Mathematics • Year 9 • Unit 1 • Lesson 16
Scientific Notation Conversion — Mixed Challenge
Pull together everything from Lessons 15–16: convert any size of number to scientific notation, fix non-standard forms, compare orders of magnitude. Spot a Year 9 mistake, then design two numbers that satisfy a given order-of-magnitude constraint.
1. Mixed problems — convert, validate, compare
Each one uses the conversion process: identify first non-zero digit $\to$ slide $\to$ count $\to$ sign. Show working. 3 marks each
1.1 Write each in scientific notation: (a) $5\,200$, (b) $1\,200\,000$, (c) $607\,000\,000$.
1.2 Write each in scientific notation: (a) $0.000\,082$, (b) $0.000\,000\,3$, (c) $0.005\,06$.
1.3 Each is NOT in valid scientific notation. Explain why each fails, then rewrite each correctly: (a) $93 \times 10^6$, (b) $0.93 \times 10^8$, (c) $10 \times 10^{10}$.
1.4 The diameter of a human hair is about $0.000\,07$ m, and Earth's radius is about $6\,370\,000$ m. (a) Write each in scientific notation. (b) State the order-of-magnitude difference. (c) Roughly how many hair-widths fit across Earth's radius (estimate to one significant figure)?
1.5 Find $n$ in each: (a) $4.5 \times 10^n = 45\,000$, (b) $7 \times 10^n = 0.000\,007$, (c) $6.07 \times 10^n = 607\,000\,000$.
1.6 Write the Earth–Sun distance $150\,000\,000\,000$ m in scientific notation, then check by reading it back as an ordinary number.
2. Find the mistake
A student is asked to write $0.000\,486$ in scientific notation. Exactly one line contains a mistake. Spot it, explain it, then redo the working correctly. 3 marks
Student's working — write $0.000\,486$ in scientific notation:
Line 1: The first non-zero digit is $4$.
Line 2: Slide the decimal right past the $4$: $0.000\,486 \to 4.86$. Moved $4$ places.
Line 3: The number is small, so $n$ is positive.
Line 4: So $0.000\,486 = 4.86 \times 10^4$.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Revisit lesson § "Small numbers $\to$ negative power". What sign should $n$ have when the original number is $< 1$?3. Open-ended challenge — design to a power
This question has many valid answers. 4 marks
3.1 Your task: invent two different ordinary numbers (one BIG, one SMALL) that meet these constraints:
• The BIG number, written in scientific notation, must have an exponent of exactly $\mathbf{n = +6}$.
• The SMALL number, written in scientific notation, must have an exponent of exactly $\mathbf{n = -5}$.
• Both numbers must come from a realistic real-world context (e.g. population, length, mass, money, volume).
For each number:
(i) State the real-world context (one sentence).
(ii) Write the ordinary number.
(iii) Convert it to scientific notation, showing the slide-and-count working.
(iv) Confirm the exponent matches the brief.
Bonus: The coefficients should be DIFFERENT — don't reuse the same $a$ for both numbers.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Big numbers
(a) $5\,200 = \mathbf{5.2 \times 10^3}$.
(b) $1\,200\,000 = \mathbf{1.2 \times 10^6}$.
(c) $607\,000\,000 = \mathbf{6.07 \times 10^8}$.
1.2 — Small numbers
(a) $0.000\,082 = \mathbf{8.2 \times 10^{-5}}$.
(b) $0.000\,000\,3 = \mathbf{3 \times 10^{-7}}$.
(c) $0.005\,06 = \mathbf{5.06 \times 10^{-3}}$.
1.3 — Fix the non-standard forms
(a) $93 \times 10^6$: coefficient $93 \geq 10$. Slide decimal one place left, add $1$ to the power: $\mathbf{9.3 \times 10^7}$.
(b) $0.93 \times 10^8$: coefficient $0.93 < 1$. Slide decimal one place right, subtract $1$ from the power: $\mathbf{9.3 \times 10^7}$.
(c) $10 \times 10^{10}$: coefficient $10$ is not strictly less than $10$. Slide decimal one place left, add $1$ to the power: $\mathbf{1 \times 10^{11}}$ (or simply $10^{11}$).
1.4 — Hair vs Earth's radius
(a) Hair: $\mathbf{7 \times 10^{-5}}$ m. Earth's radius: $\mathbf{6.37 \times 10^6}$ m.
(b) Order-of-magnitude difference: $6 - (-5) = \mathbf{11}$.
(c) Number of hairs: $\dfrac{6.37 \times 10^6}{7 \times 10^{-5}} \approx 0.91 \times 10^{11} \approx \mathbf{9 \times 10^{10}}$ — about $90$ billion hairs across the radius.
1.5 — Find $n$
(a) $4.5 \to 45\,000$ is a shift of $4$ places right; big number $\Rightarrow n = \mathbf{+4}$.
(b) $7 \to 0.000\,007$ is a shift of $6$ places left; small number $\Rightarrow n = \mathbf{-6}$.
(c) $6.07 \to 607\,000\,000$ is a shift of $8$ places right; $n = \mathbf{+8}$.
1.6 — Earth–Sun distance
$150\,000\,000\,000$ m. Slide $11$ places left to $1.5$. Big number $\Rightarrow$ positive power. $= \mathbf{1.5 \times 10^{11}}$ m.
Check: $1.5 \times 10^{11}$ shifts the decimal $11$ places right: $1.5 \to 15 \to 150 \to \ldots \to 150\,000\,000\,000$ ✓.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) For a SMALL number ($< 1$) the exponent $n$ must be NEGATIVE, not positive. The lesson rule: small numbers $\to$ negative power. The shift count of $4$ was correct, but the sign should be $-4$.
(c) Corrected working:
First non-zero digit is $4$.
Slide right past the $4$: $0.000\,486 \to 4.86$, moved $4$ places.
Number is small ($< 1$), so $n$ is negative: $n = -4$.
So $0.000\,486 = \mathbf{4.86 \times 10^{-4}}$.
3 — Open-ended challenge (sample solution)
Big number ($n = +6$):
(i) Context: Population of a mid-sized Australian city like Brisbane.
(ii) Ordinary number: $2\,560\,000$ people.
(iii) Slide $6$ places left from $2\,560\,000.$ to $2.56$. Big number $\Rightarrow$ positive power.
(iv) $2\,560\,000 = \mathbf{2.56 \times 10^6}$. Exponent matches $n = +6$ ✓.
Small number ($n = -5$):
(i) Context: Width of a single grain of fine sand.
(ii) Ordinary number: $0.000\,034$ m.
(iii) First non-zero digit is $3$. Slide $5$ places right to $3.4$. Small number $\Rightarrow$ negative power.
(iv) $0.000\,034 = \mathbf{3.4 \times 10^{-5}}$ m. Exponent matches $n = -5$ ✓.
Coefficient bonus: $2.56$ and $3.4$ are different, so the bonus is satisfied.
Marking: 2 marks for the BIG number (1 for realistic context, 1 for correct exponent matching $+6$). 2 marks for the SMALL number (1 for context, 1 for correct exponent matching $-5$). Bonus only if both coefficients are different.