Measurement error: the same desk with three instruments and what it does to the area

The idea

Every measurement is an estimate with an absolute error of half the smallest unit on the instrument for each reading, and those errors carry through into any area or volume computed from them.

What you need

• A rectangular desk about 120 cm by 60 cm
• A 30 cm ruler marked in millimetres, a builder's tape marked in millimetres, and a tape at least 1.5 m long with its millimetre marks masked so that it reads only to the nearest centimetre (a metre stick is too short to span the 120 cm edge in one reading)
• Recording table: instrument, reading, smallest unit, number of placements, absolute error, percentage error

How to do it

1. Measure the length and width of the desk with each instrument and record the readings to the instrument's smallest unit. The 30 cm ruler is shorter than both edges: lay it end to end, marking each 30 cm with a sharp pencil, and record the number of placements (four for the length, two for the width).
2. For each reading write the absolute error as half the smallest unit for each placement (one placement for each tape) and the percentage error as absolute error divided by the reading.
3. Compute the area from the millimetre readings and from the centimetre readings; then compute the largest and smallest areas consistent with each pair of readings (using reading plus and minus its absolute error).
4. Express the area's uncertainty as a percentage and compare with the sum of the two percentage errors of the sides.
5. Decide how many significant figures the area deserves from each instrument.

What you should see

A length of 120.0 cm read on a millimetre scale carries an absolute error of 0.05 cm, a percentage error of 0.04 per cent. The 30 cm ruler needs four placements for the length and two for the width, so its absolute error is at least 4 × 0.05 = 0.20 cm (0.17 per cent) and 2 × 0.05 = 0.10 cm (0.17 per cent), four and two times the builder's tape's errors before any error in lining up the placements: the same millimetre marks give a better result on one long tape than on a short ruler laid end to end. The same length read on a centimetre scale (120 cm) carries 0.5 cm and 0.42 per cent, and the 60 cm width carries 0.83 per cent. The area from the centimetre readings, 7200 cm², can be anywhere from 119.5 × 59.5 = 7110 cm² to 120.5 × 60.5 = 7290 cm², an uncertainty of about 1.25 per cent, which equals the sum of the two sides' percentage errors (0.42 + 0.83). So the area deserves two significant figures (7.2 × 10³ cm², an uncertainty of about 90 cm²) from the centimetre scale and three (7.20 × 10³ cm², an uncertainty of about 9 cm²) from the millimetre scale. The learner knows it worked when the computed range for the area matches the sum-of-percentages rule.

What changes

What you change

instrument (smallest unit)

What you measure

absolute and percentage error of the reading and of the area

What you keep the same

• same desk
• same edges measured
• readings taken at eye level

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

• A digital or finely marked instrument gives the true value.
• Errors in length and width cancel out in the area.
• A computed area can carry more significant figures than the measurements it came from.

Safety card

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

• Mathematics K–10 Syllabus (2022), Stage 5 Numbers of any magnitude; page read 2026-09-22MA5-MAG-C-01
• Australian Curriculum v9AC9M9M04AC9M10M04

Sources

The pages the author read to write this activity.

1. curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content/stage-5/fa54356e3c
2. www.amsi.org.au/ESA_middle_years/Year9/Year9_md/Year9_1e.html