Mathematics • Year 10 • Unit 2 • Lesson 7

Mixed Techniques in the Real World

Apply expand-distribute-collect to real contexts: garden bed area differences, two-room flooring, profit comparisons and a rectangle vs square comparison. Then explain (in your own words) why distributing the minus is the single most error-prone step.

Apply · Real-World Maths

1. Word problems

Each problem mixes expansion, subtraction and collection. Show your working — a final answer with no working only earns half marks.

1.1 — Garden bed area difference. A large rectangular bed measures (x + 7) m by (x + 3) m. A smaller rectangular pond inside it measures (x + 2) m by (x + 1) m.

(a) Write an expression for the bed area and the pond area (expanded).
(b) Write a simplified expression for the planting area (bed minus pond).
(c) Evaluate the planting area when x = 5 m. 4 marks

Stuck? Revisit lesson § "Distribute the Minus" — every term in the pond expansion flips sign.

1.2 — Two-room flooring. A living room measures (x + 4) m by (x + 5) m. A study measures 2 m by (x + 3) m. Tiling costs $90 per m².

(a) Write and expand an expression for the total floor area.
(b) Simplify by collecting like terms.
(c) Calculate the total tiling cost when x = 3 m. 4 marks

Stuck? Total area = (x + 4)(x + 5) + 2(x + 3). Expand each piece first, then collect.

1.3 — Profit comparison. Cafe A models its weekly profit as P_A = (x + 6)(x + 2). Cafe B models its weekly profit as P_B = (x + 5)(x + 3). Both in hundreds of dollars; x is the price set.

(a) Expand each profit expression.
(b) Write and simplify a single expression for the profit difference D = P_A − P_B. (Watch the minus.)
(c) At x = 8 which cafe is more profitable, and by how much? 4 marks

Stuck? After distributing the minus, the x² terms should cancel — that's a good sign your sign work was clean.

1.4 — Rectangle vs square. A rectangle has length (x + 3) m and width (x − 3) m. A square has side x m.

(a) Expand the rectangle's area.
(b) Write and simplify a single expression for (square area) − (rectangle area).
(c) For any value of x, what does the simplified answer tell you about which shape has a bigger area? 3 marks

Stuck? (x + 3)(x − 3) is a "difference of squares". Square area is x². The difference will be a constant.

1.5 — Recipe maths. A small loaf needs (x + 2) cups of flour. A large loaf needs (x + 5) cups. A baker makes 3 small loaves and 2 large loaves.

(a) Write a single expression for the total flour required.
(b) Simplify by collecting like terms.
(c) If x = 4, how many cups of flour are needed in total? 3 marks

Stuck? Total = 3(x + 2) + 2(x + 5). Distribute each multiplier, then add.

2. Explain your thinking

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

2.1 A classmate says: "Whenever I see something like (x² + 5x + 6) − (x² + 2x + 1), I just write x² + 5x + 6 − x² + 2x + 1 and collect. Distributing the minus is too much effort." Using everything from Lesson 7, explain (i) why the classmate's shortcut gives a wrong answer in this exact example, (ii) what the correct simplified expression actually is, and (iii) the one habit they should adopt to never make this mistake again. Use the words "distribute" and "every term" somewhere in your answer.

Stuck? Revisit lesson § "Distribute the Minus" — what does the minus actually multiply?

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Garden bed area difference

(a) Bed = (x + 7)(x + 3) = x² + 10x + 21. Pond = (x + 2)(x + 1) = x² + 3x + 2.
(b) Planting = bed − pond = (x² + 10x + 21) − (x² + 3x + 2) = x² + 10x + 21 − x² − 3x − 2 = 7x + 19.
(c) x = 5 → 7(5) + 19 = 35 + 19 = 54 m².

1.2 — Two-room flooring

(a) Total = (x + 4)(x + 5) + 2(x + 3) = (x² + 9x + 20) + (2x + 6).
(b) Simplified = x² + 11x + 26 m².
(c) x = 3 → 9 + 33 + 26 = 68 m². Cost = 68 × $90 = $6 120.

1.3 — Profit comparison

(a) P_A = (x + 6)(x + 2) = x² + 8x + 12. P_B = (x + 5)(x + 3) = x² + 8x + 15.
(b) D = P_A − P_B = (x² + 8x + 12) − (x² + 8x + 15) = x² + 8x + 12 − x² − 8x − 15 = −3.
(c) D = −3 for any x (no x in the answer), so Cafe B is more profitable by $300 — including at x = 8. Check: P_A(8) = 14 × 10 = 140. P_B(8) = 13 × 11 = 143. Difference = −3. ✓

1.4 — Rectangle vs square

(a) Rectangle = (x + 3)(x − 3) = x² − 3x + 3x − 9 = x² − 9.
(b) Square − rectangle = x² − (x² − 9) = x² − x² + 9 = 9.
(c) The simplified difference is the constant 9 m², independent of x — so the square is always 9 m² bigger than the rectangle, no matter what x is.

1.5 — Recipe maths

(a) Total = 3(x + 2) + 2(x + 5).
(b) = 3x + 6 + 2x + 10 = 5x + 16 cups.
(c) x = 4 → 5(4) + 16 = 20 + 16 = 36 cups.

2.1 — Explain your thinking (sample response)

(i) The shortcut produces "x² + 5x + 6 − x² + 2x + 1 = 7x + 7", but the correct answer is 3x + 5. The classmate has only flipped the sign of the first term (x²) and left +2x and +1 untouched — the minus must distribute across every term in the second bracket. (ii) Done correctly: (x² + 5x + 6) − (x² + 2x + 1) = x² + 5x + 6 − x² − 2x − 1 = 3x + 5. (iii) The habit to adopt is to write the sign change on its own line: rewrite "−(a + b + c)" as "−a − b − c" before doing any collecting. That single extra line guarantees that every sign flips and stops the most common sign error in Year 10 algebra.

Marking: 1 for stating the shortcut's wrong answer; 1 for the correct simplified expression 3x + 5; 1 for the "write the sign change on its own line" habit; 1 for correct use of both required phrases.