Mathematics • Year 10 • Unit 2 • Lesson 3
Expanding Binomials in the Real World
Apply FOIL to area, two-step rate problems, packaging design and a quadratic price model. Then explain why (a + b)² is NOT a² + b² in plain English — area-model style.
1. Word problems
Each problem uses FOIL or the perfect-square pattern from Lesson 3. Show your working.
1.1 — Rectangular garden. A rectangular garden has length (x + 5) m and width (x + 3) m.
(a) Write an expression for the area as a product of the two binomials.
(b) Expand using FOIL.
(c) Find the area when x = 4 m. 3 marks
1.2 — Square photo frame. A square photo frame has a side length of (x + 2) cm.
(a) Write an expression for the area of the frame as (x + 2)².
(b) Expand it using the perfect-square pattern.
(c) Find the area when x = 8 cm. 3 marks
1.3 — Phone-case packaging. A rectangular phone case has length (2x + 1) cm and width (x − 1) cm, where x is a positive whole number ≥ 2.
(a) Write the area as a product of binomials.
(b) Expand using FOIL.
(c) If x = 6 cm, what is the area? 3 marks
1.4 — Concert ticket revenue. A concert ticket costs $(x − 5) and (x + 20) people attend, where x represents the venue category index.
(a) Write an expression for the total revenue as a product of binomials.
(b) Expand using FOIL.
(c) Calculate the revenue when x = 50. 3 marks
1.5 — Extending a square room. A square room has side length x m. The owner extends one wall by 3 m and the perpendicular wall by 4 m, making a new rectangular room of (x + 3) m by (x + 4) m.
(a) Write an expression for the new room's area as a product of binomials.
(b) Expand using FOIL.
(c) Write an expression for the extra area gained (new area − x²). Simplify. 4 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A classmate confidently writes "(a + b)² = a² + b²". Using the area-grid idea from Lesson 3, explain (i) why this is wrong, (ii) what the missing piece geometrically represents on a square of side (a + b), and (iii) what the correct expansion is. Use the words "middle term" and "2ab" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Rectangular garden
(a) Area = (x + 5)(x + 3) m².
(b) FOIL: x² + 3x + 5x + 15 = x² + 8x + 15.
(c) When x = 4: 16 + 32 + 15 = 63 m².
1.2 — Square photo frame
(a) Area = (x + 2)² cm².
(b) Perfect square: x² + 2(x)(2) + 2² = x² + 4x + 4.
(c) When x = 8: 64 + 32 + 4 = 100 cm². (Check: side was 10, 10² = 100 ✓.)
1.3 — Phone case
(a) Area = (2x + 1)(x − 1).
(b) FOIL: 2x² − 2x + x − 1 = 2x² − x − 1.
(c) When x = 6: 2(36) − 6 − 1 = 72 − 7 = 65 cm².
1.4 — Concert revenue
(a) Revenue = (x − 5)(x + 20) dollars.
(b) FOIL: x² + 20x − 5x − 100 = x² + 15x − 100.
(c) When x = 50: 2500 + 750 − 100 = $3,150.
1.5 — Extending a square room
(a) New area = (x + 3)(x + 4) m².
(b) FOIL: x² + 4x + 3x + 12 = x² + 7x + 12.
(c) Extra = (x² + 7x + 12) − x² = 7x + 12 m².
The x² cancels — the extra is the two strips (3x and 4x) plus the corner (3 × 4 = 12).
2.1 — Explain your thinking (sample response)
(i) The classmate's claim "(a + b)² = a² + b²" is wrong because it's missing the middle term. (ii) Geometrically, a square with side length (a + b) splits into four pieces: an a-by-a square (area a²), a b-by-b square (area b²), and two a-by-b rectangles (combined area 2ab). The classmate's answer only counts the two corner squares and ignores the two rectangles, which is exactly 2ab of missing area. (iii) The correct expansion is (a + b)² = a² + 2ab + b² — always three terms, never two.
Marking: 1 for identifying the missing term, 1 for the area-grid description, 1 for the correct expansion, 1 for using both required words.