© Nuffield Foundation 2011 Errors Free-Standing Mathematics Activity.

© Nuffield Foundation 2011


ErrorsFree-Standing Mathematics Activity


Errors

How certain are you about the accuracy of your measurements?


ErrorsMeasurements are subject to errors.Length of swimming pool = 25 metres to nearest metre

25.5 mUpper bound

25 m 26 m24 m

24.5 mLower bound

Think about What is the shortest possible length? What is the longest possible length?

Maximum error0.5 m


4.35 kgUpper bound

4.3 kg 4.4 kg4.2 kg

4.25 kgLower bound

Maximum error0.05

Weight of package 4.3 kg to 1 decimal place

Think about: What is the smallest possible weight? What is the largest possible weight?


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.


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?


Length of journey = 250 miles to nearest 10 miles

Upper bound255 miles

250 miles

Lower bound245 miles

+ 5 miles

– 5 miles

Length of journey = 250 ± 5 miles

Think about What is the maximum error?


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 aboutWhat is the maximum error?


Temperature of furnace = 1400C to 2 significant figures

= 1400 ± 50 °C Temperature

1450°CUpper bound

1400°C 1500°C1300°C

1350°CLower bound

–50°C +50°C

Think about What is the highest possible temperature?What is the lowest possible temperature?


If temperature of furnace = 1400°C to 3 significant figures

Temperature= 1400 ± 5 °C

1405°CUpper bound

1400°C 1410°C1390°C

1395°CLower bound

– 5°C + 5°C

Think about What is the highest possible temperature now?What is the lowest possible temperature?


56 m

83 m24 m

45 mCar 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 m2

Area of B = 4648 m2

Total area = 3600 + 4648 = 8248 m2

Think about: How accurate are these estimates?

A B

= 80 45

= 83 56


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 m2

Upper bound of area of B = 4717.75 m2

Upper bound of total area = 3685.5 + 4717.75= 8403.25 m2

56.5 m

83.5 m24.5 m

45.5 m

= 81 45.5

= 83.5 56.5


55.5 m

82.5 m23.5 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 m2

Lower bound of area of B = 4578.75 m2

Lower bound of total area = 3515.5 + 4578.75 = 8094.25 m2

44.5 + 82.5 = 127 m

23.5 + 55.5= 79 m A BA B

= 79 44.5

= 82.5 55.5


56 m

83 m24 m

45 mCar park

Perimeter

Total area

Best estimate = 8248 m2

Best estimate = 416 mLower bound = 412 m

Upper bound = 420 m

Lower bound = 8094.25 m2 Upper bound = 8403.25 m2

Perimeter = 420 m (to 2 sf) Total area = 8200 m2 (to 2 sf)

Think about What final answers should be given?


Example: Find the volume and surface area of a cone with radius r = 3.5 cm, height h = 5.2 cm (to 1 dp)

hr231Volume V =

5.23.531 2 πVBest estimate

= 66.7 cm3

5.253.5531 2 πVUpper bound = 69.3 cm3

5.153.4531 2 πVLower bound = 64.2 cm3

Best Estimate of Volume= 67 cm3 (to 2 sf)

h

r

Think about What final answer should be given?


Surface area S = r(r + l)

6.26823.53.5πS = 107.41 cm2

Upper bound

22 hrl

Best estimate 22 5.23.5 l = 6.2682 cm

6.33763.553.55 πS = 110.27 cm2

22 5.253.55 l = 6.3376 cm

Surface area of cone r = 3.5 cm, h = 5.2 cm (to 2 sf)

Lower bound

6.19883.453.45 πS = 104.58 cm2

22 5.153.45 l = 6.1988 cm

Best estimate of surface area = 110 cm2 (to 2 sf)

h

r

l

Think about What final answer should be given?


• 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

© Nuffield Foundation 2011 Errors Free-Standing Mathematics Activity.

Documents

Transcript of © Nuffield Foundation 2011 Errors Free-Standing Mathematics Activity.