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AustraliaNSW
Stage 5.1-3

7.02 Accuracy and measurement

Lesson

Choosing the right scale

Which of these three numbers is larger?

$0.0025,0.00282,0.009$0.0025,0.00282,0.009

Now let's try again with these numbers:

$25,28.2,90$25,28.2,90

The second list is the same as the first, just multiplied by $10000$10000. Still, it is much easier to tell by sight that the last number of the second list is the largest, compared with the first list.

We have a better intuitive sense of the size of numbers if they are between $1$1 and $1000$1000, ideally less than $100$100. When selecting the correct scale to report a measurement, we want to lie somewhere in this range to compare it to others quickly and easily.

 

Worked example

Example 1

Are kilometres the appropriate unit of measure for the perimeter of a house?

Think: For kilometres to be appropriate, we should expect the typical house to have a perimeter between $1$1 and $100$100 kilometres.

Do: Houses are usually no more than $10$10 metres long in any dimension, so a typical perimeter would be at most $40$40 metres. This is the same as $0.04$0.04 km, which means kilometres is not an appropriate unit.

Reflect: The most appropriate unit would be metres.

 

Absolute error of measurement

The aim of a measurement is to obtain the "true" value of a quantity: the height of a tree, the temperature of a room, the mass of a rock, or whatever we want to know.

But is a tree ever exactly $5$5 metres tall? Is a room ever exactly $22^\circ C$22°C? Is a rock ever exactly $2$2 kg in weight?

We can carefully design a measurement procedure to make more and more precise measurements, which makes the number of significant figures in our measurement increase. But we can only ever report the closest marking, and at some point the object we are measuring will fall between the markings.

For any measurement tool, we say its absolute error is equal to half the distance of its smallest unit. Any measurement we make with that tool must be given as plus or minus the absolute error.

Any subsequent measurement that is more precise will fall within this range, but we can't know exactly where until we try with a better tool.

Three reported measurements taken with increasing accuracy using increasingly precise tools.

 

Worked example

Example 2

A ruler has marking every millimetre along its length. An object is placed against the ruler, and the closest mark is $17$17 mm. How should we report the measurement?

Think: The smallest unit is $1$1 mm, so the absolute error will be half of that. We should report the measurement as the closest mark plus or minus the absolute error.

Do: The measurement should be reported as $17\pm0.5$17±0.5 mm.

 

Summary

The appropriate unit of measurement to use makes the numerical value lie between $1$1 and $1000$1000, ideally less than $100$100.

The absolute error of a measuring tool is equal to half its smallest unit.

Measurements with any tool should always be reported as:

$\text{Closest mark}\pm\text{Absolute error}$Closest mark±Absolute error 

 

Practice questions

Question 1

It is most appropriate to use centimetres to measure the length of:

  1. a cup

    A

    a horse

    B

    a country

    C

    a truck

    D

    a cup

    A

    a horse

    B

    a country

    C

    a truck

    D
Question 2

A measuring tape has markings every $20$20 cm.

  1. What is the absolute error of the measuring tape?

    $10$10 cm

    A

    $20$20 cm

    B

    $19.5$19.5 cm

    C

    $1$1 cm

    D

    $10$10 cm

    A

    $20$20 cm

    B

    $19.5$19.5 cm

    C

    $1$1 cm

    D
  2. The length of an object is measured as $120$120 cm by the measuring tape. A second measurement is then taken, measuring its length to the nearest cm.

    What is the range we should expect this second measurement to lie within?

    $110$110 cm to $130$130 cm

    A

    $120$120 cm to $130$130 cm

    B

    $110$110 cm to $120$120 cm

    C

    $100$100 cm to $140$140 cm

    D

    $110$110 cm to $130$130 cm

    A

    $120$120 cm to $130$130 cm

    B

    $110$110 cm to $120$120 cm

    C

    $100$100 cm to $140$140 cm

    D
Question 3

Pauline knows that her bedroom is roughly $3$3 metres long. To what precision must she measure if she wants to know the length to:

  1. $2$2 significant figures?

    To the nearest $1$1 millimetre

    A

    To the nearest $10$10 centimetres

    B

    To the nearest $1$1 metre

    C

    To the nearest $1$1 centimetre

    D

    To the nearest $1$1 millimetre

    A

    To the nearest $10$10 centimetres

    B

    To the nearest $1$1 metre

    C

    To the nearest $1$1 centimetre

    D
  2. $4$4 significant figures?

    To the nearest $1$1 metre

    A

    To the nearest $10$10 centimetres

    B

    To the nearest $1$1 centimetre

    C

    To the nearest $1$1 millimetre

    D

    To the nearest $1$1 metre

    A

    To the nearest $10$10 centimetres

    B

    To the nearest $1$1 centimetre

    C

    To the nearest $1$1 millimetre

    D

Outcomes

MA5.1-9MG

interprets very small and very large units of measurement, uses scientific notation, and rounds to significant figures

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