Measurement and time-scale applications | Year 9

01 / LEARN

The idea, step by step

Numbers and units must travel together

$1$ kilometre is $10^{3}$ metres, $1$ millimetre is $10^{- 3}$ metres and $1$ micrometre is $10^{- 6}$ metres. A scientific-notation coefficient does not tell you which unit was used.

Before dividing two lengths to find a scale factor, express them in the same unit. Dividing a distance by a speed gives a time only when the distance unit matches the speed unit. Show the conversion rather than silently changing it.

Interpret the result, not just the exponent

A value of $2.5 \times 10^{3}$ seconds is correct but may be easier to understand as 2500 seconds or about 41.7 minutes. Keep enough digits during a calculation, then round the final result to the precision requested.

Rate models often assume constant speed or identical repeated objects. State these assumptions and consider what was left out. For example, a download-time model may ignore connection delays, and a stack-thickness model may ignore air gaps.

A common trap

Dividing measurements written in different units without converting.

Instead: Convert both quantities to compatible units first, then carry out the scientific-notation calculation.

02 / WORKED EXAMPLES

See the thinking

Example 1

Compare a drawing with a real object

A cell is $8 \times 10^{- 6}$ m long. Its drawing is 4 cm long. Find the drawing magnification.

Convert 4 cm to metres: $4 \times 10^{- 2}$ m.
Magnification $= \frac{4 \times 10^{- 2}}{8 \times 10^{- 6}} = 0.5 \times 10^{4}$ .
Normalise to get $5 \times 10^{3} = 5000$ .

5000

times

Example 2

Find travel time

A signal travels $9 \times 10^{7}$ m at $3 \times 10^{8}$ m/s. How long does the journey take?

$t = \frac{d}{v}$ ; both distances use metres.
$t = \frac{9 \times 10^{7}}{3 \times 10^{8}} = 3 \times 10^{- 1}$ s.
$3 \times 10^{- 1}$ s is 0.3 s.

0.3

Check your understanding

A sheet is $9 \times 10^{- 5}$ m thick. Find the thickness of 1000 sheets, ignoring gaps.

Show answer

$0.09$ m

$(9 \times 10^{- 5}) (10^{3}) = 9 \times 10^{- 2}$ m.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Write 3.5 km in metres using scientific notation.
2 marks
Write 0.0042 m in millimetres.
2 marks
A particle is $2.5 \times 10^{- 6}$ m wide. Express its width in micrometres.
2 marks
A beam travels $1.8 \times 10^{6}$ m in $6 \times 10^{- 3}$ s. Find its constant speed.
3 marks
A model is 15 cm long and represents a real object 30 m long. Find the scale factor from real object to model.
3 marks
A sensor records $2 \times 10^{4}$ readings each second. How many readings does it record in 5 minutes?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write 3.5 km in metres using scientific notation.
2 marks
$1$ km $= 10^{3}$ m, so multiply 3.5 by $10^{3}$ .
$3.5 \times 10^{3}$ m
Write 0.0042 m in millimetres.
2 marks
$1$ m $= 1000$ mm.
$0.0042 \times 1000 = 4.2$ .
$4.2$ mm
A particle is $2.5 \times 10^{- 6}$ m wide. Express its width in micrometres.
2 marks
$1$ micrometre $= 10^{- 6}$ m.
Divide $2.5 \times 10^{- 6}$ m by $10^{- 6}$ m per micrometre.
$2.5$ micrometres
A beam travels $1.8 \times 10^{6}$ m in $6 \times 10^{- 3}$ s. Find its constant speed.
3 marks
$v = d \div t = \frac{1.8}{6} \times 10^{6 - (- 3)} = 0.3 \times 10^{9}$ .
Normalise to $3 \times 10^{8}$ m/s.
$3 \times 10^{8}$ m/s
A model is 15 cm long and represents a real object 30 m long. Find the scale factor from real object to model.
3 marks
Convert 30 m to 3000 cm.
$15 \div 3000 = \frac{1}{200}$ .
$\frac{1}{200}$
A sensor records $2 \times 10^{4}$ readings each second. How many readings does it record in 5 minutes?
3 marks
$5$ minutes $= 300 = 3 \times 10^{2}$ seconds.
$(2 \times 10^{4}) (3 \times 10^{2}) = 6 \times 10^{6}$ .
$6 \times 10^{6}$ readings

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Year 9 numerical integer exponent laws and their extension to variables; scientific notation and practical measurement applications. Excludes fractional-index and full surd algebra taught in Year 10.

Official curriculum source ↗

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