Mathematics • Year 9 • Unit 1 • Lesson 1
Index Notation in the Real World
Apply index notation to everyday situations: viral videos, bacteria, chessboard rice, file sizes, and music streams. See how a tiny exponent describes huge numbers — then explain your method in your own words.
1. Word problems
Each problem uses index notation from Lesson 1: write the situation as $a^n$, identify the base and index, and evaluate. Show your working — a single final answer with no working only earns half marks.
1.1 — Viral video shares. Tia posts a video. After hour 1, three friends watch it. Each of those friends shares it with three more friends, so after hour 2 there are $3 \times 3 = 9$ new viewers. The "3 new each" pattern continues every hour.
(a) Write the number of new viewers in hour $5$ in index notation.
(b) Evaluate it as a normal number. 3 marks
1.2 — Bacteria on a kitchen sponge. A bacterium in a damp sponge splits in two every 30 minutes. Starting from $1$ bacterium at 8:00 am, how many will there be at 12:00 pm (noon)?
(a) How many 30-minute periods are there from 8:00 am to noon? Use this as your index.
(b) Write the answer in index notation as a power of $2$, then evaluate. 3 marks
1.3 — Chessboard rice (classic legend). A king places $1$ grain of rice on square 1, $2$ on square 2, $4$ on square 3 — doubling each square.
(a) Write the number of grains on square $n$ in index notation.
(b) How many grains are on square $10$? Show your working. 3 marks
1.4 — File sizes and powers of $2$. Computer memory is measured in powers of $2$. $1\ \text{KB} = 2^{10}$ bytes ($= 1024$ bytes).
(a) Write $2^{10}$ as a number.
(b) A photo on Sam's phone is $2^{20}$ bytes — known as "$1$ megabyte". Without using a calculator beyond doubling, show that $2^{20} = 2^{10} \times 2^{10}$ and explain what this means in plain English. 3 marks
1.5 — Music streams milestone. A song hits $10^6$ streams (one million). The artist's next song hits $10^7$ streams.
(a) Write both numbers in full digits.
(b) How many times more streams does the second song have than the first? Give a single power of $10$ as your answer. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes "$5^3 = 15$ because $5 \times 3 = 15$". They're confident they're right. In your own words, explain (i) what mistake they have made, (ii) what the index actually means, and (iii) the correct value of $5^3$. Use the words "base", "index" and "copies" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Viral video shares
(a) Each hour multiplies by $3$, so after $5$ hours of new viewers there are $3^5$ new viewers.
(b) $3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 9 \times 9 \times 3 = 81 \times 3 = \mathbf{243}$ new viewers in hour $5$.
Why this is index notation: $5$ hours of $\times 3$ each is exactly five copies of $3$ multiplied — that's $3^5$.
1.2 — Bacteria on a kitchen sponge
(a) $8:00$ am to noon $= 4$ hours $= 240$ minutes. $240 \div 30 = \mathbf{8}$ doublings.
(b) Number of bacteria $= 2^8 = 256$.
Working: $2^4 = 16$, $2^8 = 16 \times 16 = \mathbf{256}$ bacteria.
Real-world warning: this is why food left out warms up dangerously fast.
1.3 — Chessboard rice
(a) Square $n$ has $\mathbf{2^{n-1}}$ grains. (Check: square 1 has $2^0 = 1$ ✓.)
(b) Square $10$ has $2^{10-1} = 2^9 = \mathbf{512}$ grains.
Working: $2^9 = 2^4 \times 2^5 = 16 \times 32 = 512$. By square $64$ the total exceeds all rice grown on Earth in history.
1.4 — File sizes and powers of $2$
(a) $2^{10} = \mathbf{1024}$ bytes.
(b) $2^{20}$ is $20$ copies of $2$ multiplied. Split into $10$ copies $\times$ $10$ copies: $2^{20} = 2^{10} \times 2^{10} = 1024 \times 1024 = \mathbf{1{,}048{,}576}$ bytes.
In plain English: a megabyte ($2^{20}$ bytes) is "a thousand kilobytes" (close to it — really $1024$ KB). This is the product rule from Lesson 3 in disguise.
1.5 — Music streams milestone
(a) $10^6 = 1{,}000{,}000$ (six zeros). $10^7 = 10{,}000{,}000$ (seven zeros).
(b) $\dfrac{10^7}{10^6} = \mathbf{10^1 = 10}$ times more streams.
Why: each extra zero on base $10$ multiplies by $10$.
2.1 — Explain your thinking (sample response)
My classmate has confused the index with a multiplier. The index in $5^3$ does not multiply the base — it counts copies of the base. The base is $5$ and the index is $3$, so $5^3$ means three copies of $5$ multiplied together: $5 \times 5 \times 5 = 25 \times 5 = \mathbf{125}$, not $15$. A quick test: $5^2$ means two copies of $5$ multiplied — $5 \times 5 = 25$, not $5 \times 2 = 10$. The classmate's method would say $5^2 = 10$, which everyone knows is wrong, so the method itself is wrong.
Marking: 1 mark for naming the mistake; 1 for what the index really means; 1 for the correct answer $125$; 1 for clear, full-sentence explanation using "base", "index" and "copies".