1. We are learning to: - Enhance Mathematical basic skills knowledge. (Which PLT skills?) -Accurately solve inverse 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 = 2x + 5 Calculate the compound interest for amount of £5000 invested for 5 years at 4.2% interest. (a) Factorise fully 9x – 12xy (b) Simplify (4a b) x 4 ÷ -5 - 18 - c y - c = mx = y - 5 2 Write in standard form: 2) 3) Work out the volume 6m 10m 4) Write down all the integer values of x satisfying this inequality –9 ≤ 2x < 9 5) Solve: 3x - 4y = 18 2x - 5y = 19 6) 2 2 - 4 y - 5 = 2x 2 x = y - 5 2 x 2 2 = x 2 - c x = y - 5 2 x = 2 0.0000247 = 2.47x 10 -5 Volume = pi x radius x radius x length Volume = 3.14 x 3 x 3 x 10 Volume = 282.6m 3 –9 ≤ 2x < 9 –4.5 ≤ x < 4.5 This means that x is greater than or equal to -4.5 but less than 5 -4, -3, -2, -1, 0, 1, 2, 3, 4 15x - 20y = 90 8x - 20y = 76 - 7x = 14 x = 2 4 - 5y = 19 -5y = 15 y = -3 3 3 = ( ) 3 x 3 x - 4y = 64 a 9 b 3 Compound Interest = a x (1 + p) n = 5000 x(1 + 0.042 ) 5 = £6141.98

3. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills EXAMPLE 1 INVERSE PROPORTION If M is inversely proportional to R and M = 9 when R = 4. Find the value of: Means M is inversely proportional to R k is the constant of proportionality Substitute in the values of M and R 9 = Substitute k back into the original equation Substitute in R = 3 to work out M M = 12 (a) M when R = 3(b) R when M = 9 4 k M α R 1 M = R k k = 36 M = R M = 3 M = R Substitute in M = 9 to work out R 9 = R R = 9 R= 4

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 inversely proportional to M. If T = 6 and M = 2, find: (a) T when M = 6(b) M when T = 4.8 W is inversely proportional to F. If W = 5 and F = 12, find: (a) W when F = 3(b) F when W = 10 ÷ 12 INVERSE PROPORTION 5 = W =20 12 k W α F 1 W = F k k = 60 W = F W = 3 M = R 36 9 = R R = 9 R= 4 x 2 F = W = F 10 = F 60 6 F = 10 60 x 12

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 inversely with (5 – t). If Q = 8 when t = 3, find: (a) Q when t = 9(b) t when Q = 16 M varies inversely with t. If M = 9 when t = 2, find: (a) M when t = 3(b) t when M = 1.44 T α M T = kM 30 = k 5 30 = 5k ÷ - 16 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 INVERSE PROPORTION 2 x 4 Q = Q = 9 = M =4 2 k M α t 1 M = t k k = 36 M = t M = 3 F = W = F 10 = F 60 6 F = 10 60 2 2 2 9 = 4 k 2 2 M = 9 36 M =4 M = t M = 3 2 2 M = 9 1.44 = t 36 2 1.44 = t 36 2 t= t =5

6. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills EXAMPLE 2 The grant available to a section of society was inversely proportional to the number of people needing the grant. When 15 people needed the grant, they received £90 each. (a) What would the grant have been if 150 people had needed one? (b) If the grant had been £50 each, how many people would have received it? INVERSE PROPORTION Means G is inversely proportional to P k is the constant of proportionality Substitute in the values of G and P 90 = Substitute k back into the original equation Substitute in P = 150 to work out G G = £9 15 k G α P 1 G = P k k = 1350 G = P G = 150 1350 G = P Substitute in G = 50 to work out P 50 = P 1350 P = 50 1350 P= 27 people x 15

7. 2 2 Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills A new engine was being tested, but it had serious problems. The distance it went, in km, without breaking down was inversely proportional to the square of its speed in m/s. When the speed was 15m/s the engine lasted 4km. Substitute in the values of C and r 68 = k 68 = 4k (a) Find the distance covered before a breakdown, when the speed is 20m/s. (b) On one test, the engine broke down after 5.75km. What was the speed? C α r 2 C = kr 2 2 2 k =17 C = Substitute in r = 2.4 to work out W C = 17x r 2 Substitute k back into the original equation C =kr 2 2.4 2 C = 17x 5.76 C = 97.92 C = 98p C = 17 r 2 153p 153 = 17x r 2 r 2 = 9 r = 3 r = 3cm INVERSE PROPORTION EXTENSION 1 (GRADE A) Q = 9 = M =4 2 k M α t 1 M = t k k = 36 M = t M = 3 2 2 2 9 = 4 k 2 2 M = 9 M = 4 M = 3 2 M = 9 5.75 = S 900 2 5.75 = S 900 2 S= S =12.5m/s (1d.p.) 9 = M =4 2 k M α t 1 M = t k k = M = t 36 M = 3 2 2 2 9 = 4 k 2 2 M = 9 900 4 = D = 2.25km 15 k S D α 1 D = S k k = D = S 900 D = 20 900 2 4 = 225 k 2 2 D = 400 900 x 225 D = S 900 2 5.75 900

8. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills The amount of waste which a firm produces, measured in tonnes per hour, is inversely proportional to the square root of the filter beds, in square metres. The firm produces 2.25 tonnes of waste per hour, with filter beds of size 0.64m. (a) The filter beds used to be only 0.04m. How much waste did the firm produce then? (b) What are the size of the filter beds if the waste produced was 4 tonnes? x 0.8 INVERSE PROPORTION EXTENSION 2 (GRADE A) 2 2 128 = 4k W α 1.8 = k = 128 k√16 32 = k X = 32√F X = 32 x √49 X = 224 X = 32 x 7 √F 1 W = √F k 2.25 = √0.64 k 2.25 = 0.8 k W = √F 1.8 W = √0.04 1.8 W = 0.2 1.8 W = 9 tonnes W α 1.8 = k √F 1 W = √F k 2.25 = √0.64 k 2.25 = 0.8 k W = √F 1.8 W = √0.04 1.8 W = 0.2 1.8 W = 9 tonnes W = √F 1.8 F = 0.2025m 4 = √F 1.8 = 4 √F = 0.45 2

9. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills MINI-PLENARY – FUNCTIONAL MATHS QUESTION The fuel consumption, in miles per gallon (mpg) of a car is inversely proportional to its speed, in miles per hour (mph). When the car is travelling at 50mph the fuel consumption is 70mpg. How much further would the car travel on 1 gallon of fuel by travelling at 60mph instead of 70mph on a motorway? ÷ 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 INVERSE PROPORTION 70 = 50 k C α S 1 C = S k k = 3500 C = S 60mph C = S 3500 C = 60 C = 58.3 mpg (1d.p.) 70mph C = S 3500 C = 70 C = 50 mpg The car would travel 8.3 miles further by travelling at 60mph rather than 70mph.

10. 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 solve inverse proportion/variation problems. What grade are we working at?

11. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills PLENARY 1 – PROBLEM SOLVING QUESTION Which statement is represented by the graph? Give a reason for your answer. (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 (b) y α (c) y α √x It is (b) because as x increases, y decreases. This is inverse proportion. (b) INVERSE PROPORTION x 1

12. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills PLENARY 2 – EXAM QUESTION The frequency, f, of sound is inversely proportional to the wavelength, w. A sound with a frequency of 36 hertz has a wavelength of 20.25 metres. Calculate the frequency when the frequency and the wavelength have the same numerical value. edf V α h 0.5 = k C = 0.2857A 24 = 0.2857A 84 = A 2 3 =10 500 k 3 INVERSE PROPORTION (4 marks) 36 = 20.25 k f α w 1 f = w k x f k = 729 f = w This means that f = w f = w f = f f 2 = f = 27 hertz

13. Draw your brain What have you learnt? In your brain, write or draw everything you can remember about solving inverse 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?

14. . 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

15. 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

16. 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 inversely proportional to M. If T = 6 and M = 2, find: (a) T when M = 6(b) M when T = 4.8 W is inversely proportional to F. If W = 5 and F = 12, find: (a) W when F = 3(b) F when W = 10 ÷ 12 INVERSE PROPORTION 5 = W =20 12 k W α F 1 W = F k k = 60 W = F W = 3 M = R 36 9 = R R = 9 R= 4 x 2 F = W = F 10 = F 60 6 F = 10 60 x 12

17. 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 inversely with (5 – t). If Q = 8 when t = 3, find: (a) Q when t = 9(b) t when Q = 16 M varies inversely with t. If M = 9 when t = 2, find: (a) M when t = 3(b) t when M = 51.84 T α M T = kM 30 = k 5 30 = 5k ÷ - 16 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 INVERSE PROPORTION 2 x 4 Q = Q = 9 = M =4 2 k M α t 1 M = t k k = 36 M = t M = 3 F = W = F 10 = F 60 6 F = 10 60 2 2 2 9 = 4 k 2 2 M = 9 36 M =4 M = t M = 3 2 2 M = 9 1.44 = t 36 2 1.44 = t 36 2 t= t =5

18. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills A new engine was being tested, but it had serious problems. The distance it went, in km, without breaking down was inversely proportional to the square of its speed in m/s. When the speed was 15m/s the engine lasted 4km. Substitute in the values of C and r 68 = k 68 = 4k (a) Find the distance covered before a breakdown, when the speed is 20m/s. (b) On one test, the engine broke down after 5.75km. What was the speed? C α r 2 C = kr 2 2 2 k =17 C = Substitute in r = 2.4 to work out W C = 17x r 2 Substitute k back into the original equation C =kr 2 2.4 2 C = 17x 5.76 C = 97.92 C = 98p C = 17 r 2 153p 153 = 17x r 2 r 2 = 9 r = 3 r = 3cm INVERSE PROPORTION EXTENSION 1 (GRADE A) Q = 9 = M =4 2 k M α t 1 M = t k k = 36 M = t M = 3 2 2 2 9 = 4 k 2 2 M = 9 M = 4 M = 3 2 M = 9 9 = M =4 2 k M α t 1 M = t k k = M = t 36 M = 3 2 2 2 9 = 4 k 2 2 M = 9 x 225

19. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills The amount of waste which a firm produces, measured in tonnes per hour, is inversely proportional to the square root of the filter beds, in square metres. The firm produces 2.25 tonnes of waste per hour, with filter beds of size 0.64m. (a) The filter beds used to be only 0.04m. How much waste did the firm produce then? (b) What are the size of the filter beds if the waste produced was 4 tonnes? x 0.8 INVERSE PROPORTION EXTENSION 2 (GRADE A) 2 2 128 = 4k W α 1.8 = k = 128 k√16 32 = k X = 32√F X = 32 x √49 X = 224 X = 32 x 7 √F 1 W = √F k 2.25 = √0.64 k 2.25 = 0.8 k W = √F 1.8 W = √0.04 1.8 W = 0.2 1.8 W = 9 tonnes

20. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills MINI-PLENARY – FUNCTIONAL MATHS QUESTION The fuel consumption, in miles per gallon (mpg) of a car is inversely proportional to its speed, in miles per hour (mph). When the car is travelling at 50mph the fuel consumption is 70mpg. How much further would the car travel on 1 gallon of fuel by travelling at 60mph instead of 70mph on a motorway? ÷ 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 INVERSE PROPORTION

21. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills PLENARY 1 – PROBLEM SOLVING QUESTION Which statement is represented by the graph? Give a reason for your answer. (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 (b) y α (c) y α √x INVERSE PROPORTION x 1

22. Effective Participator Self Manager Independent Enquirer Creative Thinker Team Worker Reflective Learner Which ones are you using?PLT Skills PLENARY 2 – EXAM QUESTION The frequency, f, of sound is inversely proportional to the wavelength, w. A sound with a frequency of 36 hertz has a wavelength of 20.25 metres. Calculate the frequency when the frequency and the wavelength have the same numerical value. edf V α h 0.5 = k C = 0.2857A 24 = 0.2857A 84 = A 2 3 =10 500 k 3 INVERSE PROPORTION (4 marks) x f