Many difficult Mathematics questions are not difficult because the calculation is advanced. They are difficult because the mathematics is hidden inside another representation. A paragraph hides an equation. A table hides a relationship. A graph hides a rate or intersection. A diagram hides a constraint. The learner’s first job is to translate without losing meaning.
This forty-fifth Secondary 4 Mathematics Learning Guide owns that translation layer. It belongs to the Secondary Mathematics Hub and S1–S4 Capability Map.
The current SEC G3 Mathematics syllabus explicitly places translation inside AO2: candidates must interpret information, translate information from one form to another, make connections across topics, formulate problems into mathematical terms, select relevant information and interpret results in context. AO2 carries substantial assessment weight, so translation is not a decorative skill around the syllabus. It is part of what the syllabus is assessing.
Official reference: SEAB 2027 SEC G3 Mathematics K310 syllabus.
Translation is preservation
Good translation changes the representation while preserving the relationship. The words may disappear. The table may become an equation. The diagram may become a set of angle conditions. But the mathematical meaning must survive.
Change the form. Preserve the relationship.
This idea is stronger than “turn words into equations”. Sometimes the correct destination is not an equation. It may be a ratio, inequality, graph, table, geometric construction, probability tree, set diagram or labelled sketch.
The six translation questions
- What are the quantities? Name what is changing, fixed, counted or compared.
- What are the units? Dollars, kilometres, minutes, square metres and percentages are not interchangeable.
- What is the relationship? Additive, multiplicative, proportional, geometric, probabilistic or functional?
- What is the unknown? A variable should represent something specific.
- What conditions must remain true? Positivity, integer counts, parallel lines, fixed totals, non-zero denominators, time order and physical limits all matter.
- Which representation exposes the next useful step? Equation, graph, diagram, table or something else?
Words → equation: identify the verb before the algebra
Words such as “increases by”, “is twice”, “costs per”, “is shared equally”, “is 20% less than”, “travels for” and “is proportional to” encode different operations. The learner should not grab numbers and combine them before deciding what relationship the sentence describes.
Worked Example 1 | A cost sentence
A service charges a fixed $12 plus $3.50 per kilometre. Let d be distance in kilometres and C be total cost in dollars.
C=12+3.5d.
The translation preserves two structures: a fixed component and a variable component. The y-intercept 12 is not just a graph feature; it represents the fixed charge. The gradient 3.5 represents dollars per kilometre.
Worked Example 2 | “More than” is directional
Ali has 7 more books than Mei. If Mei has m books, Ali has:
m+7.
The phrase “Ali has 7 more than Mei” does not mean m−7. Translation depends on who is being described relative to whom.
Percent language → multiplier
“Increase by 15%” means multiply by 1.15. “Decrease by 15%” means multiply by 0.85. “After a 15% increase” is therefore a reverse-multiplier problem if the final value is given.
Worked Example 3 | Reverse percentage translation
After a 20% discount, a jacket costs $96. Let P be the original price.
0.80P=96.
So P=120. The key translation is not “subtract 20”. It is “the final amount is 80% of the original”.
Table → relationship: look for what stays consistent
A table is not merely a storage device. It can reveal constant differences, constant ratios, repeated multipliers or changing gradients.
Worked Example 4 | Linear pattern from a table
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 |
Each increase of 1 in x produces an increase of 3 in y, so the gradient is 3. Use y=3x+c. At x=1, y=5, so c=2.
y=3x+2.
The table has been translated into an equation. The equation can now predict values outside the table if the relationship continues.
Worked Example 5 | Direct proportion from a table
| x | 2 | 5 | 8 |
|---|---|---|---|
| y | 6 | 15 | 24 |
The ratio y/x is always 3. Therefore y is directly proportional to x:
y=3x.
A linear relationship with non-zero intercept would not be direct proportion even if its graph were straight.
Graph → meaning: read axes before shape
Before interpreting a graph, read:
- what the horizontal axis measures;
- what the vertical axis measures;
- the units;
- the scale;
- whether the graph shows a total, a rate, a distance from a point, a frequency or another quantity.
The same visual shape can mean different things under different axes.
Worked Example 6 | Distance-time graph
On a distance-time graph, a straight segment rises from (1 h,40 km) to (3 h,160 km).
Gradient=(160−40)/(3−1)=60 km/h.
Here gradient translates into speed because the axes are distance and time. The same gradient formula on a cost-distance graph would represent dollars per kilometre instead.
Graph → equation: use intercepts and gradient
A straight graph crossing the y-axis at 4 with gradient −2 translates to:
y=−2x+4.
If the graph is a real-world model, the next question is what −2 and 4 mean in context.
Graph intersection → simultaneous condition
If two lines meet at (6,20), that point satisfies both relationships. A graph of two pricing plans crossing there may mean both plans cost $20 when the usage quantity is 6.
Intersection is therefore not only a picture. It translates into “both conditions are true simultaneously”.
Diagram → constraints: do not trust appearance
A geometry diagram may look isosceles, perpendicular or parallel without actually being so. Translation must come from stated information, conventional markings or proven results.
Worked Example 7 | Parallel-line diagram
Two lines are marked parallel and a transversal creates an angle of 68°. The corresponding angle is also 68° because the parallel condition authorises that transfer.
If no parallel condition were given or proved, copying the angle would be an invalid translation from visual appearance.
Worked Example 8 | Scale drawing to actual length
A floor plan uses scale 1:100. A wall measures 6.2 cm on the plan.
Actual length=6.2×100=620 cm=6.2 m.
The translation must carry both the scale and the unit conversion.
Words → inequality: conditions create boundaries
“At least 12” means x≥12. “More than 12” means x>12. “No more than 12” means x≤12. “Fewer than 12” means x<12.
Worked Example 9 | Capacity condition
A vehicle can carry at most 850 kg. Its existing load is 310 kg. Each crate has mass 45 kg. Let n be the number of extra crates.
310+45n≤850.
Solving gives n≤12. Because n counts crates, n must also be a non-negative integer. The model therefore carries two constraints, not one.
Probability wording → sample space
Words such as “without replacement”, “independent”, “at least one”, “exactly one” and “either” change the probability structure.
Worked Example 10 | Without replacement
A bag contains 3 red and 2 blue counters. Two are drawn without replacement. Find the probability of two red counters.
P(RR)=3/5×2/4=3/10.
The second denominator changes because the first draw changes the sample space.
Text with extra information: relevance is part of translation
Real-world questions may contain details that are true but unnecessary for the requested calculation. AO2 translation includes deciding what matters.
Worked Example 11 | Select relevant data
A hall is 24 m long and 18 m wide, has a ceiling height of 6 m, contains 220 chairs and is to be carpeted. Carpet costs $35 per square metre. What is the material cost before wastage?
For floor carpet area, ceiling height and chair count are irrelevant.
Area=24×18=432 m².
Cost=432×35=$15,120.
One situation may need several representations
A strong solution may move through a chain:
words → labelled sketch → equation → graph → interpreted conclusion.
Translation is not a one-time step at the beginning. It can happen repeatedly throughout a solution.
Worked Example 12 | Two pricing plans
Plan A costs $10 plus $2 per unit. Plan B costs $25 plus $1 per unit.
Translate:
- A=10+2x
- B=25+x
Set equal to find the break-even usage:
10+2x=25+x, so x=15.
At x=15, both cost $40.
The intersection (15,40) means the plans cost the same at 15 units.
The representation audit
Before calculating, ask whether the representation preserves:
- direction — more than versus less than;
- base — percentage of what;
- unit — km/h versus m/s;
- boundary — at least versus more than;
- dependency — with or without replacement;
- geometry — given constraints rather than appearance;
- domain — permitted values of the variable;
- context — what the final number actually represents.
Common failure modes
| Failure | What was lost | Repair |
|---|---|---|
| Combines all numbers immediately | Relationship | Name quantities and operation before arithmetic |
| Uses graph shape before reading axes | Meaning of gradient/area | Read labels, scale and units first |
| Copies an angle because lines look parallel | Geometric condition | Use only stated or proved constraints |
| Writes 0.20P for “after 20% discount” | Remaining proportion | Use 0.80P |
| Writes x>12 for “at least 12” | Boundary inclusion | Translate wording precisely |
| Uses every detail in a scenario | Relevance | Link each datum to the requested quantity |
| Gets a numerical answer but cannot explain it | Return to context | Translate the result back into words and units |
Independent practice
- A taxi charges $4 plus $0.80 per kilometre. Write a cost model.
- After a 25% increase, a value is 250. Write an equation for the original x.
- A table has x=1,2,3,4 and y=7,11,15,19. Find a linear equation.
- A line crosses the y-axis at −3 with gradient 5. Write its equation.
- A container holds at most 120 L and already contains 35 L. Water enters in 8 L batches. Write an inequality for n batches.
- A bag contains 4 green and 6 yellow counters. Two are drawn without replacement. Find P(GG).
- A 1:200 map shows a path of 9 cm. Find actual distance in metres.
- Explain why a graph intersection can solve a simultaneous-equation problem.
Explained answers
1. C=4+0.8d.
2. 1.25x=250, so x=200.
3. Difference is 4, so m=4. Using (1,7), c=3. y=4x+3.
4. y=5x−3.
5. 35+8n≤120, with n a non-negative integer.
6. 4/10×3/9=2/15.
7. 9×200=1800 cm=18 m.
8. The intersection coordinate satisfies both graph equations simultaneously, so its x- and y-values solve both conditions.
The examination habit
Before the first calculation, write one line that states what the representation means. Examples:
- “Let x be the number of tickets sold.”
- “Gradient represents dollars per kilometre.”
- “The final price is 80% of the original.”
- “These lines are parallel, so alternate angles are equal.”
- “n must be a whole number because it counts boxes.”
That sentence acts as a control layer. It makes the hidden mathematics explicit before arithmetic begins.
Final thought
A learner who can calculate but cannot translate is dependent on familiar question surfaces. A learner who can translate can meet new wording, new diagrams and new contexts while preserving the mathematical relationship underneath.
The surface may change. Your job is to find the invariant relationship and carry it into a representation you can control.
Continue with AO2 Model Building or return to the Secondary Mathematics Hub.
