Mathematics • Year 7 • Unit 1 • Lesson 15
Unit 1 — Capstone Challenge
Pull every Unit 1 idea together: integers, BODMAS, fractions, decimals, percentages, ratios and rates. Six mixed problems, one "find the mistake", and one open-ended capstone planning puzzle.
1. Mixed problems — every Unit 1 topic
Each question pulls from a different lesson in Unit 1. Decide which method applies before you start writing. Show working. 2 marks each
1.1 (Integers + BODMAS) Calculate −12 ÷ 3 + (−4) × (−2).
1.2 (Fractions — three operations) Calculate 5/6 − 3/8 + 1/4. Show the LCD and all conversion steps.
1.3 (Decimals + BODMAS) Calculate 2.4 ÷ 0.3 × 0.5. Work left to right after BODMAS gives × and ÷ equal precedence.
1.4 (Percentages) A $220 jacket is reduced by 25%, then a further 10% is taken off at the checkout. What is the final price? (Hint: apply the second discount to the already-discounted price, not to the original.)
1.5 (Ratios — three-part) Share $360 in the ratio 2:3:7. List all three shares and check they add to $360.
1.6 (Rates + percentages) A bus travels 240 km in 3 hours. (a) Find its speed. (b) If the speed is increased by 25%, what is the new speed?
2. Find the mistake
Two students worked out 2.4 ÷ 0.3 × 0.5. Student A got 4. Student B got 16. Their workings are below. Read both, decide which one is correct, then explain the other's mistake. 3 marks
Student A:
2.4 ÷ 0.3 = 8. Then 8 × 0.5 = 4.
Student B:
0.3 × 0.5 = 0.15. Then 2.4 ÷ 0.15 = 16.
(a) Which student is correct?
(b) Explain in one or two sentences what mistake the other student made.
(c) State the general rule for the order of operations when only multiplication and division appear in an expression (no brackets, no +/−).
Stuck? × and ÷ have equal precedence — when only those two operations appear, work strictly left to right, not multiplication first.3. Open-ended challenge — design a backyard garden
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 — Garden capstone. A rectangular backyard has length and width in the ratio 5:3 and a perimeter of 64 m.
(a) Find the actual length and width in metres.
(b) The owner wants to plant vegetables on exactly 40% of the garden area. What is the vegetable area in m²?
(c) Fertiliser costs $12.50 per 5 kg bag and one kg covers 8 m². How many whole bags are needed for the whole garden, and what is the total fertiliser cost?
(d) GST (10%) is added to the fertiliser cost. What is the final amount the owner pays?
How did this worksheet feel?
What I'll revisit before next class:
1.1 — −12 ÷ 3 + (−4) × (−2)
BODMAS: ÷ and × before +. −12 ÷ 3 = −4. (−4) × (−2) = +8. Then −4 + 8 = 4.
1.2 — 5/6 − 3/8 + 1/4
LCD(6, 8, 4) = 24. Convert: 5/6 = 20/24, 3/8 = 9/24, 1/4 = 6/24. Calculate: 20/24 − 9/24 + 6/24 = (20 − 9 + 6)/24 = 17/24. Already in simplest form (HCF(17, 24) = 1).
1.3 — 2.4 ÷ 0.3 × 0.5
× and ÷ have equal precedence — work left to right. 2.4 ÷ 0.3: shift dots → 24 ÷ 3 = 8. Then 8 × 0.5 = 4.
1.4 — $220 jacket, 25% then 10%
After 25% off: $220 × 0.75 = $165. Then 10% off the new price: $165 × 0.90 = $148.50.
Note: this is NOT the same as 35% off the original ($220 × 0.65 = $143). Stacked discounts always cost more than the sum of the percentages.
1.5 — $360 in ratio 2:3:7
Total parts = 2 + 3 + 7 = 12. One part = $360 ÷ 12 = $30.
Shares: 2 × $30 = $60; 3 × $30 = $90; 7 × $30 = $210.
Check: $60 + $90 + $210 = $360 ✓.
1.6 — Bus speed and 25% increase
(a) Speed = 240 ÷ 3 = 80 km/h.
(b) Increase by 25%: 80 × 1.25 = 100 km/h. (Or: 25% of 80 = 20, 80 + 20 = 100.)
2 — Find the mistake
(a) Student A is correct. Answer = 4.
(b) Student B did the multiplication 0.3 × 0.5 first, instead of working left to right. × and ÷ have equal precedence — you cannot do × before ÷ just because it appears later.
(c) Rule: when an expression has only × and ÷ (no +, −, or brackets), do them strictly left to right. They have equal precedence in BODMAS.
3 — Garden capstone
(a) Perimeter = 2(L + W) = 64 → L + W = 32. Ratio 5:3 has 5 + 3 = 8 parts. One part = 32 ÷ 8 = 4 m. Length = 5 × 4 = 20 m; width = 3 × 4 = 12 m. Check: 2(20 + 12) = 2 × 32 = 64 ✓.
(b) Area = 20 × 12 = 240 m². Vegetable area = 40% of 240 = 0.40 × 240 = 96 m².
(c) Fertiliser needed for full garden: 240 m² ÷ 8 m²/kg = 30 kg. Bags: 30 ÷ 5 = 6 bags. Cost: 6 × $12.50 = $75.
(d) Final with GST: $75 × 1.10 = $82.50.
Marking: 1 for (a) length and width; 1 for (b) vegetable area; 1 for (c) bags and cost; 1 for (d) GST-inclusive total.