Did you know? If five identical notebooks cost $15, ten usually cost twice as much when the unit price stays fixed. But if five identical workers take ten hours to complete one fixed job, ten equally productive workers could theoretically finish in half the time. The first is a picture of direct proportion; the second, under stated assumptions, is inverse proportion.
This Secondary 3 G1 Mathematics tutor Bukit Timah guide explains direct and inverse proportion with step-by-step calculations, everyday examples, warning signs and a family-friendly revision plan. The quick rule is simple: direct proportion keeps a constant ratio; inverse proportion keeps a constant product. The difficult part is deciding which model actually fits the story.
The official 2027 G1 syllabuses directory identifies Mathematics as K110. Under N2, the SEAB 2027 K110 Mathematics syllabus explicitly includes direct and inverse proportion. These are whole-course skills, so a Secondary 3 learner’s current school chapter and assessment scope remain the best guide for what to practise today.
The quick answer: direct or inverse?
- Direct proportion: if y is directly proportional to x, then y/x is constant when x is non-zero. One possible model is y = kx.
- Inverse proportion: if y is inversely proportional to x, then xy is constant. A model is y = k/x, for non-zero x.
- Direct example: twice as many identical items costs twice as much if the unit price remains unchanged and there is no fixed fee.
- Inverse example: twice as many equally productive workers takes half the time for exactly the same amount of work, assuming ideal conditions.
- Important limit: a real situation is not proportional merely because two quantities increase or decrease together. Its assumptions must support the constant relationship.
Start with a unitary method before memorising formulas
The unitary method means finding the value associated with one unit and then scaling to the quantity requested. For example, if four pencils cost $6 at a constant price, one costs $1.50. Seven cost $10.50. The same arithmetic can be expressed as a direct-proportion equation, but students often understand the meaning better when they work through the one-item price first.
For inverse proportion, the natural starting point is different. If four machines complete a fixed task in six hours, total machine-hours are 4 × 6 = 24 under the ideal assumption of identical constant performance. Six machines would take 24 ÷ 6 = four hours. That calculation preserves the total work, not the hourly rate of one item.
How to tell the relationship from a table
Imagine a table with x-values 2, 4, 6 and matching y-values 5, 10, 15. The ratios y/x are each 2.5, so this is direct proportion. If the y-values were 12, 6, 4, the products xy would each be 24; that is an inverse-proportion pattern. Asking ‘what stays the same?’ is more reliable than guessing from arrows.
When students see two quantities both increasing, that is evidence of an association, not proof of direct proportion. A fictional taxi-style charge of $4 plus $2 per kilometre increases with distance, but the charge divided by distance is not constant because of the fixed starting fee. Direct proportion would require the total to be zero when distance is zero, under the model.
Worked-example studio: twenty ways to make the decision
Practice 1. Five notebooks at one unit price
Situation. Five identical notebooks cost $15 with no discount or fixed fee. What do eight cost? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- One notebook costs $15 ÷ 5 = $3.
- Eight cost 8 × $3 = $24.
- Check 15/5 = 24/8 = 3 dollars per notebook.
What the relationship means. Direct proportion: the price-per-notebook ratio stays constant. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Multiplying 15 by eight without first accounting for five notebooks produces an overestimate. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. If ten notebooks cost $30 under the same assumptions, the unit rate still remains $3. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 2. A recipe for more portions
Situation. A fictional recipe needs 450 mL of liquid for three identical servings. How much is needed for eight? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Find liquid per serving: 450 ÷ 3 = 150 mL.
- For eight servings: 8 × 150 = 1,200 mL.
- Convert if useful: 1,200 mL = 1.2 L.
What the relationship means. Direct proportion under a recipe with a fixed quantity per serving. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Using 450 × 8 assumes 450 mL is needed for each single serving. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. For six servings, 900 mL is needed. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 3. School club materials
Situation. Four identical packs contain 48 sheets in total. Under an unchanged pack size, how many sheets do seven packs contain? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- One pack contains 48 ÷ 4 = 12 sheets.
- Seven packs contain 7 × 12 = 84 sheets.
- Check 48/4 = 84/7 = 12.
What the relationship means. Direct proportion between identical packs and total contents. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Adding three sheets because there are three additional packs confuses packs with individual sheets. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. Nine packs would contain 108 sheets. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 4. Fictional fuel consumption
Situation. A fictional machine uses 2.4 litres of fluid in four hours at a constant rate. How much for ten hours? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Fluid per hour = 2.4 ÷ 4 = 0.6 litre.
- Ten hours requires 10 × 0.6 = 6 litres.
- Check units are litres, not litres per hour.
What the relationship means. Direct proportion if the operating rate really remains constant. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. The result would be unreliable if the machine has an idle period or a fixed startup use. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. For 2.5 hours under the same rate, it uses 1.5 litres. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 5. One model with a fixed fee
Situation. A fictional charge is $4 fixed plus $2 per kilometre. Does doubling distance double the total charge? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- At 2 km the charge is 4 + 2(2) = $8.
- At 4 km the charge is 4 + 2(4) = $12.
- Since $12 is not twice $8, it is not direct proportion.
What the relationship means. A useful non-example: increasing together does not prove direct proportion. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Applying a pure y = kx model ignores the starting fee. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. If the fixed fee were zero, cost = 2d would be directly proportional to distance. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 6. Compare direct tables
Situation. A table lists (x,y) as (2,7), (4,14), (6,21). Is y directly proportional to x? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Compute 7/2 = 3.5.
- Compute 14/4 = 3.5 and 21/6 = 3.5.
- The ratio y/x is constant, so yes.
What the relationship means. The constant multiplier k is 3.5. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Only looking at whether y rises misses the necessary numerical test. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. A new x = 10 would produce y = 35. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 7. A misleading growing table
Situation. A second table gives (x,y) = (1,4), (2,6), (3,8). Is it direct proportion? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Ratios are 4/1 = 4, 6/2 = 3 and 8/3 ≈ 2.67.
- The ratios are not constant.
- It is not directly proportional.
What the relationship means. The values rise by a fixed additive amount, but direct proportion needs a fixed multiplicative rate. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Mistaking equal increases in y for a constant y/x ratio. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. The table is consistent with y = 2x + 2, which has a non-zero intercept. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 8. Identical workers and one job
Situation. Four equally productive workers finish a fixed job in six hours under ideal conditions. How long for eight workers? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Total work = 4 × 6 = 24 worker-hours.
- Time with eight workers = 24 ÷ 8 = 3 hours.
- Check 4 × 6 = 8 × 3 = 24.
What the relationship means. Inverse proportion under equal productivity and no coordination losses. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Treating time as directly proportional would incorrectly double the time. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. With six workers, the ideal time would be four hours. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 9. Printing machines at constant rates
Situation. Three identical printers need 12 minutes for a fixed batch when sharing equally. Estimate time with six printers. Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Printer-minutes = 3 × 12 = 36.
- Time for six = 36 ÷ 6 = 6 minutes.
- The product printer count × time remains 36.
What the relationship means. Inverse proportion if every printer runs at the same constant rate. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Real printers may have setup, queue and transfer times that break the ideal model. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. Nine identical printers would need four minutes under the model. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 10. Distance, speed and time
Situation. A fictional distance is 120 km. At a constant average speed of 40 km/h it takes three hours. What if the speed becomes 60 km/h? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Keep distance fixed at 120 km.
- Time = distance ÷ speed = 120 ÷ 60 = 2 hours.
- Check 40 × 3 = 60 × 2 = 120 km.
What the relationship means. Inverse proportion between speed and time for one fixed distance. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. The model does not claim the vehicle travels at one speed every instant in a real trip. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. At 30 km/h the same 120 km would take four hours. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 11. Two taps and a fixed tank
Situation. One fictional tap fills a fixed tank in 12 minutes. Two identical taps together run at double the combined rate. How long? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- One tap’s work rate is one tank per 12 minutes.
- Two taps at equal rate fill two-twelfths of the tank per minute.
- Time to fill one tank is six minutes.
What the relationship means. An inverse relationship between identical taps and fill time under constant-flow assumptions. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Pressure drop when several real taps run could invalidate the exact six-minute estimate. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. Three identical taps under the model would take four minutes. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 12. Direct proportion from a constant
Situation. If y = 2.5x, find y when x = 14 and explain why this is direct. Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- The multiplier is constant at 2.5.
- At x = 14, y = 2.5 × 14 = 35.
- For any nonzero x, y/x = 2.5.
What the relationship means. The rule passes through the origin and has a constant multiplier. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Assuming the rule implies an initial fixed fee confuses it with y = 2.5x + c. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. When x doubles from 14 to 28, y doubles from 35 to 70. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 13. Inverse proportion from a constant
Situation. If xy = 48 for positive quantities, find y when x = 6. Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Rearrange to y = 48/x.
- At x = 6, y = 8.
- Check 6 × 8 = 48.
What the relationship means. The product remains the same as x changes. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Writing y = 48x confuses inverse with direct proportionality. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. At x = 12, y = 4. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 14. Detect the inverse table
Situation. A table gives (x,y): (2,18), (3,12), (6,6). What relationship is shown? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Multiply each pair: 2 × 18 = 36.
- Then 3 × 12 = 36 and 6 × 6 = 36.
- The constant product shows inverse proportion.
What the relationship means. The model may be written y = 36/x for positive x. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. The y-values fall, but falling alone is not enough; the product test is decisive. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. At x = 9 the corresponding y is 4. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 15. A decreasing non-example
Situation. A table has (x,y): (1,9), (2,8), (3,7). Is it inverse proportion? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Products are 1 × 9 = 9, 2 × 8 = 16 and 3 × 7 = 21.
- The products differ.
- It is not inverse proportion.
What the relationship means. Decreasing values do not establish a constant product. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Students may call any downward trend inverse without checking. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. This table is consistent with y = 10 − x. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 16. A factory task with changed output
Situation. Five machines complete 100 units in four hours. Ten machines must now complete 200 units. Can we automatically halve the time? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Original output rate per machine-hour = 100/(5 × 4) = 5 units.
- New required machine-hours = 200/5 = 40.
- With ten machines, time is 40/10 = 4 hours.
What the relationship means. The task doubled as machine count doubled, so the old fixed-work inverse shortcut does not apply. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Using 2 hours would ignore that twice as much work is required. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. If the task had stayed 100 units, ten machines would need two hours under the same model. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 17. Paint coverage at a constant rate
Situation. A fictional paint covers 7 m² per litre. What area will 4.5 litres cover? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Area per litre is a constant 7 m²/L.
- Multiply 4.5 × 7 = 31.5 m².
- Check units: litres cancel, leaving square metres.
What the relationship means. Direct proportion if paint thickness and surface conditions stay the same. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Coverage can vary for porous surfaces or multiple coats, so do not assume the same rate in all real situations. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. For 14 m² under this model, two litres would be needed. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 18. Geometry is not always direct
Situation. If a square’s side doubles from 3 cm to 6 cm, does its area merely double? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Original area = 3² = 9 cm².
- New area = 6² = 36 cm².
- Area becomes four times as large, not twice.
What the relationship means. Square area is proportional to the square of side length, not directly to side length itself. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Assuming every enlarged quantity grows with the same linear factor. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. The perimeter, by contrast, doubles from 12 cm to 24 cm. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 19. Find the unknown unit rate
Situation. Six identical bags of fictional grain weigh 15 kg together. How much do four bags weigh? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- One bag weighs 15/6 = 2.5 kg.
- Four bags weigh 4 × 2.5 = 10 kg.
- Check 15/6 = 10/4 = 2.5 kg per bag.
What the relationship means. Direct proportion for identical bag contents. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. Subtracting two kilograms because there are two fewer bags ignores the actual weight per bag. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. Eight such bags would weigh 20 kg. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Practice 20. A group project with diminishing returns
Situation. Four people take eight hours on a task. Will eight people always take exactly four hours? Before starting, identify what is fixed and what is changing. That question helps decide the model rather than memorise which direction an arrow points.
- Under an ideal equal-output model, total work is 4 × 8 = 32 person-hours.
- Eight people would take 32/8 = 4 hours.
- In a real collaborative task, training and coordination could break the assumption.
What the relationship means. Inverse proportion here is a conditional model, not a guarantee about people. It is worth writing the quantities and their units beside the calculation so the constant has a recognisable meaning.
Common wrong turn. An exact result should not be claimed without the constant-product assumptions. A tutor can ask for the unit rate or the product, then check which one remains unchanged.
Independent variation. Ask which parts of a school project can be divided independently, and which cannot. Invite the learner to solve it without looking back at the worked steps, and then explain the model in an ordinary sentence.
Nine diagnostic questions a tutor should ask
What, exactly, is held constant?
For prices, it may be the price of one item; for workers, it may be the fixed amount of work. Write the assumed constant before calculating. If the question changes the task itself, the inverse model can no longer be used unchanged. A useful correction then asks the student to build a tiny new table with at least three corresponding pairs and test the proposed invariant. This tests understanding beyond copying one formula.
Is there a fixed starting fee?
A fixed delivery charge turns a pure direct relationship into cost = fixed fee + rate × quantity. The total rises, but the total-to-quantity ratio is not constant. A useful correction then asks the student to build a tiny new table with at least three corresponding pairs and test the proposed invariant. This tests understanding beyond copying one formula.
Can the quantity really be divided continuously?
Six machines and 4.5 litres can be treated differently. A school group cannot employ half a person in many practical contexts, and a pack count generally needs whole numbers. Mathematical results must be interpreted. A useful correction then asks the student to build a tiny new table with at least three corresponding pairs and test the proposed invariant. This tests understanding beyond copying one formula.
Do the units make sense?
Price divided by items should give dollars per item; distance divided by speed gives time. If the resulting unit is nonsensical, inspect whether multiplication or division has been reversed. A useful correction then asks the student to build a tiny new table with at least three corresponding pairs and test the proposed invariant. This tests understanding beyond copying one formula.
Is the relationship direct or merely increasing?
For direct proportion, check multiple ratios y/x. A rising line with a nonzero intercept fails the test. Students should be able to provide a counterexample. A useful correction then asks the student to build a tiny new table with at least three corresponding pairs and test the proposed invariant. This tests understanding beyond copying one formula.
Is the relationship inverse or merely decreasing?
For inverse proportion, compare products xy at several pairs. A downward trend alone is not proof. An exact constant product matters. A useful correction then asks the student to build a tiny new table with at least three corresponding pairs and test the proposed invariant. This tests understanding beyond copying one formula.
Is the input zero?
For y = k/x, zero cannot be used as x because division by zero is undefined. Do not make claims about an inverse model at zero. A useful correction then asks the student to build a tiny new table with at least three corresponding pairs and test the proposed invariant. This tests understanding beyond copying one formula.
Are the conditions idealised?
Machine productivity, tap flow and teamwork often include setup costs, delays or uneven performance. A school problem may stipulate constant rates; in a real decision those assumptions need checking. A useful correction then asks the student to build a tiny new table with at least three corresponding pairs and test the proposed invariant. This tests understanding beyond copying one formula.
Has the learner checked the answer backwards?
For a directly proportional purchase, divide total cost by item count to recover the unit rate. For an inverse job, multiply people by hours to verify constant worker-hours. A useful correction then asks the student to build a tiny new table with at least three corresponding pairs and test the proposed invariant. This tests understanding beyond copying one formula.
A comparison table students can rebuild themselves
- Direct example: 2, 4 and 6 items at $3 each cost $6, $12 and $18. The ratios 6/2, 12/4 and 18/6 are all 3.
- Inverse example: 2, 4 and 6 identical workers performing 24 worker-hours of work need 12, 6 and 4 hours respectively. Each product is 24.
- Fixed-fee non-example: $4 plus $3 per item produces totals $10, $16 and $22 for 2, 4 and 6 items. Dividing by item count gives 5, 4 and about 3.67, not a constant.
- Decreasing non-example: pairs (1,9), (2,8) and (3,7) look like an inverse trend but have products 9, 16 and 21.
When students rebuild these tables rather than copy them, they can see why the relationship is defined by a constant rate or product, not by the appearance of a picture. It is better to understand four deliberately contrasting examples than to repeat a hundred nearly identical arithmetic questions.
A four-lesson progression for Secondary 3 G1
Lesson 1: one unit at a time
Begin with notebooks, ingredient quantities and weights of identical bags. Use the unitary method to find a single-unit rate. Ask the learner to write units at every stage, then verify by multiplying back. Introduce y = kx only after the unit rate has meaning. A student who cannot say what k represents is not ready to rely only on notation.
Lesson 2: a fixed amount of work
Move to machines, taps and time to finish one fixed task. Keep rates identical and describe every assumption. Make a table in which a larger number of helpers corresponds to less time, then calculate the constant product. Ask what would happen if task size doubled; the answer should make clear why the invariant changes.
Lesson 3: spot false proportionality
Give a price with a fixed fee, a decreasing linear table and a square whose area grows faster than its side length. Ask the learner not to calculate immediately but to justify whether direct or inverse proportion is appropriate. This makes the lesson more valuable than one containing only obvious examples. Correct decisions about which method to use matter as much as arithmetic.
Lesson 4: mixed-context assessment
Present a recipe, a constant-work machine problem, a discounted price and a travel-time problem without naming the model. Students choose the relevant relationship and label the final answer. In a small group, one learner can explain the unitary method, another the constant-product approach and a third the assumptions and checks. The tutor should still inspect each person’s individual written reasoning.
A short home routine for seven days
Day 1: Find the price of one item from a multi-item bill; then calculate a different item count and check the rate. Spend a few minutes reviewing the previous day’s first wrong decision, then ask for one independent answer in a new scenario.
Day 2: Scale a fictional recipe up and down, including a decimal number of litres; name the assumption of constant ingredients per serving. Spend a few minutes reviewing the previous day’s first wrong decision, then ask for one independent answer in a new scenario.
Day 3: Make a direct-proportion table and check that the same y/x ratio holds for three rows. Spend a few minutes reviewing the previous day’s first wrong decision, then ask for one independent answer in a new scenario.
Day 4: Make an inverse-proportion table from a fixed number of machine-hours and verify the product in three rows. Spend a few minutes reviewing the previous day’s first wrong decision, then ask for one independent answer in a new scenario.
Day 5: Compare a pure unit-price formula with a fixed-fee formula; explain why only one is direct proportion. Spend a few minutes reviewing the previous day’s first wrong decision, then ask for one independent answer in a new scenario.
Day 6: Solve an inverse task in which the number of workers changes, but clearly note the simplifying assumptions. Spend a few minutes reviewing the previous day’s first wrong decision, then ask for one independent answer in a new scenario.
Day 7: Mix direct, inverse and non-proportional stories. Choose the model before touching a calculator and explain what stays constant. Spend a few minutes reviewing the previous day’s first wrong decision, then ask for one independent answer in a new scenario.
Questions parents ask about direct and inverse proportion
Is direct and inverse proportion included in G1 Mathematics?
Yes. Both appear explicitly under N2 Ratio and Proportion in the 2027 SEAB K110 syllabus. Individual school-year sequences can vary. A useful follow-up is to change one detail—such as adding a fixed fee or doubling the total workload—and ask whether the conclusion still holds. That tests the concept rather than only one familiar calculation.
Does doubling one quantity always mean the other doubles or halves?
Only when an appropriate proportional model applies and its constant assumptions hold. Fixed fees, altered workloads and changing rates can break the rule. A useful follow-up is to change one detail—such as adding a fixed fee or doubling the total workload—and ask whether the conclusion still holds. That tests the concept rather than only one familiar calculation.
Should my child memorise y = kx and y = k/x?
The algebraic forms are useful, but first understand constant unit rate and constant product. The choice of formula should follow the meaning of the problem. A useful follow-up is to change one detail—such as adding a fixed fee or doubling the total workload—and ask whether the conclusion still holds. That tests the concept rather than only one familiar calculation.
How do I recognise inverse proportion in a table?
Multiply each x and corresponding y. If all products are the same, the table is consistent with inverse proportion, subject to the model’s domain and context. A useful follow-up is to change one detail—such as adding a fixed fee or doubling the total workload—and ask whether the conclusion still holds. That tests the concept rather than only one familiar calculation.
What is the unitary method good for?
It turns a multi-unit quantity into a one-unit rate that can be scaled. It is often clearer than rearranging a symbolic formula for beginners. A useful follow-up is to change one detail—such as adding a fixed fee or doubling the total workload—and ask whether the conclusion still holds. That tests the concept rather than only one familiar calculation.
Can inverse proportion describe a real team project exactly?
Only under strong simplifying assumptions about equal constant work rates and no coordination losses. For actual planning, treat it as a model, not a guarantee. A useful follow-up is to change one detail—such as adding a fixed fee or doubling the total workload—and ask whether the conclusion still holds. That tests the concept rather than only one familiar calculation.
How is direct proportion different from an ordinary straight-line trend?
Direct proportion has a zero intercept under the standard model and a constant ratio. A line with a starting fee can be straight yet not directly proportional. A useful follow-up is to change one detail—such as adding a fixed fee or doubling the total workload—and ask whether the conclusion still holds. That tests the concept rather than only one familiar calculation.
How should tutoring progress be measured?
Ask the learner to identify direct, inverse or neither from a new word problem, solve it, check the relevant constant and explain one modelling assumption. A useful follow-up is to change one detail—such as adding a fixed fee or doubling the total workload—and ask whether the conclusion still holds. That tests the concept rather than only one familiar calculation.
References and further learning
Read the official SEAB K110 Mathematics syllabus for the whole-course content. For complementary practice on ratios and map scales, see the previous Bukit Timah ratio guide. For numerical rates, see speed and unit conversions.
The unchanged eduKateSG Secondary 1 Mathematics tutorial reference describes the close-feedback teaching approach near Sixth Avenue MRT. It is not evidence that a particular Secondary 3 G1 timetable or vacancy exists. Families should enquire directly with eduKate Singapore for current arrangements.