Singapore School Mathematics Operating Manual · Chapter 45
Distinguish non-negotiable mathematical constraints from preferences, penalties and tie-breakers so feasibility is established before optimisation.
This chapter extends the operating manual as a cross-topic control layer. It is designed to complement, not replace, existing BTT owners for constraints, optimisation, error diagnosis, modelling, proof and examination craft.
1. Hard constraints define admissibility
The key discipline is to make hard constraints define admissibility explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
2. Soft constraints express preference
The key discipline is to make soft constraints express preference explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
3. Objectives rank feasible candidates
The key discipline is to make objectives rank feasible candidates explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
4. Domain restrictions are hard
The key discipline is to make domain restrictions are hard explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
5. Integer requirements can be hard
The key discipline is to make integer requirements can be hard explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
6. Penalties encode soft violations
The key discipline is to make penalties encode soft violations explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
7. Penalty weights are part of the model
The key discipline is to make penalty weights are part of the model explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
8. Priority rules can be lexicographic
The key discipline is to make priority rules can be lexicographic explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
9. Weighted sums allow compensation
The key discipline is to make weighted sums allow compensation explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
10. Non-compensatory gates are different
The key discipline is to make non-compensatory gates are different explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
11. Tie-breakers operate after primary conditions
The key discipline is to make tie-breakers operate after primary conditions explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
12. Constraint order can simplify reasoning
The key discipline is to make constraint order can simplify reasoning explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
13. Feasibility-first workflow
The key discipline is to make feasibility-first workflow explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
14. Hard and soft labels come from context
The key discipline is to make hard and soft labels come from context explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
15. Uncertainty can blur apparent feasibility
The key discipline is to make uncertainty can blur apparent feasibility explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
16. Slack measures distance inside a hard constraint
The key discipline is to make slack measures distance inside a hard constraint explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
17. Active constraints deserve attention
The key discipline is to make active constraints deserve attention explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
18. Conflicting soft goals create trade-offs
The key discipline is to make conflicting soft goals create trade-offs explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
19. A hierarchy audit
Ask what the problem requires, what can fail, what evidence is decisive, and which earlier operating-manual control should be invoked before calculation continues.
20. Worked mini-cases
Case A. A candidate satisfies every preference but violates a domain restriction. The candidate remains inadmissible because validity is checked before preference.
Case B. Two calculations disagree. Return to the last trusted common input, separate the branches, and identify whether the discrepancy comes from a shared dependency or a branch-specific step.
Case C. A result lies close to a threshold. Preserve precision and compare the uncertainty interval with the decision margin before classifying it.
21. Independent practice
1. Identify the admissibility condition in a problem of your choice.
2. Give one independent check that could catch a wrong intermediate result.
3. Explain why a mathematically attractive candidate may still be invalid.
4. Describe one situation where uncertainty changes the strength of a conclusion.
5. State what evidence would justify stopping the calculation.
22. Worked answers
1. The admissibility condition is the rule every candidate must satisfy before comparison.
2. Examples include substitution, a unit check, a bound, an alternative representation or an independent method.
3. Objective performance cannot compensate for violation of a hard mathematical condition.
4. A rounded or measured value near a threshold may have a plausible interval crossing both decision regimes.
5. A complete proof, verified candidate set, guaranteed bound, tolerance certificate or exhausted feasible search can justify stopping.
23. Continue through Batch 12
Return to the BTT Mathematics Hub for Batch 12 and use the neighbouring chapters as a connected control system.