One of the strongest ways to understand a problem is to change it deliberately. What if a right angle disappears? What if a percentage increase becomes a decrease? What if one equation is removed? What if the final answer is given and the question must be rebuilt?
This Secondary 2 Mathematics Learning Guide develops problem posing and reverse engineering as learning tools. The learner will unpack completed solutions, identify which conditions make each step possible, create new questions from existing structures, vary one condition at a time, predict how the solution route changes, and test whether a newly posed problem is complete and solvable.
Secondary Mathematics Hub: S1–S4 Capability Map · Secondary 2 Learning Guide, Batch 7, Guide 4. Companion guides cover functions and multiple representations, modelling and assumptions, and units, scale and conversion chains.
Course boundary. Problem posing is a cross-topic mathematical process rather than a separate syllabus chapter. The examples use existing Secondary 2 content. The purpose is to deepen control of familiar mathematics, not to accelerate into unrelated new topics.
Navigate: reverse engineering · conditions · change one thing · pose a problem · solvability · design from an answer · mixed structures · practice and answers · teaching and transfer.
1. Reverse engineering asks what each line needed in order to be true
A polished worked solution can hide the decisions that created it. Reverse engineering reads the solution backwards and asks which conditions, definitions or relationships justify each step.
Worked example 1: reverse engineer an equation solution
Completed solution:
3x + 7 = 25 → 3x = 18 → x = 6.
Reverse engineering identifies the preserved relationship: subtract 7 from both sides, then divide both sides by 3. The route depends on equality. If the original symbol were an inequality and division were by a negative number, an additional order rule would matter.
Worked example 2: reverse engineer geometry
A solution says c² = 8² + 15² = 289, so c = 17. What must have been true? The sides 8 and 15 were perpendicular legs of a right triangle and c was the hypotenuse. Without the right-angle condition, the calculation is unsupported.
Reverse engineering therefore makes hidden permissions visible.
Worked example 3: reverse engineer a percentage answer
A solution uses 80 × 1.25 = 100. This could represent a 25% increase from 80. It cannot automatically represent a 25 percentage-point increase, a reverse percentage problem, or 80 as 25% of a total. The multiplier reveals one particular relationship.
2. Identify which condition owns each conclusion
In multi-step problems, different conditions support different parts of the route. A right angle supports Pythagoras. Parallel lines may support corresponding-angle equality or triangle similarity. A total supports an equation. A constant rate supports proportional scaling.
Label each conclusion with its owner. This makes it easier to predict what breaks when one condition changes.
Worked example 4: remove one condition
Original: triangle ABC is right-angled at A, AB = 6 and AC = 8. Then BC = 10 by Pythagoras.
Change: remove the right-angle condition but keep AB = 6 and AC = 8. BC is no longer determined uniquely. Many triangles can have two sides 6 and 8 with different included angles.
One removed condition changes the problem from uniquely solvable to underdetermined.
Worked example 5: keep the route but change the numbers
Original: y = 4x + 3, find y when x = 5. Variant: y = 4x + 3, find y when x = −2. The structure and method stay the same; only sign control becomes more visible.
This is a near-transfer variant rather than a new mathematical structure.
3. Change one feature at a time to learn what is invariant
Changing several features at once can make it unclear why the solution changed. Controlled variation changes one mathematical condition while preserving the others.
Worked example 6: ratio reference changes
Original: A:B = 2:3 and total is 40. Then A = 16, B = 24. Variant 1 changes total to 55; the same part structure remains. Variant 2 changes the statement to A is 2/3 of B; the ratio remains 2:3. Variant 3 says A is 2/3 of the total; now A:total = 2:3, so A:B = 2:1.
The surface phrase 2/3 remains, but the reference quantity changes the structure.
Worked example 7: percentage direction changes
Original: increase 200 by 15% → 200 × 1.15 = 230. Variant: decrease by 15% → 200 × 0.85 = 170. Reverse variant: after a 15% increase, final amount is 230 → original = 230/1.15 = 200.
The percentage is unchanged; the relationship direction changes.
Worked example 8: graph condition changes
Original lines y = 2x + 1 and y = 10 − x intersect once. Change the second line to y = 2x + 5: equal gradients and different intercepts make the lines parallel, so there is no intersection. Change it to y = 2x + 1: the lines coincide, giving infinitely many common points.
A small coefficient change alters the solution set fundamentally.
4. Pose a problem by choosing a structure before choosing a story
Start with the mathematical relationship you want to test. Then choose quantities and a context that make the relationship natural.
Worked example 9: pose a simultaneous-equations problem
Target solution: x = 6, y = 4. Choose equations x + y = 10 and 3x + 2y = 26. Now create a context: ten items are split into two types; one type has weight 3 units and the other 2 units, total weight 26. Ask for the counts.
Before publishing the problem, solve it independently to confirm the intended answer and check that counts are meaningful whole numbers.
Worked example 10: pose a direct-proportion problem
Target structure y = 7x. A possible context: each identical pack contains 7 cards, so x packs contain y cards. A non-example would add a fixed 5 cards outside the packs; that changes the model to y = 7x + 5 and destroys direct proportionality.
Problem posing therefore tests whether the learner can distinguish the target relationship from nearby alternatives.
5. A posed problem needs enough information—but not redundant information
Too little information can leave many possible answers. Too much information may be redundant or contradictory. Good problem design asks whether the conditions determine the requested quantity uniquely within the intended domain.
Worked example 11: underdetermined problem
“Two numbers add to 20. Find the numbers.” There are many solutions: 1 and 19, 2 and 18, 7 and 13. Add the condition that their difference is 4. Now the pair is uniquely determined as 12 and 8.
Worked example 12: contradictory conditions
“Two numbers add to 10 and also add to 14.” No ordered pair can satisfy both. Contradictory conditions create an inconsistent problem rather than a difficult one.
Worked example 13: redundant information
A right rectangle has length 8 cm, width 5 cm, perimeter 26 cm, and asks for area. The perimeter statement is true but unnecessary because length and width already determine area. Redundant information can be used deliberately to test relevance selection.
6. Design from an answer to expose inverse reasoning
Instead of asking only for the answer, begin with an answer and ask what problem could produce it. This forces learners to work backwards through mathematical structure.
Worked example 14: answer 35%
Pose a percentage-increase problem with answer 35%. Choose original value 80 and final value 108. Increase = 28, and 28/80 = 0.35. The problem might ask: “A quantity rises from 80 to 108. Find percentage increase.”
Now pose a different problem whose answer is also 35%: “Find 35% of 200” gives 70, so that would not fit the target answer. The requested quantity matters.
Worked example 15: answer x = 7
Many equations can have solution 7: 2x + 3 = 17, 5(x − 2) = 25, or x² − 10x + 21 = 0 if 3 is another root. Designing the equation determines what skill is tested.
An answer alone does not identify the method. Problem posing makes this explicit.
7. Reverse engineer mixed-topic questions by locating the handoff between topics
A mixed question often contains a sequence of smaller mathematical jobs. One result becomes the input to the next. Identifying the handoff clarifies where one topic ends and another begins.
Worked example 16: scale → speed
A map scale gives an actual route of 4.5 km. A cyclist travels at 15 km/h. The first stage is scale conversion; the second is time = distance/speed = 4.5/15 h = 0.3 h = 18 minutes.
To create a variant, keep the map calculation fixed and change only the speed. To create a deeper variant, change the scale. To create a structural change, replace constant speed with two-stage travel.
Worked example 17: quadratic → geometry
A rectangle has width x and length x + 3, area 54. Solve x(x + 3) = 54, giving x = 6 or −9; geometry selects width 6 and length 9. Perimeter then becomes 30.
A useful variant asks for a different final quantity while preserving the quadratic model, such as diagonal length. Another variant changes the area while preserving the length-width relationship.
8. Ask “what must change?” before recalculating
Prediction makes variation more informative. If a directly proportional input doubles, output should double. If a square’s side doubles, area should quadruple. If a fixed cost is added, direct proportion disappears. If a denominator approaches zero, a fractional expression changes dramatically.
Predicting the direction or scale of change before calculation builds structural control.
Worked example 18: change a scale factor
A similar shape is enlarged by length factor 2, so area factor is 4. Change length factor to 3: area factor becomes 9. The relationship between length and area scaling is invariant even though the numerical factor changes.
9. Common problem-posing failures
- Numbers changed but mathematical structure unchanged unintentionally: variation too shallow for the intended goal.
- Several conditions changed at once: cause of route change becomes unclear.
- Problem lacks enough information: multiple solutions remain.
- Conditions contradict: no solution exists.
- Requested answer not aligned with data: problem ownership unclear.
- Created context violates algebraic domain: interpretation failure.
- Reverse-engineered answer accepted without re-solving: verification missing.
- Story chosen first and Mathematics forced into it: relationship becomes artificial.
10. Practice: change the problem deliberately
Questions 1–6. 1. Reverse engineer the condition needed for 6² + 8² = 10² to find a triangle side. 2. Change one condition so Pythagoras is no longer permitted. 3. Original equation 4x + 5 = 29 has x = 6. Change only the constant term so the solution becomes x = 8. 4. A direct-proportion rule is y = 7x. Add one feature that makes it linear but not directly proportional. 5. Explain what changed structurally. 6. Pose a short context for y = 7x.
Questions 7–12. 7. “Two numbers sum to 30.” Why is the problem underdetermined? 8. Add one condition so the solution is 17 and 13. 9. Create a contradictory second condition. 10. Pose a percentage-increase problem with answer 20%. 11. Pose a reverse-percentage problem whose original value is 100. 12. Explain why these two questions test different directions of reasoning.
Questions 13–18. 13. A map problem produces 6 km before a speed calculation. Give one near variant. 14. Give one structural variant. 15. A rectangle model x(x + 4) = 77 has a positive width. Find it. 16. Create a variant with the same width but different final requested quantity. 17. Explain one way to verify a problem you have posed. 18. State why changing one condition at a time can be more informative than changing everything.
Explained answers 1–6
1. The triangle must be right-angled with 6 and 8 as perpendicular legs and 10 as hypotenuse. 2. Remove or change the right-angle condition. 3. To make x = 8 in 4x + c = 29, c = −3. 4. Example: y = 7x + 5. 5. The fixed term makes the graph miss the origin and y/x no longer constant. 6. Example: seven identical items per pack, x packs, y items.
Explained answers 7–12
7. Many pairs sum to 30. 8. Add difference 4, with larger number first. 9. Example: also require their sum to be 40. 10. Example: a value rises from 50 to 60. Increase 10/50 = 20%. 11. Example: after a 20% increase the final value is 120; find the original. 12. One calculates a change relative to the original; the other reverses a multiplier.
Explained answers 13–18
13. Keep the 6 km route and change the constant speed. 14. Replace constant speed with two different speeds over two stages. 15. x² + 4x − 77 = 0 = (x + 11)(x − 7), so positive width = 7. 16. Ask for perimeter, diagonal or percentage increase in area after scaling.
17. Solve the posed problem independently and check every condition against the intended answer. 18. Controlled variation isolates which changed condition caused the route or answer to change.
11. Teaching sequence: solve → annotate → vary → pose → verify
Start with a completed problem. Ask the learner to annotate each line with the condition or rule that permits it. Then change one condition and predict which lines remain valid. Only after that ask the learner to create a new problem testing the same relationship.
Finally, exchange posed problems between learners. The solver should identify ambiguity, redundant information or missing conditions before calculating. This turns problem design into a test of mathematical communication.
Questions parents and tutors can ask
Which condition made that step possible? What happens if I remove it? Can you change the numbers without changing the method? Can you change the method by altering only one condition? Is your new problem uniquely solvable? Have you solved your own problem to verify it?
12. The transfer test: create, not copy
A learner who can solve 3x + 7 = 25 may still be following a familiar pattern. A learner who can design a different equation with solution x = 6, explain why it has that solution, change one coefficient to make the solution x = 10, and verify each version demonstrates deeper control.
The same applies in geometry, rate, graphs and probability. Creating a valid variation requires understanding which relationships are essential and which features are merely surface detail.
Reverse engineer the route. Identify the condition owned by each step. Change one feature at a time. Predict before recalculating. Pose complete problems. Verify that your new question really has the intended mathematical structure.
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