Key Stage 3 Mathematics. Level by Level. Pack E: Level 8

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1 p Key Stage Mathematics Level by Level Pack E: Level 8 Stafford Burndred ISBN Published by Pearson Publishing Limited 997 Pearson Publishing 996 Revised February 997 A licence to copy the material in this pack is granted to the purchaser strictly within their school, college or organisation. The material must not be reproduced in any other form without the epress written permission of Pearson Publishing. Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa Web site

2 Standard form Means move the decimal point 4 places to the right = Means move the decimal point places to the left = Questions Write as an ordinary number. Write as an ordinary number. Write 80 in standard form. 4 Write in standard form. Answers = = 0.04 Note: In standard form the decimal point is always placed after the first whole number The decimal point has moved places to the left. We write the number in standard form as The decimal point has moved places to the right. We write the number in standard form as Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

3 Standard form Eercises Write the following in standard form: a 87 b 460 c 5.8 d e 9000 f 0.06 g 0.78 h i j k 570 l m 0.00 n 8000 o 0.06 Write the following as ordinary numbers: a b c d e 0 5 f g h i j k l m n o Convert the following into the form z 0 n. Give the value of: (i) z (ii) n a 5800 b 4700 c 64 d 0700 e f 680 g 57. h 4.6 i 0.0 j k l m n 0.07 o Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

4 Using a calculator for powers, roots and standard form Power keys y or y Root keys y or Standard form key EXP What is the value of 5 4? y 5 4 = Calculator keys Answer: 65 What is the value of -? y + = Calculator keys Answer: 0.5 What is the value of 5? 5 = Calculator keys y Answer: Calculate Calculator keys. 8 EXP EXP 6 = The calculator display shows.67 This means.67 0 Questions Calculate the value of: Answers y 8 = 5 6 y 4 = + 7. EXP EXP 7 = Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

5 Using a calculator for powers, roots and standard form Eercises Calculate the value of the following: ( /8) -4 ( /5) - ( /7) ( /) (0.) 4 ( /8) -/ 4 ( /56) -/4 Calculate the value of the following. (Write your answer in standard form to three significant figures.) Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

6 Proportional change To calculate a 6% increase, multiply by.06 [ie ] To calculate a % increase, multiply by. [ie + 0.] To calculate a 6% decrease, multiply by 0.94 [ie ] To calculate a % decrease, multiply by 0.88 [ie - 0.] Eamples A man earns 000 per annum. He receives a 4% increase each year. How much does he earn after five years? Method: = A television costs % VAT. What is the total cost? = 5 Now try this: A television costs 5 including 7.5% VAT. Calculate the cost before VAT was added. 5 is 7.5%. We need to find 00%. It is eample reversed = 00 Question A car is bought for It depreciates by 0% each year. How much is it worth after eight years? (Give your answer to the nearest.) Answer = 6457 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

7 Proportional change Eercises A man earns per annum. He receives a 6% increase each year. How much does he earn after eight years? How many years does it take for his salary to double? A house is bought for Its value increases by % per annum. a How much is the house worth in four years? b How many years will it take for the value to reach ? There are 000 elephants in a national park. The numbers are decreasing by 5% per annum. a How many elephants will be in the park in si years? b When the number of elephants falls below 800 the elephant will be declared an endangered species. How many years will this take? 4 A meal, including 7.5% VAT cost 5.5. How much was the VAT? 5 A woman received an 8% pay rise. Her new pay was How much was her original wage? 6 The value of a car fell by 8% after one year. Its value at the end of the first year was a What was the original value of the car? b The value of the car continues to fall by 8% each year. What is its value after si years? c When the value of the car falls below 400 it is sold for scrap. How many years is the car used before it is sold for scrap? 7 A man invests 4000 at 6% per annum compound interest. How many years does it take for the sum of money to double in value? 8 The value of a car decreases by 8 each year. Its original value is How many years does it take for its value to fall below 000? Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

8 Using algebraic formulae 5a means 5 a a (5a) means 5 a 5 a 8.4 (.6) First find the value of = 5.5 Then find 5.5 =.9 (appro) Brackets Always work out the brackets first. Eample a = 4 b = 5 c = 7 a (b + c) (b + c) (5 + 7) a (b +c) = 4 = 56 Question Given a = 8.4 b =. Find the value of a +b +(a+b) ( (a - b) ) Answer a + b + (a + b) means (8.4 +.) (a - b) means (8.4 -.) = = =.58 (appro) = 04 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

9 Using algebraic formulae Eercises Given a = 8, b = 5, c =.7, d = -.74, e = -0. Evaluate the following: (4a - b) -6a(a + ab) a ( b) 4 π (a) 5 a - b 6 5a + b ab (ab) 7 c d 8 cd c - d 9 cde 0 (de) c d 4cd c (d e) c + d 4 (c - d)(c + d) cd - d c(d - e) 5 a + b 6 a (c - d) (c - d) 5cde 7 (a - b) 8 a - b - π (a + 8b) (c + d) - d c + d ( ) ( ) 9 a + b + c 0 c - d d - e de Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

10 Using algebraic formulae Questions Calculate the value of a, given b =., c = 5.4: a = ((b + c) + bc) Calculate the value of w, given y =, z = 8: w = y - z Calculate the value of r, given v = 90, h = 6: v = πr h Answers (b + c) ( ) (8.6) 7.96 bc (b + c) + bc (b + c) + bc (appro) w= - 8 w = 5 4 w = 4 5 = 4.8 v = / π r h 90 = / π r 6 90 = r r 90 = r = 4.9 r = 4.9 r =.78 (appro) Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

11 Using algebraic formulae Eercises The volume of a cylinder is given by the formula V = πr h. Calculate V given: r = 5 m h = 8 m r = 7 m h = m r = 8 m h = 6 m 4 r = 4 m h = 8 m 5 r =.5 m h =.8 m 6 r = 7.4 m h = 8. m Calculate r given: 7 V = 0 m h = m 8 V = 400 m h = 6m 9 V = 500 m h = 8 m 0 V = 4 m h = 5.4 m V = 764 m h = 7.6 m V = 478 m h =.8 m The surface area of a cylinder is given by the formula A = πr(r + h). Calculate A given: r = m h = 7 m 4 r = 7 m h = 5 m 5 r = 5 m h = 6 m 6 r =.8 m h = 7 m 7 r =.6 m h =.8 m 8 r = 8.4 m h = 6.4 m Calculate h given: 9 A = 60 m r = m 0 A = 00 m r = 4 m A = 00 m r = 5 m A = 58 m r = 7.6 m A = 40 m r = 5.4 m 4 A = 87 m r = 9.7 m The total surface area of a cone is given by the formula A = πr(l + r). l is calculated using the formula l = (h + r ). Calculate A given: 5 r = 5 m h = 7 m 6 r = 4 m h = m 7 r = 8 m h = 5 m 8 r = 4.8 m h =.6 m 9 r = 7. m h = m 0 r =.6 m h = 5. m Calculate the value of t in the formula t = r - s, given r = 7 and s =. Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

12 Transformation of formulae The following questions show several useful techniques. In each question make A the subject. Questions A = B A = B C A = B 4 C = B + A 5 C = B - A 6 A C = B 7 B Y C = 8 AB + C = D 9 B = A A Y - 7 B = 4A - B = AC + D B = C A D + A Answers A = B A = B A = B A = B C A = B 4 C = B + A B A = B + A = C C A = B A = C - B ( C ) A 5 C = B - A 6 C = B C + A = B A = C B A = B - C A = BC 7 C = B A 8 AB + C = D AC = B AB = D - C B A = A = D - C C B 9 B = Y B = A Y B + 7 = AB = A A(B + 7) = Y A = Y A = A = (B+7) Y - 7 A Y - 7 Y - 7 (B) Y - 7 6B C 4A - B = AC + D B = D + A 4A - AC = B + D B(D + A) = C C A(4 - C) = B + D D + A = B A = B + D C A = - D 4 - C B Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

13 Transformation of formulae Eercises Make Y the subject. A = B + Y A = B - Y 4A = CB - Y 4 C = Y D C 5 5A = Y 6 DE = D + Y 7 7C = 8 B + C = A + AB CY B Y D + B 9 C = 0 A + 5C = Y - 7 Y Y - 4 = C 4 - Y = Y ABC = 4 B C = D A B A + D Y c 5 b = 6 a = (y + c) y 7 a + b = y - c 8 y = a + b 9 a - b = y + c 0 d - 4 = c a +y Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

14 Equations The following questions show several useful techniques. In each question find the value of y correct to three significant figures. Questions (y + 4) = 5(y - 7) 5(y + 4) - (y - 4) = (4y - ) y = 5y y + y = = y y+ = 7 y = 8 8 y = 5. Answers (y + 4) = 5(y - 7) 5(y + 4) - (y - 4) = (4y - ) y + = 5y - 5 5y + 0-6y + = y - 6 Note: - -4 = y - 5y = -5-5y - 6y - y = y = -47 -y = y = y = - - y =.9 y =.9 y 7y y = 5y = 9 y = 7(5y + 6) What do we have on this bottom line? 4 = 8. y = 5y + 4 Therefore multiply everything by 8. y - 5y = 4 8(7y) 4 + 8(y) = 8(9) 4 -y = 4 (7y) + 4(y) = 7 4 y = - 4y + y = 7 y = -. 6y = 7 y = 7 6 y = = 7 6 = y y+ 8 = 7y 5 = (y + ) 8 = y 5 = y = y 5-9 = y -4 = y -4 = y -. = y 7 y = 8 8 y = 5. y = 8 y = 5. y =.8 y = 8. Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

15 Equations Eercises Find the value of a in each equation: 4a a = 5 6 = a = 7a + 4 a = a = 9 6 a = a - 4 = a a = Find the value of a correct to three significant figures: 9 a - 5 = 4(5a - ) 0 (a - 4) = (5a - 7) a+ 5 = 7a 6a+ = 7 a a = 6 4 5a a - 4 = 4 5 a - 8 = 5 6 a = (a + ) - (a - ) = 6 8 4(a + ) - (a + 5) = 7 9 4(a + ) + 5(a - ) = 0 0 (a - 4) - 4(a + 6) =0 6 a+ = 5 5 = a a + 4 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

16 Epansion of brackets Questions Epand the following: 5(a - ) a(5-6a) 4y 6 (y + 4y ) 4 -(a +y ) 5 4a b cd (ab 4-6ac d) 6 (a + )(5a - ) 7 (6a - 7)(4a - ) 8 (4y - )(7y + 6) 9 (6a - 4) Answers 5(a - ) a(5-6a) 0a - 5 5a - 8a 4y 6 (y + 4y ) Note: Indices are added 4 -(a +y ) 8y 9 + 6y 8 y 6 +y =y 9. -a - 6y 5 4a b cd (ab 4-6ac d) 6 (a + )(5a - ) a 4 b 6 cd - 4a 4 b c 4 d a(5a - ) + (5a - ) 5a - 9a + 0a - 6 5a + a (6a - 7)(4a - ) 6a(4a - ) - 7(4a - ) Note: -7 - = 4a - 8a - 8a + 4a - 46a + 8 (4y - )(7y + 6) 4y(7y + 6) - (7y + 6) 8y + 4y - y -8 Note: - 6 = -8 8y +y (6a - 4) This means (6a - 4)(6a - 4) 6a(6a - 4) - 4(6a -4) 6a - 4a - 4a + 6 6a - 48a +6 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

17 Epansion of brackets Eercises Epand and simplify the following epressions: 4(a + 5) 7(6 - y) a(7a - ) 4 5a(4a - ) 5 y (y + 6) 6 5 ( + + 4) 7 a b(a - b) 8 5y 6 ( - y 4 ) 9 4( + 6) + (5 - ) 0 7(4a - ) - (a - 4) 7(a - 7) - (5a + 6) 6a(a - ) + 4a (a - a) 5y z (y - yz) 4 (a + )(a + ) 5 (a - 6)(a + ) 6 (a - 5)(a - ) 7 (4a + 6y)(a - 5y) 8 (y - a)(4y + a) 9 (7a - z)(4a - 6z) 0 5abc (c - abc) 7(c - 4d) + 5d 6(7a + d) - (a-d) a - b (a) 4-9 ( + ) 5 (a + 7) 6 ( - ) 7 (a + 4) 8 (5a - ) 9 (5a - y) 0 (4c + a) Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

18 Factorisation You must do epansion of brackets before you do factorisation. Factorisation is the reverse operation to epansion of brackets. Eample Epand 4(a + ) Factorise 8a + 4 8a + 4 4(a + ) Factorising means finding common factors 6a + 5 is a factor of 6 and 5 (a + 5) Questions Factorise: 0c - c 5c d + 0c 5 d 4 6a bc + 4a 4 b d Answers 6 is the highest number that goes into 0 and (ie highest factor) c is the highest power of c that goes into c and c 6c (5c - ) c 5 c d + 0 c 5 d 4 d 5c d ( + 4c d ) c is the highest power of c that goes into c and c 5 d is the highest power of d that goes into d and d 4 is the highest number that goes into 6 and 4 a is the highest power of a that goes into a and a 4 b is the highest power of b that goes into b and b a b(c + a bd) Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

19 Factorisation Eercises Factorise: 5a + 0 y + 6 y a + 6 4a a a + 8a + 4y 9 y + 6y + 0 4y + 6a + 8c a + 6b + 9c a + 5b - 8c 0-5y a - b - 6c 5 7a + 5a 6 a - 7a 7 5y - 7y 8 0a - a 9 6ab - abc 0 7ab - 4bc ab + 8bc + 4bd a c - 8a b 5a 4 + 0a - 5a 4 5a b + 0a 5 b c 5 a c - 8a c + 8ac 6 8a b 6 c - a 5 bd 7 0a 6 b 4 c + 5a 4 bcd 8 5abc + 0abcd 9 8 yz + y 0 y z - 6y 4 z 5 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

20 Inequalities > means greater than < means less than means greater than or equal to means less than or equal to Note: The symbol always points to the smaller number. Questions Describe each of the shaded regions: y y Solve the inequalities: + < 4 6 Answers (if the line is solid it includes equal to ) y 4 y > (if the line is dotted it does not include equal to ) > y Solve as an equation. Subtract from everything: < - < Divide everything by : < 6 4 Remember, if = 6 then can equal 4 or = 6, -4-4 = 6 4, -4 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

21 Inequalities Eercises Describe each of the shaded regions: y y Solve the following inequalities: 49 4 < > < < - < -7 a and b are both integers. Find all of the possible values of b. a b < b a - Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

22 Using the straight line equation y = m + c Eamples y = + 4 y = + up m c = + c means that the line crosses the y ais at + along m = = up the y ais along the ais along - down - -4 y = - - y = - - m c = - m = - c means that the line crosses the y ais at - down the y ais along the ais Questions What is the equation of the line which passes through the points (,) and (,)? Answer Mark the points (,) and (,). Draw a straight line through the points. along down 0 4 The equation of the line is y = m + c m = - (down means -ve) c =.5 y = Note: If the line slopes up the gradient (m) is positive If the line slopes down the gradient (m) is negative Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

23 Using the straight line equation y = m + c Eercises Write down the equations of the lines shown: a What is the equation of the line which passes through the points (-,-) and (4,)? b What is the gradient of the line? 7 a What is the equation of the line which passes through the points (-,-) and (,)? b What is the gradient of the line? 8 a What is the equation of the line which passes through the points (-,5) and (,)? b What is the gradient of the line? Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

24 Graphs You should recognise these graphs. Linear graphs such as y = + 6, y = - +, etc y = y = - Quadratic graphs such as y = + - 6, y = , etc y = y = - Cubic graphs such as y = + + -, y = , etc y = y = - Reciprocal graphs such as y =, y = -, etc y = y = - 4 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

25 Graphs Eercises Label the following graphs. Choose from: y = + y = - + y = + y = - + y = y = - y = / y = - / Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

26 Graphs Eercises This graph shows the journeys made by a car and a bus. Both vehicles travelled from Dorchester to Salisbury. Salisbury Distance in kilometres Bus Car Dorchester Time What time did the bus leave Dorchester? How many times did the bus stop? How long was the first stop? 4 a Between which times did the bus travel fastest? b How did you decide? 5 Describe what happened at How many times did the car pass the bus? 7 How long did the car stop for? 8 What was the speed of the car on the first part of its journey? 9 What was the speed of the bus at 0.0? 0 What was the time of arrival of the bus? Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

27 Graphs Eercises Complete the tables and draw the graphs of the following functions for - : y = y y = y y = y What happens when = 0? 4 y = y Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

28 Shape, Space and Measures Similarity Two shapes are similar if the angles of one shape are equal to the angles of the other shape. Question A Find the length of AB and AE. 5 cm D 6 cm 6 cm E B 9 cm C Answer DE is parallel to BC. Therefore ADE is similar to ABC. Draw the two triangles separately. A A Identify the big triangle and the small triangle. D 5 cm small 6 cm E big 6 cm B 9 cm C Find two sides which are in the same position on each triangle. In this eample DE and BC. DE = 6 cm, BC = 9 cm 9 big number 4 The scale factor (SF) from small to big is. 6 small number 9 To convert any length on the small triangle to a length on the large triangle, multiply by SF 6. 9 eg AD (small triangle) SF 6 = AB 9 5 cm = 7.5 cm 6 ( ) 6 small number 5 The scale factor from big to small is. 9 big number 6 To convert any length on the large triangle to a length on the small triangle, multiply by SF. 9 6 eg AC (large triangle) SF 9 = AE 6 6 cm = 4 cm 9 ( ) Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

29 Shape, Space and Measures Similarity Eercises The triangles ABC and DEF are similar. B X 8 cm E Y 6 cm A cm C D 8 cm F a b c What is the length of BC? What is the length of DE? What is the ratio of the length AX to the length DY? Find a AD b BD A AB = 9 cm c DE AE = 6.4 cm 9 cm 6.4 cm BC = 0 cm CE =.6 cm D E.6 cm B 0 cm C d What is the ratio of length BC to length DE? Y YZ = 0 cm 9 cm V VW = cm X cm 0 cm WZ = 4 cm XV = 9 cm W 4 cm Z Find a XZ b XY Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

30 Shape, Space and Measures Trigonometry: Finding an angle Information similar to this will be given on your eamination paper. Hypotenuse Adjacent Opposite SIN = COS = TAN = OPP HYP ADJ HYP OPP ADJ To find an angle 5 HYP 5 OPP 5 Calculator keys ADJ Find Method Label the triangle Hypotenuse = the longest side, opposite the right angle Opposite = opposite the angle being used Adjacent = net to the angle being used Cross out the side not being used. In this question HYP. Look at the formulae in the bo at the top. Which uses OPP and ADJ? 4 TAN = OPP ADJ 5 = TAN = 5 ( = ) INV INV TAN TAN - Question Do not forget to press equals 8 Top left key on most calculators; it will show Shift, Inv or nd Function Find The answer displayed should be º If it is not displayed, press = Answer HYP ADJ 8 OPP COS = ADJ = HYP 8 = º Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

31 Shape, Space and Measures Trigonometry: Finding an angle Eercises Find Find y y 8. cm 5 cm 8.8 cm 5. cm 0. cm 4 a c Find c 5. cm 7. m.4 m Find a 5 z 6 8. cm 8.5 cm 7. cm 7.6 cm Find z Find d d 7 8 Find Find y y. cm.7 m 5.5 m 4. cm 9 6 cm cm d Find a 0.4 cm 4.8 cm Find d a Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

32 Shape, Space and Measures Trigonometry: Finding a side Information similar to this will be given on your eamination paper. Hypotenuse Adjacent Opposite SIN = COS = TAN = OPP HYP ADJ HYP OPP ADJ To find a side Find 0 m 8º 0 m HYP 8º ADJ OPP Method Label the triangle Hypotenuse = the longest side, opposite the right angle Opposite = opposite the angle being used Adjacent = net to the angle being used You need the side you are finding (). You need the side you know (0 m). Cross out the side not being used. In this question ADJ. Look at the formulae in the bo at the top. Which uses OPP and HYP? 4 SIN = SIN8 = 0 SIN8 = 5 Calculator keys OPP HYP SIN = This should give you an answer m Note: If this does not work ask your teacher to show you how to work your calculator. Question Answer 5 m 58º HYP OPP 5 m ADJ 58º COS = COS58 = 5 COS58 = ADJ HYP 5 = 7.95 m Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

33 Shape, Space and Measures Trigonometry: Finding a side Eercises Find the indicated side: 8 m?? 5º 7 m 4º 4 6º m? 8 m 58º 5 6? 6.8 cm º 4 m?? 8º 7 8? 40º? 5.4 cm 5 m 67º 9 58º 0 40º m??. cm Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

34 Shape, Space and Measures Trigonometry: Solving problems Solving problems This diagram shows a man at the top of a cliff looking down at a boat. 0º This is the angle of depression (looking down) CLIFF This is the angle of elevation (looking up) SEA 0º Note: The angle of depression from the top of the cliff is equal to the angle of elevation from the boat. Angles of depression and angles of elevation are measured from the horizontal. Answering questions Read the question carefully. It may help to visualise what is required. You can use objects such as pencils, rubbers, rulers to make a model of what is required. Draw a diagram. Remember you need a right-angled triangle. 4 Read the question again. Check that your diagram is correct. Question Sarah is flying a kite. The string is 80 m long and the angle of elevation is. How high is the kite? Answer Draw a diagram. º 80 m ADJ HYP? OPP OPP SIN = HYP? SIN = SIN =? = 4.4 m Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

35 Shape, Space and Measures Trigonometry: Solving problems Eercises Andrea is standing 60 m from a building. She looks up to the top. The angle of elevation of the top is 5. What is the height of the building? A helicopter is hovering directly over a police car at a height of 00 m. A criminal is spotted on the ground. The angle of depression of the criminal from the helicopter is. How far is the criminal from: a the police car? b the helicopter? A ship is 5000 m from a vertical cliff, height 000 m. a What is the angle of elevation of the top of the cliff from the ship? b What is the angle of depression of the ship from the top of the cliff? 4 A plane is flying at a height of 8000 m. The pilot looks down at the start of the runway. The angle of depression is. What is the distance from the pilot to the start of the runway? 5 A bridge has two lifting sections. The dotted line shows the position when closed. Both sections are the same length, m. A B m m 40º 50º 4 m River-bed 5 m a How far is point A above the river-bed? b How far is point B above the river-bed? c What is the angle of elevation of A from B? d What is the distance from A to B? Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

36 Shape, Space and Measures Distinguishing between formulae for length, area and volume Length has dimension Area has dimensions Volume has dimensions Length length = area Length length length = volume Length area = volume Length + length = length Area + area = area Volume + volume = volume Different dimensions cannot be added. For eample: Length cannot be added to area Volume cannot be added to area Length cannot be added to volume Numbers, eg, 7, π have no effect on the dimensions. For eample: r = radius r is a length, πr is a length r is an area, πr is an area Questions a, b, c and d are lengths. State whether each formula gives a length, area, volume or none of these. bcd ab ab + cd 4 ab + d a Answers area area volume 4 none of these Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

37 Shape, Space and Measures Distinguishing between formulae for length, area and volume Eercises a, b, c and d are lengths r = radius In each question state whether the formulae gives a length, area, volume or none of these: ab abc d a b 4 4 πr cd 5 a b 6 πr + a c 7 πr + abc 8 πr + πr d abc 9 + cd 0 a b + abd d bc + r ab + cd + r d 4 πr + πr 4 πd + r + ab d b 5 c + d abc 6ab Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

38 Handling Data Cumulative frequency Question This table shows the ages of members of a cricket club: Age Frequency What is the range of the ages? Draw a cumulative frequency diagram. What is the median age? 4 What is the upper quartile? 5 What is the lower quartile? 6 What is the interquartile range? Answer The range is 65-6 = 59 years. First complete a cumulative frequency column. Age Frequency Cumulative Frequency +0 = +0+4 = = = = 80 Note: Points are plotted at the maimum value of the class interval, eg the interval is plotted at (55,75) not (50,75) Cumulative frequency upper quartile median lower quartile Ages 5 lower quartile median 4 upper quartile is is 9 is 47 6 interquartile range 47 - = 6 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

39 Handling Data Cumulative frequency Eercises This table shows the test results (out of 50 marks) in Science. Mark Frequency Cumulative Frequency Complete the cumulative frequency column Cumulative frequency Test marks Complete the cumulative frequency diagram. What is the median mark? 4 What is the upper quartile? 5 What is the lower quartile? 6 What is the interquartile range? 7 The top 5% of pupils receive an A grade. What mark is needed for an A grade? Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

40 Handling Data Using cumulative frequency diagrams to compare distributions Question Two different makes of light bulbs were compared. The cumulative frequency diagrams show the number of hours the bulbs lasted. 00 Type A 00 Type B Cumulative frequency upper quartile median lower quartile Cumulative frequency upper quartile median lower quartile Hours Hours Use the median and interquartile range to compare the two distributions. Answer Different numbers of bulbs were used in the tests but the median and interquartile range allow comparison between the two types of bulb. The interquartile range measures the range of the middle half of the distribution. The median of bulb A is about 800 hours. The median of bulb B is about 00 hours. This implies that bulb B is better because the median bulb lasts 400 hours longer. The interquartile range of bulb A is about (00-50) 950 hours. The interquartile range of bulb B is about ( ) 400 hours. The middle half of bulb B is bunched together. The middle half of bulb A is more spread out. The information suggests that bulbs of type B are more consistent and have a longer lifetime. Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

41 Handling Data Using cumulative frequency diagrams to compare distributions Eercises Two farmers each had a flock of sheep. They decided to compare the success of the rearing methods they each used by weighing the sheep after one year Cumulative frequency Mass in kg Farmer Giles flock Cumulative frequency Mass in kg Old McDonald s flock Find the median of each distribution. Find the upper quartile of each distribution. Find the lower quartile of each distribution. 4 Find the interquartile range of each distribution. 5 Use the median, upper quartile and lower quartile of each distribution to compare the effectiveness of each farmer s methods. Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

42 Handling Data Probability AND OR means MULTIPLY means ADD Eample A bag contains three red sweets, four blue sweets and five white sweets. A boy is blindfolded. What is the probability he chooses a blue sweet, eats it, then chooses a red sweet? Method B R W W R W B B W R B W Try to rephrase the question using the key words: AND OR BLUE SWEET RED SWEET The boy needs BLUE SWEET AND RED SWEET There are four blue sweets in the bag There are sweets in the bag 4 4 = There are three red sweets in the bag Remember, a blue sweet has been removed so there are only sweets left in the bag Questions What is the probability of choosing two red sweets? What is the probability of choosing a red sweet and a white sweet in any order? Key words AND OR RED SWEET Rephrase the question using the key words. RED SWEET AND RED SWEET = Key words AND OR RED SWEET Rephrase the question using the key words. WHITE SWEET RED SWEET AND WHITE SWEET OR WHITE SWEET AND RED SWEET = 5 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

43 Handling Data Answers Probability Eercises A bag contains two blue counters, three red counters, four white counters and si yellow counters. Each time a counter is selected it is replaced. Find the probability of: W Y Y R W B W W Y Y Y B Y R R a b c d e f A white then a yellow. A white and a yellow in any order. Three reds. A red then a white. A blue, then a red, then a white, then a yellow. At least one blue in three attempts. Repeat question without replacing the counters. A coin is tossed and a die is thrown. Find the probability of: a A head and a 6. b A tail and an odd number. c A head and a number over 4. 4 The word PROBAB I L I T Y is written on cards. Cards are selected without replacement. What is the probability of choosing: a The letters P then R then O then B? b Two B s? c A P and a Y in any order? d Two identical letters, eg B and B? e Three vowels? Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

44 Handling Data Probability Questions Three coins are tossed. What is the probability of: Eactly one head? At least one head? Answers We need: HEAD AND TAIL AND TAIL OR TAIL AND HEAD AND TAIL OR TAIL AND TAIL AND HEAD + + = 8 Remember, the total probability for all of the possible ways three coins can land is. We could say: HEAD AND HEAD AND HEAD OR HEAD AND HEAD AND TAIL OR... This will work but it takes a long time! Think carefully Sometimes it is quicker to work out the probability of what we do not want. What don t we want? We don t want three tails. Any other outcome will contain at least one head. The probability of three tails is: TAIL AND TAIL AND TAIL = 8 Total probability - Probability of three tails = Probability of at least one head - = Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

45 Handling Data Probability Eercises The probability of a dry day is 0.7. The probability of a rain day is 0.. Sarah is on holiday for three days. What is the probability that: a b c d It rains on eactly one day? It rains every day? It rains on the second day? It rains on at least one day? The chance that a light bulb is faulty is. 0 a John selects a light bulb. What is the chance that it is not faulty? b c Jayne needs two light bulbs. If she buys three light bulbs what is the chance that at least two will work? Paul needs at least a 99.99% chance that one light bulb will work. How many light bulbs should he buy to ensure this probability? Show your working. The probability of Carolyn s school bus not being late each day is 0.4. If she is late five times in any week she receives a detention. What is the probability that she receives a detention in her first week of term for being late? 4 A biased coin has a 0.6 chance of landing on a head and a 0.4 chance of landing on a tail. The coin is tossed four times. a b What is the probability of at least one head? What is the probability of at least one tail? Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

46 Handling Data Tree diagrams Questions A car driver passes through two sets of traffic lights on his way to work. The lights can either be red or green. The probability of red at the first lights is 0.6. The probability of red at the second lights is 0.. Draw a tree diagram to show this and hence calculate the probability that: Both lights are red. Both lights are green. One set of lights is red and one is green. 4 At least one set of lights is red. Answers First traffic lights Second traffic lights 0. R = R 0.7 G = G 0. R = G = Red and green or green and red = Red and red or red and green or green and red = 0.7 Alternative method - green and green = 0.7 Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

47 Handling Data Tree diagrams Eercises The probability of passing a Mathematics eamination is 0.8. The probability of passing an English eamination is 0.7. a Complete this tree diagram: Mathematics English PASS 0.8 PASS FAIL FAIL PASS FAIL b c d What is the probability that a pupil passes both eaminations? What is the probability that a pupil fails both eaminations? What is the probability that a pupil passes eactly one eamination? Three hundred pupils take the eaminations. e f How many pupils would you epect to pass English and fail Mathematics? How many pupils would you epect to fail both eaminations? A fair coin is tossed three times. a Draw a tree diagram to show all of the possible outcomes. Use your tree diagram to find the probability of: b c d e Eactly two heads. At least two heads. Three tails. Eactly one head. Two dice are tossed. a Draw a tree diagram to show all of the possible outcomes. Use your tree diagram to find the probability of: b A total of. c A total of 0. d A total of less than 5. Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

48 Investigation Differences Eample Top left Bottom left Top right Bottom right Multiply top right by bottom left =? Multiply top left by bottom right =? Record the difference Going up in ones Going up in twos 4 = 8 = 7 = 5 = 5 4 Difference = Difference = 6 5 = 5 4 = = 6 = 5 4 Difference = Difference = = 4 5 = = 45 7 = 6 5 Difference = Difference = 4 Investigate Going up in threes, fours, fives. Find formulae. Try to find a general formula for: a a+t a+t a+t Where a is the top left number and the pattern goes up in t s. 4 Try square numbers. 5 Try cubed numbers. 6 Continue the investigation. Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

49 Puzzles Puzzles A mermaid was washed up on a beach. Her head is 7 cm long Her tail is as long as her head and half of her body. Her head and her tail are the same length as her body. Form equations to help you work out: a b c d The length of her body. The length of her tail. Her total length. Epress the length of head to body to tail as a ratio. A farmer has 00 animals. The animals have 6 legs between them. A chicken is worth 4. A sheep is worth 0. What is the total value of his animals? Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa

50 Puzzles A football stadium has five stands. Stand Stand Stand Stand 4 Stand 5????? people were in the stadium on Saturday. Before the match could begin a safety check had to be made. The safety regulations state: Any two adjacent stands must not hold more than 4400 people. Individual stands must not hold more than 00 people. The total number of people in stand + stand = 467 The total number of people in stand + stand = 4 The total number of people in stand + stand 4 = 45 The total number of people in stand 4 + stand 5 = 405 a b Were any safety regulations being broken? Which stands were overcrowded and by how many people? Pearson Publishing, Chesterton Mill, French s Road, Cambridge CB4 NP Tel Fa