Did you know? The statement “Every prime number is odd” sounds plausible until someone notices the number 2. Two is prime because it has exactly two distinct positive factors, yet it is even. One well-chosen example can disprove a claim containing the word *every*.
For Secondary 1 G3 Mathematics students in Bukit Timah, this is an important reasoning skill: do not merely copy rules or accept a conclusion because several examples seem to support it. Ask what the statement actually says, identify the permitted numbers or shapes, and look for a case that would make it false. That case is called a counterexample.
This Secondary 1 G3 Mathematics tutor guide introduces counterexamples through number properties, algebra, ratios, percentages, geometry, graphs and data. It includes worked demonstrations, a mini diagnostic with answers, and ways for parents to ask better questions during home practice. It supports the mathematical reasoning emphasised in the G3 curriculum without treating every example as a formal theorem.
Find a counterexample or practice question
- Quick answer: what is a counterexample?
- Worked example 1: prime-number claims
- Worked example 2: negative numbers change familiar rules
- Worked example 3: denominators do not tell the whole story
- Worked example 4: similar fractions may still represent unequal values
- Worked example 5: equal differences are not equivalent ratios
- Worked example 6: a percentage decrease does not undo an increase
- Worked example 7: successive discounts do not simply add
- Worked example 8: squaring can hide the original sign
- Worked example 9: every linear equation has exactly one solution?
- Worked example 10: the same perimeter does not imply the same area
- Worked example 11: equal area does not imply equal perimeter
- Worked example 12: a shape can look parallel without a stated fact
- Worked example 13: a horizontal graph does not always mean a stop
- Worked example 14: the mean need not be a value in the data
- Worked example 15: the arithmetic mean of speeds is not always average speed
- Ten statements for an independent claim-check diagnostic
- Answers to the claim-check diagnostic
- Frequently asked questions
Quick answer: what is a counterexample?
A counterexample is a valid example that contradicts a general statement. To disprove “All prime numbers are odd,” showing that 2 is both prime and even is enough.
To establish that a statement is true for *all* permitted values, checking a few examples is generally not enough. A proof or argument covering the entire stated set may be required.
Words such as all, every, always, never, must and cannot are useful signals. They describe broad claims that can fail if just one permitted instance does not fit.
Before testing a claim, read its restrictions. A counterexample involving negative numbers may be invalid if the statement explicitly concerns only positive values. A non-right triangle cannot disprove a fact stated only for right-angled triangles.
A claim, a test, a counterexample and a conclusion
Consider the statement “Every even whole number is composite.”
Choose the even whole number 2. It is divisible by 1 and 2, which are its only distinct positive factors. It is therefore prime, not composite.
That makes 2 a counterexample, and the original general claim is false.
A better statement is “Every even whole number greater than 2 is composite.” Such a number has at least the distinct positive factors 1, 2 and itself, so it meets the composite definition.
This demonstrates an important habit: after finding a counterexample, revise the original wording to identify whether a more careful claim might be true.
Worked example 1: prime-number claims
Claim A: “All prime numbers are odd.” Counterexample: 2, an even prime.
Claim B: “All odd whole numbers greater than one are prime.” Counterexample: 9, which is odd but has factors 1, 3 and 9.
Claim C: “A prime number has exactly two distinct positive factors.” That is the definition of a prime; a student cannot validly disprove it by inventing a different rule for the word prime.
Notice the difference between using an example and reading a definition. A learner who simply memorises a list of primes may struggle to explain *why* 9 is not prime.
For more number reasoning, see prime factorisation, HCF and LCM.
Worked example 2: negative numbers change familiar rules
Claim: “Adding a number always makes the result larger.”
Counterexample: add −4 to 7. The result is 7 + (−4) = 3, which is smaller than 7.
The claim becomes more reliable if it specifies “Adding a strictly positive number increases a real number’s value.” The word positive matters.
Another claim is “The square of a negative real number is negative.” Counterexample: (−3)² = 9, which is positive.
The original expression’s brackets also matter: −3² is −9, whereas (−3)² is +9. A tutor should require the student to write the exact expression rather than just say “negative three squared.”
Worked example 3: denominators do not tell the whole story
Claim: “The fraction with the larger denominator is always smaller.”
Compare 1/4 with 2/5. The denominator five is larger than four, but 2/5 = 0.4 is greater than 1/4 = 0.25.
That is a counterexample because the numerators differ. If the two fractions have the same positive numerator and positive denominators, then a larger denominator gives smaller equal-sized shares.
The restriction changes which claim is true.
A student who compares fraction denominators without considering the numerator may be using a rule remembered from unit fractions in a setting where it does not apply.
Worked example 4: similar fractions may still represent unequal values
Claim: “If the numerators of two positive fractions are the same, the fractions are equal.”
Take 2/3 and 2/5. The numerators are both two, but 2/3 ≈ 0.667 while 2/5 = 0.4. They are not equal.
Claim: “Two fractions are equal if their numerator and denominator are both multiplied by the same nonzero value.” For example, 3/5 = 9/15 because both terms were multiplied by three. This is an equivalence relationship.
The contrast is useful: a fraction is a relationship between two numbers, not a pair of independent labels.
Ask the student to draw equal-sized whole rectangles divided into three and five parts. That picture shows why the same numerator can represent different amounts.
Worked example 5: equal differences are not equivalent ratios
Claim: “If we add the same number to both terms of a ratio, the ratio remains unchanged.”
Begin with 2:3. Add two to both terms, producing 4:5. But 2/3 ≈ 0.667 and 4/5 = 0.8, so the relationships differ.
By contrast, multiply both original terms by two to obtain 4:6, which simplifies back to 2:3.
A valid equivalent-ratio transformation preserves the multiplicative comparison, not an arbitrary additive difference.
This counterexample can prevent errors in sharing questions where a learner adds identical amounts to two components and assumes the original ratio survives.
The equivalent ratios and tables guide gives examples of proportional and nonproportional relationships.
Worked example 6: a percentage decrease does not undo an increase
Claim: “If a price rises by 20% and then falls by 20%, it returns to its original value.”
Take an original fictional price of $100. A 20% rise gives $120. A 20% fall from the *new* $120 removes $24, leaving $96.
The amount does not return to $100. The percentage bases differed.
This is a counterexample to a common assumption about percentage changes. A different claim—“Multiplying by 1.2 and then dividing by 1.2 returns a nonzero or zero original amount”—does describe an exact inverse relationship.
A student should compare the actual multipliers, not just the two matching percentage labels.
Worked example 7: successive discounts do not simply add
Claim: “Two successive discounts of 20% and 25% mean 45% off the original marked price.”
Let the fictional original marked price be $100. After 20% off, $80 remains. Another 25% of the reduced $80 is $20, leaving $60.
The overall reduction is $40, or 40%, not 45%.
The correct remaining multiplier is 0.80 × 0.75 = 0.60. That is the reason the final price is sixty percent of the original.
A student who obtains $55 has treated both percentage reductions as portions of the same $100, contrary to the word “successive.”
Worked example 8: squaring can hide the original sign
Claim: “If a² = b², then a = b.”
Take a = 3 and b = −3. Both squares equal nine, but the original values are not equal.
The statement is false for arbitrary real numbers. A correct general relationship is that a² = b² implies a = b or a = −b.
This distinction also explains why the principal expression √49 is seven, while the equation x² = 49 has both real solutions +7 and −7.
The case is a reminder that solving an equation is not the same as evaluating a square-root expression. The permitted domain and notation matter.
Worked example 9: every linear equation has exactly one solution?
Claim: “Every equation containing x has one unique value that solves it.”
Consider 2(x + 3) = 2x + 6. Expanding gives 2x + 6 = 2x + 6, which is true for every real x. There are infinitely many solutions.
Now consider 2(x + 3) = 2x + 7. Expansion gives 2x + 6 = 2x + 7, an impossible equality. There is no solution.
Some equations have one solution, some none and some more than one. The particular structure determines the solution set.
A child who forces every exercise to end with “x = something” may need to understand equality before attempting longer equations.
Worked example 10: the same perimeter does not imply the same area
Claim: “All rectangles with the same perimeter have the same area.”
Rectangle A measures 2 cm by 8 cm. Perimeter is 2(2 + 8) = 20 cm, while area is 2 × 8 = 16 cm².
Rectangle B is a 5 cm by 5 cm square. Its perimeter is also 2(5 + 5) = 20 cm, while its area is 5 × 5 = 25 cm².
The perimeters match, but the areas differ. This is a concrete counterexample.
A physical string of fixed length can be arranged into rectangles with different widths and enclosed areas. The length around the boundary does not uniquely specify the region inside.
Worked example 11: equal area does not imply equal perimeter
Now consider a 1 cm by 12 cm rectangle and a 3 cm by 4 cm rectangle.
Both areas are twelve square centimetres. But their perimeters are 2(1 + 12) = 26 cm and 2(3 + 4) = 14 cm.
Thus equal area does not force equal perimeter either.
The two preceding counterexamples are valuable as a pair. They prevent a learner from treating perimeter and area as interchangeable formulas merely because both use the same side lengths.
In any geometry question, first ask whether the requested quantity is a boundary distance or a two-dimensional region.
Worked example 12: a shape can look parallel without a stated fact
Claim: “Lines that look parallel in a printed picture must be mathematically parallel.”
A sketch may be inaccurate or explicitly marked “not drawn to scale.” Unless parallelism is given or follows from established angle facts, appearance alone is not a proof.
For example, a transversal forming a pair of genuinely corresponding equal angles can be used under the appropriate converse property to establish parallel lines. But two arbitrary equal-looking drawn angles are insufficient.
This isn’t a counterexample with a particular number; it is a reminder that geometrical claims depend on conditions, markings and reasoning rather than visual guesswork.
Ask the pupil to distinguish the facts supplied in the text from measurements they are imagining from the illustration.
Worked example 13: a horizontal graph does not always mean a stop
Claim: “A flat line on any journey graph means the traveller has stopped.”
On a cumulative-distance-time graph, a horizontal interval means no additional distance is accumulated in the simple model.
On a speed-time graph, a horizontal line at 60 m/min means the speed is constant and positive. The traveller is not stopped in that model.
The same line shape can have a different meaning because the vertical axis measures a different quantity.
A student who reads the line before the axis label may apply a memorised picture rule to the wrong graph type.
Worked example 14: the mean need not be a value in the data
As a connected statistical extension, consider the numbers 2 and 4. Their arithmetic mean is (2 + 4)/2 = 3, although 3 is not one of the observed numbers.
Another data set, 1, 2 and 9, has mean 4 but median 2. These measures describe different features of the set.
If a student says “The average must be one of the original results,” these examples disprove that general claim.
The cited published G3 syllabus places mean, median and mode in Secondary 2. Here the example is a reasoning preview, not a claim that every Secondary 1 test includes advanced statistics.
Worked example 15: the arithmetic mean of speeds is not always average speed
Claim: “The whole-journey average speed is always the arithmetic mean of the stage speeds.”
A fictional traveller covers 6 km at 12 km/h and then 4 km at 6 km/h. Stage times are 6/12 = 1/2 hour and 4/6 = 2/3 hour.
Total distance is 10 km and total time is 7/6 hour. Whole-journey average speed is 10/(7/6) = 60/7 km/h, approximately 8.57 km/h.
The arithmetic mean of the speeds is 9 km/h, which does not equal the true journey average.
The correct quantity weights distance and elapsed time through their totals. Identical stage speeds or particular equal-time arrangements may yield a simple mean, but the general statement is false.
What does not count as a valid counterexample?
If the claim says “Every *positive* whole number greater than two that is even is composite,” choosing 2 is not a valid counterexample because it violates the “greater than two” restriction.
If a question concerns a right triangle, a non-right triangle cannot disprove a property stated only for right triangles.
If a claim concerns a *constant-speed* journey, choosing a moving-stopping-moving trip cannot disprove it unless that trip meets the stated conditions.
Always read the domain, assumptions and units. A good counterexample satisfies all the premises of the claim while contradicting its conclusion.
Why several examples don’t prove a universal rule
A pupil might test the numbers 3, 5, 7 and 11 and notice that all are odd primes. That is real evidence about those particular examples, but it does not prove the universal statement “All primes are odd.”
Finding 2 exposes the false generalisation immediately.
On the other hand, if a rule is genuinely true, proving it requires reasoning across the permitted domain. For even whole numbers greater than two, expressing one as 2k with k greater than one shows it has factors 1, 2 and itself; this supports why the composite statement applies generally.
A tutor should distinguish checking, disproving and proving as different activities, even when all use numerical examples.
A practical five-step claim-checking routine
First identify the exact statement and underline any universal wording.
Second write its assumptions, including number type, positive or negative restrictions, required diagram properties, and units.
Third look for a small, simple example near a boundary: zero, one, two, a negative number, a square, or a differently oriented shape.
Fourth calculate or reason carefully and show the contradiction clearly. A counterexample is not just an unusual-looking number; it must genuinely violate the conclusion.
Fifth explain what you can safely say instead. Sometimes a restricted version is true; other times a different relationship replaces the original.
The goal is disciplined scepticism about mathematical statements, not disputing every correct teacher explanation.
Ten statements for an independent claim-check diagnostic
For each, decide whether it is always true under the stated conditions. If false, give one counterexample.
- A. Every prime number is odd.
- B. Every odd whole number greater than one is prime.
- C. Adding any real number always increases the original value.
- D. The square of every negative real number is negative.
- E. The fraction with the larger denominator is always smaller, even when numerators differ.
- F. Adding the same positive number to both ratio terms always preserves the ratio.
- G. A 20% increase followed by a 20% decrease returns to the original amount.
- H. Rectangles with the same perimeter always have the same area.
- I. A horizontal segment on any speed-time graph means zero speed.
- J. Every equation containing x has exactly one solution.
Answers to the claim-check diagnostic
A. False. The prime number 2 is even.
B. False. Nine is odd and composite, with factor three.
C. False. Adding −4 to 7 gives 3, which is smaller.
D. False. (−3)² = 9 is positive.
E. False. 2/5 = 0.4 is greater than 1/4 = 0.25 despite having the larger denominator.
F. False. Adding two changes 2:3 to 4:5, and these ratios differ.
G. False. $100 increases to $120 then decreases to $96.
H. False. The 2 by 8 rectangle and 5 by 5 square both have perimeter 20 but different areas.
I. False. A horizontal speed-time line at 60 m/min indicates constant positive speed.
J. False. 2(x + 3) = 2x + 6 has every real x as a solution, while 2(x + 3) = 2x + 7 has none.
A student who correctly labels a statement false but cannot produce an actual counterexample may still be guessing. The contradiction is part of the answer.
How a tutor can adapt the difficulty
For a learner with insecure number foundations, begin with primes, even and odd numbers, negative arithmetic and comparing fractions. These need only a few clear values.
Once that is stable, move to algebra and ratios. The student should substitute values carefully and distinguish a numerical check from a general proof.
For a learner comfortable with these, introduce claims about geometry, graphs and statistics where the assumptions are more subtle.
A strong extension activity is to ask the pupil to rewrite a false claim as a carefully restricted true one. For example, “Every prime number is odd” becomes “Every prime number other than 2 is odd.”
This demands understanding of both the counterexample and the logical condition that excludes it.
A fifteen-minute lesson on counterexamples
Begin with three minutes of a simple claim, such as “Every even number is composite.” Allow the learner to test two, four and six and explain which one creates the contradiction.
Use five minutes for fraction and ratio claims, where the same numerator or unchanged difference may tempt overgeneralisation. Ask for calculations supporting the chosen counterexample.
Use four minutes for a geometry or graph comparison. A horizontal line has different meanings in a speed-time graph and a cumulative-distance-time graph.
Finish with a statement that is genuinely true under stated conditions. Ask why the earlier counterexample no longer applies.
The aim is not a large set of random trick questions; it is understanding precisely what has been claimed.
What mathematical reasoning looks like in three-pupil tuition
One student can propose a claim, another can look for a counterexample and a third can check whether the example meets every condition. They can then switch roles.
For instance, when testing a perimeter claim, two students might draw different rectangles with perimeter twenty while the third calculates their areas.
The teacher can observe whether each pupil is reasoning about conditions or simply guessing numbers.
The eduKateSG Secondary 1 Mathematics small-group reference describes the three-pupil, ninety-minute tutorial model near Sixth Avenue MRT. The Bukit Timah Secondary 1 G3 Mathematics tuition overview provides local programme context and class details to verify.
The MOE G2 and G3 Mathematics syllabus gives the content context for number properties, algebra, geometry, graphs and mathematical reasoning. This guide uses counterexamples as a cross-topic teaching strategy.
How a parent can use the idea at home
When a child says “this always works,” ask for one changed example. If the statement is true only for positive integers, ask whether it still works with zero, negatives or fractions.
If a student says two rectangles with the same perimeter must have the same area, draw two rectangles with different dimensions and calculate both measures.
Avoid turning every conversation into an adversarial challenge. The purpose is to help the learner understand the conditions, not to make them fear that every answer will be attacked.
A helpful parent can simply ask, “What would a counterexample have to look like?” and then let the child choose numbers.
Two counterexample challenges for an independent learner
Claim: “Multiplying a positive number by another positive number always makes it larger.” Take one-half multiplied by one-quarter. The product is one-eighth, smaller than either factor. The general claim is false because a multiplier between zero and one decreases a positive value. A more precise statement is that multiplying a positive number by a factor strictly greater than one makes it larger.
Claim: “If a graph rises from left to right, every point on it has positive coordinates.” Consider y = x − 5, an increasing straight line with gradient +1. At x = 2, the point is (2, −3), so its y-coordinate is negative even though the line increases as x increases. The sign of gradient is a statement about change, not about every plotted coordinate.
For both claims, ask the child to describe the exact mathematical condition that has been overlooked. A valid counterexample should be shown with the relevant arithmetic or coordinate substitution, not just a confident disagreement. Afterward, ask whether the learner can replace the original statement with one whose conditions are correctly specified.
A good tutor can then give the same learner a genuinely true claim and request a different form of justification. For example, the sum of two even whole numbers is even: if they can be written as 2m and 2n for whole numbers m and n, their sum is 2(m + n), another multiple of two. Testing a few sums is useful, but the algebraic representation explains why the result holds for all numbers in the specified domain.
Frequently asked questions
Is one counterexample really enough?
Yes, to disprove a universal statement, provided the example satisfies its stated conditions and contradicts its conclusion.
Do ten successful examples prove a statement always holds?
Not necessarily. A universal claim generally needs an argument covering the permitted cases, rather than only a finite collection of examples.
Can a counterexample be a negative number?
Yes, when negative values are permitted by the claim. It is not valid if the claim restricts the domain to positive values.
Is a calculator useful for testing claims?
It can calculate candidate values, but the student must still interpret the assumptions and determine whether a result is a genuine contradiction.
Are counterexamples only for advanced students?
No. Simple cases such as the prime number two or a negative number squared are accessible and useful in Secondary 1 number reasoning.
What if a student challenges a correct theorem?
Ask whether their example meets every condition. A non-right triangle cannot disprove a theorem stated for right triangles.
Do examples replace written geometry reasons?
No. Examples can check plausibility or disprove a general claim, but a correct theorem needs appropriate reasoning from its stated conditions.
How does this help with unfamiliar exam questions?
It encourages reading assumptions carefully, testing a proposed shortcut and recognising when a remembered pattern is too broad for the actual problem.
Related Secondary 1 G3 Mathematics reading
- Prime factorisation and number properties
- Equivalent ratios and proportional tables
- Index notation and negative bases
- Why familiar worksheets fail on unseen questions
- When a Mathematics question has missing information
- Explaining your mathematical reasoning aloud
A strong mathematical claim survives careful examination of its conditions. A false claim needs only one valid counterexample to reveal what was overlooked—and that discovery is a form of learning.