Mathematics • Year 10 • Unit 1 • Lesson 11
Scientific Notation — Mixed Challenge
Pull together every idea from Lesson 11: converting between decimal and a × 10ⁿ, multiplying and dividing with index laws, comparing magnitudes, and re-expressing answers in proper form (1 ≤ a < 10). Spot another student's plausible Year 10 mistake, then invent your own extreme-scale problem.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 11. Decide which rule applies before you start writing. Show your working. 3 marks each
1.1 Write 234,000 and 0.0000056 and 9,100,000,000 in scientific notation.
1.2 Calculate (6 × 10⁴) × (3 × 10⁷). Give the answer in proper scientific notation (1 ≤ a < 10).
1.3 Which is larger: 3.5 × 10⁶ or 7.2 × 10⁵? Justify in one sentence.
1.4 A newly discovered planet has mass 2.4 × 10²⁵ kg. Earth's mass is 5.972 × 10²⁴ kg. Is the new planet more or less massive than Earth, and by what factor?
1.5 Write 6.02 × 10⁻³ as an ordinary decimal, and write 0.000508 in scientific notation. Check that the two answers convert back into each other consistently.
1.6 A single grain of sand has volume about 6 × 10⁻¹⁰ m³. Bondi Beach is estimated to hold about 5 × 10¹⁵ grains. Estimate the total volume of sand on Bondi Beach in m³, in scientific notation.
2. Find the mistake
Another Year 10 student has tried to multiply two numbers in scientific notation. Their working is shown below. Exactly one line contains a mistake — and it's a mistake the lesson's Misconceptions card warns about. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — calculate (2 × 10³) × (5 × 10⁴):
Line 1: Multiply mantissas: 2 × 5 = 10
Line 2: Add exponents: 10³ × 10⁴ = 10⁷
Line 3: Combine: 10 × 10⁷
Line 4: Final answer: 10 × 10⁷
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected final answer in proper scientific notation.
Stuck? The lesson's "Heads up" box says: after multiplying or dividing, always check that 1 ≤ a < 10. Is 10 a valid mantissa?3. Open-ended challenge — design an "orders of magnitude" comparison
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Choose two real-world quantities from different fields (one very large, one very small) and compare them using scientific notation. Your two quantities must differ by at least 10 orders of magnitude (i.e. the exponents must differ by 10 or more after re-expressing in proper form).
For your two quantities:
(i) Name each one and the field it comes from (e.g. astronomy, biology, chemistry, geography).
(ii) State each value in proper scientific notation, with its unit.
(iii) Calculate the ratio of the larger to the smaller, in scientific notation, and write that ratio in plain English ("about ___ million / billion / trillion times").
Bonus: at least one of your quantities must be Australian (e.g. the height of Uluru, the area of the Great Barrier Reef, the mass of a Tasmanian devil).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Three conversions
234,000 = 2.34 × 10⁵.
0.0000056 = 5.6 × 10⁻⁶.
9,100,000,000 = 9.1 × 10⁹.
1.2 — (6 × 10⁴) × (3 × 10⁷)
Mantissas: 6 × 3 = 18. Exponents: 10⁴ × 10⁷ = 10¹¹.
Intermediate: 18 × 10¹¹. Re-express: 1.8 × 10¹².
1.3 — Which is larger?
3.5 × 10⁶ is larger because the exponent 6 > 5 — when the exponent on 10 is larger, the whole number is at least ten times larger, and the mantissa cannot bridge that gap.
1.4 — New planet vs Earth
Ratio = (2.4 × 10²⁵) ÷ (5.972 × 10²⁴) = (2.4 ÷ 5.972) × 10²⁵⁻²⁴ ≈ 0.402 × 10¹ ≈ 4.0.
The new planet is more massive than Earth — about 4 times more massive.
1.5 — Decimal ↔ scientific notation
6.02 × 10⁻³ as decimal: move decimal 3 places left → 0.00602.
0.000508 in scientific notation: move decimal 4 places right → 5.08 × 10⁻⁴.
Consistency check: converting back gives the original numbers.
1.6 — Bondi Beach sand volume
Volume = (6 × 10⁻¹⁰) × (5 × 10¹⁵) = (6 × 5) × 10⁻¹⁰⁺¹⁵ = 30 × 10⁵.
Re-express: 3 × 10⁶ m³ (= 3 million cubic metres). About the size of a small reservoir.
2 — Find the mistake
(a) The mistake is on Line 4.
(b) "10 × 10⁷" is not in proper scientific notation — the mantissa 10 violates the rule 1 ≤ a < 10. The student stopped one step too early. (Strictly the working in Lines 1–3 is correct, but Line 4 reports an intermediate value as if it were the final answer.)
(c) Re-express: 10 × 10⁷ = (1 × 10¹) × 10⁷ = 1 × 10⁸.
The lesson's "Heads up" box flags exactly this: always check the final mantissa.
3 — Open-ended challenge (sample solutions)
Many valid answers. Two examples:
Example A — Australian astronomy gap.
(i) Distance from Sydney to Uluru (geography) vs. diameter of a water molecule (chemistry).
(ii) Sydney → Uluru ≈ 2,830,000 m = 2.83 × 10⁶ m. Water molecule diameter ≈ 2.75 × 10⁻¹⁰ m.
(iii) Ratio = (2.83 × 10⁶) ÷ (2.75 × 10⁻¹⁰) ≈ 1.03 × 10¹⁶ — about 10 quadrillion times larger. Exponent gap = 6 − (−10) = 16 orders of magnitude. ✓
Example B — Australian biology gap.
(i) Adult male saltwater crocodile mass ≈ 1,000 kg = 1 × 10³ kg (biology) vs. mass of a single bacterium ≈ 1 × 10⁻¹⁵ kg.
(iii) Ratio = (1 × 10³) ÷ (1 × 10⁻¹⁵) = 1 × 10¹⁸ — about a million trillion times heavier. Exponent gap = 18. ✓
Marking: 2 marks for two valid quantities in proper sci-not form with units; 1 mark for a correct ratio calculation in sci-not; 1 mark for the order-of-magnitude gap being ≥ 10 and the plain-English description. Full marks for any pair that meets the constraints.