Mathematics • Year 10 • Unit 1 • Lesson 12
Significant Figures — Mixed Challenge
Pull together every idea from Lesson 12: counting sig figs across all four rules, rounding (including cascading carries), reporting calculations at the right precision, and resolving ambiguity with scientific notation. Spot a plausible Year 10 mistake, then design your own measurement scenario.
1. Mixed problems — choose the right rule
Each question targets a different idea from Lesson 12. State which rule(s) you used. 3 marks each
1.1 State the number of significant figures in: (a) 0.00560, (b) 20,300, (c) 100.
1.2 Round 7.856 to 2 sig figs, then round 345,900 to 3 sig figs.
1.3 Round 0.009876 to 3 sig figs. Then round 2.995 to 3 sig figs and explain the carry.
1.4 A measurement is recorded as 4.50 × 10³ metres. How many significant figures, and how would you write the same value to only 2 sig figs in scientific notation?
1.5 A rectangular room is measured as 5.2 m × 3.8 m. Calculate the area to the calculator value, then round to the correct number of sig figs and justify the choice.
1.6 Which of these has exactly 3 sig figs: 0.030, 300, 3.00, 30.0? List all that qualify, explain each, and (for any ambiguous case) show how to make it unambiguous.
2. Find the mistake
Another Year 10 student has tried to round and report a measurement. Their working is shown below. Exactly one line contains a mistake the lesson's Misconceptions card warns about. Spot it, explain why, then re-do correctly. 3 marks
Student's working — count sig figs and round 0.00450 to 2 sig figs:
Line 1: The leading zeros 0, 0, 0 are placeholders → not significant.
Line 2: The trailing 0 after the decimal point is also a placeholder → not significant.
Line 3: So 0.00450 has 2 sig figs (just the 4 and the 5).
Line 4: Rounded to 2 sig figs: 0.00450 → 0.0045 (no change to digits, drop the trailing 0).
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong, naming the rule it violates.
(c) Correct sig-fig count and corrected 2-sig-fig rounding for 0.00450.
Stuck? Rule 4: trailing zeros ARE significant when a decimal point is present. The lesson's Misconceptions card uses 0.00450 specifically.3. Open-ended challenge — design a precision dispute
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Invent a realistic measurement scenario where two different reasonable answers can be given for "how many significant figures does this measurement have?" — i.e. design a value where the trailing-zero ambiguity matters in practice.
In your scenario:
(i) Describe the real-world context (e.g. surveying a block of land, weighing a delivery).
(ii) Give the disputed measurement as written (a whole number ending in one or more zeros).
(iii) State both reasonable sig-fig interpretations and the consequence of choosing each one.
(iv) Write the measurement two different ways using scientific notation so the precision is no longer ambiguous.
Bonus: the measurement must use real Australian units (km, m, cm, kg, g, mL, L, ha) and a realistic Australian value.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Three sig-fig counts
(a) 0.00560 → leading zeros excluded; 5, 6, trailing 0 (after decimal) all count → 3 sig figs.
(b) 20,300 → 2, 0 (sandwiched), 3 all count; trailing zero(s) ambiguous → at least 3 sig figs (could be 3, 4 or 5).
(c) 100. → decimal point makes the trailing zeros significant → 3 sig figs.
1.2 — Two roundings
7.856 to 2 sig figs: first 2 digits 7, 8; next digit 5 → round 8 up to 9 → 7.9.
345,900 to 3 sig figs: first 3 digits 3, 4, 5; next digit 9 → round 5 up to 6 → 346,000.
1.3 — Including the cascade
0.009876 to 3 sig figs: first 3 sig digits 9, 8, 7; next digit 6 → round 7 up to 8 → 0.00988.
2.995 to 3 sig figs: first 3 sig digits 2, 9, 9; next digit 5 → round last 9 up → carry through both 9s and the 2 → 3.00. The two trailing zeros are essential to show 3 sig figs.
1.4 — 4.50 × 10³
All digits in a sci-not mantissa are significant: 4, 5, 0 → 3 sig figs.
Same value to 2 sig figs: 4.5 × 10³ — drop the trailing 0.
1.5 — 5.2 m × 3.8 m room
Raw area = 5.2 × 3.8 = 19.76 m². Both inputs have 2 sig figs, so report to 2 sig figs → 20 m² (or 2.0 × 10¹ m²). Reporting 19.76 implies 4 sig figs of precision the inputs don't justify.
1.6 — Which have exactly 3 sig figs?
3.00 and 30.0 both have exactly 3 sig figs (explicit decimal points make the trailing zeros count).
0.030 has only 2 sig figs (leading zeros excluded; the 3 and trailing 0 count).
300 is ambiguous: could be 1, 2 or 3 sig figs. To force 3 sig figs unambiguously: write 300. or 3.00 × 10².
2 — Find the mistake
(a) The mistake is on Line 2 (and Line 3 inherits it).
(b) The trailing 0 in 0.00450 sits after the decimal point, so by Rule 4 it IS significant — it's not a placeholder. The student wrongly applied the leading-zero rule to the trailing zero.
(c) Correct count: 0.00450 has 3 sig figs (4, 5, 0). Rounded to 2 sig figs: first 2 sig digits 4 and 5; next digit 0 < 5 → 5 stays → 0.0045. (Note: the rounded value drops the trailing 0 because it now only represents 2 sig figs.)
The lesson's Misconceptions card calls out 0.00450 explicitly.
3 — Open-ended challenge (sample solution)
(i) Real-world context: a council survey reports the area of a suburban block in Sydney as 800 m². (ii) Disputed measurement: 800 m².
(iii) Reading 1 — 1 sig fig: the surveyor measured roughly and reported to the nearest 100 m². The true area could be anywhere from 750 to 850 m². Consequence: rates calculated from this could be off by ~12.5 % either way. Reading 2 — 3 sig figs: the surveyor measured precisely and the trailing zeros are real digits. True area is 799.5 to 800.5 m². Consequence: rates are accurate to the nearest m².
(iv) Unambiguous: 8 × 10² m² (1 sig fig) or 8.00 × 10² m² (3 sig figs).
The trailing-zero ambiguity changes the rates owed by a thousand-dollar margin.
Marking: 1 for plausible Australian context with realistic units; 1 for clearly stating two sig-fig readings; 1 for naming a real-world consequence of each reading; 1 for writing both interpretations in scientific notation. Any valid scenario meeting these criteria scores full marks.