Lesson Guide
Explanation
Working backwards is the undo machine: when a puzzle tells you the END and asks for the START, reverse every operation AND reverse the order. 'Double it, add 6, get 26': undo the add first (26 − 6 = 20), then undo the double (20 ÷ 2 = 10). Two flips at once — plus becomes minus, times becomes divide, and the LAST step is undone FIRST. Always finish by running your answer forwards: 10, doubled 20, plus 6 is 26 ✓. The forward check catches every slip for free.
Your Turn
I triple a number and take away 5, getting 19. Undo it. Check: 19 + 5 = 24, ÷ 3 = 8. Forward: 8×3−5 = 19 ✓
Watch Out
Undoing in the story's original order is THE classic error: 26 ÷ 2 − 6 = 7 is wrong. Last step first, always.
Explanation
The string puzzle: mark the middle, fold in half, mark middles again — three markings in all — then unfold; neighbouring marks sit 5 cm apart. How long is the string? Forwards it's confusing; backwards it's clean. Each fold doubles the layers, so three rounds of marking split the string into 2 × 2 × 2 = 8 equal sections, separated by 7 marks. Each section is the 5 cm gap. String = 8 × 5 = 40 cm. The insight to hoard: folding n times makes 2n equal sections. Sketch two folds if you doubt it — the picture convinces instantly.
Your Turn
A strip folded twice (4 sections), marks 6 cm apart when unfolded. Length? Check: 4 × 6 = 24 cm.
Watch Out
Don't multiply by the number of MARKS (7) — sections carry the length, and sections = marks + 1. Fence-posts again!
Explanation
Money mysteries are working-backwards in disguise. Zia spends half on a book, $4 on a snack, and has $6 left — start? Undo the last event first: before the snack she had 6 + 4 = $10. Undo the halving: 'spent half' means what remains is half, so before the book she had 10 × 2 = $20. The subtle flip: 'spent half' undoes as ×2, and 'spent a third' (keeping two-thirds) undoes as ×3/2 — undo what REMAINS, not what was spent. Forward check: 20 → 10 → 6 ✓.
Your Turn
Marco spent $3 on a drink, then half of what remained on a game, leaving $5. Start? Check: 5 × 2 = 10 before the game; 10 + 3 = $13 start.
Watch Out
Track what REMAINS at each step. 'Spent half' leaves half (undo ×2); 'spent a quarter' leaves three-quarters (undo is trickier — OC sticks to halves).
Explanation
Time run backwards: the movie ENDS at 5:20 after 1 h 45 min of film, 15 minutes of ads, and a 10-minute early arrival — when did we arrive? Peel the stages off the end one at a time: film start 5:20 − 1:45 = 3:35; ads start 3:35 − 15 = 3:20; arrival 3:20 − 10 = 3:10. Subtracting 1:45 in one gulp invites errors — split it: minus 1 hour (4:20), minus 20 (4:00), minus 25 (3:35). The o'clocks are stepping stones backwards exactly as they are forwards.
Your Turn
Dinner finished at 7:05 after 50 minutes of eating and 25 minutes of cooking. When did cooking start? Check: 7:05 − 50 = 6:15; 6:15 − 25 = 5:50.
Watch Out
Backwards across the hour: 3:35 − 45 min → hop 3:35 − 35 = 3:00, then − 10 more = 2:50. Never invent '3:-10'.
Explanation
The undo toolkit: flip the operations AND the order (last step undone first); folds double sections (n folds → 2n pieces, and sections — not marks — carry the length); undo what REMAINS in spending stories ('spent half' → double the remainder); and ALWAYS run the forward check — ten seconds that catches everything. Backwards puzzles look hard and solve easily; they're confidence marks once the habit locks in. Quiz and exercises!
Your Turn
Capstone: I think of a number, halve it, add 7, and get 15. Number? Check: 15 − 7 = 8; 8 × 2 = 16. Forward: 16 → 8 → 15 ✓
Watch Out
The exam options include the forward-order-undo wrong answer every time. Your forward check is the shield.
📌 Key Points — this lesson in a nutshell
- To find a starting value, reverse each operation AND the order: undo the LAST step first.
- Operation flips: + ↔ −, × ↔ ÷, double ↔ halve.
- Folding n times makes 2n equal sections; sections (not marks) carry the length — in the example, 8 × 5 = 40 cm.
- Spending stories: undo what REMAINS — 'spent half' undoes as ×2 on the remainder.
- Time backwards: peel stages off the END one at a time, hopping through o'clocks.
- Always run the forward check — it catches every undo slip in seconds.
Step 1: Introduction
Step 1 of 5
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