Worked example · PSE / Well-being · Year 8
LNF: Numeracy → Developing mathematical proficiency → Logical reasoning
A secure Year 8 response and full answer key, annotated to show which success criterion each part meets. The key ideas: the “BEST VALUE” Hoard pack (83p per 100 gems) is worse value than the Chest (82p) and the Dragon’s Lair (77p); the cheapest way to get the 800-gem skin is 1 Pouch and 3 Handfuls for £7.46; and 20 mystery boxes do not guarantee a legendary item.
| Pack | Working | Cost per 100 gems |
|---|---|---|
| Handful | 0.99 ÷ 100 × 100 | £0.99 |
| Pouch | 4.49 ÷ 500 × 100 = 0.898 | £0.90 |
| Chest | 8.99 ÷ 1,100 × 100 = 0.817… | £0.82 |
| Hoard | 19.99 ÷ 2,400 × 100 = 0.832… | £0.83 |
| Dragon’s Lair | 49.99 ÷ 6,500 × 100 = 0.769… | £0.77 |
Check: Chest gives 1,100 ÷ 8.99 = about 122 gems per £1. 100 gems for 82p means about 122 gems for £1, so the two methods agree.
Every pack is converted to the same unit before comparing, and one answer is verified another way. Evidences criterion 1.
(a) False. The Hoard costs 83p per 100 gems. The Chest costs 82p and the Dragon’s Lair costs 77p, so two packs give cheaper gems.
(b) The Hoard (2,400 gems) is bigger than the Chest (1,100 gems) but its gems cost more: 83p per 100 compared with 82p. One counter-example is enough to prove “always” false.
(c) To persuade players to spend £19.99 instead of £8.99. Most people will not check the maths.
The pupil disproves both claims with specific numbers and understands that one counter-example disproves a claim about “always”. Evidences criterion 2.
| Way to buy | Total cost | Left over |
|---|---|---|
| One Chest | £8.99 | 300 gems |
| 8 Handfuls | 8 × 0.99 = £7.92 | 0 |
| 1 Pouch and 3 Handfuls | 4.49 + 2.97 = £7.46 | 0 |
| 2 Pouches | 2 × 4.49 = £8.98 | 200 gems |
(a) 1 Pouch and 3 Handfuls is cheapest at £7.46. The Chest has cheaper gems per 100, but you have to buy 300 gems you do not need, and they cannot be turned back into money. So the best price per gem is not always the cheapest way to get what you want.
(b) Season pass (950 gems): 2 Pouches = £8.98 (50 left over); 1 Chest = £8.99 (150 left over); 1 Pouch and 5 Handfuls = £9.44; 10 Handfuls = £9.90. 2 Pouches for £8.98 is cheapest, but only by 1p.
All the sensible options are listed and compared, so the pupil can justify that £7.46 is the cheapest rather than just the first answer found. Evidences criteria 3 and 4.
| Claim | Verdict and proof |
|---|---|
| 1 | False. Each box has a 1 in 20 chance, but that is not a promise. The simulation found about 360 players out of 1,000 got no legendary item from 20 boxes. |
| 2 | False. Every box is separate and has the same 1 in 20 chance. The 19 earlier boxes do not change the next one. |
| 3 | True, approximately. 20 × 150 = 3,000 gems. 3,000 ÷ 100 = 30, and 30 × £0.82 = £24.60, which is about £25. |
Claims about chance are tested against evidence (the simulation) and the rule that each box is separate. Evidences criteria 2 and 4.
Don’t buy the Hoard! It says “best value” but its gems cost 83p per 100, while the Chest costs 82p, so the badge is wrong. Anyway, you only need 800 gems. The Hoard costs £19.99 and leaves you 1,600 gems you might never use. Buy 1 Pouch and 3 Handfuls instead: that is exactly 800 gems for £7.46. This proves you would save £12.53, because £19.99 − £7.46 = £12.53. (70 words)
A clear argument: it disproves the badge, identifies what is actually needed, gives a cheaper option and proves the saving with a calculation. Evidences criterion 4.