Singapore School Mathematics Operating Manual · Chapter 50
Identify which terms control a result at a given scale, decide when smaller effects can be neglected, and keep approximation error subordinate to the decision being made.
This chapter is a cross-topic control layer for Primary, PSLE, SEC G1/G2/G3, E-Math, A-Math and JC Mathematics. It complements existing BTT topic owners rather than replacing them.
1. Not every term matters equally at every scale
The operating principle is to make not every term matters equally at every scale explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
2. Dominance depends on the operating region
The operating principle is to make dominance depends on the operating region explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
3. Leading terms control large-input polynomial behaviour
The operating principle is to make leading terms control large-input polynomial behaviour explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
4. Near zero a different term may dominate
The operating principle is to make near zero a different term may dominate explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
5. Relative size matters more than visual complexity
The operating principle is to make relative size matters more than visual complexity explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
6. A negligible term needs a tolerance definition
The operating principle is to make a negligible term needs a tolerance definition explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
7. Dropping a term changes equality into approximation
The operating principle is to make dropping a term changes equality into approximation explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
8. Approximation should preserve sign when sign matters
The operating principle is to make approximation should preserve sign when sign matters explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
9. Approximation should preserve threshold side when decisions matter
The operating principle is to make approximation should preserve threshold side when decisions matter explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
10. Scale separation can simplify ratios
The operating principle is to make scale separation can simplify ratios explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
11. Small-angle style approximations need domains
The operating principle is to make small-angle style approximations need domains explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
12. Rounding and term deletion are different approximations
The operating principle is to make rounding and term deletion are different approximations explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
13. Dominant balance can reveal expected magnitude
The operating principle is to make dominant balance can reveal expected magnitude explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
14. Cancellation can make a small term suddenly important
The operating principle is to make cancellation can make a small term suddenly important explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
15. Near a zero dominant terms can change owner
The operating principle is to make near a zero dominant terms can change owner explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
16. Error bounds justify neglect
The operating principle is to make error bounds justify neglect explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
17. Dimensionless ratios help judge smallness
The operating principle is to make dimensionless ratios help judge smallness explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
18. Model simplification should be reversible in principle
The operating principle is to make model simplification should be reversible in principle explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
19. Check the full expression after approximation
The operating principle is to make check the full expression after approximation explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
20. A dominant-term audit
Ask what level of claim is required, which conditions support it, how much detail is justified, and whether the conclusion survives the relevant uncertainty or model variation.
21. Worked examples
Example 1. If a conclusion depends on x>0, test what changes at x=0 and x<0 before calling it universal.
Example 2. If a small term is ignored, compare its maximum possible contribution with the decision margin or required tolerance.
Example 3. If data are grouped, do not claim exact within-group values that the representation no longer contains.
Example 4. If a model conclusion survives several plausible parameter ranges and remains on the same side of every relevant threshold, the qualitative conclusion is stronger than one supported by a single point estimate.
22. Independent practice
1. State one assumption behind a familiar school formula.
2. Give one situation where a smaller term cannot safely be ignored.
3. Explain why a grouped-data summary cannot recover exact raw observations.
4. Give one example of a conclusion that remains true under a range of input values.
5. State the check required before an approximation is used for a threshold decision.
23. Worked answers
1. Example: distance=speed×time under a constant or appropriately averaged speed model.
2. Near cancellation or near a decision threshold, a numerically small term can control the sign or classification.
3. Aggregation maps many possible raw data sets to the same grouped representation.
4. If an entire uncertainty interval lies below an upper threshold, the below-threshold conclusion is robust to every value in that interval.
5. Bound the approximation error and confirm the full plausible result remains on the required side of the threshold.
24. Continue through Batch 13
Return to the BTT Mathematics Hub for Batch 13.