Mathematics • Year 7 • Unit 2 • Lesson 8
Expanding Single Brackets in the Real World
Use the distributive law to find perimeters and total costs, to compare two-bag deals, and to expand expressions for the area of a rectangle that is described in two parts.
1. Word problems
Each problem uses ideas from Lesson 8: multiply the outside term by EVERY term inside, watching signs. Show your working — answers without working only get half marks.
1.1 — Snack packs. A snack pack contains x chips and 3 lollies. A teacher buys 4 of these snack packs.
(a) Write an expression for the total contents using a bracket: 4(x + 3).
(b) Expand the bracket. What does each part of your answer mean? 2 marks
1.2 — Rectangle area. A rectangle has width 5 cm and length (2x + 3) cm.
(a) Write an expression for the area as 5(2x + 3).
(b) Expand to find the area in expanded form.
(c) If x = 4 cm, what is the area in real numbers? Check by also working out (2 × 4 + 3) × 5 directly. 3 marks
1.3 — Family movie tickets. Each cinema ticket costs $(x − 2) (a $2 discount off the adult price of $x). A family buys 3 tickets.
(a) Write an expression for the total cost as 3(x − 2).
(b) Expand to get a simpler form.
(c) If x = 18, what is the total cost? 3 marks
1.4 — Temperature drop. A weather station records a drop of 2 degrees per hour for (x + 5) hours. The total temperature change (in degrees) is given by −2(x + 5).
(a) Expand −2(x + 5).
(b) Why are both terms in your answer negative? 2 marks
1.5 — Garden plots. A square garden plot has each side of length x metres. A second, smaller plot has each side of length (x − 3) metres. A gardener plans to fence around the smaller plot.
(a) Write an expression for the perimeter of the smaller plot as 4(x − 3) (perimeter of a square = 4 × side).
(b) Expand the bracket.
(c) If x = 8 m, how many metres of fence are needed? 3 marks
2. Explain your thinking
This question is about communication, not just symbols. Use full sentences. 4 marks
2.1 A classmate writes: "3(x + 2) = 3x + 2 — because you multiply 3 by x, then the +2 just comes along." Use your own words to explain (i) why this answer is wrong, (ii) what the correct expansion is, (iii) the distributive law in plain English. Use a specific value (like x = 5) to demonstrate that 3x + 2 and 3x + 6 give different answers.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Snack packs
(a) Total = 4(x + 3).
(b) Expanded: 4 × x + 4 × 3 = 4x + 12. Meaning: 4x = total chips (4 packs × x chips each), 12 = total lollies (4 packs × 3 lollies each).
1.2 — Rectangle area
(a) Area = 5(2x + 3).
(b) Expanded: 5 × 2x + 5 × 3 = 10x + 15 cm².
(c) Substitute x = 4: 10 × 4 + 15 = 40 + 15 = 55 cm². Direct check: (2 × 4 + 3) × 5 = 11 × 5 = 55 ✓.
1.3 — Family movie tickets
(a) Total = 3(x − 2).
(b) Expanded: 3 × x + 3 × (−2) = 3x − 6.
(c) Substitute x = 18: 3 × 18 − 6 = 54 − 6 = $48.
1.4 — Temperature drop
(a) −2(x + 5) = −2 × x + (−2) × 5 = −2x − 10 degrees.
(b) Both terms are negative because we are multiplying a negative outside (−2) by two positive inside terms (x and 5). Neg × pos = neg in both cases.
1.5 — Garden plots
(a) Perimeter = 4(x − 3).
(b) Expanded: 4 × x + 4 × (−3) = 4x − 12 m.
(c) Substitute x = 8: 4 × 8 − 12 = 32 − 12 = 20 m of fence.
2.1 — Explain your thinking (sample response)
(i) The student is wrong because they only multiplied the 3 by the x and forgot to multiply the 3 by the +2 as well. The 3 outside must visit BOTH terms inside.
(ii) The correct expansion is 3(x + 2) = 3 × x + 3 × 2 = 3x + 6.
(iii) The distributive law in plain English: "When a number is multiplied by a bracket, share that number with every term inside the bracket." You can draw arrows from the outside term to each inside term — every arrow becomes a multiplication.
Check with x = 5: Original 3(5 + 2) = 3 × 7 = 21. Student's 3x + 2 = 17 ✗. Correct 3x + 6 = 15 + 6 = 21 ✓.
Marking: 1 for spotting "didn't multiply the 2"; 1 for stating correct answer 3x + 6; 1 for clear plain-English explanation of distributive law; 1 for a sensible numerical check.