Mathematics • Year 10 • Unit 1 • Lesson 12
Significant Figures in the Real World
Apply significant figures to engineering, building, manufacturing and laboratory scenarios — exactly where Lesson 12's real-life link says precision matters. Then explain in your own words why "more digits" does not mean "more accurate".
1. Word problems
Each problem uses the four sig-fig rules and the principle "an answer cannot be more precise than its least precise measurement". Show every step.
1.1 — Tiling a bathroom floor. A bathroom floor is measured as 5.2 m by 3.8 m (both to 2 sig figs).
(a) Calculate the area without rounding intermediate values.
(b) Write the area to the appropriate number of significant figures, and explain in one sentence why a tiler should not report 19.76 m². 3 marks
1.2 — Concrete strength on a building site. Australian Standard AS 1379 specifies concrete grade N32 as "32 MPa to 2 significant figures" — the lesson explicitly references this kind of building standard.
(a) A batch tests at 31.847 MPa. State this measurement rounded to 2 significant figures.
(b) Another batch tests at 32.5 MPa. Does this meet the N32 spec to 2 sig figs? Justify in one sentence.
(c) Why does the lesson say "too many significant figures means unnecessary expense, too few means potential failure"? 3 marks
1.3 — Lab measurement of mass. A student weighs a chemistry sample on a balance and records the mass as 0.00450 g.
(a) State the number of significant figures.
(b) The student then says "the leading zeros mean the balance reads 5 significant figures". Explain why this is wrong.
(c) Re-express the mass in scientific notation so the precision is unambiguous. 3 marks
1.4 — Engineering tolerance on a rivet. A Boeing engineer drills a rivet hole to a diameter of 4.762 mm. The original measurement was specified to 3 significant figures.
(a) Round 4.762 mm to 3 sig figs.
(b) The engineer reports the hole as 4.76200 mm. Why is this misleading?
(c) What range of true diameters could the 3-sig-fig value of part (a) represent? 3 marks
1.5 — Reporting an experiment. A Year 10 physics student measures a metal block as 5.2 cm × 3.8 cm × 2.0 cm.
(a) Calculate the volume, leaving the full calculator display first.
(b) Round to the appropriate number of significant figures and explain your choice.
(c) Write your final volume in scientific notation. 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A friend says: "If I write more digits in my answer, my answer is more accurate." Using everything from Lesson 12, explain (i) what's wrong with that reasoning, (ii) the difference between accuracy and precision as defined in the Key Terms card, and (iii) why a builder reporting a wall as "10.0000 m long" is making an honesty mistake. Refer to "trailing zeros" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Bathroom area
(a) Area = 5.2 × 3.8 = 19.76 m² (calculator value, no rounding).
(b) Both inputs have 2 sig figs, so the answer must be reported to 2 sig figs → 20 m² (or 2.0 × 10¹ m²). Reporting 19.76 implies 4 sig figs of precision, but the inputs only justify 2.
The lesson's SAQ 3 anchor: a calculation can't be more precise than its least precise measurement.
1.2 — Concrete strength
(a) 31.847 to 2 sig figs: first two digits 3 and 1, next digit 8 ≥ 5 → round 1 up to 2. → 32 MPa. Meets spec.
(b) 32.5 to 2 sig figs: first two 3 and 2, next digit 5 → round 2 up to 3. → 33 MPa. Above spec, still meets N32 (concrete only needs to meet OR exceed the rated strength).
(c) Over-specifying (e.g. requiring 32.000 MPa, 5 sig figs) demands more expensive cement and more testing than needed; under-specifying could put a building below its rated load and risk failure. 2 sig figs is what quality control can reliably verify — this is exactly the lesson's Real-Life Link argument.
1.3 — Lab mass 0.00450 g
(a) 3 sig figs (the 4, the 5, and the trailing 0 after the decimal).
(b) Leading zeros are NEVER significant — Rule 3. The student has confused place-value zeros with measured zeros.
(c) In scientific notation: 4.50 × 10⁻³ g — now every digit in the mantissa is unambiguously significant.
1.4 — Rivet hole
(a) 4.762 mm to 3 sig figs: first 3 digits 4, 7, 6. Next digit 2 < 5 → 6 stays. → 4.76 mm.
(b) Reporting 4.76200 implies 6 sig figs of precision — but the drill was only specified to 3. The extra zeros are dishonest precision.
(c) 3-sig-fig value 4.76 mm represents any true diameter in the half-open interval [4.755, 4.765) mm — a ±0.005 mm tolerance.
1.5 — Block volume
(a) Volume = 5.2 × 3.8 × 2.0 = 39.52 cm³ (raw calculator).
(b) All inputs have 2 sig figs, so answer should have 2 sig figs → 40 cm³. Reporting 39.52 cm³ overstates precision.
(c) In scientific notation: 4.0 × 10¹ cm³.
2.1 — Explain your thinking (sample response)
(i) Writing more digits doesn't make the underlying measurement more accurate — it only adds artificial precision that the measuring instrument never actually had. A 30 cm school ruler can't deliver 0.0001 cm of real information no matter how many zeros you tack on. (ii) The Key Terms card distinguishes precision (how detailed the recorded value is, indicated by significant figures) from accuracy (how close that value is to the true value); a number can be very precise but very inaccurate, or accurate but imprecise. (iii) A builder writing 10.0000 m is claiming 6 sig figs of trailing zeros after the decimal point, which implies measurement to the nearest 0.0001 m (= 0.1 mm) — far beyond what any tape measure on a building site can deliver. The honest report is "10.0 m" or "10 m to the nearest 0.1 m".
Marking: 1 for naming "extra digits don't add real info", 1 for accurate distinction of precision vs accuracy, 1 for using "trailing zeros" correctly, 1 for a worked example of dishonest precision.