Mathematics • Year 10 • Unit 2 • Lesson 2
Expanding Brackets in the Real World
Apply the distributive law to area problems, group discounts, ticket bundles and area-comparison setups. Then explain why brackets aren't optional — they tell you exactly what gets multiplied through.
1. Word problems
Each problem uses the distributive law a(b + c) = ab + ac. Show your working — final answer with no working only earns half marks.
1.1 — Area of a garden bed. A rectangular vegetable bed has length (x + 7) m and width 4 m.
(a) Write an expression for the area in m².
(b) Expand the brackets so the expression has no brackets left.
(c) If x = 3, calculate the area in m². 3 marks
1.2 — Concert ticket bundle. A concert sells tickets at $(x + 15) each (with a $15 booking fee on top of the base price x). A school orders 25 tickets.
(a) Write an expression for the total cost the school pays.
(b) Expand the brackets.
(c) If the base price x = $80, what does the school pay in total? 3 marks
1.3 — Two paddocks, fenced together. A farmer has two square paddocks: one with side (3x) m and one with side (2x + 5) m. He plans to fence the perimeter of each.
(a) Write expressions for the perimeter of each paddock (perimeter of a square = 4 × side).
(b) Expand both expressions.
(c) Write a single simplified expression for the total length of fencing needed. 4 marks
1.4 — Group discount comparison. A gym charges (x + 12) dollars per person for a class. A neighbouring gym charges (x + 8) dollars per person. Both classes have 15 people.
(a) Write an expression for the total revenue from each gym.
(b) Expand both.
(c) Write a simplified expression for the difference in revenue between the two gyms. Is it ever zero? 4 marks
1.5 — Path around a swimming pool. A rectangular pool measures x m by (x + 4) m. It's surrounded by a path 2 m wide on every side. The total outer rectangle (pool + path) therefore measures (x + 4) m by (x + 8) m.
(a) Write an expression for the total outer area, before expanding.
(b) Write an expression for the pool's area.
(c) Write an expression for the area of just the path (outer area − pool area), then expand and simplify. 4 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A classmate writes the working "−3(x − 5) = −3x − 15" and asks you to check it. Using everything from Lesson 2, explain (i) what is wrong with their answer, (ii) what the correct expansion is, and (iii) the one mental check the classmate could do every time to avoid this mistake. Use the words "distribute" and "negative" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Garden bed
(a) Area = 4(x + 7) m².
(b) = 4x + 28 m².
(c) When x = 3: 4(3) + 28 = 12 + 28 = 40 m².
1.2 — Concert ticket bundle
(a) Total = 25(x + 15) dollars.
(b) = 25x + 375.
(c) When x = 80: 25(80) + 375 = 2000 + 375 = $2,375.
1.3 — Two paddocks
(a) Paddock 1: P = 4(3x). Paddock 2: P = 4(2x + 5).
(b) Paddock 1: 12x. Paddock 2: 8x + 20.
(c) Total fencing = 12x + 8x + 20 = 20x + 20 m.
1.4 — Group discount comparison
(a) Gym 1 = 15(x + 12). Gym 2 = 15(x + 8).
(b) Gym 1 = 15x + 180. Gym 2 = 15x + 120.
(c) Difference = (15x + 180) − (15x + 120) = $60. Gym 1 always brings in $60 more than Gym 2, regardless of x. It is never zero, because the per-person difference ($4) times 15 people = $60 of constant gap.
1.5 — Path around a pool
(a) Outer area = (x + 4)(x + 8) m².
(b) Pool area = x(x + 4) = x² + 4x m².
(c) Path = (x + 4)(x + 8) − (x² + 4x). Expand the first: x² + 8x + 4x + 32 = x² + 12x + 32. Subtract: (x² + 12x + 32) − (x² + 4x) = 8x + 32 m².
The pool's x² cancels neatly — the path's area grows linearly with x.
2.1 — Explain your thinking (sample response)
(i) The classmate's answer is wrong because they only applied the negative sign to the first inner term (x) and kept the second inner term's sign (−15) intact. (ii) When you distribute −3 across (x − 5), the multiplier hits every inner term with its full sign: (−3)(x) = −3x and (−3)(−5) = +15 (a negative times a negative is positive). The correct expansion is −3x + 15. (iii) The one mental check is: "Did the outside multiplier — sign and all — hit every inner term?" If yes, the signs of the inner terms should have flipped wherever the multiplier itself was negative.
Marking: 1 for identifying the −15 (sign error), 1 for the correct expansion, 1 for the mental check, 1 for using both required words.