Working Backwards from Averages | OCR GCSE

Working backwards from averages — given the mean, median, mode and range, find the data. OCR GCSE Foundation, Year 10 Unit 7 Lesson 3. Includes the impossible-puzzle moment (Q1) that quietly introduces proof by exhaustion.

The problem-solving capstone of the calculation work in L01–L02. Every clue is a constraint on a position: median → middle, mode → repeats, mean × n → total, range → split the ends. Pupils who can reverse the mean here will handle the “the mean of n values is m; one more is added…” exam question type without re-teaching.

What’s included:

• §2 Three-Integer scaffold — I Do (mean 8, median 10, range 12 → 1, 10, 13) and We Do, both with the box-grid layout that makes the method visible
• §3 Four-Integer scaffold — adds the mode constraint; I Do (mode = median, clean) and We Do (mode = median with range 14 — extreme split)
• Three-misconception Spot the Mistake — median-without-sorting, range-anchored-to-median (instead of spanning the ends), mode-as-“appears once”
• Four independent practice tasks — A, B (three-integer pair sharing the ends-sum step), C (mode = median, clean), D (mode at the bottom end — pupils who force the mode into the middle break the median)
• Going Deeper Q1 — an impossible puzzle. Mean 11, median 11, mode 9, range 4. The arithmetic forces a bimodal {9, 9, 13, 13}, so the mode isn’t 9 alone. Pupils discover that doing the method correctly and reaching a contradiction is itself the answer. Early exposure to proof by exhaustion at GCSE Foundation level.
• Q2 — design your own puzzle for a partner to rebuild. Strong pupils notice some clue sets give multiple answers (when no mode is fixed) or none (like Q1).
• Exit Ticket — Jenny’s golf scores. A classic textbook question, often printed with the wrong answer (69, 75, 75, 79 has mode 75, not 76). Correct answer: 69, 76, 76, 79 — note for colleagues using older resources.

Teacher key (4 pages):

• Full box-grid working for all 11 puzzles
• Rough pacing guide (~55 min lesson) — starting point, not script
• Per-Sam diagnostic notes: Sam 1 wrong → forward-averages gap (median), needs L01–L02 retrieval, not just today’s method
• Marking summary table with “mode-position” patterns called out (middle in C, bottom in D, impossible in Q1)
• Bridge note on the slotcell scaffold as a candidate for reuse in any future work-backwards or constraint lesson

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