G2 Mathematics K210 multi-step problem solving becomes more reliable when a learner can see which intermediate quantities must exist before the final target can be calculated. Many difficult questions are not difficult because the final formula is unknown. They are difficult because the learner has not identified the dependency chain that connects the givens to the target.
This seventieth Learner’s Guide develops dependency graphs for K210 Mathematics. It differs from Vol 0047 on backsolving, which starts from the target and reasons backwards. Dependency graphs can be built forwards, backwards or from both ends: which quantities are known, which quantities can be derived, and which derived quantities unlock the requested answer?
The current 2027 K210 syllabus includes short-answer work, longer Paper 2 questions, real-world applications, reasoning and communication. The dependency-graph method below is an eduKateSengkang training framework for organising multi-step work; the official syllabus remains the authority for assessed content and paper structure.
The central rule: every target depends on something
If the final answer cannot be calculated directly, do not search randomly for a formula. Ask which quantity the target depends on, then which earlier quantity that depends on. A problem becomes easier when the dependency chain is visible.
The five-node model
- Target quantity.
- Immediate formula or relationship containing the target.
- Missing quantities inside that relationship.
- Earlier relationships that can produce those missing quantities.
- Given information that anchors the chain.
The learner does not need to draw a literal graph on every question. Training uses visible nodes and arrows so that, under time, the same structure can run mentally.
1. Start by naming the target
Write the quantity, not merely x. A variable without meaning can drift during a multi-step solution.
Use labels such as total cost, final speed, angle ABC, probability of event E or mean score. The name constrains which intermediate quantities can be relevant.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
2. Separate target from intermediate quantity
A correct intermediate value is not automatically the answer.
When a number appears, ask whether it matches the noun and unit requested in the question. If not, label it and continue the dependency chain.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
3. Use units to predict missing nodes
The target unit often suggests which operations or quantities are required.
A speed target needs distance and time; an area target needs dimensions; a rate target needs numerator and denominator quantities. Units can reveal dependency structure before arithmetic begins.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
4. Use formula structure as a graph
A formula tells you which nodes feed the target.
If target T depends on A and B, check which of A and B are given and which must be derived. Do not calculate unrelated quantities simply because they are available.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
5. Use word relationships as graph edges
Not every dependency is written as a formula. ‘After a 20% discount’, ‘twice as long’, ‘three more than’, ‘shared equally’ all define edges between quantities.
Translate the relationship before substituting numbers.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
6. Use diagrams as dependency maps
A geometry diagram can show that one angle unlocks another, one length unlocks an area, or one radius unlocks circumference and area.
Mark derived versus given quantities. Do not treat every visible measurement as independent.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
7. Use tables as dependency maps
A table can reveal which row or column supplies the quantities needed for a rate, mean or comparison.
Name ownership before calculating so the right values feed the target node.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
8. Use graphs as dependency maps
A graph can supply a coordinate, gradient, intercept or trend needed for a later calculation.
Identify which graphical property is the required intermediate quantity rather than extracting random points.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
9. Do not calculate every available number
A common weak strategy is to perform arithmetic on all visible numbers and hope one result becomes useful.
Dependency graphs reverse that habit: calculate only quantities that lie on a route to the target.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
10. Do not let calculator availability create fake nodes
A calculator can make almost any arithmetic easy, but easy arithmetic can still be irrelevant.
The question is not ‘What can I calculate?’ but ‘What must be known before the target is available?’
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
11. Mark given nodes differently from derived nodes
In training, circle givens and box derived values.
This makes it obvious whether a later step depends on evidence from the question or on your own previous calculation, which improves error containment.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
12. Label fragile nodes
If one intermediate quantity is uncertain, mark it.
Later subparts may still be solvable using the learner’s own value or an independent route. This supports the error-containment work from Vol 0027.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
13. Algebra: define the unknown before the equation
A word problem may contain several quantities, but only one should initially be represented by x.
Choose the variable that makes the dependency structure simplest, then express other quantities from it.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
14. Algebra: derived expressions are nodes too
If one quantity is x + 4 and another is 2x, those expressions are intermediate nodes before any numeric solution exists.
The graph can be symbolic. Numeric values are not required for a dependency to be useful.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
15. Algebra: equation formation closes the graph
Once two expressions describe the same quantity or a total condition links them, the equation becomes the edge that allows x to be solved.
The equation is not the beginning of the problem; it is the point where the dependency map becomes solvable.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
16. Algebra: substitute back only where needed
After finding x, identify which target quantity depends on x.
Do not automatically compute every expression in the problem. Follow the target dependency only.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
17. Percentage: identify the base node
A percentage is always a percentage of something.
Before multiplying or dividing, name the base: original price, final price, population, total marks or another quantity. Wrong-base errors are dependency errors.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
18. Reverse percentage: final depends on original
If sale price is a percentage of original price, the original is the missing upstream node.
Represent final = fraction × original, then solve for the upstream quantity instead of trying random percentage additions.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
19. Successive percentage change: base changes between stages
The second percentage depends on the result after the first change, not the original amount unless the question says otherwise.
Create a stage node after each change so the dependency order remains visible.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
20. Ratio: distinguish part nodes from whole node
A part-to-part ratio does not directly give part-to-whole fraction until the total ratio units are constructed.
The whole is a derived node equal to the sum of the ratio parts.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
21. Ratio sharing: unit value is an intermediate node
If a total is divided in ratio a:b:c, one ratio unit often unlocks every part.
Calculate total ratio units, then one-unit value, then the required share. The dependency chain is short and transparent.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
22. Rate: name numerator and denominator
Speed, cost per item and density have different dependency directions.
Write the verbal unit before the formula. The unit tells you which quantity sits above and below the division.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
23. Rate: total depends on rate and exposure
A rate alone does not give a total without time, quantity or another denominator-related measure.
Make the missing exposure variable visible rather than assuming the rate is the final answer.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
24. Speed: distance can be intermediate
A problem may give speed and time for one stage, then ask for an average across a whole journey.
Individual stage distances become intermediate nodes feeding total distance and total time.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
25. Average speed: do not average speeds blindly
The final target depends on total distance divided by total time.
Unless the time or distance conditions justify it, the individual speed values are not the immediate inputs to the average-speed node.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
26. Geometry: radius may be hidden upstream
A circumference or diameter may be given while area is asked.
Radius is the required intermediate node. Identify it before substituting into the area formula.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
27. Geometry: angle dependencies form chains
One angle can unlock another through straight-line, triangle or polygon relationships.
Write small arrows in training: known angle → derived angle → target angle. This prevents random theorem application.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
28. Geometry: similar shapes require scale-factor nodes
A length scale factor can unlock missing lengths; area and volume use different powers of the scale factor.
The scale factor is the intermediate node, but its transformation depends on the quantity type.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
29. Geometry: Pythagoras needs the correct side roles
The target side depends on the two other sides through the right-triangle relationship.
Identify hypotenuse and legs before calculating. Side identity is part of the dependency graph.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
30. Trigonometry: angle or length depends on chosen ratio
The target may be connected to two known quantities through sine, cosine or tangent.
Choose the ratio from side roles, not from a memorised formula list. The dependency edge is the right-triangle relationship.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
31. Coordinate geometry: gradient depends on two points
If gradient is required, two coordinates are upstream nodes.
If an equation is required, gradient and an intercept or point become intermediate inputs to the line model.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
32. Coordinate geometry: distance and midpoint are different targets
Both use the same endpoint coordinates but create different dependency structures.
Name the target first so the coordinates are combined in the correct way.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
33. Statistics: mean depends on total and count
If one value is missing, the total may need to be reconstructed from the given mean first.
Mean × count creates an upstream total node that can then reveal the missing value.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
34. Statistics: weighted mean depends on frequencies
Values alone are not enough when observations occur multiple times.
Frequency × value products feed the total; total frequency feeds the denominator. The graph prevents a simple unweighted average.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
35. Statistics: median depends on order and position
The median is not directly calculated from raw unsorted values.
Order is an operation node; position is an intermediate rule; only then does the target emerge.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
36. Statistics: range depends on extremes
You do not need every internal value to find range.
Maximum and minimum are the only upstream nodes. Dependency thinking reduces unnecessary processing.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
37. Probability: event probability depends on favourable and total outcomes
Before dividing, identify the event and the sample space.
Wrong sample-space size is an upstream dependency error, not a division error.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
38. Probability: complement can create a shorter graph
Sometimes P(not A) is easier to calculate than P(A) directly.
The complement relationship creates an alternate dependency path. Choose the shorter valid graph.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
39. Probability: multi-stage events need stage nodes
A tree or organised table makes conditional branches visible.
Each stage outcome becomes a node feeding the joint event. This reduces double counting and missing cases.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
40. Probability: overlap changes the graph
When events are not mutually exclusive, intersection is an explicit node.
If the overlap node is ignored, addition produces a double-counted result.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
41. Real-world modelling: quantities come before equations
A modelling problem may contain prices, capacities, distances and constraints.
List the quantities that influence the decision, then connect them. Equations should emerge from the graph rather than from keyword matching.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
42. Real-world modelling: constraints are gates
A mathematical value may exist but fail a capacity, budget, integer or minimum condition.
Treat the constraint as a gate after the numerical node. The final target is the feasible answer, not merely the raw calculation.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
43. Real-world modelling: assumptions change edges
Vol 0064 separates stated conditions, derived facts and invented assumptions.
In a dependency graph, an assumption creates or removes an edge. If the assumption is unjustified, the entire route to the target may be invalid.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
44. Real-world modelling: compare alternatives through shared nodes
Two options may depend on common quantities such as distance, usage or time.
Calculate shared upstream nodes once, then feed both alternative branches. This avoids duplicate work and inconsistent values.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
45. Financial context: total cost can have fixed and variable branches
A model may contain a fixed fee plus a per-unit charge.
Represent both branches explicitly before summing. This prevents using the variable rate as though it were the whole cost.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
46. Financial context: discount and tax can be sequential
If both are applied, each stage may use a different base depending on wording.
Create a node after each stage rather than combining percentages casually.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
47. Data interpretation: derived change depends on baseline
A final value alone may not reveal absolute or percentage change.
Initial value is an upstream node; difference comes next; percentage change then depends on the original base.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
48. Graph modelling: prediction depends on the model first
To predict a value, the learner may first need a gradient, intercept or relationship from data.
Do not jump directly from a graph picture to a numeric prediction. Build the model node first.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
49. Checking: dependency graphs locate error propagation
If the final answer is wrong, trace upstream nodes until the first wrong value appears.
This is faster than re-solving every line and complements Vol 0055 error signatures.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
50. Checking: units can validate graph edges
When one node feeds another, units should transform consistently.
A unit mismatch often identifies the exact edge where multiplication, division or conversion went wrong.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
51. Checking: magnitude can validate nodes
Estimate the expected scale of important intermediate values.
A wildly implausible node can be corrected before it contaminates the rest of the graph.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
52. Checking: substitute target backwards
Once a final value is found, reverse through one or two edges where possible.
This links dependency graphs with Vol 0047 backsolving and provides an independent verification route.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
53. Error containment: downstream work can remain valid
If one upstream node is wrong but later reasoning uses it consistently, later method may still demonstrate mathematical structure.
In training, distinguish the source error from downstream follow-through so the repair targets the first weak link.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
54. Return protocol: leave the graph state visible
If you skip a problem, mark the target, known nodes and next missing node.
When you return, Vol 0066 becomes faster because you resume the dependency state instead of rereading from zero.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
55. Question choice: compare graph accessibility
In Paper 2 Section B, one option may reveal a clearer dependency chain than the other.
Question choice can therefore consider method visibility, not only topic preference.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
56. Time control: stop when the next node is unknown
If you have no idea what quantity is needed next, continued arithmetic is unlikely to help.
Mark the unresolved dependency, move on if necessary, and return later with a fresh model.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
57. Time control: easy node first can unlock hard node
Some questions feel difficult because the learner looks directly at the final target.
Find the nearest derivable intermediate quantity. One simple node can make the rest of the graph obvious.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
58. Mixed-topic transfer: graph structure can survive topic changes
Ratio, rate, geometry and statistics look different on the surface but share the logic of upstream and downstream quantities.
Training dependency thinking across topics builds a portable planning skill rather than another chapter-specific trick.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
59. Practice with missing-node questions
Give a partly completed solution and ask which quantity must be found next.
This isolates planning from calculation and makes the dependency skill visible.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
60. Practice with extra-information questions
Include data that are not needed.
The learner should exclude nodes with no path to the target. This trains resistance to ‘use every number’ behaviour.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
61. Practice with two valid routes
Design questions solvable by different dependency graphs.
Compare length, error risk and transparency. The shortest route is not always best if it hides working or creates fragile arithmetic.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
62. Practice with broken graphs
Provide a dependency map containing one unjustified edge.
The learner must identify which relationship does not follow from the givens, strengthening the assumption audit.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
63. Practice with graph-to-solution translation
Give a correct dependency diagram and require full working.
This shows whether the learner can convert planning structure into mathematical communication.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
64. Practice with solution-to-graph translation
Give a worked solution and ask the learner to reconstruct its dependency graph.
This reveals hidden intermediate quantities and helps the learner see why the method works.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
65. Practice under time
Begin untimed until the learner can name nodes and edges accurately.
Then add mixed timed sets so planning becomes rapid enough for examination use.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
66. Use the Mathematics Hub
If the learner cannot create an edge because the underlying formula or concept is missing, dependency mapping is not the main repair.
Return to the Mathematics Hub, learn the relationship, then bring it back into mixed problems.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
67. Use the PSLE bridge
Earlier word-problem work already teaches that an intermediate quantity often unlocks the final answer.
G2 extends the same idea across algebra, graphs, statistics, probability and real-world modelling.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
68. Use Examination Craft
The Examination Craft hub helps decide when to pause, skip, return and check.
Dependency graphs make those control decisions more informed because the learner knows exactly which node is missing.
In a dependency graph, the key exam question is always: Does this quantity lie on a valid path to the target? If yes, calculate or derive it carefully. If not, leave it alone. This prevents busy arithmetic from replacing mathematical planning.
A compact dependency-graph routine
- Name the target quantity and unit.
- Write the immediate relationship that contains the target.
- Mark which required inputs are known and which are missing.
- Find relationships that produce the missing inputs.
- Work from the givens through the shortest reliable path.
- Label important intermediate quantities.
- Check the final target against units, constraints and context.
Readiness criteria
- You name the target before calculating.
- You distinguish given and derived quantities.
- You identify intermediate quantities before random arithmetic.
- You use units to test dependency edges.
- You contain upstream errors instead of restarting everything.
- You can resume skipped questions from a visible graph state.
- You compare alternate solution routes by clarity and risk.
- Your planning survives mixed-topic questions.
Official-source discipline
For current K210 structure and assessment requirements, use the official 2027 G2 Mathematics K210 syllabus. Dependency graphs are a training framework for planning mathematical work, not an SEAB-prescribed notation.
Final rule: find what unlocks the target
A multi-step question becomes manageable when the learner stops seeing one large problem and starts seeing a chain of dependent quantities.
Name the target, identify the missing upstream node, derive only what the target needs and keep the path visible. The solution then becomes a sequence of justified unlocks rather than a search through every formula you remember.
