Repairing the chapter weightage in an 80-mark Mathematics paper
A teacher is finalising the HSC Mathematics and Statistics paper: 80 marks, three hours. The board's chapter weightage and the teacher's draft are:
| Chapter | Required | Draft |
|---|---|---|
| Differentiation and Applications | 20 | 26 |
| Integration and Applications | 16 | 16 |
| Differential Equations | 8 | 4 |
| Vectors | 8 | 8 |
| Three-dimensional Geometry | 8 | 8 |
| Linear Programming | 8 | 4 |
| Probability Distributions | 12 | 14 |
The paper has ten 2-mark questions, eight 3-mark questions and nine 4-mark questions, and every 4-mark question is paired with an internal-choice alternative from the same chapter. The teacher must repair the mismatch with the fewest changes, without altering the total marks, the number of questions or the mark pattern. A colleague suggests simply deleting one 4-mark differentiation question and adding a 4-mark question on linear programming, saying “the marks will balance out”. The head of department asks for a written justification before signing the paper.
1. Quantify the mismatch and give a minimal set of changes that restores the weightage.
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Deviations (draft − required): Differentiation +6, Integration 0, Differential Equations −4, Vectors 0, 3-D Geometry 0, Linear Programming −4, Probability Distributions +2. The deviations sum to zero, so the total is right but the chapter balance is not.
Each change moves marks between two chapters, so at least three swaps are needed to correct four deviating chapters. A minimal set:
- Replace one 4-mark Differentiation question by a 4-mark Differential Equations question: Differentiation 22, DE 8.
- Replace one 2-mark Differentiation question by a 2-mark Linear Programming question: Differentiation 20, LP 6.
- Replace one 2-mark Probability question by a 2-mark Linear Programming question: LP 8, Probability 12.
Result: 20, 16, 8, 8, 8, 8, 12 - total 80, still 27 questions, still ten 2-mark, eight 3-mark and nine 4-mark items.
2. Assess the colleague's suggestion that one 4-mark swap will balance the paper.
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It does not. Removing a 4-mark differentiation question takes Differentiation from 26 to 22, still 2 above the required 20, and adding 4 marks to Linear Programming fixes only that chapter, taking it from 4 to 8. Differential Equations stays at 4 against 8, and Probability stays at 14 against 12.
It also breaks the paper's structure: the total falls to 79 unless something is added back, the number of questions falls to 26, and the count of 4-mark questions changes, disturbing the mark pattern. The internal-choice pairing is disrupted as well, because the deleted differentiation question had an OR partner left without a pair. Corrections must be made mark for mark in both directions, keeping the total, the question count and the mark pattern fixed.
3. What must a replacement question satisfy, apart from the chapter weightage?
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The internal-choice pairing must be preserved. If the differentiation question being removed was one of an OR pair, the partner must also be replaced, and both alternatives must come from the same chapter as each other; otherwise a candidate who chooses by chapter can get an unintended advantage or no choice at all. The new pair should be of comparable difficulty and length, carry the same mark distribution, and be solvable within the share of the 3-hour budget that the blueprint allows.
The replacement must be from the prescribed syllabus, must not duplicate an existing question, and must preserve the paper's balance of knowledge, understanding and application items. The teacher should also check that the answer key and the expected time per question are consistent with the new item before signing.