Mathematics • Year 10 • Unit 1 • Lesson 11
Scientific Notation in the Real World
Apply scientific notation to extreme-scale measurements from astronomy, cell biology, nanotechnology and demography — exactly the contexts where decimal form becomes unworkable. Then explain in your own words why the form a × 10ⁿ matters.
1. Word problems
Each problem uses the form a × 10ⁿ (1 ≤ a < 10) and the index laws from Lesson 11. Show every conversion and combine carefully — a final number with no working only earns half marks.
1.1 — Comparing planetary masses. The mass of the Sun is approximately 1.989 × 10³⁰ kg and the mass of Earth is approximately 5.972 × 10²⁴ kg (the lesson's anchor figures).
(a) Calculate how many times more massive the Sun is than Earth. Give your answer in scientific notation, to 3 significant figures.
(b) Write that ratio in plain English (e.g. "the Sun is about ___ thousand / million times more massive"). 3 marks
1.2 — Nanoparticles at CSIRO. Researchers measure a nanoparticle with diameter 4 × 10⁻⁹ m. They line up 250 such particles end to end on a slide.
(a) Calculate the total length of the line in metres. Give your answer in scientific notation.
(b) Convert that length to millimetres. 3 marks
1.3 — Speed-of-light travel time. Light travels at 3 × 10⁸ m/s. The Sun is 1.5 × 10¹¹ m from Earth.
(a) Calculate the time (in seconds) for light to travel from the Sun to Earth. Use time = distance ÷ speed.
(b) Convert your answer to minutes (1 minute = 60 s). Why is "8 minutes" a famous textbook number? 3 marks
1.4 — Australian population and budget. Australia's population is approximately 2.6 × 10⁷ people (lesson anchor). The federal budget for a recent year was approximately $6.8 × 10¹¹.
(a) Calculate the per-person share of the budget. Give your answer in scientific notation.
(b) Express that per-person figure in ordinary dollars (rounded to the nearest dollar). 3 marks
1.5 — SKA radio waves. The Australian Square Kilometre Array Pathfinder telescope (lesson anchor) detects signals from distances of about 10²⁵ m. A typical radio wavelength is about 0.21 m.
(a) How many wavelengths fit into the travel distance of 10²⁵ m? Give your answer in scientific notation.
(b) Why does this calculation explain the need for scientific notation in radio astronomy? 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A student looks at 6,000,000,000,000,000,000,000 kg (the rough mass of Earth) and 0.00000000000000000000000167 kg (the rough mass of a hydrogen atom) and says "they're both just numbers — why bother with scientific notation?". Using everything from Lesson 11, explain (i) what's hard about comparing those two numbers in decimal form, (ii) what scientific notation gives you that decimal form doesn't, and (iii) why scientists choose the form 1 ≤ a < 10 instead of allowing any value for a. Refer to "orders of magnitude" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Sun vs Earth mass
(a) Ratio = (1.989 × 10³⁰) ÷ (5.972 × 10²⁴) = (1.989 ÷ 5.972) × 10³⁰⁻²⁴ = 0.333 × 10⁶ = 3.33 × 10⁵.
(b) The Sun is about 333 thousand times more massive than Earth.
Note the adjustment: 0.333 × 10⁶ is not in proper scientific notation, so we shift the decimal one place right and reduce the exponent by 1.
1.2 — Nanoparticle line
(a) Length = 250 × (4 × 10⁻⁹) = (2.5 × 10²) × (4 × 10⁻⁹) = 10 × 10⁻⁷ = 1 × 10⁻⁶ m.
(b) 1 × 10⁻⁶ m × 10³ mm/m = 1 × 10⁻³ mm = 0.001 mm.
The lesson's CSIRO anchor: nanoparticles live around 10⁻⁹ m, so 250 of them just reach the micrometre scale.
1.3 — Light from the Sun
(a) time = (1.5 × 10¹¹) ÷ (3 × 10⁸) = (1.5 ÷ 3) × 10¹¹⁻⁸ = 0.5 × 10³ = 5 × 10² s (= 500 s).
(b) 500 s ÷ 60 ≈ 8.3 minutes. This is why astronomy textbooks say "the Sun you see is 8 minutes old" — the light you're seeing left the Sun about 8 minutes ago.
1.4 — Per-person share
(a) Per person = (6.8 × 10¹¹) ÷ (2.6 × 10⁷) = (6.8 ÷ 2.6) × 10¹¹⁻⁷ ≈ 2.615 × 10⁴ = 2.6 × 10⁴ (2 sig figs).
(b) That is about $26,000 per person per year.
1.5 — Radio wavelengths across the cosmos
(a) Number of wavelengths = (1 × 10²⁵) ÷ (2.1 × 10⁻¹) = (1 ÷ 2.1) × 10²⁵⁻⁽⁻¹⁾ ≈ 0.476 × 10²⁶ = 4.76 × 10²⁵.
(b) Writing 4.76 × 10²⁵ takes seconds; writing it in ordinary decimal form requires twenty-six digits. Scientific notation makes radio-astronomy arithmetic tractable.
This is exactly why the lesson cites the SKA as the anchor example.
2.1 — Explain your thinking (sample response)
(i) Comparing two long decimals is almost impossible because the eye cannot count fifty zeros accurately, and any miscount throws the size estimate off by a factor of ten. (ii) Scientific notation strips a number down to its essential order of magnitude — the exponent on 10 — so 6 × 10²⁴ kg (Earth) and 1.67 × 10⁻²⁷ kg (a hydrogen atom) sit on a single comparable scale, and the difference in exponents (24 − (−27) = 51) tells me Earth is roughly 10⁵¹ times heavier than a hydrogen atom. (iii) The rule 1 ≤ a < 10 makes the form unique: every number has exactly one valid representation, so "4.56 × 10⁷" is unambiguous, while 45.6 × 10⁶ and 0.456 × 10⁸ describe the same value but break the rule.
Marking: 1 for naming the comparison difficulty, 1 for what sci-not gives (order of magnitude), 1 for using "orders of magnitude" correctly, 1 for explaining the 1 ≤ a < 10 uniqueness rule.