Mathematics • Year 7 • Unit 2 • Lesson 6
Multiplying Algebraic Terms in the Real World
Use the rules of algebraic multiplication to find rectangle areas, box volumes, and total costs where the dimensions are written as letters instead of numbers.
1. Word problems
Each problem uses ideas from Lesson 6: multiply coefficients, multiply variables, add powers when the same variable appears twice. Show your working — answers without working only get half marks.
1.1 — Garden bed. A rectangular garden bed has width 3x metres and length 4y metres.
(a) Write an expression for the area of the garden (area = width × length).
(b) If x = 2 m and y = 5 m, what is the area in real numbers? 2 marks
1.2 — Square mosaic tile. A square mosaic tile has each side of length 5a centimetres.
(a) Write an expression for the area of one tile (area of a square = side × side).
(b) Write an expression for the total area of 3 of these tiles. 3 marks
1.3 — Lunchbox stack. A box of lunchboxes is 2x lunchboxes wide, 3y lunchboxes long, and the stack is 4 lunchboxes high. The total number of lunchboxes is width × length × height.
(a) Write an expression for the total number of lunchboxes.
(b) If x = 5 and y = 2, how many lunchboxes are there? 3 marks
1.4 — Fence posts. A school fence needs (−2x) posts on the front edge and 5x posts on the side. To find a "post-area" estimate, the groundskeeper multiplies these two numbers together.
(a) Write and simplify the product (−2x) × (5x).
(b) Why does it make sense that the answer is negative if x is positive? 2 marks
1.5 — Phone covers. A factory makes phone covers in rectangular shapes that are 2a cm wide and 3a cm tall. Each batch contains 5 covers.
(a) Write an expression for the area of one cover.
(b) Write an expression for the total area of plastic used in one batch.
(c) If a = 4 cm, how much plastic is in one batch in square centimetres? 3 marks
2. Explain your thinking
This question is about communication, not just symbols. Use full sentences. 4 marks
2.1 A classmate writes: "x × x = 2x — because the x appears twice." Use your own words to explain (i) why this answer is wrong, (ii) what the correct answer is, (iii) the difference between x + x and x × x. Use one concrete example with a number (e.g. let x = 5) to back up your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Garden bed
(a) Area = 3x × 4y = 12xy square metres.
(b) Substitute x = 2 and y = 5: Area = 12 × 2 × 5 = 120 m².
1.2 — Square mosaic tile
(a) Area of one tile = 5a × 5a = (5 × 5)(a × a) = 25a² cm².
(b) Total area of 3 tiles = 3 × 25a² = 75a² cm².
1.3 — Lunchbox stack
(a) Total = 2x × 3y × 4 = (2 × 3 × 4)(xy) = 24xy lunchboxes.
(b) Substitute x = 5, y = 2: Total = 24 × 5 × 2 = 240 lunchboxes.
1.4 — Fence posts
(a) (−2x) × (5x) = (−2 × 5)(x × x) = −10x².
(b) When you multiply a negative quantity by a positive one, the result is negative. Here −2x is negative (assuming x is positive), and 5x is positive, so the product is negative.
1.5 — Phone covers
(a) Area of one cover = 2a × 3a = (2 × 3)(a × a) = 6a² cm².
(b) Total area in one batch = 5 × 6a² = 30a² cm².
(c) Substitute a = 4: Total = 30 × 4² = 30 × 16 = 480 cm².
2.1 — Explain your thinking (sample response)
(i) Saying x × x = 2x is wrong because the rule "the letter appearing twice means a coefficient of 2" only works for addition (x + x = 2x), not multiplication.
(ii) The correct answer is x × x = x² ("x squared"). When you multiply the same variable by itself, the power goes up by 1 each time.
(iii) x + x means "x plus another x", which gives 2x. But x × x means "x times another x", which gives x². They look similar but mean completely different operations.
Concrete check with x = 5: x + x = 5 + 5 = 10 (this equals 2 × 5, so 2x ✓). x × x = 5 × 5 = 25 (this equals 5², so x² ✓). The two answers are very different.
Marking: 1 for spotting the wrong rule; 1 for stating the correct answer x²; 1 for explaining the + vs × difference; 1 for a sensible numerical check.