Mathematics • Year 7 • Unit 2 • Lesson 15
Bracket Equations in the Real World
Use brackets to model "groups of …" situations — apples per box, pages per chapter, sides of a polygon — and solve with the distributive law.
1. Word problems
For each: choose a letter, write an equation with brackets, then solve (either by expanding OR by dividing first — whichever is cleaner). Always check.
1.1 — Boxes of apples. A grocer packs apples into boxes. Each box holds (x + 7) apples. Three identical boxes contain 36 apples in total.
(a) Write an equation in x.
(b) Solve for x and write the answer in a sentence. 2 marks
1.2 — Triangle perimeter. An isosceles triangle has two equal sides of length (x + 4) cm and a base of 6 cm. Its perimeter is 26 cm.
(a) Write an equation. (Perimeter = sum of all sides.)
(b) Solve for x. 2 marks
1.3 — Pages per chapter. A novel has 5 chapters of equal length. Each chapter has (x − 2) pages. The whole novel has 90 pages.
(a) Write an equation in x.
(b) Solve for x and answer: how many pages are in each chapter? 2 marks
1.4 — Rectangle area. A rectangle has width 4 cm and length (x + 3) cm. Its area is 32 cm². (Recall area = length × width.)
(a) Write an equation in x.
(b) Solve for x and write the actual length in a sentence. 2 marks
1.5 — Birthday lolly bags. Each lolly bag holds 2x lollies and a 5-cent novelty toy. Eight bags together cost $1.36 — but this question is about lollies, not money. Each bag has (2x + 3) lollies after the host added 3 extras to every bag. Eight bags hold 88 lollies in total.
(a) Write an equation in x.
(b) Solve for x. 2 marks
2. Explain your thinking
This question is about communication, not just symbols. Use full sentences. 4 marks
2.1 A classmate expanding 3(x + 4) writes "3x + 4". In your own words, explain (i) what mistake they've made, (ii) what the distributive law a(b + c) = ab + ac is telling us to do, (iii) what 3(x + 4) actually expands to with all working, and (iv) give one real-life "3 groups of …" example where ignoring the second multiplication would change the answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Boxes of apples
(a) Equation: 3(x + 7) = 36.
(b) Divide by 3: x + 7 = 12. Subtract 7: x = 5. Each box holds 5 + 7 = 12 apples. Check: 3(5 + 7) = 3(12) = 36 ✓.
1.2 — Triangle perimeter
(a) Equation: 2(x + 4) + 6 = 26.
(b) Subtract 6: 2(x + 4) = 20. Divide by 2: x + 4 = 10. Subtract 4: x = 6. (Equal sides are 6 + 4 = 10 cm.) Check: 2(10) + 6 = 26 ✓.
1.3 — Pages per chapter
(a) Equation: 5(x − 2) = 90.
(b) Divide by 5: x − 2 = 18. Add 2: x = 20. Each chapter has 20 − 2 = 18 pages. Check: 5(18) = 90 ✓.
1.4 — Rectangle area
(a) Equation: 4(x + 3) = 32.
(b) Divide by 4: x + 3 = 8. Subtract 3: x = 5. The length is 5 + 3 = 8 cm. Check: 4 × 8 = 32 ✓.
1.5 — Birthday lolly bags
(a) Equation: 8(2x + 3) = 88.
(b) Divide by 8: 2x + 3 = 11. Subtract 3: 2x = 8. Divide by 2: x = 4. Each bag has 2(4) + 3 = 11 lollies. Check: 8(11) = 88 ✓.
2.1 — Explain your thinking (sample response)
(i) The student multiplied the 3 only by the first term (x) and forgot to multiply it by the second term (+4). They wrote 3x + 4 when the correct answer needs the 3 to "reach" every term inside.
(ii) The distributive law says the number outside the brackets must multiply EACH term inside: a(b + c) = a × b + a × c = ab + ac.
(iii) 3(x + 4) = 3 × x + 3 × 4 = 3x + 12.
(iv) "Three identical lunchboxes each contain x sandwiches and 4 grapes." Total grapes = 3 × 4 = 12, NOT just 4. If you only multiply once, you'd say there are only 4 grapes in total across all three lunchboxes — clearly wrong.
Marking: 1 for naming the missed multiplication; 1 for stating the distributive law; 1 for the correct expansion 3x + 12; 1 for a sensible "groups of" example.