Mathematics • Year 9 • Unit 1 • Lesson 12

Product Rule in the Real World

Use the algebraic product rule and one-step expansions in real contexts: rectangle areas, garden plots, lab dilutions, bake-sale takings and a phone screen. Then explain a classmate's slip in full sentences.

Apply · Real-World Maths

1. Word problems

Each problem uses the product rule (one or more variables) or a one-step expansion $a(b+c)$. Show every step — a final answer alone earns half marks.

1.1 — Rectangle area. A rectangle has length $2x^2 y$ cm and width $5xy^3$ cm.

(a) Write its area as a single simplified term.
(b) State the coefficient and the index of each variable in your answer. 3 marks

Stuck? Area = length $\times$ width. Coefficients multiply, indices on the same variable add.

1.2 — Square garden plot. Nia is planning a square garden bed with side length $4 a^3$ metres.

(a) Write its area as a single simplified term.
(b) If she doubles the side length to $8 a^3$ m, write the new area and say how many times bigger it is. 3 marks

Stuck? Squaring means multiplying it by itself: $(4a^3) \times (4a^3)$. Same as the product rule with two identical terms.

1.3 — Lab dilution (signs matter). A chemistry student records two scaling factors as $-3 m^2 n$ and $-4 m n^3$. To get the combined effect she multiplies them together.

(a) Calculate the combined scaling factor, simplified.
(b) Will the combined factor be positive or negative? Justify in one sentence. 3 marks

Stuck? Decide the sign first — two negatives multiplied give a positive.

1.4 — Bake-sale takings. At the Year 9 bake sale, $3x$ stall-helpers each sell $(2x^2 + 5x)$ items in an hour.

(a) Write the total number of items sold in an hour by expanding $3x(2x^2 + 5x)$.
(b) If $x = 4$, evaluate your expanded expression to find the actual number of items. 3 marks

Stuck? Distribute, then sub $x = 4$ at the very end — neat substitution avoids slips.

1.5 — Phone screen pixels. Sam's phone screen is $2 x^4$ pixels wide and $3 x^2 y$ pixels tall (for some pixel scale $x$ and aspect factor $y$).

(a) Write the total pixel area as a single simplified term.
(b) Why does the variable $y$ keep its own index of $1$ in your answer (not combine with $x$)? 3 marks

Stuck? Three-column rule: coefficient column, $x$ column, $y$ column. Different bases never merge.

2. Explain your thinking

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

2.1 A classmate writes "$3 a^2 \times 2 a = 5 a^3$". They're confident they're right. In your own words, explain (i) which two parts of the calculation they have mixed up, (ii) the correct answer with working, and (iii) what advice you'd give them to avoid this mistake next time. Refer to "coefficient" and "index" in your answer.

Stuck? Revisit lesson § "Common Pitfalls — Forgetting to multiply coefficients". Their $3 + 2$ should have been $3 \times 2$.

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 area

(a) Area $= 2x^2 y \times 5 x y^3 = (2 \times 5) \cdot x^{2+1} \cdot y^{1+3} = \mathbf{10 x^3 y^4}$ cm$^2$.
(b) Coefficient $= 10$; index of $x = 3$; index of $y = 4$.

1.2 — Square garden plot

(a) Area $= (4a^3)^2 = (4 \times 4)(a^3 \times a^3) = \mathbf{16 a^6}$ m$^2$.
(b) New area $= (8a^3)^2 = (8 \times 8)(a^{3+3}) = 64 a^6$ m$^2$. Ratio $= \dfrac{64 a^6}{16 a^6} = \mathbf{4}$. Doubling the side gives 4 times the area.

1.3 — Lab dilution

(a) $-3 m^2 n \times -4 m n^3$. Sign: $(-)(-) = (+)$. Coefficients: $3 \times 4 = 12$. Indices: $m^{2+1} = m^3$, $n^{1+3} = n^4$. Answer: $\mathbf{12 m^3 n^4}$.
(b) Positive — two negative factors multiplied always give a positive product.

1.4 — Bake-sale takings

(a) $3x(2x^2 + 5x) = 3x \cdot 2x^2 + 3x \cdot 5x = 6 x^3 + 15 x^2$.
(b) Sub $x = 4$: $6(4)^3 + 15(4)^2 = 6(64) + 15(16) = 384 + 240 = \mathbf{624}$ items.

1.5 — Phone screen pixels

(a) Area $= 2 x^4 \times 3 x^2 y = (2 \times 3)(x^{4+2})(y) = \mathbf{6 x^6 y}$ pixels$^2$.
(b) The product rule $a^m \times a^n = a^{m+n}$ only applies when the BASE is the same. $x$ and $y$ are different bases (different letters), so the $y$ keeps its own index of $1$ and sits as a separate factor.

2.1 — Explain your thinking (sample response)

My classmate has confused the coefficient step with the index step. They added the coefficients ($3 + 2 = 5$) when they should have multiplied them ($3 \times 2 = 6$). They got the index part right ($2 + 1 = 3$), because the product rule says indices on the same base ADD. The correct working is: coefficients $3 \times 2 = 6$, $a$ indices $2 + 1 = 3$, so $3a^2 \times 2a = \mathbf{6 a^3}$. My advice would be to always run the three-column check from the lesson — sign, coefficient (multiply), and each variable (add the indices) — so you don't accidentally swap operations. A quick check with $a = 1$: $3(1)^2 \times 2(1) = 3 \times 2 = 6$, which matches $6a^3 = 6$, not $5$.

Marking: 1 mark for spotting the swap; 1 for the correct answer; 1 for clearly using "coefficient" and "index"; 1 for sensible advice.