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CLASS VIII 1) Readiness Programme & 2) Academic Year 2020-21 CHAPTER - I : RATIONAL NUMBERS STATE COUNCIL OF EDUCATIONAL RESEARCH & TRAINING, TELANGANA, HYDERABAD. MATHEMATICS WORKSHEETS LEVEL-1 LEVEL-2
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DEVELOPMENT TEAM Chairperson Smt. B. Seshu Kumari, Director, SCERT, Telangana, Hyderabad. Subject In-charges Sri K. Rajender Reddy, Retd. Teacher, Hyderabad. Sri S. Dharmender Singh, ZPHS, Dhannur, Adilabad. Members Sri Nagula Ravi, ZPSS, Beervally, Sarangapur, Nirmal. Sri Penta Ashok, ZPSS, Manjulapur, Nirmal, Adilabad. Sri N.Shekar Verma, ZPSS, Ponkal, Mamada, Nirmal. Sri D.Girish Kumar, G.H.S, Thirpally, Adilabad (Mdl.), Adilabad. Sri D. Sanjay Deshpande, G.H.S, RPL, Adilabad (Mdl.), Adilabad. Sri N. Srinivas, ZPHS, Depaiguda, Jainath, Adilabad. Smt. R. Nivedita, ZPHS, Polkampally, Moosapet, Mahabubnagar. Sri Raju, Model Basic High School, Mahabubnagar. Editors Sri K. Rajender Reddy, Retd. Teacher, Hyderabad. Sri Fasiuddin, GHS, Shashabgutta, Mahabubnagar. Coordinators Prof. Tahseen Sultana, Head, C&T Dept., SCERT, Telangana, Hyderabad. Dr. P. Revathi Reddy, Deputy Director, TET HOD, Planning Department, SCERT, Telangana, Hyderabad. Assistant Coordinator Sri T. Manohara Chary, SCERT, TS, Hyderabad. Technical Support Sri G. Srinivas Reddy, SCERT, Telangana, Hyderabad. Sri Wasim Akram, SCERT, Telangana, Hyderabad. Sri D. Kannaiah, SCERT, Telangana, Hyderabad. Sri S. Koteshwar Rao, Vidyanagar, Hyderabad. Sri Zakiuddin Liyaqat Mumtaz Computers, Hyderabad.
MATHEMATICS (EM) - CLASS VIII (Class VII Basics) Level 1 Sl. No. Name of the chapter Topic Page 1 Fractions, Decimals and Rational Numbers Addition and Subtract of unlike fraction. 1 2 Fractions, Decimals and Rational Numbers Multiplication of fraction with whole number and another fraction. 3 3 Fractions, Decimals and Rational Numbers Divisions of fraction with whole number and another fraction 5 4 Fractions, Decimals and Rational Numbers Convert rational number into decimal form. 7 5 Integers Addition and Subtraction of Integers on number line. 9 6 Integers Multiplication and Division of Integers 11 7 Integers Properties of Integers under addition 13 8 Integers Properties of Integers under Multiplication 15 9 Integers Properties of Integers under Subtraction and Division 17 10 Exponents Exponential form, Expanded form & prime factorization 19 11 Exponents Laws of Exponents and related problems 21 12 Exponents Laws of Exponents and related problems 23 13 Algebraic Expressions Addition & Subtraction of Algebraic Expressions 25 14 Simple Equations Solving Equations by Transposing the terms 27 Index
Mathematics – Class VIII Aecfeoke Yect 2222/23 *Nexen 2) Sl.No. Name of the unit/ chapter Topic Page I Rational Numbers 1. Introduction of rational numbers 1 2. Rational Numbers - Closure property 3 3. Rational Numbers - Commutative property 6 4. Rational Numbers - Associative property 9 5. Rational Numbers - Identity property 12 6. Rational Numbers - Inverse property (in addition) 14 7. Finding Reciprocal of Rational Numbers 16 8. Representation of Rational Numbers on Number Line 18 9. Rational number between two rational numbers using like fraction method 20 10. Rational number between two rational numbers using average method 23 11. Decimal representation of rational numbers (terminating) 26 12. Decimal representation of rational numbers (non terminating) 28 13. Conversion of decimal number to fraction 30 Index
Readiness Programme LEVEL-1
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FRACTIONS, DECIMALS AND RATIONAL NUMBERS MULTIPLICATION OF A FRACTION WITH WHOLE NUMBER AND FRACTIONS Worksheet No : L1-2 Class : VIII Subject : Mathematics Learning outcomes : After completing this work sheet you are able to do 1. Multiplication of fraction with whole number. 2. Multiplication of fraction with another fraction. Conceptual understanding You know that 2 x 3 means we have to add 3, 2 times i.e., 3+3=6 In 7th class we learnt how to multiply a fraction with a whole number i.e, 3 2x3 6 2x = = 5 5 5 You also learn that multiply 2 fractions is nothing but multiplying the numerators & denominators. i.e., 2 3 2x3 6 x = = 5 7 5x7 35 , Product of two fractions= Product of Numerator Product of Denominator Let us recall the knowledge once again by solving some problems. Model problem solving Ex:1) Find the value of 3 4x 5 ? Sol: Here we know that 4 is a whole number and 3 5 is a fraction. As per multiplication how many times we have to add 3 5 ? Yes, 4 times Then, 3 3 3 3 3 12 4x 5 5 5 5 5 5 (How?) Or 4 3 4x3 12 Product of Numerator x = = 1 5 1x5 5 Product of Denominator 3
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Roll of 10 :- If denominator of a rational number consists of 10,100,1000,.... etc. then shift the decimal point to the left side of numerator as many zeros are there in 10’s. Thousands Hundreds Tens Unit Tenths Hundredths Thousandths 1000 100 10 1 1 0.1 10 1 0.01 100 1 0.001 1000 Example: 1. 132 132 1 2. 132 13.2 10 3. 132 1.32 100 4. 13.2 1.32 10 Assignment :- Convert following rational numbers into decimal form 1. 23 6 2. 53 5 3. 17 4 4. 14.2 3 5. 52 11 6. 2794 10 7. 583.62 100 8. 0.627 1000 Instruction :- Write some more examples for rational numbers and convert them into decimal form. What I have learnt :- After completing this work sheet I can convert any rational number into perfectly partially can’t do decimal form. 8
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INTEGERS MULTIPLICATIONS AND DIVISIONS OF INTEGERS Worksheet No : L1-6 Class : VIII Subject : Mathematics Learning outcomes : After completing this work sheet you are able to 1. Solve the problems involving multiplications and divisions of integers. 2. Uses the negative symbol in different contexts. Conceptual understanding You have already learnt about multiplications and divisions of integers. Let us review what we have already learnt. When you multiply or divide two integers of some sign the result is always positive, if they are opposite signs the result is always negative. Product of even number of negative integers is positive, product of odd number of negative integers is negative. Let us result that knowledge once again by solving some problems. Model problem solving Computer the following. 1. 4 x -2 Sol: We know that product of opposite signs the result is negative So, 4 x -2 = -8 2. -5 x -6 Sol: We know that product of two integers of same sign the result is always positive. So, -5 x -6 = +30 11
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INTEGERS PROPERTIES OF INTEGERS UNDER ADDITION Worksheet No : L1-7 Class : VIII Subject : Mathematics Learning outcomes : After completing this work sheet you are able to Expressing the number properties of integers under addition in general form and gives counter examples. Conceptual understanding & Model problem solving I. Closure Property The sum of two integers is always an integer. For any two integers a & b, a+b is also an integer Study the following : i) 16 + 3 = 19 The sum is integer ii) (-5) + (-5) = -10 The sum is integer iii) (7) + (-6) = -1 The sum is integer Therefore, integers are closed under addition. II. Commutative Property Study the following : i) (3) + (-5) = (-5) + (3) = -2 ii) (-2) + (-6) = (-6) + (-2) = -8 Did you find any pair of integers for when the sum is different when the order is changed? You would have not. Therefore, addition is commutative for integers. In general for any two integers a & b, a+b=b+a III. Associative Property Study the following : i) (-2+3) + (-4) = (-2) + [3+(-4)] = -3 ii) (-6) + [(-1)+(-5)] = [(-6]+(-1)]+(-5) = -12 13
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III. Associative Property i) Study the following : (2 - 7) - 5 2 - (7-5) -5 - 5 2 -2 -10 0 Therefore, subtraction is not associative. ii) Study the following : [-12 4] 3 is not equal to (-12) [4 3] Thus, division of integers is not associative. IV. Divsion by zero and 1 i) Any integer divided by zero is meaningless and zero divided by a non-zero integer is equal to zero. Ex:- i) 5 0 =is not defined ii) 0 5 =0 Therefore, for any integer a, a 0 is not defined, but 0 a=0 for a 0 ii) When we divide a integer by 1 it gives the same. Ex:- i) 5 1 = -5 ii) 11 1 =11 Thus a negative or positive integer divided by 1 gives the same integer as quotient. In general for any integer a, a 1 = a Practice Problems and Instructions : Collect the some more similar problems and discuss with friends which properties are followed by integer which are not under subtraction and division. What I have learnt :- After completing this work sheet 1. I can counter examples. perfectly partially can’t do 18
POWERS AND EXPONENTS Worksheet No : L1-10 Class : VIII Subject : Mathematics Learning outcomes : After completing this work sheet you are able to know (i) exponential form (ii) writing a number in exponential form by prime factorization (iii) convert the exponent form to expansion form. Conceptual understanding & Model problem solving You know that 23 means multiplying 2, three times i.e, 23=2x2x2 In 23 , 2 is base and 3 is exponent. For 57 ; expansion form is 5x5x5x5x5x5x5 For 3x3x3x3; exponent form is 34 In 7th class you have already learnt that “Any number can be written in exponent form by prime factorization. Let us do some problems for revision. Ex:1) Which is greater 23 or 32 Sol: 23=2x2x2=8 (why) 3 2=3x3=9 so, 32 > 23 Ex:2) Write the expanded form of p7 Sol: In p7 , what is the base ? How many times we have to multiply p ? yes, we have to multiply 7 times. so, p7=pxpxpxpxpxpxp Ex:3) What is the exponent form of 5x5x5x5 ? Sol: Here how many times 5 multiplied ? so, 4 is the power What is the base for this exponent form ? yes, 5 is base here so, 5x5x5x5=54 19
Ex:4) Write 450 in exponential form using prime factorization? Sol: 2 450 5 225 5 45 3 9 3 3 1 so, 450=21x32x52 ASSIGNMENT 1) Write 2500 in exponential form using prime factorization. 2) Write base, pwoer and write in expand form a) (2x)5 b) 34 c) (5m)3 d) m3 3) Which is bigger 34 or 43 ? 4) Write exponent form of 3x3x3x5x5x5x5 Instructions :- Collect some more problems solve them and share with your friends. What I learnt : After completing this worksheet, I can do 1. Exponential form, Expanded form perfectly partially can’t do 2. Writing a number in exponential form by perfectly partially can’t do prime factorization 3. Conversion of exponent to expanded form perfectly partially can’t do so, 450 can be written as 450=2x3x3x5x5 Here, 2 come 1 time 3 come 2 times 5 come 2 times 20
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SIMPLE EQUATIONS SOLVING EQUATIONS BY TRANSPOSING THE TERMS Worksheet No : L1-14 Class : VIII Subject : Mathematics Learning outcomes : After completing this work sheet you are able to Solve the equations by transposing the terms. Conceptual understanding & Model problem solving You have already learnt how to solve the equations using trial and error method and solving equations in much lesser time by transposing terms. And you know how to check the result by substituting the value of (x) variable. Let us review what we have already learnt. For balancing on equation we add / substract / multiply / divide the same number on both sides, so that equality remains undisturbed. An equation remains same if the LHS and the RHS are interchanged. Let us recall that knowledge once again by solving some problems. Model Problems : 1. Solve the equation 12 = x - 3 by transposing the terms and check. Sol: Here, LHS = 12 and RHS = x-3 Total value of RHS is 3 less than x To find the value of what we have to add both sides Yes, we have to add 3 Then, we get 12 + 3 = x - 3 + 3 15 = x 27
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Academic Year 2020-21 CHAPTER-I : RATIONAL NUMBERS LEVEL-2
Mathematics – Class VIII Aecfeoke Yect 2222/23 *Nexen 2) Sl.No. Name of the unit/ chapter Topic Page I Rational Numbers 1. Introduction of rational numbers 1 2. Rational Numbers - Closure property 3 3. Rational Numbers - Commutative property 6 4. Rational Numbers - Associative property 9 5. Rational Numbers - Identity property 12 6. Rational Numbers - Inverse property (in addition) 14 7. Finding Reciprocal of Rational Numbers 16 8. Representation of Rational Numbers on Number Line 18 9. Rational number between two rational numbers using like fraction method 20 10. Rational number between two rational numbers using average method 23 11. Decimal representation of rational numbers (terminating) 26 12. Decimal representation of rational numbers (non terminating) 28 13. Conversion of decimal number to fraction 30 Index
1 I. Learning Outcomes : After completing this worksheet students are able to know the defination of Rational numbers. 1. Rational Numbers Introduction. II. Conceptual understanding /Model Problem / Example / Activities : l Temperature at Jammu on a certain day is 0 1 5 2 below zero degrees. The temperature there is - 0 1 5 2 . This number is not an integer. Do you know what we call this type of numbers. Def:-A number which can be written in the form of p q where p and q are integers and q ¹ 0 is called a Rational number. For example 2 3 - , 6 7 are all rational numbers. Can we tell 0, -2, 4 are rational ? Yes, Because 0 2 0 ,2 1 1 - = -= and 4 = 4 1 can be express as p q form, these 0, -2 and 4 also rational numbers. 1. Find the values of p and q in the following numbers. When they are expressed in the form p q and tell weather they are Rational numbers. STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 1 Topic / Concept: Introduction of Rational Numbers ACADEMIC YEAR 2020-21 (LEVEL 2)
2 Sol: Since 2 = 2 1 comparing with p q . p = 2, q = 1 are integers, q ¹ 0. \ 2 is a Rational umber. 2. 9 since 9 = +3 means 3 or -3 \ 3 = 3 1 or -3 = 3 1 - p = 3 & q = 1 or p = -3 & q = 1 (or) \ 9 is a Rational Number. 3. 0.5 since 0.5 = 5 10 p = 5, q = 10 are integers. \ 0.5 is a rational number. III. Worksheet 1. 2 5 25 61 ,; , 39 6 2 -- - Find the values of p, q when they are expressed in the form of p q and say wether they are rational numbers are not ? IV. What I have learnt Partially Perfectly Can't do I have learnt 1. About the defintion of Rational Numbers 2. Insertion of Rational Numbers Instructions 1. Go through the example and exercise sums similar to the learned concept and practice regularly. 2. Collect similar problems. Share and discuss with your friends
3 I. Learning Outcomes : After completing this worksheet students are able to 1. Rational numbers are closed under the operations of addition, substraction and multiplication. II. Conceptual understanding /Model Problem / Example/ Activities : l We have seen that intgers are closed under addition, substraction, and multiplication but not under division, Now we shall check, For rational numbers. 1. Find the values of (i) 3 5 8 7 æ ö + -ç ÷ è ø 3 5 21 ( 40) 19 8 7 56 56 æ ö +- - +- = = ç ÷ è ø (a rational numbers) (ii) 3 4 8 5 - -æ ö + ç ÷ è ø 3 4 15 ( 32) 47 8 5 40 40 - - - +- - æ ö += = ç ÷ è ø (a rational numbers) (iii) 4 6 44 42 86 7 11 77 77 + += = (a rational numbers) \ sum of two rational numbers is again a rational number. \ rational numbers are closed under addition. If a and b are rational Numbers, therefore a + b is also a Rational Number. STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 2 Topic / Concept: Rational Numbers - Closure Property ACADEMIC YEAR 2020-21 (LEVEL 2)
4 The difference of two rational numbers be again a rational number. (i) Find the value of 5 2 . 7 3 - - 5 2 5 3 2 7 15 14 7 3 21 21 - - -´ - ´ - - = -= = 29 21 - = (a rational number). (ii) 5 4 5 5 8 ( 4) 25 32 8 5 8 5 40 - ´ - ´- + -= = ´ 57 40 = (a rational number). (iii) 3 8 7 5 æ ö - - ç ÷ è ø 3 8 15 ( 56) 7 5 35 æ ö - -- =- = ç ÷ è ø 15 56 35 + = 71 35 = (a rational number) We find that rational numbers are closed under substraction. That is, for any two rational numbers a and b, a - b is also a rational number. Let us now see the product of two numbers. Find :- (i) 2 4 3 5 - ´ (ii) 3 2 7 5 ´ = 24 8 3 5 15 - - ´ = 32 6 . 7 5 35 = ´= both the products are rational number. (iii) 4 6 5 11 - - ´ 24 55 = is a rational number.
5 We say that rational numbers are closed under multiplication. That is for any two rational numbers a and b, a ´ b is also a rational number. 1. Find the value of 5 2 3 5 - ¸ 5 2 3 5 - = ¸ 5 5 3 2 - = ´ 25 6 - = is a rational number. We find that for any rational number a, a ¸ 0 is not defined. So rational numbers are not closed under division. III. Worksheet Find the value of following and verify the closure property under the given operation. (i) 2 5 3 7 - + (ii) 6 8 5 3 - -æ ö + ç ÷ è ø (iii) 2 5 3 4 - (iv) 1 3 2 5 - (v) 7 6 3 5 - (vi) 5 0 4 - ¸ IV. What I have learnt Partially Perfectly Can't do 1. I understood Rational numbers are closed under the operations of addition, substraction and multiplication. Instructions 1. For more practice of concepts collect some more examples problems, share among your friends and discuss.
6 I. Learning Outcomes : After completing this worksheet students understood Rational number are commutative under the operations addition and multiplication. II. Conceptual understanding /Model Problem / Example / Activities / : We have seen that integers satisfy commutative property under addition and multiplication. Now we shall check far Rational numbers. 1. Find the value of 2 5 3 7 - + and 5 2 7 3 æ ö - + ç ÷ è ø compare these two values what do you observe? 2 5 ( 2 7) (5 3) 3 7 (3 7) - -´ + ´ + = ´ 14 15 21 - + = 1 21 = 5 2 7 3 æ ö - + ç ÷ è ø (5 3) ( 2 7) (7 3) ´ +-´ = ´ 15 ( 14) 21 + - = 15 14 21 - = STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 3 Topic / Concept: Rational Numbers - Commutative property ACADEMIC YEAR 2020-21 (LEVEL 2)
7 1 21 = \ We can say 25 5 2 377 3 - -æ ö +=+ ç ÷ è ø 2. Find the values of 12 16 10 6 - -æ ö + ç ÷ è ø and 16 12 6 10 - -æ ö + ç ÷ è ø Compare these two values. What do you inger ? 12 16 10 6 - -æ ö + ç ÷ è ø ( 12 3) ( 16 5) 30 - ´ +- ´ = 36 ( 80) 30 - +- = 116 30 - = 16 12 6 10 - -æ ö = + ç ÷ è ø ( 16 5) ( 12 3) 30 - ´ +- ´ = 80 ( 36) 30 - +- = 116 30 - = 12 16 16 12 10 6 6 10 -- -- æö æö \+ =+ ç÷ ç÷ èø èø For two rational numbers a and b, a + b = b + a. 3. Find the value of 2 5 3 4 - and 5 2 4 3 - 2 5 3 4 - = 8 15 12 - = 7 12 - 5 2 4 3 - 15 8 12 - = 7 12 = 25 52 34 43 \- ¹ - \ Rational numbers are not commutative under substraction.