Mathematics • Year 9 • Unit 1 • Lesson 2
Evaluating Powers in the Real World
Use sign rules, fraction powers and BIDMAS in everyday contexts: a savings account that halves, a board-game elevation, a tournament bracket, a recipe scale-down, and a bank-account fine. Then explain your thinking in your own words.
1. Word problems
Each problem uses ideas from Lesson 2: negative bases with brackets, fraction powers, $1^n$ and $0^n$, or BIDMAS with indices. Show your working — a single final answer with no working only earns half marks.
1.1 — Halving a chocolate bar. Jess starts with one full chocolate bar. She eats $\dfrac{1}{2}$, leaving $\dfrac{1}{2}$. Her brother eats half of what's left, leaving $\dfrac{1}{2}$ of $\dfrac{1}{2}$. Their mum does the same, then Jess does it again — four halvings in total.
(a) Write the fraction of the bar remaining after $4$ halvings as $\left(\dfrac{1}{2}\right)^4$.
(b) Evaluate it as a single fraction. 3 marks
1.2 — Board-game elevation. In a strategy board game, every "fall" card subtracts $2^3$ metres from your altitude. Theo's character starts at $50$ m and draws three "fall" cards in a row.
(a) Write the total altitude lost as a single calculation involving $2^3$.
(b) What altitude is Theo's character at after the three falls? Use BIDMAS — indices before subtraction. 3 marks
1.3 — Esports knockout bracket. A knockout tournament has $2^6$ players. Each round, exactly half the remaining players are eliminated, so the number of players left after $n$ rounds is $\dfrac{2^6}{2^n}$ — a power-of-$2$ pattern.
(a) How many players start the tournament?
(b) How many are still in after $4$ rounds? (Use the pattern; no need for the quotient rule yet.) 3 marks
1.4 — Scaling down a recipe. A recipe for $8$ people uses $\left(\dfrac{3}{4}\right)^2$ kg of flour per person (yes, a fraction squared — strange but true). Mia is cooking for just herself, so wants the per-person amount.
(a) Evaluate $\left(\dfrac{3}{4}\right)^2$ as a fraction in lowest terms.
(b) Express the answer as a decimal to $4$ decimal places. Is it less than half a kilogram per person? 3 marks
1.5 — Bank account overdraft fine. Each time a bank account is overdrawn, the bank charges $-(\$2^4)$ — that is, a fine of $\$2^4$ subtracted from the balance. Jamal's account is overdrawn $3$ times in one week.
(a) Write the total fine using one power of $2$ multiplied by $3$.
(b) If Jamal's starting balance was $\$80$, what is the balance after the three fines (assume no other transactions)? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate evaluates $-4^2$ and gets $16$. They argue: "The negative is part of the base, so $(-4)^2 = 16$ either way — the brackets don't matter." In your own words, explain (i) what mistake they have made about brackets, (ii) what $-4^2$ actually means, and (iii) the correct value of $-4^2$. Refer to "what the index applies to" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Halving a chocolate bar
(a) Each halving multiplies by $\dfrac{1}{2}$. Four halvings: $\left(\dfrac{1}{2}\right)^4$.
(b) $\left(\dfrac{1}{2}\right)^4 = \dfrac{1^4}{2^4} = \mathbf{\dfrac{1}{16}}$ of the bar remains.
Why a power of a fraction: each halving multiplies the remaining fraction by another $\dfrac{1}{2}$, four times total.
1.2 — Board-game elevation
(a) Total lost $= 3 \times 2^3$ metres.
(b) BIDMAS: $2^3 = 8$ first; then $3 \times 8 = 24$; then $50 - 24 = \mathbf{26}$ m. Theo's character is at $\mathbf{26}$ m altitude.
Why BIDMAS matters: doing $50 - 3$ first would give $47 \times 2^3 = 376$, which is nonsense.
1.3 — Esports knockout bracket
(a) $2^6 = 64$ players start.
(b) After $4$ rounds, $2^{6-4} = 2^2 = \mathbf{4}$ players remain.
So one more round eliminates 2 players, leaving the final 2 — and one final to decide the winner.
1.4 — Scaling down a recipe
(a) $\left(\dfrac{3}{4}\right)^2 = \dfrac{3^2}{4^2} = \mathbf{\dfrac{9}{16}}$ kg per person.
(b) $9 \div 16 = \mathbf{0.5625}$ kg. Yes — $0.5625 > 0.5$, so it's slightly MORE than half a kilogram per person, not less. (A typical Year 9 surprise — squaring a number less than 1 makes it smaller, but $\dfrac{9}{16}$ is still just over $\dfrac{1}{2}$.)
1.5 — Bank account overdraft fine
(a) Total fine $= 3 \times 2^4$ dollars.
(b) BIDMAS: $2^4 = 16$; $3 \times 16 = 48$; $80 - 48 = \mathbf{\$32}$ balance.
Real-world warning: fixed fines based on powers add up fast — three small fines wiped out $60\%$ of the balance.
2.1 — Explain your thinking (sample response)
My classmate has forgotten that the index only applies to what is directly attached to it. In $-4^2$ there are no brackets, so the index $2$ applies only to the $4$ — not to the negative sign. The expression means $-(4^2) = -(16) = \mathbf{-16}$, NOT $16$. If they wanted the answer $16$, they would need to write $(-4)^2$ — the brackets are what include the negative in the base. A quick check: BIDMAS says indices are calculated before the negative sign is applied (the negative behaves like a "multiply by $-1$" out the front), so $4^2$ happens first, then negate.
Marking: 1 mark for naming what the index applies to; 1 for what $-4^2$ actually means; 1 for the correct answer $-16$; 1 for a clear, full-sentence explanation.