ans: Number 15008 is divisible by 8 only. vii. 9370 Ans: Number 9370 is divisible by 10 only. viii. 55008 Ans: Number 55008 is divisible by 6, 8 and 9 only. ix. 104800 Ans: Number 104800 is divisible by 8 and 10 only. x. 13830 Ans: Number 13830 is divisible by 6 and 10 only. xi. 414 Ans: Number 414 is divisible by 6 and 9 only. Q 3: Using the tests of divisibility, find out which of the following numbers are divisible by 11,12,15,25. i. 478300 Ans: Number 478300 is divisible by 25 only. ii. 384730 Ans: Number 384730 is divisible by none. iii. 709500 Ans: Number 709500 is divisible by all 11, 12,15 and 25. iv. 3107556 Ans: Number 3107556 is divisible by 12 only. v. 32820 Ans: Number 32820 is divisible by 12 and 15 only. vi. 406857 Ans: Number 406857 is divisible by 11 only. vii. 5293440 Ans: Number 5293440 is divisible by 12 and 15 only. viii. 100010 Ans: Number 100010 is divisible by none. ix. 252525 Ans: Number 252525 is divisible by 15 and 25 only. x. 2838 Ans: Number 2838 is divisible by 11 only. xi. 3104316 Ans: Number 3104316 is divisible by 12 only. xii. 450000 Ans: Number 450000 is divisible by 12, 15 and 25 only. Q 4: Using the test of divisibility answer of the following. i. Is 8775 divisible by 15? Ans: Yes ii. Is 1300010 divisible by 10? Ans: Yes iii. Is 4128 divisible by 12? Ans: Yes iv. Is 8235 divisible by 8? Ans: No v. Is 97250 divisible by 25? Ans: Yes vi. Is 155376 divisible by 6? Ans: Yes

CSS Middle Standard “Mathematics” 33 vii. Is 8884351 divisible by 10? Ans: No viii. Is 25896 divisible by 3? Ans: Yes ix. Is 29520 divisible by 3? Ans: Yes x. Is 2536987 divisible by 2? Ans: No Lesson # 3 Procedure: Begin by asking about factors. Explain the Prime Factorization method to find the given number by factor tree and repeated division method. Express the prime factorization in the index notation as explained on page # 46, 47, and 48 Tell them that Twin primes are the prime numbers differ by 2 for example 3and 5, 5 and 7 etc. List the Twin number between 50 to 80. Invite the students for book reading. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (3c) Q.1 Write prime factorization of given numbers with factor tree. i. 60 60 2 30 2 2 15 2 2 3 5 So, prime factors are of 60 = 2235 ii. 400 400 2 200 2 2 100 2 50 2 25 2 2 2 2 5 5 So, prime factors of 400 = are 222255 iii. 750

CSS Middle Standard “Mathematics” 34 750 2 375 2 3 125 2 3 5 25 2 3 5 5 5 So, prime factors of 750= 23555 iv. 800 800 2 400 2 2 200 2 100 2 50 2 2 2 25 2 2 2 2 2 5 5 So, prime factors of 80 = 222255 v. 3423 3423 3 1141 3 7 163 So, prime factors of 3423 = 37163 vi. 84368 84368 2 42184 2 21092 2 10546 2 2 2 2 5273 So, prime factors of 84368 = 22225273 vii. 7345 7345 5 1469 5 13 113 So, prime factors of 7345 = 513113

CSS Middle Standard “Mathematics” 35 Q.2 Find the prime factorization of the given numbers by repeated division method. i. 230400 2 230400 2 115200 2 57600 2 28800 2 14400 2 7200 2 3600 2 1800 2 900 2 450 5 225 5 45 3 9 3 3 1 So, 230400 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3 ii. 64878 2 64878 3 32439 11 10813 983 So, 64878 = 2 × 3 × 11 × 983 iii. 31515 5 31515 5 10505 11 2101 191 191 1 So, 31515 = 3 × 5× 11 × 191 iv. 36005 5 36005 19 7201 379 379 1 So, 36005 = 5 × 19 × 379 v. 705642 2 705642 3 352821 7 117607 53 16801 317 317 1 So, 705642 = 2 × 3 × 7 × 53 × 317 vi. 136004 2 136004 2 68002 11 34001 11 3091 281 281 1 So, 136004 = 2 × 2 × 11 × 11 × 281 vii. 756 2 756 2 378 3 189 3 63 3 21 7 7 1 So, 756 = 2 × 2 × 3 × 3 × 3 × 7 viii. 333 3 333 3 111 37 37 1 So, 333 = 3 × 3 × 37 ix. 42894 2 42894 3 21447 3 7149 2383 2383 1 So, 42894 = 2 × 3 × 3 × 2383

CSS Middle Standard “Mathematics” 36 x. 3000 2 3000 2 1500 2 750 3 375 5 125 5 25 5 5 1 So, 3000 = 2 × 2 × 2 × 3 × 5 × 5 × 5 xi. 1935 3 1935 3 645 5 215 43 43 1 So, 1935 = 3 × 3 × 5 × 43 Q 3: Write in index notation. i. 2 × 2 × 3 × 3 × 3 × 5 Ans: 2 2× 33× 51 ii. 2 × 5 × 5 × 5 × 8 × 8 Ans: 2 1× 53× 82 iii. 3 × 4 × 4 × 4 × 4 × 5 × 5 × 5 × 7 × 7 Ans: 3 1× 44× 53×72 Q.4 Factorize the given numbers and express their factors in the index notation. i. 756 2 756 2 378 3 189 3 63 3 21 7 7 1 So, 756 = 2 × 2 × 3 × 3 × 3 × 7= 22 × 33 × 71 ii. 630 2 630 3 315 3 105 5 35 7 7 1 So, 630 = 2 × 3 × 3 × 5 × 7= 21 × 32 × 51 × 71 iii. 665 5 665 7 133 19 19 1 So, 665 = 5 × 7 × 19= 51 × 71 × 191 iv. 840 2 840 2 420 2 210 3 105 5 35 7 7 1 So, 840 = 2 × 2 × 2 × 3 × 5 × 7 = 23 × 31 × 51 × 71 v. 1159 19 1159 61 61 1 So, 1159 = 19 × 61 = 191 × 611 vi. 1225 5 1225 5 245 7 49 7 7 1 So, 1225 = 5 × 5 × 7 × 7 = 52 × 72 vii. 1482 2 1482 3 741 13 247 19 19 1 So, 1482 = 2 × 3 × 13 × 19 = 21 × 31× 131 × 191 viii. 3996 2 3996 2 1998 3 999 3 333 3 111 37 37 1 So, 3996 = 2 × 2 × 3 × 3 × 3 × 37 = 22 × 33× 371

CSS Middle Standard “Mathematics” 37 ix. 9090 2 9090 3 4545 3 1515 5 505 101 101 1 So, 9090 = 2 × 3 × 3 × 5 × 101= 21 × 32× 51 × 1011 x. 8700 2 8700 2 4350 3 2175 5 725 5 145 29 29 1 So, 8700 = 2 × 2 × 3 × 5 × 5 × 29 = 22 × 31× 52 × 291 Lesson # 4 Procedure: Begin by writing the word HCF on the board and ask the students about it. Put the chart of prime numbers between 1 and 1000 in class. Use the explanation as given on pages # 49, 50, 51, and 52 to calculate the H.C.F of two or more than two, 2-digits number. Explain the Prime Factorization method to find the H.C.F of given numbers as explained on page # 50, 51. Explain the Long Division method to find the H.C.F of given numbers as explained on page # 51 & 52. Invite the students for book reading. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part Iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (3d) Q.1 Find H.C.F by Prime factorization method. i. 54, 99 Prime factors of 54 = 2 × 33 , Prime factors 99 = 32 × 11 Common factors of 54 and 99 = 3 × 3 = 9 Hence the highest common factor (HCF) = 9 ii. 25, 100 Prime factors of 25 = 5 × 5, Prime factors 100 = 2 × 2 × 5 × 5 2 54 3 27 3 9 3 3 1 3 99 3 33 11 11 1 5 25 5 5 1 2 100 2 50 5 25 5 5 1

CSS Middle Standard “Mathematics” 38 Common factors of 25 and 100 = 5 × 5 = 25 HCF of 25 and 100 = 25 iii. 44, 88, 176 Ans: Prime factors of 44 = 2 × 2 × 11, Prime factors 88 = 2 × 2 × 2 × 11 Prime factors of 176 = 2 × 2 × 2 × 2 × 11 Common factors of 44, 88 and 176 = 2 × 2 × 11 = 44 HCF of 44, 88 and 176 = 44 iv. 184, 230, 276 Prime factors of 184 = 2 × 2 × 2 × 23 Prime factors of 230 = 2 × 5 × 23 Prime factors of 276 = 2 × 2 × 3 × 23 Common factors of 184, 230 and 276 = 2 × 23 = 46 HCF of 189, 230 and 276 = 46 v. 22, 77, 55, 99 Prime factors of 22 = 2 × 11, Prime factors of 77 = 7 × 11 Prime factors of 55 = 5 × 11, Prime factors of 99 = 3 × 3 × 11 Common factors of 22, 77, 55 and 99 = 11 HCF of 22, 77, 55 and 99 = 11 Q.2 Find the H.C.F of the following numbers by long division method. i. 180, 270 Ans: 1 180 270 180 2 90 180 180 0 Therefore, HFC of 180 and 270 is 90. ii. 852, 1065 1 852 1065 852 4 213 852 852 0 Therefore, HCF of 852 and 1065 is 213. 2 184 2 92 2 46 23 23 1 2 230 5 115 23 23 1 2 276 2 138 3 69 23 23 1 2 22 11 11 1 7 77 11 11 1 5 55 11 11 1 3 99 3 33 11 11 1 2 44 2 22 11 11 1 2 88 2 44 2 22 11 11 1 2 176 2 88 2 44 2 22 11 11 1

CSS Middle Standard “Mathematics” 39 iii. 300, 396 1 300 396 300 3 96 300 288 8 12 96 96 0 Therefore, HCF of 300 and 396 is 12. iv. 735, 840, 1050 1 3 840 1050 210 735 840 4 630 2 210 840 105 210 840 210 0 0 Therefore, HCF of 735, 840 and 1050 is 105. v. 11, 77, 300 3 11 77 300 1 11 231 1 11 69 77 0 69 8 8 69 64 1 5 8 5 1 3 5 3 1 2 3 2 2 1 2 2 0 Therefore, HCF of 11, 77 and 300 is 1. vi. 399, 665 and 1463 2 3 665 1463 133 399 1330 5 399 133 665 0 665 0 Therefore, HCF of 399, 665 and 1463 is 133. vii. 44, 132 and 66 2 1 66 132 44 66 132 44 2 0 22 44 44 0 Therefore, HCF of 44, 132 and 66 is 22.

CSS Middle Standard “Mathematics” 40 viii. 36, 180 and 200 Ans: 1 1 180 200 20 36 180 9 20 1 20 180 16 20 180 16 4 0 4 16 16 0 Therefore, HCF of 36, 180 and 200 is 4. ix. 100, 1000, 4850, 245 Ans: 19 10 245 4850 100 1000 4655 1 1000 195 245 0 195 3 50 195 150 1 45 50 45 9 5 45 45 HCF of 245 and 8450 is 5. 0 HCF of 100 and 1000 is 100. Now HCF of 5 and 100 is 5 because: 20 5 100 100 0 Therefore, HCF of 245, 4850, 100 and 1000 is 5. x. 145, 540, 675, 765 1 4 540 765 145 675 540 2 580 1 225 540 95 145 450 2 95 1 90 225 50 95 180 2 50 1 45 90 45 50 90 45 9 0 5 45 45 HCF of 545, 765 is 45. HCF of 145, 675 is 5. 0 Now of HCF of 45 and 5 is 5, because: 9 5 45 45 0 Therefore, HCF of 145, 540, 675 and 765 is 5.

CSS Middle Standard “Mathematics” 41 Q.3 Find the highest number which exactly divides these numbers. i. 24, 240, 304 Prime factors of 24 = 2 × 2 × 2 × 3 Prime factors of 240 = 2 × 2 × 2 × 2 × 3 × 5 Prime factors of 304 = 2 × 2 × 2 × 2 × 19 HCF of 24, 240 and 304 is 23 = 8 ii. 196, 490, 1190 Prime factors of 196 = 2 × 2 × 7 × 7 Prime factors of 490 = 2 × 5 × 7 × 7 Prime factors of 1190 = 2 × 5 × 7 × 17 HCF of 196, 490 and 1190 = 2 × 7 = 14 iii. 147, 217, 3514 Prime factors of 147 = 3 × 7 × 7 Prime factors of 217 = 7 × 31 Prime factors of 3514 = 2 × 7 × 251 HCF of 147, 217 and 3514 = 7 iv. 225, 525, 675 Prime factors of 225 = 5 × 5 × 3 × 3 Prime factors of 525 = 5 × 5 × 3 × 7 Prime factors of 675 = 5 × 5 × 3 × 3 ×3 HCF of 225, 525 and 675 = 3 × 5 × 5 = 75 2 24 2 12 2 6 3 3 1 2 240 2 120 2 60 2 30 3 15 5 5 1 2 304 2 152 2 76 2 38 19 19 1 2 196 2 98 7 49 7 7 1 2 490 5 245 7 49 7 7 1 2 1190 5 595 7 119 17 17 1 7 147 7 21 3 3 1 7 217 31 31 1 2 3514 7 1757 251 251 1 5 225 5 45 3 9 3 3 1 5 525 5 105 3 21 7 7 1 5 675 5 135 3 27 3 9 3 3 1

CSS Middle Standard “Mathematics” 42 Lesson # 5 Procedure: Introduce the word LCM on the board and ask the students about it. Use the explanation as given on page # 53to calculate the L.C.M of two or more than two, 2-digits number. Explain the Prime Factorization method to find the L.C.M of given numbers as explained on page # 53 & 54. Explain the Division method to find the L.C.M of given numbers as explained on page# 54&55. Invite the students for book reading. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part i Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (3e) Q.1 Find the L.C.M of the following numbers by Prime factorization method. Note: LCM = Common factors × non-common factors i. 100, 400 100 = 2 × 2 × 5 × 5 = 22 × 52 400 = 2 × 2 × 2 × 2 × 5 × 5 = 24 × 52 L.C.M = 24 × 52 = 400 ii. 70, 98, 175 70 = 2 × 5 ×7 98 = 2 × 7 × 7 = 2 × 72 175 = 5 × 5 × 7 = 52 × 7 L.C.M = 2 × 52 × 72 = 2450 iii. 8, 12, 18 8 = 2 × 2 × 2 = 23 12 = 2 × 2 × 3 = 22 × 3 18 = 2 × 3 × 3 = 2 × 32 2 100 2 50 5 25 5 5 1 2 400 2 200 2 100 2 50 5 25 5 5 1 2 70 5 35 7 7 1 2 98 7 49 7 7 1 5 175 5 35 7 7 1 2 8 2 4 2 2 1 2 12 2 6 3 3 1 2 18 3 9 3 3 1

CSS Middle Standard “Mathematics” 43 L.C.M = 23 × 32= 8 × 9= 72 iv. 77, 33, 55 77 = 7 × 11 33 = 3 × 11 55 = 5 × 11 L.C.M = 3 × 5 × 7 × 11 = 1155 v. 49, 70, 149 49 = 7 × 7 70 = 2 × 5 × 7 149 = 149 L.C.M = 2 × 5 × 72 × 149 = 73010 vi. 32, 36, 48 32 = 2 × 2 × 2 × 2 × 2 = 25 36 = 2 × 2 × 3 × 3 = 22 × 32 48 = 2 × 2 × 2 × 2 × 3 = 24 × 3 L.C.M = 25 × 32= 288 vii. 320, 480, 720 320 = 2 × 2 × 2 × 2 × 2 × 2× 5 = 26× 5 480 = 2 × 2 × 2 × 2 × 2 × 3 × 5 = 25 × 3 × 5 720 = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 24 × 32 × 5 L.C.M = 26 × 32× 5 = 2880 viii. 21, 28, 35, 77 21 = 3 × 7 28 = 2 × 2 × 7 = 22 × 7 35 = 5 × 7 77 = 7 × 11 L.C.M = 22 × 3 × 5 × 7 × 11 = 4620 Q.2 Find the L.C.M of the following numbers by division Method. 7 77 11 11 1 3 33 11 11 1 5 55 11 11 1 7 49 7 7 1 2 70 5 35 7 7 1 149 149 1 2 32 2 16 2 8 2 4 2 2 1 2 36 2 18 3 9 3 3 1 2 48 2 24 2 12 2 6 3 3 1 2 320 2 160 2 80 2 40 2 20 2 10 5 5 1 2 480 2 240 2 120 2 60 2 30 3 15 5 5 1 2 720 2 360 2 180 2 90 3 45 3 15 5 5 1 3 21 7 7 1 2 28 2 14 7 7 1 7 35 5 5 1 7 77 11 11 1

CSS Middle Standard “Mathematics” 44 i. 20, 24, 45 Ans: 2 × 2 × 2 × 3 × 3 × 5 = 360 Therefore, L.C.M = 360 ii. 12, 15, 18, 21 Ans: L.C.M = 2 × 2 × 3 × 3 × 5 × 7 = 1260 iii. 72, 240, 196 Ans: LCM = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 7 × 7 = 35, 280 iv. 60, 75, 80 Ans: L.C.M = 2 × 2 × 2 × 2 × 3 × 5 × 5 = 1200 v. 16, 24, 30, 36 Ans: LCM = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720 vi. 72, 96, 144, 168 LCM = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 7 = 2016 vii. 75, 125, 175, 225 LCM = 3 × 3 × 5 × 5 × 5 × 7 = 7875 3 75, 125, 175, 225 3 25, 125, 175, 75 5 25, 125, 175, 25 5 5, 25, 35, 5 5 1, 5, 7, 1 7 1, 1, 7, 1 1, 1, 1, 1 2 20, 24, 45 2 10, 12, 45 2 5, 6, 45 3 5, 3, 45 3 5, 1, 15 5 5, 1, 5 1, 1, 1 2 12, 15, 18, 21 2 6, 15, 9, 21 3 3, 15, 9, 21 3 1, 5, 3, 7 5 1, 5, 1, 7 7 1, 1, 1, 7 1, 1, 1, 1 2 72, 240, 196 2 36, 120, 98 2 18, 60, 49 2 9, 30, 49 3 9, 15, 49 3 3, 5, 49 5 1, 5, 49 7 1, 1, 49 7 1, 1, 7 1, 1, 1 2 60, 75, 80 2 30, 75, 40 2 15, 75, 20 2 15, 75, 10 3 15, 75, 5 5 5, 25, 5 5 1, 5, 1 1, 1, 1 2 16, 24, 30, 36 2 8, 12, 15, 18 2 4, 6, 15, 9 2 2, 3, 15, 9 3 1, 3, 15, 9 3 1, 1, 5, 3 5 1, 1, 5, 1 1, 1, 1, 1 2 72, 96, 144, 168 2 36, 48, 72, 84 2 18, 24, 36, 42 2 9, 12, 18, 21 2 9, 6, 9, 21 3 9, 3, 9, 21 3 3, 1, 3, 7 7 1, 1, 1, 7 1, 1, 1, 1

CSS Middle Standard “Mathematics” 45 Lesson # 6 Begin by asking the importance of HCF and LCM in the real life. Tell students that HCF is important to find the largest size of tiles that may fit into rooms, while constructing a building. Any such activity is not merely useful in its mathematical field, but increases the power of reasoning and thinking. Ask do we need to find LCM of numbers? Tell them that Subtraction / Addition of fractions cannot be done accurately without finding the LCM of denominators. LCM is important in everyday life for time and speed, and time and work problems, different people running around circular race tracks, timing of bells and flashing lights, such as from a lighthouse. Invite the class for book reading. Note for the teacher: Focus on all examples given in the book. Always solve few questions on the board by yourself. Then encourage students to solve other questions there. For notebooks work, begin by pair work and finally ending with individual work. Suppose you have an exercise consisting of 5 questions. Each question further consists of 5 parts then your strategy should be: Question No. Done by the teacher Board practice by the students Pair work Individual work Home work 1 Part i Part ii Part iii Part iv Part v Focus on the definitions and ask students to learn the definitions with understanding. Once in a week, homework can be assigned based on the definition given in the chapter / lesson / unit going on in the classroom. Exercise (3f) Q1. Find the greatest number that can completely divide the number 252, 441, 504 and 315. Ans: Prime factors of 252 = 2 × 2 × 3 × 3 × 7 Prime factors of 441 = 3 × 3 × 7 × 7 Prime factors of 504 = 2 × 2 × 2 × 3 × 3 × 7 HCF = 3 × 3 × 7 = 63 Q2. Find the greatest capacity of a measuring cylinder that can exactly measure the liquids of 16 , 175 and 200 . Ans: Amounts of liquids 165 , 175 , 200 LCM = Highest Capacity = 2 × 2 × 2 × 3 × 5 × 5 × 7 × 11 = 46, 200 liters. 2 252 2 126 3 63 3 21 7 7 1 3 441 3 147 7 49 7 7 1 2 504 2 252 2 126 3 63 3 21 7 7 1 2 165, 175, 200 2 165, 175, 100 2 165, 175, 50 3 165, 175, 25 5 55, 175, 25 5 11, 35, 5 7 11, 7, 1 11 11, 1, 1 1, 1, 1

CSS Middle Standard “Mathematics” 46 Q3. Find the least length of a rope which can be cut into whole number of pieces of lengths 45cm, 75cm and 81cm. Ans: Lengths of pieces = 45 cm, 75 cm, 81 cm Least length = LCM Least length = 3 × 3 × 3 × 3 × 5 × 5 = 2025 Q4. Ibrahim art class held 4 days in a month and Quran class held 5 days in a month. What is the next day his both classes will be held if the last common class was held on 20th of April? Ans: To find the next common class, we find LCM of 4 and 5. LCM = 2 × 2 × 5 = 20 So, next common class will be held on 20th April + 20 = 10th may Q5. Find the smallest number that is exactly divisble by 84,78,144,174. Ans: Least length = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 13 × 29 = 380,016 Q6. The lengths of four ropes are, 115m, 125m, 145m, and 175m. Find the length of the road which covers the four ropes completely. Ans: Length of four ropes = 115m, 125m, 145m, 175m. Length of rope that covers completely = LCM Length of rope that covers completely = 5 × 5 × 5× 7 × 23 × 29 = 583, 625 3 45, 75, 81 3 15, 25, 27 3 5, 25, 9 3 5, 25, 3 5 5, 25, 1 5 1, 5, 1 1, 1, 1 2 4, 5 2 2, 5 5 1, 5 1, 1 2 84, 78, 144, 174 2 42, 39, 72, 87 2 21, 39, 36, 87 2 21, 39, 18, 87 3 21, 39, 9, 87 3 7, 13, 3, 29 7 7, 13, 1, 29 13 1, 13, 1, 29 29 1, 1, 1, 29 1, 1, 1, 1 5 115, 125, 145, 175 5 23, 25, 29, 35 5 23, 5, 29, 7 7 23, 1, 29, 7 23 23, 1, 29, 1 29 1, 1, 29, 1 1, 1, 1, 1

CSS Middle Standard “Mathematics” 47 Q7. Find the least number of students exactly in groups of 45, 40 and 60 to participate in the exhibition. Ans: Least number of students = LCM LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360 Review Exercise 3 Q 1: Choose the correct answer and fill the circle: i. A number which divides the ___completely having no remainder is called a ______: dividend, factor dividend, addition dividend, multiples dividend,subtraction ii. The number which in not the multiple of 2 is called a __________: composite number even number odd number prime number iii. The only even prime number is: 4 0 2 10 iv. 13 is a ____________ because it is divisible by it self and _____________: odd number,1 prime number ,1 even number, 0 none v. Highest number which is a common _________of two or more number is HCF: factor greatest multiple all vi. The smallest number which is a common _______of two or more _______ is called LCM: multiple, factor multiple, number factor, number factor, dividend vii. 317 is a _________number: even composite negative prime viii. 191 is a/an _______number, but also a _________number: even, prime even, composite odd, prime none Q 2: i. Without dividing, find the numbers written on the books which are exactly divisible by 3 and 6. Ans: 960, 2460, 3030, 6480, 1740, 8100, 9510, 4290, 6690. ii. Explain how you found the numbers. Ans: We find these numbers by using divisibility test of 3 and 6. iii. How can you tell, without dividing, that a number is exactly divisible by 6? Ans: If the last digit of a number is 0, 2, 4, 6, 8 and sum of the digits is divisible by 3, than that number is exactly divisible by 6. Q 3: Without dividing List the numbers from 430 to 440 which are; i. Exactly divisible by 8 Ans: 432, 440 ii. Not exactly divisible by 6 Ans: 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440 Q 4: Table number card: 32 50 85 70 95 56 62 48 80 37 65 92 (a) List the table of numbers which are: (i) Multiples of 4 (ii) Multiples of 5 (b) List the first ten : (i) Multiple of ..7.. (ii) Multiple of ..9.. (c) Which multiples of 7 in part (b) are exactly divisible by 3? 2 45, 40, 60 2 45, 20, 30 2 45, 10, 15 3 45, 5, 15 3 15, 5, 5 5 5, 5, 5 1, 1, 1

CSS Middle Standard “Mathematics” 48 (d) Which multiples of 9 in part (b) are exactly divisible by 4? Ans: (a) (i) 32, 56, 48, 80, 92 (ii) 50, 85, 70, 95, 80, 65 (b) (i) 7, 14, 21, 8, 35, 42, 49, 56, 63, 70 (ii) 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 (c) 21, 42, 63 (d) 36, 72 Q 5: List the first 20 multiples of 2 & 5. Which is the smallest numbers that is a common multiple of 2 and 5? Ans: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 Q 6: List the Smallest number that is a common multiple of 22, 48, 88. Ans: 2 × 2 × 2 × 2 × 3 × 11 = 528 Q 7. Find the prime factorization of the following: i. 180 Ans: Prime Factors of 180 = 2 × 2× 3 × 3 × 5 22 × 32 × 5 ii. 240 Ans: Prime Factors of 240 = 2 × 2× 2 × 2 × 3 × 5 24 × 3 × 5 iii. 200 Ans: Prime Factors of 200 = 2 × 2× 2 × 5 × 5 23 × 52 iv. 550 Ans: Prime Factors of 550 = 2×5× 5× 11 2 × 52 × 11 v. 770 Ans: Prime Factors of 770 = 2 × 5 × 7 × 11 vi. 1175 Ans: Prime Factors of 1175 = 5× 5× 47 52 × 47 2 22,48,88 2 11,24,44 2 11,12,22 2 11,6,11 3 11,3,11 11 11,1,11 1,1,1 2 180 2 90 3 45 3 15 5 5 1 2 240 2 120 2 60 2 30 3 15 5 5 1 2 200 2 100 2 50 5 25 5 5 1 2 550 5 275 5 55 11 11 1 2 770 5 385 7 77 11 11 1 5 1175 5 235 47 47 1

CSS Middle Standard “Mathematics” 49 vii. 10000 Ans: viii. 10005 Ans: Factor of 10005 = 3 × 5 × 23 × 29 Prime Factors of 10000 = 2× 2× 2× 2× 5× 5× 5× 5 24 × 54 Q 8. Find the H.C.F by Prime factorization. i. 48, 56, 72 Ans: Prime factors of 48 = 2 × 2 × 2 × 2 × 3 Prime factors of 56 = 2 × 2 × 2 × 7 Prime factors of 72 = 2 × 2 × 2 × 3 × 3 HCF = 2 × 2 × 2 = 8 ii. 102, 68, 136 Ans: Prime factors of 102 = 2 × 3 × 17 Prime factors of 68 = 2 × 2 × 17 Prime factors of 136 = 2 × 2 × 2 × 17 HCF = 2 × 17 = 34 iii. 405, 783, 513 Ans: Prime factors of 405 = 3 × 3 × 3 × 3 × 5 Prime factors of 783 = 3 × 3 × 3 × 29 Prime factors of 513 = 3 × 3 × 3 × 19 HCF = 3 × 3 × 3 = 27 2 48 2 24 2 12 2 6 3 3 1 2 56 2 28 2 14 7 7 1 2 72 2 36 2 18 3 9 3 3 1 2 102 3 51 17 17 1 2 68 2 34 17 17 1 2 136 2 68 2 34 17 17 1 3 405 3 135 3 45 3 15 5 5 1 3 783 3 261 3 87 29 29 1 3 513 3 171 3 57 19 19 1 2 10000 2 5000 2 2500 2 1250 5 625 5 125 5 25 5 5 1 3 10005 5 3335 23 667 29 29 1

CSS Middle Standard “Mathematics” 50 iv. 198, 360 Ans: Prime factors of 198 = 2 × 3 × 3 × 11 Prime factors of 360 = 2 × 2 × 2 × 3 × 3 × 5 HCF = 2 × 3 × 3 = 18 v. 1024, 576 Ans: Prime factors of 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 Prime factors of 576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 HCF = 2 × 2 × 2 × 2 × 2 × 2 = 64 Q9. Find the HCF by long division method. i. 84, 144 Ans: 1 84 144 84 1 60 84 60 2 24 60 48 2 12 24 24 HCF = 12 0 ii. 120, 168 1 120 168 120 2 48 120 96 2 24 48 48 0 HCF = 24 2 198 3 99 3 33 11 11 1 2 360 2 180 2 90 3 45 3 15 5 5 1 2 1024 2 512 2 256 2 128 2 64 2 32 2 16 2 8 2 4 2 2 1 2 576 2 288 2 144 2 72 2 36 2 18 3 9 3 3 1