FreeStanding Mathematics Activity Errors Nuffield Foundation 2011 Errors

Free-Standing Mathematics Activity Errors © Nuffield Foundation 2011

Errors How certain are you about the accuracy of your measurements? © Nuffield Foundation 2011

Errors Measurements are subject to errors. Length of swimming pool = 25 metres to nearest metre Think about What is the shortest possible length? What is the longest possible length? 24. 5 m Lower bound 24 m © Nuffield Foundation 2011 25. 5 m Upper bound 25 m Maximum error 0. 5 m 26 m

Weight of package 4. 3 kg to 1 decimal place Think about: What is the smallest possible weight? What is the largest possible weight? 4. 25 kg Lower bound 4. 2 kg 4. 35 kg Upper bound 4. 3 kg Maximum error 0. 05 © Nuffield Foundation 2011 4. 4 kg

Errors in measurements Lower bound = lowest possible value Upper bound = highest possible value When a measure is expressed to a given unit, the maximum error is half a unit. © Nuffield Foundation 2011

Errors in measurements Accuracy Maximum error Nearest 100 50 Nearest 10 5 Nearest whole number 0. 5 To 1 decimal place(nearest 0. 1) 0. 05 To 2 decimal places(nearest 0. 01) 0. 005 Think about What is the maximum error? © Nuffield Foundation 2011

Length of journey = 250 miles to nearest 10 miles Think about What is the maximum error? Upper bound 255 miles Lower bound 245 miles 250 miles – 5 miles +5 miles Length of journey = 250 ± 5 miles © Nuffield Foundation 2011

Winning time in a race 36. 32 seconds to nearest 0. 01 second Think about What is the maximum error? Maximum error = 0. 005 seconds Upper bound = 36. 32 + 0. 005 = 36. 325 seconds Lower bound = 36. 32 – 0. 005 = 36. 315 seconds Winning time = 36. 32 ± 0. 005 seconds © Nuffield Foundation 2011

Temperature of furnace = 1400°C to 2 significant figures Think about What is the highest possible temperature? What is the lowest possible temperature? 1350°C Lower bound 1450°C Upper bound 1400°C 1300°C – 50°C +50°C Temperature = 1400 ± 50 °C © Nuffield Foundation 2011 1500°C

If temperature of furnace = 1400°C to 3 significant figures Think about What is the highest possible temperature now? What is the lowest possible temperature? 1395°C Lower bound 1405°C Upper bound 1400°C 1390°C – 5°C + 5°C Temperature= 1400 ± 5 °C © Nuffield Foundation 2011 1410°C

45 m Car park Best estimate of perimeter = 45 + 24 + 83 + 56 + 128 + 80 24 + 56 = 80 m = 416 m Best estimate of area Area of A = 80 45 = 3600 m 2 Area of B = 83 56 = 4648 m 2 Total area = 3600 + 4648 = 8248 m 2 © Nuffield Foundation 2011 24 m 83 m A B 56 m 45 + 83 = 128 m Think about: How accurate are these estimates?

Car park upper bounds Upper bound of perimeter 45. 5 m = 45. 5 + 24. 5 + 83. 5 + 56. 5 + 129 + 81 24. 5 m = 420 m Upper bound of area 24. 5 + 56. 5 = 81 m 83. 5 m A B 56. 5 m 45. 5 + 83. 5 = 129 m Upper bound of area of A = 81 45. 5 = 3685. 5 m 2 Upper bound of area of B = 83. 5 56. 5 = 4717. 75 m 2 Upper bound of total area = 3685. 5 + 4717. 75 = 8403. 25 m 2 © Nuffield Foundation 2011

Car park lower bounds Lower bound of perimeter = 44. 5 + 23. 5 + 82. 5 + 55. 5 + 127 + 79 = 412 m Lower bound of area 44. 5 m 23. 5 + 55. 5 = 79 m 82. 5 m A B 44. 5 + 82. 5 = 127 m Lower bound of area of A = 79 44. 5 = 3515. 5 m 2 Lower bound of area of B = 82. 5 55. 5 = 4578. 75 m 2 Lower bound of total area = 3515. 5 + 4578. 75 = 8094. 25 m 2 © Nuffield Foundation 2011 55. 5 m

Car park 45 m Perimeter 24 m 83 m Best estimate = 416 m Lower bound = 412 m 56 m Upper bound = 420 m Total area Think about What final answers should be given? Best estimate = 8248 m 2 Lower bound = 8094. 25 m 2 Upper bound = 8403. 25 m 2 Perimeter = 420 m (to 2 sf) Total area = 8200 m 2 (to 2 sf) © Nuffield Foundation 2011

Example: Find the volume and surface area of a cone with radius r = 3. 5 cm, height h = 5. 2 cm (to 1 dp) Volume V = Best estimate = 66. 7 cm 3 h Upper bound = 69. 3 cm 3 Lower bound = 64. 2 cm 3 Think about What. Estimate final answer should 3 (to 2 sf) Best of Volume = 67 becmgiven? © Nuffield Foundation 2011 r

Surface area of cone r = 3. 5 cm, h = 5. 2 cm (to 2 sf) Surface area S = r(r + l) Best estimate = 6. 2682 cm = 107. 41 cm 2 Upper bound = 6. 3376 cm = 110. 27 Lower bound r cm 2 = 6. 1988 cm = 104. 58 cm 2 Think about What final answer should be given? Best estimate of surface area = 110 cm 2 (to 2 sf) © Nuffield Foundation 2011 l h

At the end of the activity 12. 0 0. 1 cm • What is the maximum value for the diameter of this CD? • What is the minimum value for the diameter? • What are the maximum and minimum values for the radius? • Write the radius in the form a b • Work out a best estimate for the area of the top of the CD. How accurately do you think you should give the answer? • Work out the upper and lower bounds for the area. Was the answer you gave reasonable? • In general, what accuracy should you give in answers to calculations involving measurements? © Nuffield Foundation 2011