Mathematics • Year 9 • Unit 1 • Lesson 15
Introduction to Scientific Notation
Build fluency RECOGNISING and READING scientific notation $a \times 10^n$ with $1 \leq a < 10$. Watch one worked example, fill in a guided one, then complete eight independent practice problems.
1. I do — fully worked example
Reading a positive power means shifting the decimal point RIGHT. The power tells you how many places.
Problem. Write $3.2 \times 10^5$ as an ordinary number.
Step 1 — Read the power.
$n = 5$ — positive, so the number is BIG.
Reason: positive power $\Rightarrow$ shift the decimal right $\Rightarrow$ the number gets bigger.
Step 2 — Decide direction and distance.
Shift the decimal point $5$ places to the RIGHT.
Reason: multiplying by $10^5$ is the same as multiplying by $10$ five times — each $\times 10$ shifts the decimal one place right.
Step 3 — Shift, adding zeros as you go.
$3.2 \to 32 \to 320 \to 3\,200 \to 32\,000 \to 320\,000$
Reason: once you run out of digits, fill with zeros to hold the place value.
Step 4 — Write the answer.
$3.2 \times 10^5 = 320\,000$
Answer: $\mathbf{320\,000}$.
2. We do — fill in the missing steps
Same structure, but the working is faded. Fill in each blank. 4 marks
Problem. Write $4.7 \times 10^{-4}$ as an ordinary number.
Step 1 — Read the power: $n = \_\_\_\_$ . The sign of $n$ is __________________ , so the number is __________________ (big / small).
Step 2 — Direction and distance: shift the decimal point $\_\_\_$ places to the __________________ (right / left).
Step 3 — Shift, adding leading zeros as you go:
$4.7 \to 0.47 \to 0.0\_7 \to 0.00\_\_ \to 0.\_\_\_\_\_\_$
Step 4 — Write the answer:
$4.7 \times 10^{-4} = \_\_\_\_\_\_\_\_$
3. You do — independent practice
Show working under each problem. Foundation = read or recognise. Standard = compare or rewrite. Extension = real-world readings.
Foundation — read or recognise
3.1 Write $2.5 \times 10^4$ as an ordinary number. 1 mark
3.2 Write $9 \times 10^{-3}$ as an ordinary number. 1 mark
3.3 Write $6.02 \times 10^7$ as an ordinary number. 1 mark
3.4 Circle each one that IS in scientific notation. For each rejected one, say why in 5 words or fewer: $42 \times 10^3$ • $4.2 \times 10^4$ • $0.42 \times 10^5$ • $6 \times 10^6$. 1 mark
Standard — compare and reason
3.5 Which is bigger: $7 \times 10^5$ or $3 \times 10^6$? Justify your choice using the powers. 2 marks
3.6 Is $0.5 \times 10^6$ in scientific notation? Why or why not? 2 marks
Extension — read real measurements
3.7 The Sun is about $1.5 \times 10^{11}$ m from Earth. Write that distance as an ordinary number, and state its order of magnitude (the power of $10$). 2 marks
3.8 A hydrogen atom is about $1 \times 10^{-10}$ m wide. Write that width as an ordinary number, and say in one sentence why scientific notation is more useful here than the ordinary form. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded $4.7 \times 10^{-4}$)
Step 1: $n = \mathbf{-4}$. Sign is negative, so the number is small (less than $1$).
Step 2: shift $\mathbf{4}$ places to the left.
Step 3: $4.7 \to 0.47 \to 0.047 \to 0.0047 \to \mathbf{0.000\,47}$.
Step 4: $4.7 \times 10^{-4} = \mathbf{0.000\,47}$.
3.1 — $2.5 \times 10^4$
Shift 4 places right: $2.5 \to 25 \to 250 \to 2\,500 \to \mathbf{25\,000}$.
3.2 — $9 \times 10^{-3}$
Shift 3 places left: $9 \to 0.9 \to 0.09 \to \mathbf{0.009}$.
3.3 — $6.02 \times 10^7$
Shift 7 places right: $6.02 \to 60.2 \to 602 \to 6020 \to 60\,200 \to 602\,000 \to 6\,020\,000 \to \mathbf{60\,200\,000}$.
3.4 — Which are in scientific notation?
Valid: $\mathbf{4.2 \times 10^4}$ and $\mathbf{6 \times 10^6}$.
Rejected: $42 \times 10^3$ — coefficient too big. $0.42 \times 10^5$ — coefficient too small.
3.5 — Which is bigger: $7 \times 10^5$ vs $3 \times 10^6$?
$\mathbf{3 \times 10^6}$ is bigger. The larger power wins: $10^6 = 10 \times 10^5$, so even though $3 < 7$, $3 \times 10^6 = 3\,000\,000$ vs $7 \times 10^5 = 700\,000$.
3.6 — Is $0.5 \times 10^6$ in scientific notation?
No — the coefficient $a = 0.5$ is less than $1$, but scientific notation requires $1 \leq a < 10$. Correct form: shift the decimal one place right and reduce the power by $1$: $0.5 \times 10^6 = 5 \times 10^5$.
3.7 — Sun distance
$1.5 \times 10^{11} = \mathbf{150\,000\,000\,000}$ m. Order of magnitude: $10^{11}$ (about a hundred billion metres).
3.8 — Hydrogen atom width
$1 \times 10^{-10} = \mathbf{0.000\,000\,000\,1}$ m. Scientific notation is more useful because writing ten zeros is error-prone — it's easy to mis-count and write nine or eleven by mistake. $1 \times 10^{-10}$ tells you the exact power instantly.