HOW SINGAPORE EDUCATION WORKS · Recognising Proportional Relationships in Tables and Graphs
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To check direct proportion, ask whether one quantity is always the same multiple of the other. In a table, compare y ÷ x for non-zero x-values. In a graph, look for the straight-line relationship y = kx, with its line passing through the origin. Check the axes and the situation before deciding.
Full chapter index · Practice and checking · Sources and next reading · How Singapore Education Works
“Both quantities increase” is not enough. Even a straight line can include a fixed starting amount, making it linear without being directly proportional.
This guide is for school learners and parents. It develops a mathematical distinction through original practice cases, rather than specify what every subject level or examination requires. Use the actual task, current syllabus and teacher's guidance for assessed work.
The choices at a glance
| What you notice | Possible issue | Useful next move |
|---|---|---|
| Both quantities rise and are called proportional | Increasing together is mistaken for a constant ratio | Compare corresponding y divided by x |
| Equal differences are treated as enough | A linear rule may include a fixed amount | Check the starting value |
| The graph corner is called the origin | The displayed axes may begin elsewhere | Read both axis labels |
| A few matching rows become a universal rule | The model's scope has been skipped | State the assumption or limit the claim |
Full chapter index
Define the quantities and test the relationship
Inspect a mass model and read its graph
Repair an inference and practise independently
Return later, transfer and answer questions
CHAPTER 1 OF 8 · Define the quantities and test the relationship
1. Define the quantities before comparing numbers
Write what x and y represent, including their units. A row reading x = 4 and y = 300 means little until you know whether the quantities are packets and grams, minutes and metres, or something else.
In a direct-proportion model, y = kx. The same constant k links every permitted pair. For positive quantities, doubling x doubles y, and tripling x triples y.
The units of k follow from the quantities. If x counts packets and y is the mass of the packets in grams, k has units of grams per packet. The number alone does not describe the relationship fully.
The Australian Mathematical Sciences Institute's proportion module explains this constant-multiple relationship and its graph. That is a useful international mathematical reference, not a Singapore examination specification.
Start with the meaning of the quantities, then choose the calculation that tests the relationship.
CHAPTER 2 OF 8 · Define the quantities and test the relationship
2. Separate a constant ratio from a constant increase
Suppose the pairs are (1, 6), (2, 10) and (3, 14). As x increases by 1, y increases by 4 each time. That is a constant increase.
But the ratios are 6 ÷ 1 = 6, 10 ÷ 2 = 5 and 14 ÷ 3, which is about 4.67. They are not the same, so these values do not fit direct proportion.
They fit the linear rule y = 4x + 2. The extra 2 is a fixed starting amount. It remains when x = 0 under that model.
Now compare y = 4x. The pairs (1, 4), (2, 8) and (3, 12) have constant ratio 4. There is no added starting amount.
A learner may notice equal differences and stop too early. Ask which property the question requires: a constant increase, a constant multiple, or some other relationship. Those are different claims.
CHAPTER 3 OF 8 · Inspect a mass model and read its graph
3. A worked case: packet mass and combined mass
Assume a fictional training setup uses identical sealed craft packets. Each sealed packet, including its wrapper, has mass 75 g. An empty tray has mass 120 g. Ignore measurement variation for this exact mathematical model.
Let x be the number of packets. Let N be the total packet mass in grams, and G be the combined mass of the tray and those packets. The model gives N = 75x and G = 120 + 75x.
| Number of packets x | Packet mass N in g | Tray plus packets G in g |
|---|---|---|
| 0 | 0 | 120 |
| 2 | 150 | 270 |
| 4 | 300 | 420 |
| 6 | 450 | 570 |
For the packets, 150 ÷ 2 = 75, 300 ÷ 4 = 75 and 450 ÷ 6 = 75. The constant is 75 g per packet. Packet mass is directly proportional to packet count under the stated model.
For tray plus contents, the corresponding ratios are 135, 105 and 95 g per packet. These are not constant because the fixed tray mass is included.
Doubling from two packets to four doubles packet mass from 150 g to 300 g. It does not double the combined mass: 270 g becomes 420 g, rather than 540 g. Both relationships have equal increases of 75 g per added packet, but only N = 75x has direct proportion.
CHAPTER 4 OF 8 · Inspect a mass model and read its graph
4. Read the graph's structure and domain
A graph of N against x follows a straight line through (0, 0) in the model. A graph of G against x follows a parallel straight line crossing the vertical axis at 120.
That intercept has a meaning: the tray is present before any packets are added. Naming that meaning is more useful than memorising that an intercept “makes it wrong.” It makes the relationship a different kind.
Check the displayed axes. If a graph window begins at x = 2, the origin may not be visible. The line touching the corner of that window does not mean it passes through (0, 0).
Check the permitted inputs too. In the packet example, x counts whole packets. Plot the allowed points. A continuous line can represent the underlying formula, but a point at x = 2.5 does not describe a permitted whole-packet arrangement unless the situation is changed to allow part packets.
Passing through the origin alone is also insufficient: y = x² passes through it but is curved and does not keep y ÷ x constant. The line's form and the context matter together.
CHAPTER 5 OF 8 · Repair an inference and practise independently
5. Diagnose the first incorrect inference
If the learner says “they both go up, so they are proportional,” compare the packet and combined-mass columns. Both increase, yet the ratio test distinguishes them.
If they use differences instead of ratios, ask what happens when x doubles. Keep the original pair visible so they compare corresponding values, not unrelated rows.
If they divide x by y when the question asks for y per unit x, inspect the unit. Reciprocal ratios can also reveal a constant relationship when defined, but they give a different constant with different units. State which quantity is being expressed per one unit of the other.
If they calculate 0 ÷ 0 to find k, use a non-zero x-value. The origin is consistent with y = kx, but that undefined quotient cannot identify the constant.
If a few table rows fit, avoid claiming they prove an unlimited real-world rule. A stated mathematical model can justify extending the pattern; a handful of observations may only support a provisional model. The guide to examples and mathematical proof explains that boundary.
CHAPTER 6 OF 8 · Repair an inference and practise independently
6. Practise a ratio check and a changed starting amount
For guided practice, a model has y = 18x, where x counts whole craft strips and y is their total length in centimetres. Calculate y for x = 2, 5 and 8, then identify the constant and its unit.
Checking answer: the lengths are 36 cm, 90 cm and 144 cm. The constant is 18 cm per strip. At x = 3, the model gives 54 cm. The whole-strip interpretation restricts the allowed x-values.
For independent practice, a different exact model has a fixed 30 cm of material reserved before adding the same 18 cm for each strip. Write the total-length rule and decide whether total length is directly proportional to strip count.
Checking answer: y = 30 + 18x. It is linear but not directly proportional to x. At x = 2 the total is 66 cm; at x = 4 it is 102 cm. Doubling x has not doubled y.
Now suppose a table gives (2, 14), (4, 28) and (6, 45). The ratios are 7, 7 and 7.5, so the listed pairs do not all fit one direct-proportion rule. Do not discard the last row without a reason. Check whether the data or the proposed model needs attention.
CHAPTER 7 OF 8 · Return later, transfer and answer questions
7. Return later and transfer the test
At a later study session, ask the learner to explain the difference between y = 6x and y = 6x + 10 before reopening the worked example. Look for a constant multiple in the first and an added fixed amount in the second.
For changed-context practice, consider a fictional tank receiving water at a constant 12 litres per minute. If it starts empty and that rate remains constant, volume added after t minutes is 12t litres.
If the tank already contains 40 litres, total volume is 40 + 12t. The filling rate has not changed, but the total-volume relationship now has a starting amount. This is a model for mathematical practice, not operating instructions for a real installation.
Mathematical notation helps learners compare reasoning across countries. The assumptions still need words: constant rate, empty or partly filled at the start, and the range of times for which the model applies.
The guide to mathematical modelling develops the next question: whether the chosen rule describes the real situation well enough for the intended purpose.
CHAPTER 8 OF 8 · Return later, transfer and answer questions
8. Common questions about direct proportion
Is every straight-line graph directly proportional?
No. A straight line with a non-zero vertical intercept includes an added starting amount. For direct proportion y = kx, the line passes through the origin.
Can a direct-proportion graph have a negative constant?
Mathematically, it can where the domain and quantities allow it. The positive mass and count examples here use a positive constant. Do not impose the restrictions of one example on every mathematical setting.
Does a constant ratio in three rows prove the rule forever?
It shows that those rows fit the rule. Extending beyond them requires a stated model, a mathematical argument or justified evidence appropriate to the task.
What if the measured values are only approximately proportional?
Inspect measurement precision, the range and the purpose of the model. Do not turn small differences into exact equality without explanation. For a school task, follow the stated assumptions and required treatment of data.
How do I check my constant?
Multiply it by a permitted x-value and see whether it gives the corresponding y, then state its unit. Check another pair as well; one pair can fit many different models.
Sources and next reading
Source checked on 22 October 2026: Australian Mathematical Sciences Institute, Proportion. The packet, craft-strip and tank cases are original fictional teaching models.
Continue with mathematical modelling and examples, counterexamples and proof, or return to How Singapore Education Works.
Learning at other stages
For early-years learning, explore membership under a clear sorting rule. Sorting by Two Attributes in Kindergarten.
For university writing, connect sources without overstating their evidence. Synthesising Sources in University Writing.
For work and later-life learning, practise reading intervals, units and starting values. Reading Measuring Scales in Adult Life.