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STUDENT'S SECTION MATHEMATICS Integers The CBSE-International is grateful for permission to reproduce and/or translate copyright material used in this publication. The acknowledgements have been included wherever appropriate and sources from where the material may be taken are duly mentioned. In case any thing has been missed out, the Board will be pleased to rectify the error at the earliest possible opportunity. All Rights of these documents are reserved. No part of this publication may be reproduced, printed or transmitted in any form without the prior permission of the CBSE-i. This material is meant for the use of schools who are a part of the CBSE-International only. The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the learning process in harmony with the existing personal, social and cultural ethos. The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged. The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in view. The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand, appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations and additions wherever and whenever necessary. The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink their approaches for knowledge processing by their learners. In this context, it has become imperative for them to incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of these requirements. The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all subject areas to cater to the different pace of learners. The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some nonevaluative components in the curriculum which would be commented upon by the teachers and the school. The objective of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives, SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'. The Core skills are the most significant aspects of a learner's holistic growth and learning curve. The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework (NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to millions of learners, many of whom are now global citizens. The Board does not interpret this development as an alternative to other curricula existing at the international level, but as an exercise in providing the much needed Indian leadership for global education at the school level. The International Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The Board while addressing the issues of empowerment with the help of the schools' administering this system strongly recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to their peers through the interactive platforms provided by the Board. I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr. Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the development and implementation of this material. The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion forums provided on the portal. Any further suggestions are welcome. Vineet Joshi Chairman Advisory Conceptual Framework Shri Vineet Joshi, Chairman, CBSE Dr. Sadhana Parashar, Director (Training), Shri G. Balasubramanian, Former Director (Acad), CBSE Ms. Abha Adams, Consultant, Step Dr. Sadhana Parashar, Director (Training), Ideators VI-VIII Ms Aditi Mishra Ms Guneet Ohri Ms. Sudha Ravi Ms. Himani Asija Ms. Neerada Suresh Ms Preeti Hans Ms Neelima Sharma Ms. Gayatri Khanna Ms. Urmila Guliani Ms. Anuradha Joshi Ms. Charu Maini Dr. Usha Sharma Prof. Chand Kiran Saluja Dr. Meena Dhani Ms. Vijay Laxmi Raman Material Production Groups: Classes VI-VIII English : Physics : Mathematics : Ms Neha Sharma Ms. Vidhu Narayanan Ms. Deepa Gupta Ms Dipinder Kaur Ms. Meenambika Menon Ms. Gayatri Chowhan Ms Sarita Ahuja Ms. Patarlekha Sarkar Ms. N Vidya Ms Gayatri Khanna Ms. Neelam Malik Ms. Mamta Goyal Ms Preeti Hans Ms. Chhavi Raheja Biology: Ms Rachna Pandit Mr. Saroj Kumar Ms Renu Anand Hindi: Ms. Rashmi Ramsinghaney Ms Sheena Chhabra Mr. Akshay Kumar Dixit Ms. Prerna Kapoor Ms Veena Bhasin Ms. Veena Sharma Ms Trishya Mukherjee Ms. Seema Kapoor Ms. Nishi Dhanjal Mr. Manish Panwar Ms Neerada Suresh Ms. Kiran Soni Ms. Vikram Yadav Ms Sudha Ravi Ms. Monika Chopra Ms Ratna Lal Ms Ritu Badia Vashisth Ms. Jaspreet Kaur CORE-SEWA Ms Vijay Laxmi Raman Ms. Preeti Mittal Ms. Vandna Ms. Shipra Sarcar Ms.Nishtha Bharati Chemistry Ms. Leela Raghavan Ms.Seema Bhandari, Ms. Poonam Kumar Ms. Seema Chopra Mendiratta Ms. Madhuchhanda Ms. Rashmi Sharma MsReema Arora Ms. Kavita Kapoor Ms Neha Sharma Ms. Divya Arora Ms. Sugandh Sharma, E O Mr. Navin Maini, R O (Tech) Shri Al Hilal Ahmed, AEO Ms. Anjali, AEO Shri R. P. Sharma, Consultant (Science) Mr. Sanjay Sachdeva, S O Coordinators: Dr. Srijata Das, E O Dr Rashmi Sethi, E O (Co-ordinator, CBSE-i) Ms. Madhu Chanda, R O (Inn) Mr. R P Singh, AEO Ms. Neelima Sharma, Consultant (English) Ms. Malini Sridhar Ms. Leela Raghavan Dr. Rashmi Sethi Ms. Seema Rawat Ms. Suman Nath Bhalla Geography: Ms Suparna Sharma Ms Aditi Babbar History : Ms Leeza Dutta Ms Kalpana Pant Ms Ruchi Mahajan Political Science: Ms Kanu Chopra Ms Shilpi Anand Economics : Ms. Leela Garewal Ms Anita Yadav CORE-Perspectives Ms. Madhuchhanda, RO(Innovation) Ms. Varsha Seth, Consultant Ms Neha Sharma Ms.S. Radha Mahalakshmi, EO 4. Study Material 1 5. Student's support material (Student's worksheets) 19 v SW 1: Warm Up Activity W1 20 Recall Integers v SW 2: Warm Up Activity W2 22 Appreciate Your Knowledge v SW 3: Pre Content Worksheet P1 23 Addition of Integers v SW 4: Pre Content Worksheet P2 25 Addition and Subtraction of Integers 1 v SW 5: Content Worksheet C1 27 Addition and Subtraction of Integers 2 v SW 6: Content Worksheet C2 28 Multiplication of Integers v SW 7: Content Worksheet C3 30 Skill Drill v SW 8: Content Worksheet C4 31 Multiplication of more than two integers v SW 9: Content Worksheet C5 33 Properties of multiplication of Integers v SW 10: Content Worksheet C6 38 Distributive Property of Integers v SW 11: Content Worksheet C7 41 Properties of Integers SW 12: Content Worksheet C8 v Division of Integers 44 v SW 13: Content Worksheet C9 46 Properties of division of Integers v SW 14: Content Worksheet C10 48 BODMAS- Simplification of brackets v SW 15: Content Worksheet C11 51 Independent Practice v SW 16: Content Worksheet C12 53 Word problems on Integers v SW 17: Post Content Worksheet PC1 57 Appreciate your knowledge v SW 18: Post Content Worksheet PC2 60 Evaluate your knowledge 6. Acknowledgments 62 7. Suggested videos/ links/ PPT's 63 STUDY MATERIAL 1 Introduction In Class VI, you studied about a special type of numbers called integers. You also learnt how to compare integers and to arrange them in ascending or descending order. Concepts of absolute value of an integer, operations of addition and subtraction on integer were also discussed. In this unit, we shall first recall these concepts in brief and learn more about integers regarding their multiplication and division. Finally, we shall discuss properties of operations on integers and solve some word problems involving integers. Simplification of expressions using BODMAS rules will also be explained. 1. Basic Concepts: A recall Integers The numbers ….., –4, –3, –2, –1, 0,1, 2, 3, 4, ….. are called integers. Numbers ….., –4, –3, –2, –1, are called negative integers and numbers 1, 2, 3, 4, ….. are called positive integers. ‘0’ is neither negative nor positive number. Integers can be represented on a number line: Negative integers positive integers Fig 1. Ordering of Integers An integer a lying to the left of another integer b on the number line is smaller i.e, a < b or b > a. (i) Every positive integer is greater than every negative integer i.e, 5 > –3, 2 > –100, 1 > –999 etc. 2 (ii) Zero (0) is less than every positive integer and greater than every negative integer. For example, 0 < 1 < 2, 0 < 5 etc. 0 >–1, 0 >–5 etc (iii) For any two integers a and b, if a > b then –a < –b, and if a < b then –a > –b. For example, 4 > 3 and –4 < –3, 9 > 7 and –9 < –7 6 < 8 and –6 > –8 and so on. Absolute value of an Integer Absolute value of an integer is its numerical value regardless of its sign. For example, absolute value of –4, denoted as |–4|, is 4. Absolute values of 3, denoted as |3|, is 3. Similarly, |–99| = 99, |2| = 2 |–5| = 5 |18| = 18 Addition of integers Rules (i) When two integers are of the same sign, i.e., both positive or both negative, then we add their absolute values and assign the sign common to both. For example, (+ 19) + (71) = + (|19| + |71|) + (|19| + |71|) = + (19 + 71) = + 90 or 90 3 (–35) + (–13) = – (|–35| + |–13|) = – (35 + 13) = – 48 (ii) When two integers are of different signs, i.e., one integer is positive and the other is negative, then to find their sum, we first ‘find’ absolute values of the integers, subtract smaller absolute value from the greater absolute value and assign the sign of the integer with greater absolute value. For example, (–92) + 98 = + (|98| – |–92|) = + (98 – 92) = 6 Similarly, (+91) + (–98) = – (|–98| – |91|) = – (98 – 91) =–7 (iii) When 0 is added to any integer, we get the integer itself. For example, (–3) + 0 = –3 0 + (4) = 4 Additive inverse If the sum of two integers is 0, then each integer is called the additive inverse of the others. For example, as 4 + (–4) = 0, so, –4 is the additive inverse of 4 and 4 is the additive inverse of –4. Similarly, 481 is the additive inverse of – 481 and – 481 is the additive inverse of 481 as 481 + (–481) = 0 Additive inverse of 0 is 0 as 0 + 0 = 0 4 Subtraction of Integers Subtraction is considered as inverse operation of addition. Therefore, to subtract an integer b from another integer a, we add additive inverse of b i.e. (–b) to a. For example, 3 – (–7) = 3 + (+ 7) = 10 Similarly, (Rule 1 of addition) –53 – (42) = –53 + (–42) = –95 (Rule 1 of addition) –51 – (–10) = –51 + (+ 10) = –41 2. (Rule 2 of addition) Multiplication of Integers Recall that multiplication of whole numbers is repeated addition. For example 3 x 7 = 7 + 7 + 7 = 21, 10 x 3 = 3 + 3+ 3 +3 +3 +3 +3 +3+3 +3 = 30 In the same way, we can find 3 x (–7) = (–7) + (–7) + (–7) = –21 10 x (–3) = (–3) + (–3) + (-3) + (–3) + (–3) + (–3) + (–3) + (–3) + (–3) + (–3) = –30 6 x (–3) = (–3) + (–3) + (–3) + (–3) + (–3) + (–3) = –18 and so on. From the above discussion, we can say that If a and b are two positive integers then a x b = ab [positive integers x positive integers = positive integer] and a x (–b) = – (ab) [Positive integers x negative integer = negative integer] 5 What about the product (–3) x 6? To find this, observe the following pattern: 3 x 6 = 18 2 x 6 = 12 1x6=6 0x6=0 –1 x 6 = ? –2 x 6 = ? –3 x 6 = ? Observe that in the above case, first number is decreasing by 1 at each step, and the corresponding product is decreasing by 6 (= 2nd integer). 3 x 6 = 18 (3 –1) 2 x 6 = 12 (18 – 6) (2 –1) 1x6=6 (12 – 6) (1 –1) 0x6=0 (6 – 6) (0 –1) –1 x 6 = –6 (0 –6) (–1 –1) –2 x 6 = –12 (–6 –6) (–2 –1) –3 x 6 = –18 (–12 –6) Thus, –3 x 6 = –18 = – (3 x 6) What is the product –4 x 6? It will be –18–6 = –24 (see the above pattern) We have (–1) x 6 = –6 = – (1 x 6) (–2) x 6 = –12 = – (2 x 6) (–3) x 6 = –18 = – (3 x 6) (–4) x 6 = –24 = – (4 x 6) 6 In general, we can say that for two positive integers a and b, (–a) x b = – (ab) Negative integer x positive integer = negative integer What is the product (–3) x (–6)? Observe the following: (–3) x 6 = –18 5=6–1 (–3) x 5 = –15 (–18+3) 4=5–1 (–3) x 4 = –12 (–15+3) 3 = 4 –1 (-3) x 3 = – 9 (–12+3) 2=3–1 (–3) x 2 = –6 (–9+3) 1=2–1 (–3) x 1 = –3 (–6+3) 0 = 1 –1 (–3) x 0 = 0 (–3+3) –1 = 0 –1 (–3) x (–1) = ? –2 = –1 –1 (–3) x (–2) = ? –3 = –2 –1 (–3) x (–3) =? Do you see any pattern? Observe how do the products change. You can see that 2nd number in each step is decreasing by 1 and product is increasing by 3, From the pattern, (–3) x (–1) = 0+3 = 3 = 3 x 1 (–3) x (–2) = 3+3 = 6 = 3 x 2 (–3) x (–3) = 6+3 = 9 = 3 x 3 and so on. 7 Thus, (–3) x (–6) = 3 x 6 = 18 In general, for positive integers a and b (–a) x (–b) = ab Negative integer x negative integer = positive integer We can now summarise the discussion on multiplication of two integers as follows: 1. To find the product of two integers of like signs, find the product of their absolute values and assign the ‘+’ sign to the product. 2. To find the product of two integers of unlike signs, find the product of their absolute values and assign ‘–’sign to the product. Let us consider some examples illustrating product of integers. Example 1: Find the following products: (i) (–4) x 6 (ii) 3 x 11 (iii) 2 x (–19) (iv) (–11) x (–16) (v) (–5) x (–6) x (–3) (vi) (–5) x (–4) x (–3) x (–4) Solution (i) (–4) x 6 = – (4 x 6) = –24 [ (–a) x b = – (ab)] (ii) 3 x 11 = 33 (iii) 2 x (–19) = – (2 x 19) = –38 [a x (–b ) = –ab] (iv) (–11) x (–6) = +(11 x 6) = 66 (v) (–5) x (–6) x (–3) [(–a) x (–b) = ab] = [(–5) x (–6)] x (–3)] = [+ (5 x 6)] x (–3) 8 = 30 x (–3) = – (30 x 3) = –90 (vi) (–5) x (–4) x (–3) x (–4) = [(–5) x (–4)] x (–3) x (–4) = (20) x (–3) x (–4) = [20 x (–3)] x (–4) = (–60) x (–4) = 240 From the above products, you can observe that (i) Product of integers involving even number of negative integer is positive. (ii) Product of integer involving odd number of negative integer is negative. Example 2: Find the sign of the following products: (i) (–1) x (–1) x (–1) x (–1) x (–1) (ii) (–20) x (–31) x (–1001) x (43) (iii) (124) x (–108) x 99 x (–99) x (–1241) x (–100) Solution: (i) There are five (odd) negative integers. So, their product will be negative. (ii) Number of negative integer is 3 (odd) So, the product will be negative (iii) Number of negative integer is 4 (even) So, the product will be positive. 3. Division of integers You know that division is the inverse operation of multiplication. Recall that for a multiplication fact in whole numbers, there are two corresponding division facts (i) 5 x 8 = 40, 9 The division facts are: 40 ÷ 5 = 8 40 ÷ 8 = 5 Similarly, for 4 x 9 = 36, we have the following division facts. 36 ÷ 4 = 9 36 ÷ 9 = 4 We can extend it to integers as follows: Multiplication fact Division facts (–2) x (–3) = 6 6 ÷ (–2) = –3 and (–8) x (9) = –72 6 ÷ (–3) = –2 (–72) ÷ (–8) = 9 and (10) x (–5) = –50 (–72) ÷ 9 = –8 (–50) ÷ 10 = –5 and (–10) x (–5) = 50 (–50) ÷ (–5) = 10 50 ÷ (–10) = –5 and 3 x (–11) = –33 50 ÷ (–5) = –10 (–33) ÷ 3 = –11 and – 4 x 5 = –20 (–33) ÷ (–11) = 3 (–20) ÷ (–4) = 5 and (–20) ÷ 5 = –4 From the above division facts, observe that (i) If the dividend and divisor are integers of like signs, then the quotient is an integer with a positive sign (+), provided the absolute value of the dividend is divisible by the divisor. For example, 40 ÷ 5 = 8 36 ÷ 4 = 9 10 (–72) ÷ (–8) = + =9 (–50) ÷ (–5) = 10 (ii) If the dividend and divisor are integers of unlike signs, then the quotient is an integer with a negative sign (–), provided the absolute value of the dividend is divisible by the divisor. For example, 6 ÷ (–2) =– = –3 50 ÷ (–5) = – = –10 (–50) ÷ 10 = – = –5 (–72) ÷ 9 = – = –8 Example 3: Find (i) 18 ÷ (–3) (ii) (–54) ÷ 18 (iii) (–18) ÷ (–6) (iv) (–8) ÷ (–2) (v) 16501 ÷ (–16501) (vi) (–729) ÷ (–81) (vii) 190 ÷ (–1) Solution: (i) 18 ÷ (–3) = – = –6 (ii) (–54) ÷ 18 = – (iii) (–18) ÷ (–6) = + (iv) (–8) ÷ (–2) = + = 4 (v) 16501 ÷ (–16501) = – (vi) (–729) ÷ (–81) = + = –3 =3 = –1 =9 11 (vii) 190 ÷ (–1) = – = –190 Division of an Integer by 0 Recall that division is repeated subtraction as multiplication is repeated addition. Let us try to find –23 ÷ 0, using repeated subtraction We have (–23)– 0 = –23 (–23) – 0 = –23 (–23) – 0 = –23 (–23) – 0 = –23 --------This process is never ending. So, we say that division of an integer by 0 is meaningless. What about division of 0 by a non zero integer? Like whole numbers, the quotient is 0. 4. Properties of Operations on Integer Addition (i) The sum of two integers is an integer For example 5 + (–3) = 2 (integer) (–3) + (–5) = –8 (integer) 8+ (–8) = 0 (integer) This property is called closer property for addition (ii) For any two integers a and b, a+b=b+a For example, 5 + (–3) = 2, and (–3) – 5 = 2 5 + (–3) = (–3) + 5 12 Similarly, (–9) + (2) = 2 + (–9) This property is called commutative property of addition. (iii) For any three integers a, b and c a + (b+c) = (a+b) + c For example, 3 + [(–2) + (–5)] = 3 + (–7) = –4 and [3 + (–2)] + (5) = 1 + (–5) = –4 So, 3 + [(–2) + (–5)] = [3+ (–2)] + (–5) This property is called associative property of addition (iv) For any integer a, a+0=0+a=a 0 is called the identity element (or additive identity) Subtraction (i) For any two integers a and b a – b or b – a is an integer For example, 3 – (–2) = 5 (integer) This property is called closure property for subtraction (ii) Subtraction is not commutative For example, (–8) – (–7) = –1 (–7) – (–8) = 1 So, (–8) – (–7) ≠ (–7) – (–8) (iii) Subtraction is not associative For example (–5) – [(–2) + (–3)] ≠ [(–5) – (–2)] + (–3) (iv) 0 is not an identity element for subtraction because a – 0 = but 0 – a = -a i.e. a – 0 ≠ 0 – a Multiplication (i) For any two integers a and b, a x b or ab is an integer. 13 For example, (–2) x (–1) = 2 (integer) (–11) x 5 = –55 (integer) This is called closure property for multiplication (ii) For any two integers a and b, axb=bxa This is called commutative property of multiplication (iii) For any three integers a, b and c a x (b x c) = (a x b) x c For example, (–3) x [(8 x 5)] = (–3) x 40 = –120 [(–3) x 8] x 5 = –24 x 5 = –120 i.e. (–3) x [(8 x 5) = [(–3) x 8] x 5 This is called associative property of multiplication. (iv) 1 is the multiplicative identity (or identity element) of any integer a i.e. 1 x a = a x 1 = a Property of zero (0) For any integer a, ax0=0xa=0 Distributive property of multiplication over addition For any three integers a, b and c a x (b + c) = a x b + a x c (a + b) x c = a x c + b x c You may verify it by taking any three different integers a, b and c. Division None of the above properties is true for division of integers. 14 You may satisfy yourself by taking different integers a, b, c etc. Example 4: Fill in the blanks: (i) 2 + (–15) = -------- +2 (ii) –3785 ÷ ---------- = 1 (iii) ------- ÷ 555 = –1 (iv) -------- ÷ 578 = 0 (v) – 6 + --------- = 0 (vi) 19 + -------- = 0 (vii) 15 + (–15) = ------ (viii) (–2) x [3 x (–5)] = [(―) x 3] x (–5) (ix) –578 x ----- = –578 (x) (–5) x [(–2) + 3] = (–5) x (―) + (―) x 3 Solution: (i) –15 (ii) –3785 (vi) –19 (vii) 0 (iii) –555 (iv) 0 (v) 6 (viii) –2 (ix) 1 (x) –2 , –5 Example 5: Which of the following statements are true and which are false? (i) Addition of integers is commutative (ii) Integers are closed for division (iii) 1 is the identity element for addition of integers (iv) Multiplication is association for integers (v) 0 is the identity element for subtraction (vi) Quotient of two integers is an integer (vii) Addition is distributive over multiplication for integers (viii) 1 is the identity element for multiplication 15 Solution : 5. (i) True (ii) False (iii) False (iv) True (v) False (vi) False (vii) False (viii) True BODMAS Rule In BODMAS, B stands for ‘brackets’, O for ‘of’, D for ‘Division’, M for Multiplication’, A for ‘Addition’ and S for ‘Subtraction’. In simplifying expression, we perform these operations in the order, they are listed above. We explain this through some examples: Example 6: Find the value of (i) 30 – 5 x 2 of 3 + (19 – 3) ÷ 4 (ii) 81 of [59 – {7 x 8+ (17–2 of 5)}] Solution: (i) 30 – 5 x 2 of 3 + (19 – 3) ÷ 4 = 30 – 5 x (2 of 3) + 16 ÷ 4 (Removing the brackets) = 30 – 5 x 2 x 3 + 16 ‚ 4 (Performing ‘of’) = 30 – 5 x 2 x 3 + 4 (Performing D) = 30 – 30 + 4 (Performing M) = 34 – 30 (Performing A) =4 (Performing S) (ii) 81 of [59 – {7 x 8+ (17–2 of 5)}] = 81 of [59 – {7 x 8 + (17 – 2 x 5)}] = 81 of [59 – {7 x 8 + (17 –10)}] = 81 of [59 – {56 + 7}] = 81 of [59 – 63] = 81 of (–4) = 81 x (–4) = –324 16 6. Applications of Integers We now explain the use of operations on integers in solving some daily life problems. Example 7: A question paper contains 15 questions. 4 marks are given for every correct answer and –1 marks are given for every correct answer. Hamid attempts all the questions. What is his total score, if only 10 of his answers are correct? Solution: Marks given for 1 correct answer = 4 So, marks for 10 correct answers = 4 x 10 = 40 Marks for 1 incorrect answer = –1 So, marks for (15–10) = 5 incorrect answers = –1 x 5 = –5 So, total score of Hamid = 40 + (–5) = 35 Example 8: A shopkeeper earns a profit of 80 per bag of wheat sold and a loss of 50 per bag of rice sold. The company sold 2500 bags of wheat and 3400 bags of rice in a month. Calculate the profit of loss or the shopkeeper. Solution: Profit on 1 bag of wheat = 80 So, profit on 2500 bags of a wheat = 80 x 2500 = 200000 Loss on 1 bag of rice = 50 i.e., profit on 1 bag of rice =– 50 So, profit on 3400 bags of rice =– 50 x 3400 Total Profit = (–50 x 3400) = (–170000) = [200000 + (–170000)] = 30000 17 Example 9: A certain freezing process requires that room temperature be lowered from 45ºC at the rate of 5ºC every hour. What will be the room temperature at the end of 12 hours? Solution: Decrease in temperature in 1 hour = 5ºC Room temperature after 12 hours = 45ºC + (–5ºC) x 12 = 45ºC + (–60 ºC) = –15ºC 18 19 Student’s Worksheet – 1 Warm Up (W1) Recall Integers Name of the student ______________________ Date ____________ Activity - Numbers and more Numbers. . . 1. What are whole numbers? Can they be negative? ________________________________________________________________________ ________________________________________________________________________ 2. What are integers? Write 15 integers of your choice in the space given below. _______________________________________________________________________ _______________________________________________________________________ OUT 20 Put these numbers in the appropriate boxes. One Number may be put in more than one box. Natural Numbers Whole Numbers Integers Write True or False on the basis of your observations. All whole numbers are natural. __________ All whole numbers are integers. __________ All natural numbers are whole numbers. __________ 21 Student’s Worksheet – 2 Warm Up (W2) Appreciate Your Knowledge Name of the student ______________________ Date ____________ Activity - Mirror Numbers on Number Line. Recall that negative integers are the images of positive integers and vice-a-versa. Integer ‘0’ is neither positive nor negative. Fill up the given table. One example is done for you. Integer Write image of (A) integer (B) 5 -5 Give an integer > Image Give an integer < Image Find (A) + (B) -2 -8 5 + (-5) = 0 (-2 > -5 ) ( -8 < -5 ) -3 7 0 -4 -2 9 -15 23 -16 -3 -41 Write your observation. Is a + (-a) =0 ?(where a is any integer) ______________________________________________________________________________ ______________________________________________________________________________ 22 Student’s Worksheet – 3 Precontent (P1) Addition of Integers Name of the student ______________________ Date ____________ Activity - Integer Grid Find 3 integers in a row, column or a diagonal, where the third number is the sum of the first and the second number. One is already done for you. Here, (-7) + 2 = -5. Find at least 5 more such relations and record them here. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ 23 Activity -Addition of Integer on Number line Add on number line 1. ( – 5 ) + 7 = _______ 2. 5 + ( – 4 ) = _______ 3. – 7 – 2 = ________ 4. – 8 + 6 – 3 = _______ 5. 5 + 3 – 8 = ________ 24 Student’s Worksheet – 4 Precontent (P2) Addition and Subtraction of Integers 1 Name of the student ______________________ Date ____________ Activity - The Number SPIDER Start with an integer of your choice and encircle it to form the body of a SPIDER . Draw eight ‘legs’ rediating from the body and write another way of writing the selected integer , using addition and subtraction of integers. An example is provided here. Go ahead and complete it . Make SIX NUMBER SPIDERS of your choice. 25 Make your own CRAWLY SPIDERS. 26 Student’s Worksheet – 5 Content Worksheet (C1) Addition and Subtraction of Integers - 2 Name of the student ______________________ Date ____________ Task 1: Find the hidden number. Solve the expression in each box.Colour it red if answer is -3 and blue if answer is - 4. Identify the hidden number. 7-11 3-6 10-13 -2-2 5-9 6-10 -3-1 4-7 16-20 0-4 13-17 -1-2 31-35 -42+36 59-63 27-30 11-14 76-79 -11+7 -5+1 -9+5 3-7 12-16 21-25 -8+4 The hidden number is……………….. Task 2:Add or subtract the integers. Help the rat to perform the operation correctly and move forward and eat the cheese. 27 Student’s Worksheet – 6 Content Worksheet (C2) Multiplication of Integers Name of the student ______________________ Date ____________ Task 1: Complete the multiplication table. Multiply each integer in the row with corresponding integer in the column: X -3 -2 -3 0 3 -2 4 -1 2 0 0 1 -1 -3 -2 2 0 1 2 3 4 5 -3 -6 -9 -12 -15 -2 -4 -3 -4 -5 0 0 0 0 -1 1 2 2 3 -9 3 4 -8 5 -10 6 -18 0 -6 0 5 10 6 12 28 3 5 6 8 9 12 12 16 15 6 12 25 Task 2: Solve and then join the dots in ascending order. Colour the figure obtained. Task 3: Write the integer that makes the following statement true. 1. 4 x 2. 3. 4. 5. = 44 x ( 5) = 45 2x 6 x ( 5) 9x =8 = = 63 29 Student’s Worksheet – 7 Content Worksheet (C3) Skill Drill Name of the student ______________________ Date ____________ Activity - Integer Grid with multipication Find 3 integers in a row, column or a diagonal, where the third number is the product of the first and the second number. One is already done for you. Here, (–2) x (–1) = 2 Find at least 5 more such relations and record them here. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ _______________________________________________________________________ 30 Student’s Worksheet – 8 Content Worksheet (C4) Multiplication of Three or More Integers Name of the student ______________________ Date ____________ Task 1: Multiply the following integers: 3 × ( -5) × 9 =_____________ (-9) × (-2) × 8 =_____________ (-7 ) × 3 × 3 =_____________ 7 × (-6) × (-8) =_____________ (-8) × 5 × 0 =_____________ (-5) × (-5) × 7 =_____________ 7 × (-6) × 8 =_____________ (-2)×(-10)×(-8) =_____________ (-2) × 2 × (-9) =_____________ 8 × (-1) × 4 Task 2: If ∆ = - 5 , a) b) c) =_____________ = -2 ,evaluate the following: = ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ _______________________________________________________________ 31 Task 2: Solve and place the integers in order starting with the least or smallest.(the first one is done for you.) -2 X 3 X3 -4 X 2 X 4 2X5X3 (-6) X (-2) X 3 3 X (-4)X 2 4 X 4 X -3 -5 X 0 X 2 5 X (-1) X 3 -2 X 2X1 -2 X -3 X -7 5 X 2 (-3) 2 X -3 X -1 6 X -3 X 2 -4 X -2 X 3 -3 X -7 X 0 32 Student’s Worksheet – 9 Content Worksheet (C5) Properties of Multiplication of Integers Activity 1: Choose the correct property of multiplication from the cloud and write in the blank. Commutative Distributive Associative Closure Multiplicative identity i) 6 X 13 = 13 X 6 …………………................................. ii) 2X1=2 …………………................................. iii) -6 X 3 = -18 …………………................................. iv) (9 X 12) X 3 = 9 X (12 X 3) …………………….............................. v) 7 X (3 + 10) = 7 X 3 + 7 X 10 …………………………...................... vi) 1X9=9 vii) 13 X (-9) = (-9) X 13 viii) -13 X (8 + 10) = -13 X 8 + (-13) X 10…………........................................... ix) -4 X (-3) = -3 X (-4) x) -2 X (45 +33) = (-2) X 45 + (-2) X 33........................................................... …………………….............................. …………………….............................. ………………..................................... 33 Activity 2: In numbers 1-9, select the property that is being illustrated and encircle it. 1) -3 + 6 = 6 + -3 Associative Property of Addition Commutative Property of Addition Distributive Property Property of Additive Inverse 2) 3) 3 + (5 + 7) = 3 + (7 + 5) a) Distributive Property b) Property of Additive Inverse c) Associative Property of Addition d) Commutative Property of Addition 7•3=3•7 a) Property of Multiplicative Inverse b) Associative Property of Multiplication c) Commutative Property of Multiplication d) Distributive Property 34 4) 5) 6) 7) 8) (7 • 5) • 2 = 7 • (5 • 2) a) Identity Property for Multiplication b) Associative Property of Multiplication c) Property of Multiplicative Inverse d) Commutative Property of Multiplication 2(7 + 3) = 2(7) + 2(3) a) Commutative Property of Addition b) Associative Property of Multiplication c) Property of multiplicative inverse d) Distributive Property -2(x + 5) = -2x - 10 a) Property of multiplicative inverse b) Associative Property of Multiplication c) Distributive Property d) Identity Property for Multiplication -3(x - 3) = -3x + 9 a) Identity Property for Multiplication b) Commutative Property of Multiplication c) Property of multiplicative inverse d) Distributive Property x+0=x a) Commutative Property of Addition b) Associative Property of Addition c) Property of Additive Inverse d) Identity Property of Addition 35 9) a=a•1 a) Associative Property of Multiplication b) Identity Property of Multiplication c) Property of Multiplicative Inverse d) Commutative Property of Multiplication Activity 3: In numbers 10-16, choose the best answer that answers the question and encircle it. 10) 11) 12) Which of the following is an illustration of the associative property? A) a(b + c) = ab + ac B) ab + 0 = ab C) a + (b + c) = (a + b) + c D) a+b=b+a Which sentence is an example of the distributive property? A) a (b + c) = ab + ac B) a•1=a C) ab = ba D) a(bc) = (ab)c What number is the additive identity element? A) 1 13) B) 0 D) 2 Which statement best illustrates the additive identity property? A) 6 + 2 = 2 + 6 14) C) -1 B) 6 + 0 = 6 C) 6 + (-6) = 0 What number is the multiplicative identity element? A) ½ B) 0 C) -1 D) 1 36 D) 6(2) = 2(6) 15) 16) Which statement best illustrates the inverse property of addition? A) 3 + 3= 6 B) -3 + 3= 0 C) 3+0=3 D) 3(1) = 3 Which statement best illustrates the inverse property of multiplication? A) 2 • (-½) = -1 B) 2•1=2 C) 2•0=0 D) 2•½= 37 Student’s Worksheet – 10 Content Worksheet (C6) Distributive Property of Integers 1. Draw tens-sticks and ones-dots to illustrate the numbers. Then use the distributive property to multiply. One is done for you. Illustrate the next three sums on its basis. 5 × 23 5 × 20 = 5×3= 5 × 23 = 100 + 15 = 115 2 × 65 38 3× 58 4 × 27 39 2. Break the second number (factor) into tens and ones. Multiply in parts (tens and ones separately), and add. 5 × 17 = 5 × (10 + 7). 8 × 41 = 8 × (__ + _). 5 × 10 = 5×7= 8 × 40 = 8×1= + 5 × 17 = 5 × (__ + _). 5× 5× + 6 × 73 = 6 × (__ + _). + 3. Now break the second number (factor) into hundreds, tens and ones. Multiply in parts (hundreds, tens, and ones separately), and add. a. 5 × 123 5 × 100 = 5 × 20 = 5×3= b. 8 × 115 8 × 100 = 8 × 10 = 8×5= + g. 5 × 149 + h. 7 × 106 + + 40 Student’s Worksheet – 11 Content Worksheet (C7) Independent Practice Task 1: Match the following Additive inverse of 9 1 +0 Multiplicative identity -2 x 1 10 Additive inverse of -6 0 x 10 Multiplicative inverse of -5 -3 + 9 Additive inverse of 0 3 x (-3) Task 2 :Solve and arrange in ascending orderAdditive inverse of -2, additive inverse of 3, multiplicative identity of 19, additive identity of -5 and multiplicative inverse of -1. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ _______________________________________________________________ ________________________________________________________________ 41 Task 3 : Sunny and Bunny are two rabbits. They want to pick pairs of two apples such that number on one apple is additive inverse of the number on the other apple. Colour such pairs of apples with same shade and help them. How many such pairs are their ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 42 Task 5: Answer the following questions. Also write 5 more examples of your own. Question Answer 1. (50/3) 2. 3. 4. 5. 6. 7. = 8. 1 43 Student’s Worksheet – 12 Content Worksheet (C8) Division of Integers Name of the student ______________________ Date ____________ Activity -Integer Grid does Division Trickkkk…. Find 3 integers in a row, column or a diagonal, such that when you divide the second number with the first one then you get the third number. One is already done. Here, (–90) (–30) = 3 Find at least 5 more such relations and record them here. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 44 Recall that absolute value of a Task: Absolute Star number n is |n|. The absolute value of a number is always a positive number (or zero). Solve the following absolute values: a) b) c) d) e) f) g) h) -|-6| |-12|-|-5| |-12|+ |-2| |-21| + |5-12| |0| ÷|-6| |11-5| ×|1-5| |-21| +|21| |-1-6| -|-17| i) Write your observation about absolute value: The absolute value of an integer is always ______________________than the integer. The absolute value of an integer is never _______________________ than the number. Integers having absolute value 5 are ______________________. 45 Student’s Worksheet – 13 Content Worksheet (C9) Properties of Division of Integers Name of the student ______________________ Date ____________ Activity - Make Me Happy Apply your knowledge of properties of division of integers and find which statements are true. I am Happy : Write the serial number of true statements in the happy face. I am Sad : Write the serial number of false statements in the sad face. Make Me Happy : Correct the false statements and write them in the Big Happy Face and thus make them happy again. 1. The collection of integers is commutative under division. 2. Division is closed in integers. 3. Division is associative in integers. 4. There is no division identity for integers. 5. The distributive prorety of division over addition holds good in integers. 6. (−a) ÷ b = −(a ÷ b). 7. (−4) ÷ 0 = 0. 8. a ÷ (−a) = −1 = (−a) ÷ a 9. Division is distributive over subtraction in integers. 10. 0 ÷ (−a) = 0. 11. 1 is the division identity element. 12. Commutative Property of division does not hold good in integers. 13. {(−8) ÷ (−4)} ÷(−2) ≠ (−8) ÷ {(−4) ÷(−2)}. 46 14. 1 ÷ 0 is not defined. Dear friends, now Make Me Happy again by correcting the false statements. Recall all PROPERTIES OF INTEGERS. 47 Student’s Worksheet – 14 Content Worksheet (C10) Bodmas – Simplification of Brackets Name of the student ______________________ Date ____________ Activity - Find the Mystry Number (A) You get me if 24 is added to, 15 divided by 3. Find me. Now observe that, Mystry Number = (15 ÷ 3) + 24 = 5 + 24 = 29 (B) Based on the above illustration find the BODMAS expression for the following statements and solve them using BODMAS rules. 1. The difference of eight, and four multiplied by twenty. 2. Twenty eight is divided by, four subtracted fromnegative three. 3. The sum of five ,and two divided by negative two. 4. Negative five multipied by negative six, is subtractedfrom quotient of forty fivedivided by negative nine. 5. Four divided by, the difference between ten and six. 6. The additive inverse of fifteen is multiplied by the sum of twelve and additive inverse of forty two. 7. Negative four multiplied by negative ten is subtracted from eight multiplied by negative one. 8. Four divided by, the difference between ten and six. 48 Bodmas expression Working / Solution 49 (C) Solve using the order of operations. Inner most brackets ÷ x + − 1) 3 + 6 − (7 − 8) 2) {(12 − 7) x (2 x 4)} ÷ 8 3) 56 − 24 ‚ (11 − 3) x 8 + 4 4) −7 x 6 − 18 ‚ 3 + 11 (36 ‚ 9 − 6) − 5 5) (15) x (−4) ÷ {(−5) x (−3) x (−2)} _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 50 Student’s Worksheet – 15 Content Worksheet (C11) Independent Practice Name of the student ______________________ Date ____________ Task 1 Solve the following expressions:1. 10 – [ 9 – { 16 ÷ 2 –( 6 – 9)}] 2. 18 – 8 ÷ 4 x 3 3. 7 (14 – 3 – 4) + (–2) 4. (–4) x (–16) ÷ (–4) + (–3) 5. 9 + { 6 + 5 x 3 – ( 7 ÷ (–7))} 6. – 3 –[(–7) x {( –4) + 7 – (–1)}] 7. – 8 – [7 – {2 + 5 – (6 – 7 – 3)}] 8. 9 ÷ (15 – 7 – 5) + 6 9. 25 – 12 ÷ 6 – 3 x 8 10. 8 + [6 – 3 –{ 5 + 2 + 8 ÷ (–2)}] 11. 15 x 4 + 12 ÷ 6 - 9 x 4 _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 51 Task 2 :Place parenthesis in the following problem to make the solution correct. I think …I know who should be treated first. 1. 8 x 0 −7 + 7 −5 = −19 ________________________________ 2. 9 – 4 + 7 – 6 x 2 = 3 ________________________________ 3. 3 – 4 – 18 ÷ – 6 + 12 = 2 ________________________________ 4. –13 + 52 ÷ – 4 – 9 = – 3 ________________________________ 5. – 6 x – 2 ________________________________ –3x–4=1 Task 3: Make bodmas expressions on your own using { } , ( ) , at least five integers having answers: 5 –4 –13 17 52 Student’s Worksheet – 16 Content Worksheet (C12) Word Problems on Integers Name of the student ______________________ Date ____________ Activity - Reality Report 1. In a class test containing 15 questions, 5 marks are given for every correct answer and (–3) marks given for every incorrect answer and 0 marks for question not attempted. Find Martin’s scores if, Gosh!....I will get –3 for every incorrect answer…. It will bring my score down… a) he gets 9 correct and 6 incorrect answers. b) he gets 6 correct and 5 incorrect answers. 2. A shopkeeper earns a profit of Re 1 by selling one pen and incurs a loss of 40p per pencil of her old stock. a) In a particular month she incurs a loss of Rs 5. In this period, she sold 45 pens. How many pencils did she sell in this period. b) In the next month she neither earned profit nor loss. If she sold 70 pens, how many pencils did she sell? _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 53 _______________________________________________________________________ _______________________________________________________________________ 3. If a toy company earns a profit of Rs 20 on sale of a Barbie doll but suffers a loss of Rs 12 on sale of a Drum. a) If company sold 20 dolls and 25 drums in a month then find the profit or loss as the case may be. b) If company sold 15 dolls in a month then find the number of drums it must sell so as to have neither profit no loss. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 4. A submarine was situated 800 feet below sea level. If it descends 250 feet per hour, what is its new position after 6 hours? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 5. The temperature was 14°F at 4P.M. If the temperature dropped by 2°F per hour, what will be the temperature at midnight? __________________________________________________________________________ __________________________________________________________________________ 54 __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 6. An earthworm is 30 feet deep in the well. Every day he climbs 3 feet up the walls of the well but slips down 2 feet at night, as the walls becomes damp. How many days will it take him to climb out of the well? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 7. A submarine left the surface of the water at the rate of 5 m per second. How long will it take the submarine to reach 535m below the sea level? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 8. In a game of ‘Table Polo’, marbles have to be shot into a hole. All players decided that each marble in the hole will be awarded 5 points where as each marble missed will have a penalty of 2 points. Kevin shoots 5 in the hole out of his 12 chances. Find his final score. __________________________________________________________________________ __________________________________________________________________________ 55 __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 9. A water tanker contains 750 liters of water but due to a small hole, water is leaking at the rate of 7 liters per hour. What will be the quantity of water left in the tanker after 15 hours? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 10. An elevator descends into a mine shaft at the rate of 14m/minute. If the descent starts from 15 m above the ground level, how long will it take to reach 265m below the ground level? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 56 Student’s Worksheet – 17 Post Content Worksheet (PC1) Appreciate Your Knowledge Name of the student ______________________ Activity - Speed Drill 1. (-7) x (-5) = 2. (-8) x (10) = 3. (+2) x (-15) = 4. (0) x (-9) = 5. (-13) x (+3) = 6. (-9) x (+2 – 6) = 7. (-12) x (+7) = 8. (+30) (-6) = 9. (-24) (+6) = 57 Date ____________ 10. (-7) (-7) = 11. (+30) x (-2) x (-4) = 12. (+7) x (-9) x (+2) x (-2) = 13. (-24) (-8) = 14. (+8) (+2) = 15. (+15) x (+3) = 16. (+9 – 4 + 2) x (+1) = 17. (1 – 4 +7) x (–5 +8) = 18. (-6) x (-7 + 1) = 19. (-5) x (0) x (+2) x (+8) = 20. {(-6) x (-2)} {(-3) x (-4)} = 58 Activity - Multi Magic (A) Complete the MAGIC SQUARE so that the product of the numbers in each row,column or diagonal is –1728. (B) Complete this multiplication table by using positive and negative numbers. (C) 1. Simplify and name the property used. ( – 756) ( – 131) + 756 2. Verify that a (b+c)=(a ( – 31). b ) + (a where a = – 3, b = –2 and c = 5 . Name the property verified. 59 c) Student’s Worksheet – 18 Post Content Worksheet (PC2) Evaluate Your Knowledge Name of the student ______________________ Task 1: Complete the crossword. 60 Date ____________ Task 2: Match the following:_____ 5x(3+8)=(5x3)+(5x8) 1) Associative Property of Multiplication _____ 0/5=0 2) Associative Property of Addition _____ 4x5=5x4 3) Commutative Property of Addition _____ (3x5)x7=3x(5x7) 4) Identity Property of 0 _____ (6+8)+(4+6)=(4+6)+(6+8) 5) Multiplication and Division Property of 0 _____ (2+5)+5=2+(5+5) 6) Commutative Property of Addition _____ 24/1=24 7) Multiplication and Division Property of 0 _____ 2x9=9x2 8) Associative Property of Multiplication _____ 0+5=5 9) Distributive Property _____ (6x35)x4=6x(35x4) 10) Multiplication and Division Property of 0 _____ 3x0=3 11) Distributive Property _____ (8x5)-(6x5)=5x(8-6) 12) Associative Property of Multiplication _____ 48x1=48 13) Associative Property of Multiplication _____ 15-15=0 14) Identity Property of 1 _____ 19+12=12+19 15) Identity Property of 0 _____ (4x10)x10=4x(10x10) 16) Commutative Property of Multiplication _____ 3x(2+7)=(3x2)+(3x7) 17) Distributive Property _____ (5+12)+18=5+(12+18) 18) Identity Property of 1 _____ 28x0=0 19) Associative Property of Addition _____ (16x10)x10=16x(10x10) 20) Commutative Property of Multiplication 61 Acknowledgments Websites Referred to: 1. http://www.helpingwithmath.com/resources/math_facts.htm 2. http://www.math-drills.com/integers.shtml 3. http://www.ixl.com/math/grade/seventh/ 4. http://www.softschools.com/quizzes/math/multiply_integers/quiz3215.html 5. http://www.col.org/stamp/JSMath3.pdf 6. http://www.softschools.com/grades/6th_and_7th.jsp 7. http://www.homeschoolmath.net/teaching/md/distributive.php 8. http://www.discoveryeducation.com/free-worksheets/properties-of-numbermatching.fun 9. http://www.kwiznet.com Reference Books: 1. NCERT Mathematics for class 7 62 Suggested Video Links Name Title/Link Area and perimeter in real contexts. Video Clip 1 http://www.youtube.com/watch?v=1cuN8e-y-fI 2+1 math rocks introduction to perimeter and area Video Clip 2 http://www.youtube.com/watch?v=D5jTP-q9TgI definition of perimeter and finding perimeter Video Clip 3 http://www.youtube.com/watch?v=Ceqc-Q-tqkc&feature=related Defining perimeter and calculating for irregular shapes http://www.youtube.com/watch?v=XgDC1qb48ps&feature=channel Introducing & Teaching Perimeter & Area Using Graph Paper Video Clip 4 http://www.youtube.com/watch?v=K4I27chrYHc Area of rectangle Video Clip 5 http://www.youtube.com/watch?v=ZTQWVsJ2cxw&feature=related Area and perimeter part 1 http://www.youtube.com/watch?v=YMdRhsdOeZI Video introducing a Perimeter, Area and Volume Unit Video Clip 6 http://www.youtube.com/watch?v=iBtVLnTmEYU Area and perimeter of composite shapes Video Clip 7 http://www.youtube.com/watch?v=97gIIl0NPBY&feature=related http://matti.usu.edu/nlvm/nav/category_g_2_t_4.html Weblink1 http://www6.funbrain.com/poly/index.html Weblink 2 http://www.aaamath.com/geo78-perimeter-rectangle.html - section2 Weblink 3 63 CENTRAL BOARD OF SECONDARY EDUCATION Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India