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Enhancing Mathematical Skills: Direct Proportion Problems

Learn to accurately solve direct proportion/variation problems and enhance your mathematical skills. Practice writing equations, solving for unknowns, and using standard form. Explore real-life cross/curricular connections.

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Enhancing Mathematical Skills: Direct Proportion Problems

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  1. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Real life cross/curricular links? Where are we in our journey? Which ones are you using? PLT Skills LESSON OBJECTIVES Always aim high! We are learning to: • Enhance Mathematical basic skills knowledge. (Which PLT skills?) • Accurately solve direct proportion/variation problems. (Grade A). AUTHOR www.mistrymaths.co.uk

  2. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager STARTER PLT Skills TASK 2) 3) Write in standard form: 1) Make x the subject of: y = mx + c 8m 0.0067 15 3 a b y - c -3 6.7 x 10 = m y - c mx = 20m Work out the volume Solve: 7x + 5y = 22 3x - 2y = 26 5) Volume = pi x radius x radius x length x = x x x 3.14 4 Volume = 20 4 3 1004.8cm Volume = 4) Write down all the integer values of x satisfying this inequality –8 ≤ 2x < 8 14x + 10y = 44 6) 2 (a) Factorise fully 8x – 12xy (b) Simplify (5ab) 15x - 10y = 130 + 174 = 29x ( ) - 2 y x 4 x 3 = –8 ≤ 2x < 8 x 3 6 5 = ÷ m - 18 x 5 - c ≤ 4 –4 x < 18 - 2y = 26 This means that x is greater than or equal to -4 but less than 3 125 = -2y = 8 -4, -3, -2, -1, 0, 1, 2, 3 -4 y = EXTENSION Calculate the compound interest for amount of £4000 invested for 4 years at 3.2% interest. n 4 ) = Compound Interest = a x (1 + p) x ( 1 = 0.032 £4537.10 4000 +

  3. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION EXAMPLE 1 If W varies directly with F and when W = 24 , F = 6. Find the value of W when F = 10. W α F Means W is directly proportional to F W = kF k is the constant of proportionality Substitute in the values of W and F = 6 24 k 24 = 6k k 4 = Substitute k back into the original equation W =kF = W F 4 Substitute in F = 10 to work out W = 10 W 4 x = W 40

  4. T α M T = 6M T = kM 10 = 6M = 5 30 k 30 = 5k 1.67 = M M = 1.67 (2d.p.) Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager 6 = k PLT Skills DIRECT PROPORTION TASK 1 (GRADE A) 1) T is directly proportional to M. If T = 30 and M = 5, find: (a) T when M = 3 (b) M when T = 10 W α F W = 12F W = kF 90 = 12F = 3 36 k 36 = 3k 7.5 = F F = 7.5 12 = k 2) W is directly proportional to F. If W = 36 and F = 3, find: (a) W when F = 5 (b) F when W = 90 ÷ 12 W α F W = kF W = 12 x 5 = 3 36 k W = 60 36 = 3k 12 = k

  5. T α M T = 6M T = kM 10 = 6M = 5 30 k 30 = 5k 1.67 = M M = 1.67 (2d.p.) Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager 6 = k PLT Skills DIRECT PROPORTION TASK 2 (GRADE A) 1) Q varies directly with P. If Q =100 when P = 2, find: (a) Q when P = 4 (b) P when Q = 400 X α Y X = 2.5Y X = kY X = 2.5 x 11 = 7 17.5 k Q = 27.5 17.5 = 7k 2.5 = k 2) X varies directly with Y. If X = 17.5 when Y = 7, find: (a) X when Y = 11 (b) Y when X = 40 X α Y X = 2.5Y X = kY ÷ 2.5 40 = 2.5Y = 7 17.5 k 17.5 = 7k 16 = Y Y = 16 2.5 = k W α F W = kF W = 12 x 5 = 3 36 k W = 60 36 = 3k 12 = k

  6. T α M T = 6M T = kM 10 = 6M = 5 30 k 30 = 5k 1.67 = M M = 1.67 (2d.p.) Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager 6 = k PLT Skills DIRECT PROPORTION TASK 3 (GRADE A) 1) The distance covered by a train is directly proportional to the time taken for the journey. The train travels 105 miles in 3 hours. C α W (b) How much time will it take for the train to cover 315 miles? (a) What distance will the train cover in 4 hours? C = 0.19W C = kW 34.20 = 0.19W = 250 47.50 k 47.50 = 250k 180 = W W = 180kg 0.19 = k 2) The cost of fuel delivered to your door is directly proportional to the weight received. When 250kg is delivered, it costs £47.50. X α Y X = 2.5Y X = kY ÷ 0.19 (a) How much will it cost to get 450kg delivered? (b) How much would be delivered if the cost were £34.20 40 = 2.5Y = 7 17.5 k 17.5 = 7k 16 = Y Y = 16 2.5 = k W α F W = kF W = 12 x 5 = 3 36 k W = 60 36 = 3k 12 = k

  7. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION EXAMPLE 2 The cost of a circular badge is directly proportional to the square of it’s radius. The cost of a badge with a radius of 2cm is 68p. Find: (b) The radius of a badge costing £1.53 (a) The cost of a badge of radius 2.4cm 153p C α r 2 2 C = kr Substitute in the values of C and r Substitute in the values of C and r = 153 17 x 2 = 2 68 k 68 = 4k 9 = k 17 = ÷ 17 3 r = 2 Substitute k back into the original equation C =kr = r 3cm 2 2 = C r r 17 = C 17 Substitute in r = 2.4 to work out W 2 2 2 = 2.4 r r C 17 x = 5.76 C 17 x = C = 98p C 97.92

  8. T α M T = 6M T = kM 10 = 6M = 5 30 k 30 = 5k 1.67 = M M = 1.67 (2d.p.) Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager 6 = k PLT Skills DIRECT PROPORTION X α √F EXTENSION 1 (GRADE A) X = k√F X = 32√F 2 1) W is directly proportional to M . If W = 12 when M = 2, find: (a)W when M = 4 (b) M when W = 75 X = 32 x √49 X = 32 x 7 128 = 4k X = 224 32 = k 2 W α M 2 W = kM X = 32√F k√16 2 128 12 = = 2 k 48 = 32√F 1.5 = √F 2) X is directly proportional to √F. If X = 128 and F = 16, find: (a) X when F = 49 (b) F when X = 48 2.25 = F 2 2 2 2 2 2 2 2 W = 3M 75 = 3M 25 = M 25 = M W = 3M W = 3 x 4 75 = 3M W = 3M 5 = M ÷ 32 F = 2.25 M = 5 square 12 = 4k W = 48 3 = k 5 = M M = 5

  9. T α M T = 6M T = kM 10 = 6M = 5 30 k 30 = 5k 1.67 = M M = 1.67 (2d.p.) Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager 6 = k PLT Skills DIRECT PROPORTION 3 y α √b EXTENSION 2 (GRADE A) 3 3 y = k√b y = 20√b 3 1) P is directly proportional to f . If P = 400 when f = 10, find: (a)P when f = 6 (b) f when P = 50 3 3 100 = k√125 40 = 20 x √b 100 = 5k 3 2 = √b 3 20 = k P α f 3 P = kf 8 = b 3 400 400 = = 1000k 10 k b = 8 3 2) y is directly proportional to √b. If y = 100 when b = 125, find: (a) y when b = 8 (b) b when y = 40 F = 2.25 2.25 = F 3 2 2 3 3 2 3 3 125 = f 50 = 0.4f P = 0.4f 75 = 3M P = 0.4 x 6 25 = M P = 0.4f W = 3M 5 = M cube ÷ 20 M = 5 X = k√F X = 32√F X = 32 x √49 X = 32√F X = 32 x 7 P = 86.4 0.4 = k 5 = f f = 5

  10. T α M 37.50 = 2.5√p 15 = √p Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills 225 = p DIRECT PROPORTION p = 225 people EXTENSION 3 (GRADE A) 1) The cost of serving tea and biscuits varies directly with the square root of the number of people at the buffet. It costs £25 to serve tea and biscuits to 100 people. (b) For a cost of £37.50, how many could be served tea and biscuits? (a) How much will it cost to serve tea and biscuits to 900 people? 3 m α r 3 m = kr 3 115.2 115.2 = = 64k 4 k 3 = r 1.8 = k r = 3mm 2) The mass, in grams, of ball bearings varies directly with the cube of the radius, measured in millimetres. A ball bearing of radius 4mm has a mass of 115.2g. 27 = r 3 2 2 3 2 3 3 m = 1.8r P = 0.4f 48.6 = 1.8r 25 = M W = 3M 75 = 3M ÷ 1.8 (a) What will be the mass of a ball bearing of radius 8mm? (b) A ball bearing has a mass of 48.6g. What is the radius? f = 5 W α F X = 32√F W = kF 48 = 32√F W = 12 x 5 = 3 36 k W = 60 1.5 = √F 36 = 3k 2.25 = F 12 = k 5 = M F = 2.25 M = 5

  11. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION MINI-PLENARY – FUNCTIONAL MATHS QUESTION The number of children who can play safely in a playground is directly proportional to the area of the playground. A playground with an area of 210m is safe for 60 children. 2 C α A (a) How many children can play safely in a playground of area 154m ? (b) A playgroup has 24 children. What is the smallest playground area in which they could safely play? 2 C = 0.2857A C = kA 24 = 0.2857A = 210 60 k 60 = 210k 84 = A 0.2857 = k 2 A = 84m ÷ 0.2857

  12. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager • LINK BACK TO OBJECTIVES • Accurately solving direct proportion/variation problems. What grade are we working at? DISCOVERY PLT Skills

  13. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION PLENARY 1 – PROBLEM SOLVING QUESTION The sketched graphs show each of these proportion statements. (a) y α x (b) y α x 2 (c) y α √x C α A C = 0.2857A C = kA 24 = 0.2857A = 210 60 k 60 = 210k 84 = A 0.2857 = k 2 A = 84m y y y ÷ 0.2857 (a) (b) (c) Match each statement to the correct sketch. 0 0 0 x x x

  14. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION 3 V α h PLENARY 2 – EXAM QUESTION The volume, V cubic metres, of a hot-air balloon is proportional to the cube of its height, h metres. A balloon with a height of 10m has a volume of 500 cubic metres. (a) Find an equation connecting V and h. (3 marks) (b) Find the volume of a hot-air balloon which has a height of 30 metres. (1 mark) 3 V α h 3 C = 0.2857A V = kh 3 3 24 = 0.2857A 500 500 = = 10 10 k k 500 = 1000k V = 0.5 x 27000 84 = A V = 13500m 3 0.5 = k 0.5 = k 2 A = 84m 3 3 3 3 3 5000 = 0.5h V = 0.5h V = 0.5h V = 0.5h V = 0.5 x 30 3 10000 = h edf (c) Another hot-air balloon has a volume of 5000 cubic metres. Find it’s height. (3 marks) h = 21.5m 21.5 = h 3

  15. What grade are we working at? Where are we in our journey? What have you learnt? Draw your brain In your brain, write or draw everything you can remember about solving direct proportion problems. It can be a skill or a reflection, or something else that might be prominent in your brain.

  16. Team Worker Positive Thinker Creative Entrepreneur Independent Learner Reflective Learner Responsible Citizen SELF ASSESSMENT Enterprise Skills Which ones are you using? Plenary Activity How well do you understand the task? . I fully understand I don’t understand I nearly understand www.mistrymaths.co.uk

  17. Team Worker Positive Thinker Creative Entrepreneur Independent Learner Reflective Learner Responsible Citizen SELF ASSESSMENT Enterprise Skills Which ones are you using? Plenary Activity WWW (What Went Well) EBI (Even Better If) On your post it notes… Think about how you can improve your work. www.mistrymaths.co.uk

  18. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION TASK 1 (GRADE A) 1) T is directly proportional to M. If T = 30 and M = 5, find: (a) T when M = 3 (b) M when T = 10 2) W is directly proportional to F. If W = 36 and F = 3, find: (a) W when F = 5 (b) F when W = 90

  19. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION TASK 2 (GRADE A) 1) Q varies directly with P. If Q =100 when P = 2, find: (a) Q when P = 4 (b) P when Q = 400 2) X varies directly with Y. If X = 17.5 when Y = 7, find: (a) X when Y = 11 (b) Y when X = 40

  20. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION TASK 3 (GRADE A) 1) The distance covered by a train is directly proportional to the time taken for the journey. The train travels 105 miles in 3 hours. (b) How much time will it take for the train to cover 315 miles? (a) What distance will the train cover in 4 hours? 2) The cost of fuel delivered to your door is directly proportional to the weight received. When 250kg is delivered, it costs £47.50. (a) How much will it cost to get 450kg delivered? (b) How much would be delivered if the cost were £34.20

  21. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION EXTENSION 1 (GRADE A) 2 1) W is directly proportional to M . If W = 12 when M = 2, find: (a)W when M = 4 (b) M when W = 75 2) X is directly proportional to √F. If X = 128 and F = 16, find: (a) X when F = 49 (b) F when X = 48 2 2 2 W = 3M 75 = 3M 25 = M 5 = M M = 5

  22. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION EXTENSION 2 (GRADE A) 3 1) P is directly proportional to f . If P = 400 when f = 10, find: (a)P when f = 6 (b) f when P = 50 3 2) y is directly proportional to √b. If y = 100 when b = 125, find: (a) y when b = 8 (b) b when y = 40

  23. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION EXTENSION 3 (GRADE A) 1) The cost of serving tea and biscuits varies directly with the square root of the number of people at the buffet. It costs £25 to serve tea and biscuits to 100 people. (b) For a cost of £37.50, how many could be served tea and biscuits? (a) How much will it cost to serve tea and biscuits to 900 people? 2) The mass, in grams, of ball bearings varies directly with the cube of the radius, measured in millimetres. A ball bearing of radius 4mm has a mass of 115.2g. (a) What will be the mass of a ball bearing of radius 8mm? (b) A ball bearing has a mass of 48.6g. What is the radius? M = 5

  24. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION MINI-PLENARY – FUNCTIONAL MATHS QUESTION The number of children who can play safely in a playground is directly proportional to the area of the playground. A playground with an area of 210m is safe for 60 children. 2 (a) How many children can play safely in a playground of area 154m ? (b) A playgroup has 24 children. What is the smallest playground area in which they could safely play? 2

  25. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION PLENARY 1 – PROBLEM SOLVING QUESTION The sketched graphs show each of these proportion statements. (a) y α x (b) y α x 2 (c) y α √x y y y Match each statement to the correct sketch. 0 0 0 x x x

  26. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills DIRECT PROPORTION PLENARY 2 – EXAM QUESTION The volume, V cubic metres, of a hot-air balloon is proportional to the cube of its height, h metres. A balloon with a height of 10m has a volume of 500 cubic metres. (a) Find an equation connecting V and h. (3 marks) (b) Find the volume of a hot-air balloon which has a height of 30 metres. (1 mark) (c) Another hot-air balloon has a volume of 5000 cubic metres. Find it’s height. (3 marks)

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