Mathematics • Year 9 • Unit 1 • Lesson 16
Converting to Scientific Notation in the Real World
Convert real measurements into scientific notation: a hair-width, Earth's radius, a red blood cell, ocean volumes and population sizes. Compare orders of magnitude, then explain why "sign of $n$" is the whole story.
1. Word problems
Each problem converts an ordinary number into scientific notation. Show every step. The sign of the power tells you which way you're going.
1.1 — Human hair vs Earth's radius. A human hair is about $0.000\,07$ m thick. Earth's radius is about $6\,370\,000$ m.
(a) Write each in scientific notation.
(b) State the order of magnitude (the power of $10$) of each.
(c) Roughly how many hair-widths fit across Earth's radius? (Use the powers — coefficients are close enough to estimate.) 4 marks
1.2 — Red blood cell vs sports field. A red blood cell is about $0.000\,008$ m wide. A football field is about $100$ m long.
(a) Write each in scientific notation.
(b) By how many orders of magnitude does the field exceed the cell? 3 marks
1.3 — Ocean volumes. The Indian Ocean holds about $264\,000\,000\,000\,000\,000$ litres of water.
(a) Write that volume in scientific notation.
(b) Explain in one sentence why the scientific notation form is more useful when comparing it to "the Atlantic at about $3.1 \times 10^{17}$ L". 3 marks
1.4 — World population. The world population in $2026$ is roughly $8\,100\,000\,000$ people.
(a) Write this in scientific notation.
(b) Australia's population is about $2.7 \times 10^7$. Roughly what fraction of the world's population is Australian? Show the calculation using powers. 3 marks
1.5 — A small drop. A single droplet of mist is about $0.000\,000\,3$ L of water.
(a) Write the droplet's volume in scientific notation.
(b) If a cloud contains $5 \times 10^{12}$ such droplets, write the total water content in scientific notation. (Multiply the two values together; use the product rule on the powers.) 3 marks
2. Explain your thinking
This question is about communication. Use full sentences. 4 marks
2.1 A classmate writes "$0.000\,486$ in scientific notation is $4.86 \times 10^4$" — they shifted the decimal the right number of places but used a positive power. In your own words, explain (i) what they got right, (ii) the specific rule they broke about the sign of $n$, (iii) the correct answer with working, and (iv) a simple "size check" they could have done to catch the slip. Use the phrase "small numbers have a negative power" somewhere in your reply.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Hair vs Earth's radius
(a) Hair: $0.000\,07 = \mathbf{7 \times 10^{-5}}$ m. Earth's radius: $6\,370\,000 = \mathbf{6.37 \times 10^6}$ m.
(b) Hair order of magnitude: $\mathbf{10^{-5}}$. Earth's radius: $\mathbf{10^6}$.
(c) Number of hairs: $\dfrac{6.37 \times 10^6}{7 \times 10^{-5}} = \dfrac{6.37}{7} \times 10^{6 - (-5)} \approx 0.91 \times 10^{11} \approx \mathbf{9 \times 10^{10}}$ — about $90$ billion hairs across the radius.
1.2 — Red blood cell vs sports field
(a) Cell: $0.000\,008 = \mathbf{8 \times 10^{-6}}$ m. Field: $100 = \mathbf{1 \times 10^2}$ m.
(b) Orders apart: $2 - (-6) = \mathbf{8}$ orders of magnitude. The field is about $10^8$ ($100$ million) times longer than a cell is wide.
1.3 — Ocean volumes
(a) $264\,000\,000\,000\,000\,000 = \mathbf{2.64 \times 10^{17}}$ L.
(b) Both oceans are now expressed with the same power $10^{17}$, so the comparison is immediate: $2.64$ vs $3.1$ — the Atlantic is slightly bigger. Writing 17-digit numbers side by side would force you to line up zeros, which is error-prone.
1.4 — World population
(a) $8\,100\,000\,000 = \mathbf{8.1 \times 10^9}$ people.
(b) Australian share: $\dfrac{2.7 \times 10^7}{8.1 \times 10^9} = \dfrac{2.7}{8.1} \times 10^{7-9} = \dfrac{1}{3} \times 10^{-2} \approx \mathbf{3.3 \times 10^{-3}}$ — about $0.0033$, or roughly $0.33\%$ of the world's population.
1.5 — A small drop
(a) $0.000\,000\,3 = \mathbf{3 \times 10^{-7}}$ L.
(b) Total = $3 \times 10^{-7} \times 5 \times 10^{12} = (3 \times 5) \times 10^{-7+12} = 15 \times 10^5$. Fix the coefficient (must be $1 \leq a < 10$): $15 \times 10^5 = 1.5 \times 10^6$. Answer: $\mathbf{1.5 \times 10^6}$ L (1.5 million litres of water in the cloud).
2.1 — Explain your thinking (sample response)
My classmate got the COEFFICIENT right: they correctly slid the decimal to land on $4.86$. The mistake is the SIGN of the power. The rule from Lesson 16 is that small numbers have a negative power — and $0.000\,486$ is less than $1$, so $n$ must be negative. They used $+4$ but should have used $-4$. The correct working is: identify the first non-zero digit ($4$), slide the decimal $4$ places to the right ($0.000\,486 \to 4.86$), and because the original number is less than $1$, use $n = -4$. So $0.000\,486 = \mathbf{4.86 \times 10^{-4}}$. A simple size check: $4.86 \times 10^4 = 48\,600$, which is a HUGE number — completely the wrong size compared to the original tiny $0.000\,486$. That would have caught the slip immediately.
Marking: 1 mark for noting what was right; 1 for naming the rule; 1 for the correct answer; 1 for a sensible size check.