3. I would like to give my sincere thanks to everyone who all supported me, for I have completed my project effectively and, moreover, on time. I am equally grateful to my teacher Ms. Ridhi Khanna. She gave me moral support and guided me in different matters regarding the topic. She had been very kind and patient while suggesting me the outlines of this project and correcting my doubts. I thank her for her overall support. Last but not the least, I would like to thank my parents who helped me a lot in gathering different information, collecting data and guiding me from time to time in making this project. Despite their busy schedules, they gave me different ideas in making this project unique. Thank you

4. Direct And Inverse Proportion Mathematics Class VIII Duration: 35 minutes

5. 1) Know the basic algebraic equation that represent direct and inverse variation. 2) Apply techniques for classifying the types of variation.

6. At the end of this lesson you will be able to : Define Direct and Indirect proportion. Calculate values directly or indirectly proportional to each other using different method.

7. Mathematics is a creative endeavor. It is a human activity which arises from experience and becomes an integral part of our culture and society, of everyday work and life. Teaching mathematics effectively is quit hard to attain, sometimes student find mathematics as a boring subject they had, but as a pupil teacher, we must prepare ourselves in these problems, we have to be flexible and creative to achieve our goals. Methods is a procedure that one follows in order to achieve goals, it stands for a specified course which serves as a guide in order “ not to get lost on the way.” As a teacher someday, we must have a lots of method know in teaching our subject in order to be more effective and creative in the process of learning.

8. Teaching Aids : Whiteboard, Marker, Pointer, Graphs, Ilustrated books, Projector. Teaching Methods: Inductive Method, Problem solving Method, Lecture Method

9. The act of changing The amount of a change. The extent or degree to which something varies is called variation. The act of changing The amount of a change. The extent or degree to which something varies is called variation.

10.  When we talk about a direct proportion, we are talking about a relationship where, two given units x and y are directly related to each other. If, x increases, y increases Or, x decreases, y decreases at a constant rate.  When we talk about inverse proportion, we are talking about a relationship where, two given units x and y are in inverse proportion. If, x increases, y decreases Or, vice versa.  When we talk about a direct proportion, we are talking about a relationship where, two given units x and y are directly related to each other. If, x increases, y increases Or, x decreases, y decreases at a constant rate.  When we talk about inverse proportion, we are talking about a relationship where, two given units x and y are in inverse proportion. If, x increases, y decreases Or, vice versa.

11. More articles will cost more. More is the articles we have, more heavier it will be. More is the distance covered by a car, more is the petrol consume d by it. More is the money deposited in a bank, more is the interest earned in a fixed period. More is the speed of car, less is the time taken to cover a fixed distance. More is the number of workers, less is the time taken to complete the work. More pipes we have, less time it will take to fill a tank. More is the populatio n of country, less is the area available per person. Examples from daily life

12. Direct Proportion Direct Variation Directly Proportional Direct Proportion Direct Variation Directly Proportional L T o

13. In direct proportion we generally use this formula : Thus, x and y are in direct proportion, if = k, where k is a constant, i.e., In direct proportion we generally use this formula : Thus, x and y are in direct proportion, if = k, where k is a constant, i.e., Quantity (x) x1x1 x2x2 x3x3 Cost (y)y1y1 y2y2 y3y3

14. Ask your friend to fill the following table and find the ratio of his age to the corresponding age of his mother. What do you observe? Do F and M increase (or decrease) together? Is F and M same every time? No! You can repeat this activity with other friends and write down your observations. Thus, variables increasing (or decreasing) together need not always be in direct proportion. For example: (i) physical changes in human beings occur with time but not necessarily in a predetermined ratio. (ii) changes in weight and height among individuals are not in any known proportion and (iii) there is no direct relationship between the height of a tree and the number of leaves growing on its branches. Think of some more similar examples. Age 5 year agoPresent ageAge after 5 year Friend’s age (F) Mother’s age (M) F/M

15. Example 1 If it costs Rs. 85 for 5 bars, what is the cost of 3 bars ? Solution. Cost of 1 bar :Find the cost of 1 bar ? 85  5 = Rs. 17 Find the cost of 3 bars ? 17 x 3 = Rs. 51 3 bars cost Rs 51.

16. The Cross-Multiplication Method. This method is a more sophisticated way of solving direct proportion questions but it has two advantages: It establishes a very strong routine to solve the problem. It makes Inverse Proportion questions easier to handle. Using cross multiplication, we get Let

17. Proportion Method Example 2 If y varies directly to x,and y=6 when x=5, then find y when x=15. Example 2 If y varies directly to x,and y=6 when x=5, then find y when x=15. Solution

18. Now lets solve using the equation Either method gives the correct answer, choose the easiest for you.

19. Eg. 3) Observe the table given below and find whether x and y are directly proportional: Sol.) Clearly, therefore, x and y are directly proportional x 357912 Y 610141824

20. Example 4) If x and y are directly proportional, find the values of x 1, x2 and y1. in the table given below. Sol. since x and y are directly proportional, we have : Now, X3x1x1 x2x2 10 Y 366096 y1y1

21. Direct Proportion (i) Two quantities x and y are said to be in direct proportion if whenever the value of, then the value of in such a way that the ratio remains constant. (ii) When x and y are in direct proportion, we have:

22. Q1) Observe the table given below and find whether x and y are in proportional: (i) (ii) Q2) If x and y are directly proportional, find the values of x 1, x 2 and y 1 in the table given below: Q3) If 18 dolls cost Rs 630, how many dolls can be bought for Rs 455 ? Q4) The cost of 15 metres of a cloth is Rs 981. What length of this cloth can be purchased for Rs 1308 ? Q1) Observe the table given below and find whether x and y are in proportional: (i) (ii) Q2) If x and y are directly proportional, find the values of x 1, x 2 and y 1 in the table given below: Q3) If 18 dolls cost Rs 630, how many dolls can be bought for Rs 455 ? Q4) The cost of 15 metres of a cloth is Rs 981. What length of this cloth can be purchased for Rs 1308 ? X3581126 Y915243378 x5791518 Y1521276072 X3 x 1 x2 10 Y72120192 y1

23. Q4) A taxi charges a fare of Rs1275 for a journey of 150 km. How much would it charge for a journey of 124km? Q5) A car is travelling at the average speed of 50 km/hr. How much distance would it travel in 1 hour and 12 minutes ? Q6) If the thickness of a pile of 12 cardboards is 64 mm, find the thickness of a pile of 312 such cardboards ? Q7) Reena types 540 words during half an hour. How many words would she type in 8 minutes ? Q8) A vertical pole 5 m 60 cm high casts a shadow 3 m 20 cm long. Find at the same time (i) the length of shadow cast by another pole 10 m 50 cm high. (ii) the height of a pole which casts a shadow 5 m long. Q9) A mixture of paint pigment is prepared by mixing 1 part of red pigments (x) with 8 parts of base (y). In the following table, find the parts of base that need to be added, Q10) Suppose 2 kg of sugar contains 9 x 10 6 crystals. How many sugar crystals are there in (i) 5 kg of sugar ? (ii) 1.2 kg of sugar ? Q4) A taxi charges a fare of Rs1275 for a journey of 150 km. How much would it charge for a journey of 124km? Q5) A car is travelling at the average speed of 50 km/hr. How much distance would it travel in 1 hour and 12 minutes ? Q6) If the thickness of a pile of 12 cardboards is 64 mm, find the thickness of a pile of 312 such cardboards ? Q7) Reena types 540 words during half an hour. How many words would she type in 8 minutes ? Q8) A vertical pole 5 m 60 cm high casts a shadow 3 m 20 cm long. Find at the same time (i) the length of shadow cast by another pole 10 m 50 cm high. (ii) the height of a pole which casts a shadow 5 m long. Q9) A mixture of paint pigment is prepared by mixing 1 part of red pigments (x) with 8 parts of base (y). In the following table, find the parts of base that need to be added, Q10) Suppose 2 kg of sugar contains 9 x 10 6 crystals. How many sugar crystals are there in (i) 5 kg of sugar ? (ii) 1.2 kg of sugar ? x1471220 Y8????

24. Take a clock and fix its minute hand at 12. Record the angle turned through by the minute hand from its original position and the time that has passed, in the following table: Take a clock and fix its minute hand at 12. Record the angle turned through by the minute hand from its original position and the time that has passed, in the following table: Time passed (T) (in minutes) (T1) 15 (T2) 30 (T3) 45 (T4) 60 Angle turned (A) (in degree) (A1) 90 (A2)(A3)(A4)

25. What do you observe about T and A? Do they increase together? Is same every time? Is the angle turned through by the minute hand directly proportional to the time that has passed? Yes! From the above table, you can also see T1 : T2 =A1 : A2, because T1 : T2 = 15 : 30 = 1:2 A1 : A2 = 90 : 180 = 1:2 Check if T2 : T3 = A2 : A3 and T3 : T4 = A3 : A4 You can repeat this activity by choosing your own time interval.

26. National Council Of Educational Research And Training, Mathematics, ( Textbook for class VIII) R.S Aggarwal, Msc. PhD, Mathematics for class VIII http://www.learnnext.com/CBSE/Class- 8/Maths/Direct-and-Inverse- Proportions/Direct-and-Inverse- Proportions/L-2242.htm

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