Did you know that doubling one quantity can halve another without producing a straight line? If a fixed job takes twelve days with three equally productive workers, an idealised model says six workers would take six days. Worker count rises while time falls, and their product stays at 36 worker-days. That fixed product is the clue to inverse proportion.
A student may see the word ‘more’ and multiply the answer by two, even when the problem describes sharing a fixed task among more people. Another sees a falling graph and declares it inverse proportion without checking any numerical relationship. The repair is simple: ask what stays constant—ratio, product or neither—before selecting a formula.
For a parent looking for a Secondary 2 G2 Mathematics tutor in Bukit Timah, the first useful question is what the student can explain independently. This guide answers the topic question first, then works through examples, checks, common errors and short home practice. The aim is to help readers act on one specific gap rather than collect another intimidating pile of worksheets.
Quick answer for busy parents and students
| Question | Short answer |
|---|---|
| What will I learn? | Inverse Proportion Curves and the Constant Product in the context of Secondary 2 G2 Mathematics. |
| What do I do first? | Read the question and name the mathematical relationship before selecting a rule. |
| How should I practise? | Complete one explained example, then a new version without prompts. |
| How can a parent help? | Ask why the first line is valid and how the result can be checked. |
| Will this guarantee a grade or subject-level change? | No. Schools determine assessments and subject-level pathways, while tutoring supports learning. |
How this fits Singapore’s G2 Mathematics level
Did you know? G2 is a subject level under Full Subject-Based Banding, not a label for a child’s entire ability. In Secondary 2, schools use mathematical reasoning, algebra, graphs, geometry and data to build more independent problem solving. A child’s current school worksheets, schemes of work and marked papers should guide the order of tuition even when an outside source offers a different chapter sequence.
Yishun Secondary School’s published 2026 Secondary 2 G2 Mathematics outline includes this chapter area, but it is one school’s outline, not a uniform lesson calendar for every Singapore secondary school. Likewise, the 2027 Singapore-Cambridge Secondary Education Certificate syllabus gives broader upper-secondary context; a student who is in Secondary 2 in 2026 would normally progress to Secondary 3 in 2027, rather than sit the 2027 Secondary 4 examination.
Official references: 2026 school Mathematics content outlines and SEAB’s G2 SEC syllabus listings. For immediate WA or school examination preparation, follow the student’s own teacher’s assessment scope.
The idea behind inverse proportion curves and the constant product
For simple inverse proportion, y=k/x and therefore xy=k, where k is a constant and x is nonzero. When quantities represent positive counts or measurements, the corresponding graph is a decreasing curve in the positive quadrant. The same mathematical rule can be described through tables, equations, graphs and carefully stated real-world assumptions.
A tutor should ask a question that exposes the reasoning: ‘What does this symbol represent?’, ‘Which sides or values correspond?’, or ‘What stays unchanged?’ If the student can answer clearly, the arithmetic usually becomes easier to organise. If they cannot, more difficult questions often hide rather than fix the missing idea.
Ten concepts to make the topic clear
1. The product, not the ratio, stays constant
For direct proportion y=kx, the ratio y/x stays constant. For inverse proportion y=k/x, the product xy stays constant.
How to teach this: Check a table by multiplying each paired x and y; if every product matches, that supports inverse proportion.
What to look out for: A decreasing relationship alone does not prove the product is constant.
2. The equation y=k/x records a simple inverse model
If x=3 and y=8, then k=xy=24 and the rule is y=24/x. This lets us find y for other nonzero x.
How to teach this: First calculate k from the given pair; only then substitute the new input.
What to look out for: Writing y=24x would describe direct proportion instead.
3. Doubling x halves y in a fixed-product model
When k stays the same, changing x from 4 to 8 changes y from k/4 to k/8, half as large.
How to teach this: Explain the multiplication: 4×6 and 8×3 both equal 24.
What to look out for: This is not a universal rule for every quantity that goes down when another goes up.
4. Tripling x divides y by three
If an inverse model initially has y=15 at x=2, then at x=6 the new y is 5, provided k remains fixed.
How to teach this: The original product is 30; the new product is 6×5=30.
What to look out for: Do not subtract a fixed difference from y rather than divide by the scale factor.
5. The graph bends rather than making a straight line
For y=12/x with positive x, the points (1,12),(2,6),(3,4),(4,3),(6,2) lie on a smooth decreasing curve.
How to teach this: Plot positive table values at even axis scales and connect smoothly where the domain permits.
What to look out for: Do not join all points with one straight ruler line.
6. Zero input is excluded from the formula
The expression k/x is undefined at x=0. In positive contextual models, a zero worker count or zero speed is not a valid way to complete the task under this formula.
How to teach this: Record x>0 when the context requires positive quantities.
What to look out for: The curve can approach an axis without giving a valid point at x=0.
7. Speed and time depend on fixed distance
For a fixed 180 km trip at constant speed v, time t=180/v. If v=60 km/h, t=3 h; if v=90 km/h, t=2 h.
How to teach this: Write fixed distance=speed×time before forming the inverse relation.
What to look out for: Traffic, breaks and changing speed can make this simple model unrealistic outside the question.
8. Worker-day models have explicit simplifying assumptions
If four equally productive workers need nine days, work is 36 worker-days. Six such workers would ideally need six days.
How to teach this: State ‘same fixed job, equal productivity and no overhead’ before calculation.
What to look out for: Real projects may not scale exactly as worker counts change.
9. Not every descending straight line is inverse proportion
The rule y=10−x decreases as x increases, but xy=x(10−x) is not constant. Its graph is a straight line, not y=k/x.
How to teach this: Test at x=2 and x=4: products are 16 and 24.
What to look out for: Do not classify by the direction of change alone.
10. More complex inverse powers are extensions
A rule y=k/x² uses constant x²y rather than xy. This resembles inverse proportion reasoning but is a different model.
How to teach this: Use it only when assigned or as enrichment, after the basic y=k/x relation is secure.
What to look out for: Do not use a squared rule for an ordinary constant-product problem.
Ten worked examples with checks
Read the question first, attempt your own opening step and only then compare the explanation. The examples are original learning exercises, not predicted school examination questions. Focus on the reason for choosing a method and on checking the answer against the original statement.
Worked example 1: Find a constant product
Question: y is inversely proportional to x, and y=8 at x=3. Find k.
Method: By xy=k, 3×8=24, so k=24 and y=24/x.
Check: Original pair gives 24/3=8.
Common wrong turn: Do not call 8/3 the constant in an inverse model.
Worked example 2: Find a new output
Question: Using y=24/x, find y when x=6.
Method: y=24/6=4.
Check: 6×4=24 agrees with the original constant.
Common wrong turn: Writing 24×6=144 changes inverse into direct proportion.
Worked example 3: Make an inverse table
Question: Make a table for y=12/x at x=1,2,3,4,6.
Method: Outputs are 12,6,4,3,2 respectively.
Check: Every product xy is 12.
Common wrong turn: The x=0 row is not defined.
Worked example 4: Apply the doubling rule
Question: y is inversely proportional to x. When x=4, y=9. Find y at x=8.
Method: k=4×9=36. New y=36/8=4.5.
Check: Doubling x has halved y from 9 to 4.5.
Common wrong turn: Subtracting four from y would not keep the product.
Worked example 5: Workers and days
Question: Three equally productive workers need twelve days for a fixed job. How long for six such workers?
Method: Work=3×12=36 worker-days. At six workers, time=36/6=6 days.
Check: 6×6=36; the ideal model assumes fixed work and equal productivity.
Common wrong turn: The answer is not 24 days simply because worker count doubled.
Worked example 6: A different worker model
Question: Eight identical machines complete a fixed process in six hours. Under ideal scaling, how long for twelve machines?
Method: Total machine-hours=8×6=48. Divide by twelve to get four hours.
Check: 12×4=48.
Common wrong turn: This is a teaching model, not a guarantee that real machines have no coordination or capacity limits.
Worked example 7: Speed and time
Question: A fixed journey is 180 km. Find time at 60 km/h and 90 km/h under constant-speed assumptions.
Method: t=d/v. At 60, time=3 h; at 90, time=2 h.
Check: 60×3=180 and 90×2=180.
Common wrong turn: Do not ignore units: kilometres divided by km/h gives hours.
Worked example 8: Check a table for inverse proportion
Question: Pairs are (2,9),(4,4.5),(8,2.25). Is y inversely proportional to x?
Method: Products are 18,18,18, so yes, for these values they fit y=18/x.
Check: Every pair satisfies the proposed constant-product rule.
Common wrong turn: The observations support the model but cannot alone prove every future situation follows it.
Worked example 9: Reject a misleading decreasing rule
Question: Is y=10−x inverse proportion for positive x less than ten?
Method: At x=2, y=8 and xy=16. At x=4, y=6 and xy=24. Products differ, so it is not inverse proportion.
Check: Graph is a descending straight line, not the basic inverse curve.
Common wrong turn: The direction of change is not enough to classify the relation.
Worked example 10: Scale the input by three
Question: In an inverse model, when x=2, y=15. What is y when x=6?
Method: Constant k=2×15=30. At x=6, y=30/6=5.
Check: Tripling x divided y by three.
Common wrong turn: Do not assume subtraction of the same amount from both quantities.
Worked example 11: Find an unknown input
Question: For y=60/x, find x when y=4.
Method: 4=60/x, so 4x=60 and x=15.
Check: 15×4=60.
Common wrong turn: Write x=60/4, not 4/60.
Worked example 12: An optional inverse-square contrast
Question: In the extension y=k/x², y=9 at x=2. Find y at x=3.
Method: k=x²y=4×9=36, then y=36/9=4.
Check: x²y remains 36; xy does not remain constant.
Common wrong turn: Do not present this as the standard y=k/x rule.
How to recognise a genuine error rather than guess
One wrong first line can create several later wrong lines. An error log should identify the earliest incorrect decision, such as matching unrelated sides, substituting a negative number without brackets, using a ratio in place of a product, or collecting unlike algebraic terms. That is a far more useful description than simply writing ‘careless’.
When the first cause is known, the learner should correct the principle and solve a changed problem independently. A copied perfect model answer is not evidence of the same skill. We also distinguish calculation errors from misunderstanding: a student who selects the right rule but multiplies incorrectly needs different practice from one who cannot explain why the rule applies.
Twenty short practice questions with answers
Use these for retrieval rather than a race. Attempt without the solution visible, check your reasoning, and save missed ideas for a brief later session. Parents can ask for the explanation rather than take over the calculation.
| Question | Answer | Why |
|---|---|---|
| Inverse product for x=3,y=8 | 24 | 3×8. |
| If y=24/x, x=6 gives y= | 4 | 24/6. |
| y=12/x at x=2 | 6 | Constant product. |
| y=12/x at x=3 | 4 | 12/3. |
| y=12/x at x=6 | 2 | 12/6. |
| Can x=0 in y=12/x? | No | Division by zero undefined. |
| If x doubles under inverse proportion, y does what? | Halves | Product fixed. |
| If x triples, y does what? | Divides by three | Product fixed. |
| x=4,y=9 gives k | 36 | 4×9. |
| Same k, x=8 gives y | 4.5 | 36/8. |
| Three workers, twelve days: work in worker-days | 36 | 3×12. |
| Same job, six workers: days | 6 | 36/6. |
| Distance 180 km at 60 km/h: time | 3 hours | 180/60. |
| Distance 180 km at 90 km/h: time | 2 hours | 180/90. |
| For y=60/x, y=4 gives x | 15 | 60/4. |
| Product of 4 and 4.5 | 18 | Check inverse table. |
| Does y=10−x have constant xy? | No | Products differ. |
| Is basic inverse graph straight for x>0? | No | It’s curved. |
| For direct proportion y=kx, which is constant? | y/x | Nonzero x. |
| For inverse proportion y=k/x, which is constant? | xy | Nonzero x. |
A practical 90-minute three-pupil tutorial model
eduKateSG’s established small-group reference describes a three-pupil format. In such a group, every learner should still receive an independent diagnostic start, a chance to explain and specific feedback. The suggested timings below illustrate an educational method; they are not a claim about class vacancies, individual tutoring arrangements or guaranteed grades.
| Stage | Approximate time | Learning evidence |
|---|---|---|
| Diagnose the first wrong step | 10 minutes | Student attempts one unfamiliar question and explains the opening decision. |
| Make the principle visible | 15 minutes | Tutor builds a rule with a small contrast example. |
| Two guided worked questions | 20 minutes | Student explains why each method fits the problem. |
| Independent variations | 25 minutes | Learner solves changed questions with progressively fewer prompts. |
| Retrieve an earlier skill | 10 minutes | Student connects the idea to prior number, algebra or geometry learning. |
| Finish with a check and plan | 10 minutes | Student corrects the first error and identifies a follow-up task. |
A group size of three only becomes helpful when the tutor can hear each student’s reasoning. One quick learner completing the solution should not stand in for the others. Encourage all three to write the first line before discussion, then compare valid approaches respectfully.
Four weeks of purposeful home revision
Week 1: Identify the starting gap
Sort numerical tables into constant-ratio, constant-product and neither, explaining each decision.
Keep one school example, one learner’s first attempt and one corrected attempt. Record the exact prerequisite that was uncertain; do not repeat the whole chapter when one small concept is causing the mistake.
Week 2: Stabilise the mathematical move
Find k in y=k/x, complete value tables and sketch the curved positive branch accurately.
Space practice across short sessions. Cover the model answer before trying an altered example and ask the learner to explain why the first step still works.
Week 3: Mix the representations
Solve fixed-work and fixed-distance stories while stating the modelling assumptions and units.
Include two similar-looking questions that require different decisions. In an unlabelled assessment, recognising the mathematical structure is part of the skill.
Week 4: Test transfer independently
Mix direct and inverse contexts and check unfamiliar questions without being told the proportion type.
Use a fresh mixed mini-assessment. Compare method choice, accuracy, explanation and self-checking with the initial diagnostic rather than simply counting completed pages.
What parents can ask without teaching the whole syllabus
- What does the question ask for, and what information is given?
- Which relationship or condition tells you what to do first?
- How would you explain this method to someone in your class?
- Which check would make an incorrect answer obvious?
- What changes if I alter one number, label or diagram?
- Could you start a fresh version without this answer key?
Keep the routine realistic. A few focused questions over several days often reveal more than a long session filled with copied corrections. Allow the child time to think and protect schoolwork, sleep and rest. Tutoring should increase clarity, not replace the learner’s own decisions.
How to tell whether progress is real
| Skill | Early stage | Developing | Independent |
|---|---|---|---|
| Read the question | Needs help identifying what matters | Identifies information after a prompt | States goal and conditions alone |
| Choose a method | Copies the preceding example | Selects with a small cue | Explains the choice unprompted |
| Calculate accurately | Repeated structural errors | Mostly correct with checking | Reliable on changed numbers |
| Explain reasoning | Recites a formula | Describes one or two steps | Justifies the essential steps |
| Transfer learning | Stalls when presentation changes | Handles familiar variations | Handles unfamiliar related problems |
Marks are valuable feedback but a single mark compresses many skills. A better short-term sign is a student who can begin a fresh question, show relevant reasoning, correct an identified first mistake and explain the result in the context of the original task.
Secondary 3 readiness and subject-level decisions
Strong Secondary 2 G2 Mathematics provides worthwhile knowledge in its own right and can support future subject choices. Families considering a more demanding subject level should consult the school about its current eligibility criteria, the student’s sustained performance, workload and interests. A tuition provider cannot promise movement to G3 or a particular grade.
Frequently asked questions
1. What does ‘inversely proportional’ actually mean?
In the basic model, two quantities multiply to a constant, so y=k/x for nonzero x.
2. How can my child recognise inverse proportion in a table?
Multiply each paired x and y. Equal products are the numerical signature of the basic inverse rule.
3. Are all decreasing graphs inverse proportion?
No. A decreasing straight line can have a changing product xy; inverse proportion has a particular curve.
4. Why cannot x equal zero?
The formula divides k by x, and division by zero is undefined.
5. When is the worker-days model realistic?
Only under stated simplifying assumptions about fixed work and equal productivity. Real work may involve overhead and constraints.
6. Why do speed and time vary inversely?
For a fixed distance d travelled at constant speed, speed×time=d, so time=d/speed.
7. What is the difference from direct proportion?
Direct has constant ratio y/x; inverse has constant product xy. Their typical graphs and scaling effects differ.
8. Is inverse-square work compulsory at Secondary 2 G2?
Follow the student’s school scope. A squared inverse relation can be used as a clearly marked extension after the simple constant-product model.
A thoughtful closing idea
Did you know? A single mathematical decision that a student understands independently can become the foundation for several future questions. Teach the reason, practise the method, check it against the original task and try a changed example later. That is a more durable learning goal than finishing pages as quickly as possible.
Related reading: Direct and Inverse Proportion Made Clear and Map Scales, Distances and Area Ratios. Also see the Bukit Timah Secondary 2 Mathematics overview.
The original Clementi three-pupil Mathematics tutorial reference remains unchanged. For current teaching arrangements and class enquiries, contact eduKate Singapore directly; available places and individual outcomes should be confirmed.
Read the other three Secondary 2 G2 Mathematics guides
This four-part collection connects function tables, quadratic expressions, inverse proportion and similar triangles. Choose the chapter that matches your current school work and try an independent changed question after reading.