1. © Nuffield Foundation 2011 Errors Free-Standing Mathematics Activity

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

3. © Nuffield Foundation 2011 Errors Measurements are subject to errors. Length of swimming pool = 25 metres to nearest metre 25.5 m Upper bound 25 m 26 m 24 m 24.5 m Lower bound Think about What is the shortest possible length? What is the longest possible length? Maximum error 0.5 m

4. © Nuffield Foundation 2011 4.35 kg Upper bound 4.3 kg 4.4 kg 4.2 kg 4.25 kg Lower bound Maximum error 0.05 Weight of package 4.3 kg to 1 decimal place Think about: What is the smallest possible weight? What is the largest possible weight?

5. © Nuffield Foundation 2011 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.

6. © Nuffield Foundation 2011 Errors in measurements Nearest 100 50 Accuracy Maximum error Nearest 10 5 Nearest whole number 0.5 To 1 decimal place 0.05 (nearest 0.1) To 2 decimal places 0.005 (nearest 0.01) Think about What is the maximum error?

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

8. © Nuffield Foundation 2011 Winning time Maximum error = 0.005 seconds Lower bound = 36.32 – 0.005= 36.315 seconds Upper bound = 36.32 + 0.005= 36.325 seconds Winning time in a race 36.32 seconds to nearest 0.01 second = 36.32 ± 0.005 seconds Think about What is the maximum error?

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

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

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

12. © Nuffield Foundation 2011 Car park upper bounds Upper bound of perimeter = 45.5 + 24.5 + 83.5 + 56.5 + 129 + 81 45.5 + 83.5= 129 m 24.5 + 56.5 = 81 m = 420 m Upper bound of area Upper bound of area of A A B = 3685.5 m 2 Upper bound of area of B = 4717.75 m 2 Upper bound of total area = 3685.5 + 4717.75 = 8403.25 m 2 56.5 m 83.5 m 24.5 m 45.5 m = 81  45.5 = 83.5  56.5

13. © Nuffield Foundation 2011 55.5 m 82.5 m 23.5 m 44.5 m Car park lower bounds Lower bound of perimeter = 44.5 + 23.5 + 82.5 + 55.5 + 127 + 79 = 412 m Lower bound of area Lower bound of area of A = 3515.5 m 2 Lower bound of area of B = 4578.75 m 2 Lower bound of total area = 3515.5 + 4578.75 = 8094.25 m 2 44.5 + 82.5 = 127 m 23.5 + 55.5 = 79 m A B A B = 79  44.5 = 82.5  55.5

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

15. © 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 Upper bound = 69.3 cm 3 Lower bound = 64.2 cm 3 Best Estimate of Volume = 67 cm 3 (to 2 sf) h r Think about What final answer should be given?

16. © Nuffield Foundation 2011 Surface area S =  r ( r + l ) = 107.41 cm 2 Upper bound Best estimate = 6.2682 cm = 110.27 cm 2 = 6.3376 cm Surface area of cone r = 3.5 cm, h = 5.2 cm (to 2 sf) Lower bound = 104.58 cm 2 = 6.1988 cm Best estimate of surface area = 110 cm 2 (to 2 sf) h r l Think about What final answer should be given?

17. © Nuffield Foundation 2011 Write the radius in the form a  b 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? 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? 12.0  0.1 cm At the end of the activity