Singapore School Mathematics Operating Manual · Chapter 30
Some Mathematics problems are difficult because several correct operations are available, but only some of them are useful now.
A later calculation may depend on a side length that has not yet been found. A percentage change may require an original value that is still hidden. A tangent equation may require both a point and a gradient. A probability branch may need the state after an earlier draw. A multi-part examination question may deliberately build one usable result for the next part.
This chapter develops dependency order: the structure that determines which mathematical tasks must happen before others. It also develops solving sequence: the deliberate order in which the learner executes those tasks.
The central habit is: do not ask only “what can I calculate?” Ask “what does the target depend on, and which dependency is ready to solve?”
1. A dependency is a mathematical prerequisite
If area A=1/2bh and the height h is unknown, the area depends on h.
If h itself depends on an angle found from an angle sum, then the dependency chain is:
angle → height → area.
The chain determines the natural solving order.
2. Dependency order is not the same as textbook order
This chapter does not describe Primary-to-Secondary curriculum progression.
It describes the local architecture inside one problem.
A Secondary 3 question can still have a dependency chain of four or five internal steps even though all topics are already known.
3. A target tree reveals the workflow
Suppose the target is total cost of tiling a floor.
Cost depends on number of tiles and price per tile.
Number of tiles depends on floor area and tile area.
Floor area depends on room dimensions.
The target tree becomes:
dimensions → floor area → tile count → total cost.
4. Independent branches can be solved in parallel
Some targets depend on two separate quantities.
A line equation needs a point and a gradient.
The point and gradient may be found independently from different data.
The dependency graph branches, then recombines.
5. A dependency graph is not always a straight chain
In a probability tree, later branch probabilities depend on earlier outcomes.
In simultaneous geometry, two separate lengths may both feed one cosine-rule step.
In optimisation, the objective and feasible set can be developed separately before candidate comparison.
Strong solving sequence respects this graph structure.
6. The first useful calculation is the one that unlocks another required calculation
A printed number may be available immediately but irrelevant.
A hidden angle may be more valuable because it unlocks a side, which unlocks the target.
Usefulness is determined by dependency, not by visual prominence.
7. “Hence” often signals a dependency handoff
If part (a) establishes an expression and part (b) says “hence”, the question is signalling that the earlier result is intended as an input.
The chapter on Linked Question Parts, Hence and Result Handoffs develops that examination structure.
Here the broader point is that mathematical results can become prerequisites for later work.
8. Solving too early can create unnecessary complexity
Suppose a symmetric expression can be simplified before numerical substitution.
Substituting complicated values first may produce long arithmetic.
Simplifying the structure first can reduce later work.
The dependency is not “numbers first”; it may be “structure first, values later”.
9. Exact form should often be preserved through early dependencies
If one stage gives √3 and the next multiplies by 2√3, keeping exact form gives 6 directly.
Rounding √3 too early creates unnecessary approximation error.
The handoff format matters to downstream stability.
10. Unit conversion should occur before incompatible quantities are combined
If speed is in km/h and time is in minutes, convert time before multiplication.
The correct dependency is:
convert unit → combine quantities.
Combining first and “fixing units later” can produce a wrong numerical scale.
11. Domain restrictions should be established before candidate filtering
For log(x−2)=3, establish x>2.
Then solve the transformed equation.
Finally filter candidates through the original domain.
The restriction belongs early in the workflow even if its final effect appears late.
12. Case splitting creates conditional workflows
For |x| or a piecewise function, the workflow branches according to conditions.
Each branch has its own local solving sequence.
The branches then recombine into the complete solution set.
13. Probability without replacement has state-dependent sequence
The second draw depends on the first because the composition changes.
You cannot calculate the second-draw probability correctly without specifying the first branch.
The problem’s dependency structure is temporal and conditional.
14. Recurrence relations are explicit dependency chains
uₙ₊₁ depends on uₙ.
To find u₅ from u₁, the learner may need u₂, u₃ and u₄ unless a direct formula is available.
A closed-form expression can compress a long dependency chain into one direct evaluation.
15. A direct formula changes workflow without changing the object
For an arithmetic sequence, repeated addition finds later terms.
The nth-term formula uₙ=a+(n−1)d bypasses intermediate terms.
Different workflows can solve the same dependency problem at different levels of compression.
16. Algebraic elimination chooses a dependency order
For simultaneous equations, one variable can be eliminated first, then the other reconstructed.
The chosen elimination determines the local workflow.
A good sequence minimises awkward fractions or large coefficients where possible.
17. Back-substitution is a dependency reversal
After solving x, substitute it into an earlier equation to recover y.
The forward system linked x and y together.
The solving workflow isolates one variable, then reconstructs the other.
18. Geometry constructions have order constraints
You cannot draw a circle of a required radius until the radius is known.
You cannot locate the intersection of two loci until both loci have been defined.
Construction problems therefore make dependency order visible spatially.
19. Calculus workflows have standard dependency patterns
To find a tangent equation:
find the point, differentiate, evaluate gradient, then form the line.
To classify a stationary point:
differentiate, solve f′(x)=0, then use sign change or second derivative where appropriate.
Each target carries a characteristic dependency graph.
20. Optimisation should establish feasibility before declaring the optimum
A stationary point can be mathematically elegant but inadmissible.
The workflow should include constraints before final comparison.
Best among all real values is not the same as best among feasible values.
21. Errors propagate downstream
If an early radius is wrong, every later circumference, area or volume using it may also be wrong.
This is why early high-dependency quantities deserve careful checking.
The more downstream steps depend on a result, the more valuable an early verification becomes.
22. Some intermediate results are high-leverage nodes
A midpoint in coordinate geometry may feed a gradient, line equation and later intersection.
A reconstructed original value may feed several percentage changes.
These nodes should be labelled clearly and checked independently.
23. Dependency awareness supports partial-credit strategy
If an earlier part is uncertain but a later question gives or implies a result, use the available result where permitted rather than abandoning the entire workflow.
Examination structure often allows later mathematical reasoning to continue.
24. Cyclic dependency signals a modelling problem
If A requires B and B requires A with no independent relation to break the cycle, the problem may be underdetermined.
Additional information or simultaneous solving is required.
A dependency cycle cannot always be resolved by simply “starting somewhere”.
25. Simultaneous equations resolve coupled dependencies
x depends on y and y depends on x.
Instead of treating this as an impossible cycle, solve the coupled system as one module.
Dependency analysis helps identify when separate sequential solving should be replaced by joint solving.
26. A practical solving-sequence audit
Ask:
What does the target require? Which required quantities are already known? Which are hidden? What does each hidden quantity depend on? Which subproblem is ready now? Which result will unlock the most downstream work? Where should I check before passing a value forward?
27. Independent practice
1. A tangent equation is required for y=x² at x=3. List the dependency sequence.
2. A room’s tile cost requires room area, tile area, price per tile and whole tile count. Write a dependency chain.
3. Why should 30 minutes be converted before using a speed in km/h?
4. A recurrence gives uₙ₊₁=uₙ+4 and u₁=3. Find u₄ by dependency order.
5. Explain why exact √2 may be a safer handoff than 1.414.
6. What is a high-leverage intermediate result?
7. Why can simultaneous equations be viewed as coupled dependencies?
8. Give one example where two subproblems can be solved in parallel.
9. Why should feasibility be checked before final optimisation?
10. What should happen if A depends on B and B depends on A with no additional relation?
28. Worked answers
1. Find point (3,9); differentiate to get dy/dx=2x; evaluate gradient 6; form y−9=6(x−3).
2. Dimensions → room area; tile dimensions → tile area; room area/tile area → minimum whole tile count; tile count×price → total cost.
3. The quantities must share compatible time units before multiplication or division.
4. u₂=7, u₃=11, u₄=15.
5. It preserves exact information and avoids early rounding error.
6. An intermediate value used by many downstream steps, making its accuracy especially important.
7. Each variable’s value is constrained by the other, so they must be solved jointly or one eliminated before reconstruction.
8. For a line equation, point and gradient may be derived independently before recombination.
9. An inadmissible candidate cannot be the constrained optimum.
10. Seek another independent relation, solve jointly if possible, or conclude the information is insufficient.
29. Continue through Batch 08
Use Decomposition and Recomposition to define the modules, Normalisation and Rescaling when different modules use incompatible scales, and Sampling, Resolution and Aliasing when dependencies are inferred from incomplete observations.
Return to the BTT Mathematics Hub for Batch 08.