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

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

3. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills EXAMPLE 1 DIRECT PROPORTION 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 24 = k 6 24 = 6k k =4 Substitute k back into the original equation W =kF W = 4 F Substitute in F = 10 to work out W W = 4x 10 W = 40

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

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

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

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

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

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

10. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills EXTENSION 3 (GRADE A) 1) 2) 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. DIRECT PROPORTION T α M ÷ 1.8 W α F W = kF 36 = k 3 36 = 3k 12 = k W = 12 x 5 W = 60 (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? 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. (b) A ball bearing has a mass of 48.6g. What is the radius? (a) What will be the mass of a ball bearing of radius 8mm? W = 3M 2 75 = 3M 2 M = 5 25 = M 2 5 = M X = 32√F 48 = 32√F F = 2.25 2.25 = F 1.5 = √F 37.50 = 2.5√p p = 225 people 15 = √p 225 = p m α r 3 m = kr 3 =4 115.2 k 3 1.8 = k m = 1.8r 3 3 = r r = 3mm = 64k 115.2 P = 0.4f 3 f = 5 48.6 = 1.8r 3 27 = r 3

11. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills MINI-PLENARY – FUNCTIONAL MATHS QUESTION DIRECT PROPORTION (b) A playgroup has 24 children. What is the smallest playground area in which they could safely play? 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 ? 2 ÷ 0.2857 C α A C = kA 60 = k 210 60 = 210k 0.2857 = k C = 0.2857A A = 84m 24 = 0.2857A 84 = A 2

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

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

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

15. Draw your brain What have you learnt? 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. What grade are we working at? Where are we in our journey?

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

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

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

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

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

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

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

23. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills EXTENSION 3 (GRADE A) 1) 2) 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. DIRECT PROPORTION (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? 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. (b) A ball bearing has a mass of 48.6g. What is the radius? (a) What will be the mass of a ball bearing of radius 8mm? M = 5

24. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills MINI-PLENARY – FUNCTIONAL MATHS QUESTION DIRECT PROPORTION (b) A playgroup has 24 children. What is the smallest playground area in which they could safely play? 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 ? 2

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

26. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills PLENARY 2 – EXAM QUESTION DIRECT PROPORTION (b) Find the volume of a hot-air balloon which has a height of 30 metres. (1 mark) 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) (c) Another hot-air balloon has a volume of 5000 cubic metres. Find it’s height. (3 marks)