Mathematics • Year 7 • Unit 2 • Lesson 9

Expanding and Simplifying in the Real World

Use the two-step "expand, then collect" method to find the perimeter of shapes whose sides are bracketed expressions, to add up multi-part purchases, and to combine multiple discounts.

Apply · Real-World Maths

1. Word problems

Each problem uses ideas from Lesson 9: expand every bracket FIRST, then collect like terms. Show your working — answers without working only get half marks.

1.1 — Two snack bundles. A small bundle has 2 chips and 3 lollies. A large bundle has 4 chips and 1 lolly. Sam buys x small bundles and y large bundles.

(a) Total chips = 2x + 4y. Total lollies = 3x + y. Total items = 2x + 4y + 3x + y. Simplify the total.
(b) Now Sam buys (x + 2) small bundles instead. Write the total chips as 2(x + 2) and expand. 2 marks

Stuck? For (a), collect like terms: x-terms together, y-terms together.

1.2 — Triangle perimeter. A triangle has three sides: (x + 3), (2x + 1), and (x − 2) centimetres.

(a) Write the perimeter as the sum of the three sides.
(b) Simplify by collecting like terms.
(c) Find the perimeter when x = 4. 3 marks

Stuck? Perimeter = (x + 3) + (2x + 1) + (x − 2). No brackets needed to multiply here — just collect like terms.

1.3 — Two-group movie booking. A class of 2(x + 3) students and a teacher group of 3(x + 1) parents all book movie tickets.

(a) Write the total number of tickets as 2(x + 3) + 3(x + 1).
(b) Expand and simplify.
(c) If x = 5, how many tickets in total? 3 marks

Stuck? 2(x + 3) = 2x + 6. 3(x + 1) = 3x + 3. Add: 5x + 9.

1.4 — Cake batter recipe (less an estimate). A baker prepares 3(2x − 1) cups of flour, then removes 2(x + 4) cups for a different recipe.

(a) Write the remaining flour as 3(2x − 1) − 2(x + 4).
(b) Expand and simplify. (Be careful with the second bracket — it's subtracted.) 3 marks

Stuck? 3(2x − 1) = 6x − 3. −2(x + 4) = −2x − 8. Combine: 6x − 3 − 2x − 8 = 4x − 11.

1.5 — Three-store shopping. Ben spends 2(x + 5) dollars at Store A, 3(x − 2) dollars at Store B, and returns one item worth (x + 1) dollars (a refund, so subtract).

(a) Write the total net spend as 2(x + 5) + 3(x − 2) − (x + 1).
(b) Expand and simplify.
(c) If x = 10, what is Ben's net spend in dollars? 3 marks

Stuck? Three brackets — expand each, then collect everything at the end. The third bracket has a minus in front, so flip its signs.

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) − (x − 1) = 3x + 6 − x − 1 = 2x + 5". Use your own words to explain (i) which step is wrong, (ii) what the correct expansion is, (iii) the rule for handling a "subtracted bracket". Use a value (like x = 4) to demonstrate that the correct answer gives a different number than 2x + 5.

Stuck? When you subtract a bracket, ALL signs inside the bracket flip. So −(x − 1) = −x + 1, not −x − 1.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Two snack bundles

(a) Total = 2x + 4y + 3x + y = (2x + 3x) + (4y + y) = 5x + 5y.
(b) 2(x + 2) = 2 × x + 2 × 2 = 2x + 4 chips.

1.2 — Triangle perimeter

(a) Perimeter = (x + 3) + (2x + 1) + (x − 2).
(b) Collect: x-terms x + 2x + x = 4x; constants 3 + 1 − 2 = 2. Perimeter = 4x + 2 cm.
(c) Substitute x = 4: 4 × 4 + 2 = 18 cm.

1.3 — Two-group movie booking

(a) Total = 2(x + 3) + 3(x + 1).
(b) Expand: 2x + 6 + 3x + 3. Collect: 5x + 9.
(c) Substitute x = 5: 5 × 5 + 9 = 25 + 9 = 34 tickets.

1.4 — Cake batter recipe

(a) Remaining flour = 3(2x − 1) − 2(x + 4).
(b) Expand: 6x − 3 − 2x − 8 (note −2 × x = −2x and −2 × 4 = −8). Collect: x-terms 6x − 2x = 4x; constants −3 − 8 = −11. Answer: 4x − 11 cups.

1.5 — Three-store shopping

(a) Net spend = 2(x + 5) + 3(x − 2) − (x + 1).
(b) Expand: 2x + 10 + 3x − 6 − x − 1. Collect: x-terms 2x + 3x − x = 4x; constants 10 − 6 − 1 = 3. Answer: 4x + 3 dollars.
(c) Substitute x = 10: 4 × 10 + 3 = $43.

2.1 — Explain your thinking (sample response)

(i) The mistake is in the very first expansion step. The student wrote −(x − 1) as −x − 1, but the minus sign in front of the bracket should flip BOTH signs inside. So −(x − 1) is actually −x + 1.
(ii) Correct expansion: 3(x + 2) − (x − 1) = 3x + 6 − x + 1. Collect: x-terms 3x − x = 2x; constants 6 + 1 = 7. Answer: 2x + 7 (not 2x + 5).
(iii) Rule for subtracted brackets: think of "−(...)" as "−1 × (...)". When you multiply every term inside by −1, every sign inside flips. + becomes −, and − becomes +.
Check with x = 4: Original 3(4 + 2) − (4 − 1) = 18 − 3 = 15. Student's 2x + 5 = 13 ✗. Correct 2x + 7 = 15 ✓.

Marking: 1 for spotting which step is wrong; 1 for stating correct answer 2x + 7 with working; 1 for the "every sign flips" rule explained clearly; 1 for a sensible numerical check.