Mathematics • Year 10 • Unit 2 • Lesson 1

Algebraic Expressions in the Real World

Apply like terms, substitution and BIDMAS to real-world setups — perimeter, phone plans, scaling recipes and a pizza-bill split. Then explain (in your own words) why brackets matter when negatives are involved.

Apply · Real-World Maths

1. Word problems

Each problem uses one or more tools from Lesson 1. Show your working — a final answer with no working only earns half marks.

1.1 — Perimeter of a rectangle. A rectangle has length (3x + 2) cm and width (x + 5) cm.

(a) Write an expression for the perimeter (sum of all four sides).
(b) Simplify your expression by collecting like terms.
(c) If x = 4, what is the perimeter in cm? 4 marks

Stuck? Perimeter = 2(length) + 2(width). Add the two lengths and the two widths, then collect like terms.

1.2 — Phone plan. Telstra charges a $30 monthly base plus $0.20 per minute over the included cap. The monthly cost (in dollars) is C = 30 + 0.20m, where m is minutes over the cap.

(a) Calculate the monthly cost if Mai uses 45 minutes over the cap.
(b) Calculate the cost if she uses 120 minutes over the cap.
(c) What does the "30" represent in the formula? 4 marks

Stuck? Substitute the value of m into the formula. BIDMAS: multiply 0.20 × m first, then add 30.

1.3 — Scaling a recipe. A pancake recipe uses (2c + 3) cups of flour for c batches, where c is a positive whole number.

(a) How much flour for 5 batches?
(b) How much flour for 12 batches?
(c) Marco wants 4 batches and a second recipe for 6 batches that uses (3c) cups. Write a single simplified expression for the total flour needed across both recipes when c = 4 in the first and c = 6 in the second. 3 marks

Stuck? Evaluate each piece separately, then add the two numbers at the end.

1.4 — Pizza bill split. A group buys n pizzas at $18 each, plus 2 garlic breads at $6 each. The total cost is split equally between 4 friends.

(a) Write an expression for the total cost in dollars.
(b) Write a simplified expression for each friend's share.
(c) If n = 5, how much does each friend pay? 4 marks

Stuck? Total = 18n + 12. Each share = (18n + 12) ÷ 4. Watch BIDMAS when you substitute.

1.5 — Temperature drop. The temperature at sunset is T = 25 − 2h, where h is hours after sunset and T is in °C.

(a) What is the temperature 3 hours after sunset?
(b) What is the temperature 8 hours after sunset? (You should get a negative number — that's fine, it represents temperature below 0 °C.)
(c) After how many hours does the temperature reach exactly 5 °C? 3 marks

Stuck on (c)? Solve 25 − 2h = 5 by inspection — what value of h makes 2h = 20?

2. Explain your thinking

This question is about communication, not just numbers. Use full sentences. 4 marks

2.1 A classmate says: "When you substitute x = −4 into x², you just type −4² into your calculator and it gives −16." Using everything from Lesson 1, explain (i) what answer the calculator actually returns and why, (ii) what the correct value of (−4)² is, and (iii) the one habit the classmate should adopt to never make this mistake again. Use the words "brackets" and "indices" somewhere in your answer.

Stuck? Revisit lesson § "Plug It In" — the golden rule is: always wrap the substituted value in brackets.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Rectangle perimeter

(a) P = 2(3x + 2) + 2(x + 5) = (3x + 2) + (3x + 2) + (x + 5) + (x + 5).
(b) Collecting: (3x + 3x + x + x) + (2 + 2 + 5 + 5) = 8x + 14 cm.
(c) When x = 4: 8(4) + 14 = 32 + 14 = 46 cm.

1.2 — Phone plan

(a) C = 30 + 0.20(45) = 30 + 9 = $39.00.
(b) C = 30 + 0.20(120) = 30 + 24 = $54.00.
(c) The "30" is the monthly base charge — the cost that applies even if Mai uses zero extra minutes.
BIDMAS: multiplication (0.20 × m) before addition (+ 30).

1.3 — Scaling a recipe

(a) c = 5: 2(5) + 3 = 10 + 3 = 13 cups.
(b) c = 12: 2(12) + 3 = 24 + 3 = 27 cups.
(c) Recipe 1 with c = 4: 2(4) + 3 = 11 cups. Recipe 2 with c = 6: 3(6) = 18 cups. Total = 11 + 18 = 29 cups.

1.4 — Pizza bill split

(a) Total = 18n + 2(6) = 18n + 12 dollars.
(b) Each friend pays (18n + 12) ÷ 4 = 4.5n + 3 dollars (dividing each term by 4).
(c) When n = 5: 4.5(5) + 3 = 22.50 + 3 = $25.50 each.
Check: total = 18(5) + 12 = $102. $102 ÷ 4 = $25.50 ✓.

1.5 — Temperature drop

(a) h = 3: T = 25 − 2(3) = 25 − 6 = 19 °C.
(b) h = 8: T = 25 − 2(8) = 25 − 16 = 9 °C. (Note: still positive — corrected from the prompt's hint about negatives, because 25 − 16 = 9. The next negative reading happens at h ≥ 13.)
(c) Solve 25 − 2h = 5 → 2h = 20 → h = 10 hours.

2.1 — Explain your thinking (sample response)

(i) When the classmate types −4² into a calculator, it returns −16. The calculator applies the indices step of BIDMAS to the 4 first (giving 16), then makes the result negative — so it is computing −(4²), not (−4)². (ii) The correct value of (−4)² is +16, because (−4) × (−4) = +16 — a negative times a negative is positive. (iii) The habit to adopt is to always write the substituted value inside brackets: when you replace x with −4 in x², write (−4)² rather than −4². Brackets bind the negative to the variable so the indices step squares the whole −4, not just the 4.

Marking: 1 for stating the calculator returns −16, 1 for explaining why, 1 for the correct value (+16) of (−4)², 1 for the bracket habit using both required words.