A word problem is not difficult because it contains words. It becomes difficult when the learner cannot see which quantities matter, how they are related, what is fixed, what may vary, and which mathematical representation makes the structure easier to control.
This Secondary 2 Mathematics Learning Guide develops word problems as a modelling process. The learner moves from language to variables, diagrams, tables, equations, graphs or geometric constraints, then returns to the original situation to test whether the mathematical answer is meaningful. The aim is not to collect more templates. It is to make representation itself a deliberate mathematical skill.
Secondary Mathematics Hub: S1–S4 Capability Map · Secondary 2 Learning Guide, Batch 4, Guide 1. Companion guides cover accuracy and estimation, mixed-topic method selection, and error analysis and transfer.
Course boundary. This is a cross-topic learning guide rather than a separate chapter. It supports the problem-solving and mathematical-process demands that run through algebra, proportion, graphs, geometry, mensuration, statistics and probability. Use the current school course to decide which content examples are immediately assessable.
Navigate: Read the situation · Name quantities · Choose a representation · Build equations · Use constraints · Track units · Verify the model · Practice and answers · Teaching and transfer.
1. Read for relationships before reading for operations
Many learners scan for trigger words: total means add, difference means subtract, of means multiply. Those associations sometimes help, but they are too weak to control an unfamiliar problem. The same word can appear in different structures, and the same structure can be expressed with different words.
A stronger first reading asks four questions: What are the quantities? Which are known? Which are unknown? What relationship connects them?
Worked example 1: the word more can describe several structures
“A has 8 more than B” means A = B + 8. “A has 20% more than B” means A = 1.2B. “A produces 8 more items per hour than B” compares rates. The surface word more remains, but the mathematical relationship changes.
This is why operation hunting fails. The learner must identify whether the comparison is additive, multiplicative or rate-based.
Mark givens, unknowns and conditions separately
In a long question, underline numerical givens, box the required quantity and circle structural conditions such as parallel, equal, directly proportional, at most, constant speed or selected without replacement. The visual coding is optional; the separation of roles is essential.
A condition is not merely another number. It grants permission to use a relationship. “Right-angled” permits Pythagoras or right-triangle trigonometry. “Directly proportional” permits y = kx. “At most” creates an inequality boundary.
2. Name quantities before assigning symbols
A variable is useful only when its meaning remains stable. Let x be the number of adult tickets is stronger than writing x without definition. The short sentence prevents the learner from later treating x as money, total tickets or price.
Worked example 2: ticket model
A fictional event sells 50 tickets. Adult tickets cost 12 dollars and student tickets cost 7 dollars. Total revenue is 475 dollars. Let a be the number of adult tickets and s the number of student tickets.
The count relationship is a + s = 50. The revenue relationship is 12a + 7s = 475. These equations come from different units: tickets and dollars. That distinction helps prevent mixing the conditions.
Solving gives a = 25 and s = 25. Check both conditions: 25 + 25 = 50 and 12(25) + 7(25) = 475.
Choose variables that simplify the model
Suppose two consecutive integers have product 156. Let the smaller be n and the larger n + 1. This variable choice turns the relationship directly into n(n + 1) = 156.
Letting the two integers be x and y is not wrong, but then another equation y = x + 1 is needed. Good representation reduces unnecessary complexity without changing the mathematics.
3. Choose the representation that exposes the relationship
Words are one representation. Equations, diagrams, tables and graphs are others. A strong learner can move between them while preserving the same relationship.
- Use a diagram when spatial relationships matter.
- Use a table when corresponding values or repeated cases matter.
- Use an equation when equality or an unknown relationship can be compressed symbolically.
- Use a graph when change, intersection or overall behaviour matters.
- Use a number line when order and inequality boundaries matter.
Worked example 3: direct proportion in three forms
A machine-like process produces 18 units in 3 minutes at a constant rate. A table gives 1 minute → 6 units, 2 → 12, 3 → 18, 4 → 24. The equation is y = 6t. The graph is a straight line through the origin with gradient 6.
All three represent the same constant-rate relationship. The table is convenient for selected values, the equation for calculation, and the graph for seeing proportionality across the domain.
Worked example 4: geometry needs a labelled sketch
A pole, ground and line of sight form a right triangle. The pole height is unknown, the ground distance is 12 m and the angle of elevation is 35°. A sketch immediately shows the unknown is opposite the 35° angle and the 12 m side is adjacent.
The diagram therefore selects tangent: tan 35° = h/12. Without the representation, a learner may guess among sine, cosine and tangent.
4. An equation should express a sentence that remains true
Before solving an equation, read it back in words. If x + 7 = 25 represents a story, what does x mean? Why is 7 added? Why does the total equal 25? If the equation cannot be explained, it may have been assembled from numbers rather than relationships.
Worked example 5: perimeter model
A rectangle has length 3 cm more than twice its width. Its perimeter is 42 cm. Let width be w. Then length is 2w + 3.
Perimeter gives 2[w + (2w + 3)] = 42. Divide by 2: 3w + 3 = 21. Hence 3w = 18 and w = 6. Length = 15. Check: 2(6 + 15) = 42.
The bracket matters because perimeter doubles the sum of both dimensions. Writing 2w + 2w + 3 = 42 would double the width twice but only one copy of the extra 3, changing the geometry.
Worked example 6: percentage model
After a 20% reduction, a hypothetical price is 96 dollars. Let original price be P. The final price is 80% of the original, so 0.8P = 96. Hence P = 120.
Adding 20% of 96 would use the reduced amount as the reference and does not reverse the original operation. The model must preserve the stated reference quantity.
5. Constraints decide which algebraic answers belong to the problem
Mathematics often produces candidate values before context filters them. A quadratic equation may have a negative root that cannot represent a physical length. An inequality may give real values while the context requires a whole-number count. A probability model must stay between 0 and 1.
Worked example 7: quadratic length model
A rectangle has width x cm and length x + 5 cm. Its area is 84 cm². The model is x(x + 5) = 84, giving x² + 5x − 84 = 0.
Factorise: (x + 12)(x − 7) = 0. Algebra gives x = −12 or 7. The length condition requires x > 0, so the physical rectangle is 7 cm by 12 cm.
Worked example 8: whole-number constraint
A budget model gives n ≤ 8.6, where n is the number of identical whole boxes. The greatest permitted count is 8, not 9 and not 8.6. The algebraic boundary must be interpreted in the discrete domain.
This is not ordinary rounding. It is choosing the greatest whole number that still satisfies the condition.
6. Units are part of the model, not decoration
Units reveal what operations mean. Speed in km/h multiplied by time in hours gives kilometres. Area in cm² multiplied by prism length in cm gives cm³. A price rate in dollars/kg multiplied by kilograms gives dollars.
Worked example 9: rate with incompatible time units
A vehicle travels at 72 km/h for 25 minutes. Convert 25 minutes to 25/60 hours. Distance = 72 × 25/60 = 30 km.
Multiplying 72 by 25 without changing units treats 25 minutes as 25 hours. The arithmetic is easy; the model is wrong.
Dimension can check a formula
If a purported area calculation ends in metres rather than square metres, inspect the formula or unit conversion. If volume is computed by multiplying only two lengths, a dimension is missing. Units can expose structural errors before the numerical answer is judged.
7. Multi-step word problems need a route map
Longer questions often contain several relationships. Instead of calculating immediately, write a short route: find scale factor → find missing length → find area → apply percentage. This protects the logic when many numbers appear.
Worked example 10: map, distance and rate
A map scale is 1:50,000. Two places are 7.2 cm apart on the map. The represented distance is 7.2 × 50,000 cm = 360,000 cm = 3.6 km.
If a cyclist covers the route at an average speed of 18 km/h under the simplified model, time = 3.6/18 h = 0.2 h = 12 minutes. The first relationship is scale; the second is speed-time-distance. One formula cannot solve both stages.
Worked example 11: composite geometry and cost
A rectangular board is 1.2 m by 0.8 m, with a circular hole of radius 0.1 m removed. Area to be covered = 0.96 − π(0.1²) ≈ 0.9286 m².
If material costs 25 dollars per m², estimated material cost = 25 × 0.9286 ≈ 23.22 dollars before any additional commercial conditions. The geometry produces area; the rate converts area to cost.
8. Verification asks whether the model still matches the original situation
Checking should not mean only repeating arithmetic. Return to the original relationships. Does the pair satisfy both simultaneous equations? Does the calculated hypotenuse remain longest? Does a percentage decrease produce less than the original? Does a count remain whole and non-negative?
Worked example 12: verify through the original sentence
Two numbers sum to 56 and differ by 14. Solving gives 35 and 21. Check the original sentence: 35 + 21 = 56 and 35 − 21 = 14. This is stronger than checking only the final algebraic line.
Estimate before trusting precision
If a 20% discount is applied to about 100 dollars, the final price should be around 80 dollars. A calculator answer of 800 suggests a place-value or percentage-entry error. Estimation creates an independent expectation before the device returns a number.
9. Common modelling errors and what they reveal
- Every number is used: repair relevance selection.
- Variable changes meaning halfway through: repair quantity definition.
- Diagram is measured by eye: repair dependence on stated conditions.
- Equation assembled from keywords: repair relationship reading.
- Units conflict: repair dimensional consistency.
- Negative length accepted: repair contextual constraints.
- Correct calculation answers the wrong quantity: repair question ownership.
- Final answer not checked in original condition: repair verification.
10. Mixed practice: represent before solving
Questions 1–6. 1. A number is 7 more than twice another number. Let the smaller be x; write the larger. 2. Two numbers sum to 40 and differ by 8; form simultaneous equations. 3. A rectangle’s length is 4 cm more than its width and its perimeter is 36 cm; form an equation. 4. A quantity after a 25% reduction is 90; form an equation for the original P. 5. A car travels at 60 km/h for 45 minutes; state the unit conversion needed. 6. Explain why a variable definition should be written before a long solution.
Questions 7–12. 7. Solve the rectangle problem from Question 3. 8. Solve the reverse-percentage problem from Question 4. 9. A map scale is 1:25,000 and a distance is 8 cm on the map; find the real distance in km. 10. A bag model gives P(red) = 3/10. What information must be true for favourable/total counting to be valid? 11. A quadratic length model gives x = −5 or 8. Which root is valid for a positive length? 12. An inequality gives n ≤ 6.7 for a count. What is the greatest valid whole number?
Questions 13–18. 13. A triangular prism has cross-sectional area 24 cm² and length 15 cm. Build and evaluate the volume model. 14. A right triangle has angle 30° and adjacent side 10 cm; write the trigonometric equation for the opposite side. 15. A dataset has total 84 across 7 observations; write the mean model and answer. 16. A fictional cost is 5 dollars fixed plus 3 dollars per item. Write C in terms of item count n. 17. Explain why this cost is not directly proportional to n. 18. Give two independent checks for any one answer above.
Explained answers: questions 1–6
1. 2x + 7. 2. x + y = 40 and x − y = 8 if x is the larger. 3. 2[x + (x + 4)] = 36. 4. 0.75P = 90. 5. Convert 45 minutes to 0.75 hours. 6. It keeps the symbol tied to one quantity and prevents meaning drift.
Explained answers: questions 7–12
7. 2(2x + 4) = 36 gives 4x + 8 = 36, so x = 7; dimensions 7 cm by 11 cm. 8. P = 120. 9. 8 × 25,000 cm = 200,000 cm = 2 km. 10. The counted elementary outcomes must be equally likely.
11. 8. 12. 6.
Explained answers: questions 13–18
13. V = area of cross-section × length = 24 × 15 = 360 cm³. 14. tan 30° = x/10. 15. Mean = 84/7 = 12. 16. C = 5 + 3n. 17. The non-zero fixed term means C/n is not constant and the graph does not pass through the origin.
18. Examples include substitution into the original equation, unit analysis, estimation, an alternative method, checking geometric constraints, or verifying all simultaneous conditions.
11. Teaching sequence: separate representation from execution
Give learners short word problems and ask only for a representation—no solving. One question might require an equation, another a labelled sketch, another a table. This isolates the modelling decision from arithmetic fluency.
Next provide two possible representations and ask which is more useful. Then ask students to solve and verify. Finally change the surface of the problem while preserving the same relationship. A ticket problem can become a container problem; a proportional printing problem can become a recipe-scale problem.
Questions parents and tutors can ask
What does each number represent? What is the unknown? Which relationship is fixed? What representation would make that easier to see? Which information is irrelevant? What unit should the answer have? Which condition will you use to check the result?
12. The transfer test: remove the chapter label
A fictional project uses rectangular panels. Each panel has length 0.4 m more than its width. The area is 1.4 m². Material costs 18 dollars per square metre, and 8 panels are needed.
Let width be x. Then x(x + 0.4) = 1.4. Solving the resulting quadratic gives candidate dimensions. The positive width is used. The area per panel is already given as 1.4 m², so material area for 8 panels is 11.2 m² and basic material cost is 201.60 dollars under the simplified rate model.
The question contains algebra, geometry and rate language, but no chapter heading. The learner succeeds by identifying quantities and relationships rather than waiting for the worksheet to announce a method.
Read the quantities. Name the relationship. Choose a representation. Preserve units and constraints. Solve. Return the answer to the original situation.
Use Algebraic Factorisation and Structural Control, Ratio, Proportion, Rate and Percentage, and Geometry, Similarity and Mathematical Constraints when a particular representation needs repair.
Continue to Accuracy, Estimation, Rounding and Calculator Discipline · Return to the Secondary Mathematics Hub.
