Mathematics • Year 9 • Unit 1 • Lesson 15
Scientific Notation — Mixed Challenge
Mix reading, validating and comparing scientific notation. Identify the only valid form, spot a Year 9 mistake involving leading zeros, then design two different scientific-notation expressions for the same target number.
1. Mixed problems — read, validate, compare
Each one uses the definition $a \times 10^n$ with $1 \leq a < 10$. Show working. 3 marks each
1.1 Write each as an ordinary number: (a) $3.2 \times 10^5$, (b) $4.7 \times 10^{-4}$, (c) $9 \times 10^{-3}$.
1.2 From this list, write down the ones that ARE in scientific notation, and rewrite the ones that AREN'T: $42 \times 10^3$, $4.2 \times 10^4$, $0.42 \times 10^5$, $10 \times 10^3$, $6 \times 10^6$.
1.3 Order these from smallest to largest: $7 \times 10^5$, $3 \times 10^6$, $9 \times 10^4$, $1 \times 10^6$. Justify the order using the powers.
1.4 The mass of an electron is about $9.11 \times 10^{-31}$ kg. The mass of a proton is about $1.67 \times 10^{-27}$ kg. By how many orders of magnitude does the proton outweigh the electron?
1.5 Decide if each statement is TRUE or FALSE, and explain in one sentence each: (a) $5 \times 10^0 = 5$. (b) $0.6 \times 10^7$ is in scientific notation. (c) $10 \times 10^4 = 10^5$.
1.6 Find the value of $n$ such that $3.2 \times 10^n = 32\,000\,000$. Show how you got there.
2. Find the mistake
A student is asked to write $4.7 \times 10^{-4}$ as an ordinary number. Exactly one line contains a mistake. Spot it, explain it, then redo the working correctly. 3 marks
Student's working — write $4.7 \times 10^{-4}$ as an ordinary number:
Line 1: The power is $-4$, so I shift the decimal $4$ places to the LEFT.
Line 2: $4.7 \to 0.47 \to 0.047 \to 0.0047$
Line 3: That's $3$ shifts so far. One more: $0.0047 \to 0.00047$.
Line 4: So $4.7 \times 10^{-4} = 0.0047$.
(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? Count the shifts in Line 2 carefully. Then check whether Line 4 used the result FROM Line 3 or only from Line 2.3. Open-ended challenge — same number, two forms
This question has many valid answers. 4 marks
3.1 The ordinary number $\mathbf{45\,000\,000}$ can be written in scientific notation as $4.5 \times 10^7$. But it can ALSO be written in other (non-standard) "$a \times 10^n$" forms that aren't valid scientific notation.
(i) Write $45\,000\,000$ in correct scientific notation.
(ii) Write two other $a \times 10^n$ forms of $45\,000\,000$ where the coefficient is NOT between $1$ and $10$ (one with $a \geq 10$, one with $a < 1$). Show each is equal to $45\,000\,000$.
(iii) For each of your two non-standard forms, explain in one sentence which rule of scientific notation it breaks.
Bonus: Convert each of your two non-standard forms back to the proper scientific notation form.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Ordinary numbers
(a) $3.2 \times 10^5 = \mathbf{320\,000}$.
(b) $4.7 \times 10^{-4} = \mathbf{0.000\,47}$.
(c) $9 \times 10^{-3} = \mathbf{0.009}$.
1.2 — Validate scientific notation
Valid: $\mathbf{4.2 \times 10^4}$ and $\mathbf{6 \times 10^6}$.
Not valid (with correct form):
• $42 \times 10^3$ — coefficient too big; correct: $4.2 \times 10^4$.
• $0.42 \times 10^5$ — coefficient too small; correct: $4.2 \times 10^4$.
• $10 \times 10^3$ — coefficient too big (the rule says $a < 10$, strictly); correct: $1 \times 10^4$ (or just $10^4$).
1.3 — Order smallest to largest
Compare powers first; tie-break with coefficients.
$9 \times 10^4 = 90\,000$ < $7 \times 10^5 = 700\,000$ < $1 \times 10^6 = 1\,000\,000$ < $3 \times 10^6 = 3\,000\,000$.
Order: $\mathbf{9 \times 10^4, \; 7 \times 10^5, \; 1 \times 10^6, \; 3 \times 10^6}$. The bigger power dominates; when powers tie, the coefficient decides.
1.4 — Proton vs electron mass
Difference in powers: $-27 - (-31) = \mathbf{4}$ orders of magnitude. The proton is about $10^4 = 10\,000$ times more massive than the electron (the coefficients are roughly similar, so the power difference dominates).
1.5 — True or false
(a) TRUE — $10^0 = 1$, so $5 \times 10^0 = 5 \times 1 = 5$.
(b) FALSE — the coefficient $0.6$ is less than $1$, breaking the rule $1 \leq a < 10$. Correct form: $6 \times 10^6$.
(c) TRUE — $10 \times 10^4 = 10^{1+4} = 10^5$ (product rule for powers of the same base).
1.6 — Find $n$ in $3.2 \times 10^n = 32\,000\,000$
To go from $3.2$ to $32\,000\,000$, shift the decimal $\mathbf{7}$ places to the right. So $n = \mathbf{7}$.
Check: $3.2 \times 10^7 = 32\,000\,000$. ✓
2 — Find the mistake
(a) The mistake is on Line 4.
(b) Line 4 used the result from Line 2 (which was only $3$ shifts) instead of the result from Line 3 (which completed the $4$th shift). Even though Line 3 correctly added one more shift, the student then wrote the wrong number in Line 4.
(c) Corrected working:
The power is $-4$, so shift the decimal $4$ places left.
$4.7 \to 0.47 \to 0.047 \to 0.0047 \to 0.00047$.
So $4.7 \times 10^{-4} = \mathbf{0.000\,47}$ (or equivalently $0.00047$).
3 — Open-ended challenge (sample solution)
(i) Standard form: $45\,000\,000 = \mathbf{4.5 \times 10^7}$.
(ii) Two non-standard forms:
Form A (coefficient too big): $45 \times 10^6$. Check: $45 \times 1\,000\,000 = 45\,000\,000$ ✓.
Form B (coefficient too small): $0.45 \times 10^8$. Check: $0.45 \times 100\,000\,000 = 45\,000\,000$ ✓.
(iii) Form A breaks the "$a < 10$" rule — the coefficient $45$ is too big. Form B breaks the "$a \geq 1$" rule — the coefficient $0.45$ is less than $1$.
(Bonus) Convert back: Form A: shift one place left and add $1$ to the power: $45 \times 10^6 = 4.5 \times 10^7$. Form B: shift one place right and subtract $1$ from the power: $0.45 \times 10^8 = 4.5 \times 10^7$.
Marking: 1 mark for (i); 2 marks for two correct non-standard forms with verification; 1 mark for naming the rule each breaks. Bonus only if both conversions back to standard form are correct.