Sec 1 G1 Maths Bukit Timah | Counterexamples Explained

Secondary 1 G1 Mathematics Tutor Bukit Timah | Why One Counterexample Can Disprove a Maths Rule. Did you know? Nine alone disproves the claim that every odd number is prime. Test an ‘always’ statement before memorising it as a rule.

For Secondary 1 G1 Mathematics students in Bukit Timah, a counterexample is a practical way to think more carefully about definitions, signed numbers, fractions, ratios, geometric shapes and number patterns. This is a reasoning skill illustrated with G1 content, not a claim that formal proof is a separate examination topic.

The original eduKateSG Secondary 1 Mathematics tuition model near Sixth Avenue MRT describes three-pupil groups, weekly 1.5-hour lessons and guided corrections. The situations below are illustrative questions a tutor or parent can use, not reported student cases or guarantees of grades.

The quick answer: what a counterexample does

Read the full claim and identify whether it says every, always or never
Choose a case that satisfies the claim’s stated conditions
Check whether it contradicts the claimed result
If it does, the universal claim is false
If it does not, look for another case; one successful example does not establish a universal rule

Seventeen G1 Maths claims worth testing

1. Did you know? One small exception can defeat an ‘always’ claim

Did you know? To show that ‘every odd number is prime’ is false, you need just one properly chosen example: 9. Nine is odd but has positive factors 1, 3 and 9, so it is not prime. That single example is enough to refute the word every.

In Mathematics, an example that satisfies the conditions of a claim but contradicts its conclusion is a counterexample. For a Secondary 1 G1 learner, the idea is an accessible reasoning habit, not a requirement to attempt advanced formal proof. It helps pupils test rules that sound reasonable but are incomplete.

Parents can ask: ‘Does this sentence say always, every or never? Can we find one valid case where it fails?’ The process encourages thinking about definitions rather than merely collecting familiar correct answers.

2. The counterexample must fit the claim’s conditions

Suppose someone claims, ‘Every prime number is odd.’ The number 2 disproves the statement, because 2 is prime and is even. It fits the condition ‘prime number’ and contradicts ‘odd’.

Now consider the different statement, ‘Every prime number greater than 2 is odd.’ The number 2 cannot disprove this revised claim because it does not satisfy the stated condition ‘greater than 2’. A counterexample must qualify for the claim being tested.

This is a useful reason to read an entire mathematical sentence. One added condition can change whether a familiar example is relevant. In a G1 Mathematics question, words like positive, nonzero, greater than and equal to control the answer.

3. A single correct example does not prove a rule for every case

Suppose a pupil notices that 3 × 4 is larger than both three and four, and claims that ‘multiplication always makes a number bigger’. The example is correct, but the universal claim is false.

Take 0.5 × 0.5 = 0.25. The product is smaller than either positive factor. This one example shows that the claim needs conditions. Multiplying a positive number by a positive fraction below one makes it smaller, not larger.

An important distinction follows: confirming a rule for three convenient numbers may suggest a pattern, but it does not necessarily establish that the rule holds for all possible values. Searching for an exception is a powerful beginner’s check.

4. Division does not always make quantities smaller

Some pupils hear ‘divide’ and expect a smaller numerical answer. Yet 1 ÷ 0.5 = 2. There are two half-sized groups in one whole. The answer is larger because the divisor is a positive quantity smaller than one.

Similarly, 1.2 ÷ 0.3 = 4. Four groups of 0.3 fit into 1.2. Both calculations disprove the unqualified slogan ‘division always makes numbers smaller’.

The better lesson is to ask what the divisor represents. If you divide a positive quantity by a number greater than one, the result is smaller; if you divide it by a positive number between zero and one, the result is larger. The conditions matter more than the slogan.

5. Not every pair of negative signs gives a positive result

Here is another misleading shortcut: ‘two negatives make a positive.’ Under multiplication, (−3) × (−4) = 12 is correct. But under addition, (−3) + (−4) = −7.

The addition example is a counterexample to the slogan being used without naming the operation. The pupil should not conclude that the multiplication rule itself is wrong. The correct rule depends on which operation is being performed.

This is a strong example of mathematical precision. Replacing an inaccurate slogan with a statement that includes its conditions gives students a better habit for signed fractions and algebra.

6. A bigger denominator does not always make a fraction smaller

If the numerator is fixed at one and the whole is unchanged, 1/8 is smaller than 1/4. This may tempt a learner into claiming that every fraction with a larger denominator is smaller.

Compare 5/8 with 1/3. Even though eight exceeds three, 5/8 = 0.625, which is greater than 1/3. The numerators are different, so the simple unit-fraction comparison does not apply.

The repaired statement should specify its condition: when comparing positive unit fractions of the same whole, a larger denominator means a smaller piece. When both numerators and denominators change, compare the actual values.

7. Same area does not imply same perimeter

Imagine a square with side 4 cm. Its area is 4 × 4 = 16 cm², and its perimeter is 4 + 4 + 4 + 4 = 16 cm.

Now consider a rectangle measuring 2 cm by 8 cm. Its area is also 16 cm², but its perimeter is 2(2 + 8) = 20 cm. The two shapes cover equal surfaces but have different boundaries.

This is a counterexample to the claim ‘equal areas always mean equal perimeters’. A tutor can ask a pupil to draw both on square paper. The separate area and boundary measurements are easy to see, and the units make the distinction explicit.

8. Equal volume does not guarantee equal surface area

A 3 cm by 4 cm by 2 cm cuboid has volume 24 cm³ and surface area 2(12 + 6 + 8) = 52 cm².

A second cuboid, measuring 2 cm by 2 cm by 6 cm, also has volume 24 cm³, but surface area 2(4 + 12 + 12) = 56 cm².

The pair disproves the claim that equal-volume solids must have equal surface area. They contain equal three-dimensional volumes, but the amount of exposed outside surface differs. This also reinforces why pupils must identify what a geometry question measures before selecting a formula.

9. Not every rectangle’s diagonal is a line of symmetry

A square is a special rectangle, and its diagonals are lines of symmetry. But for a non-square rectangle of length 6 cm and width 3 cm, folding along a diagonal does not align its two halves as mirror images.

Therefore the claim ‘both diagonals of every rectangle are lines of symmetry’ is false. A single non-square rectangle is a counterexample. The rectangle still has two lines of symmetry, through its central horizontal and vertical axes.

The student should test the reflection rather than count corners or diagonals by sight. Mathematical categories have conditions: being a rectangle does not require all four sides to be equal.

10. The first term matching does not establish an nth-term rule

Consider the sequence 3, 5, 7, 9. A pupil proposes the rule 3n because at position n = 1 it produces three. That single agreement does not establish the proposed rule.

At n = 2, 3n gives six, whereas the second term is five. The second position is a counterexample to the claim that 3n describes the stated terms. The simpler expression 2n + 1 matches all four provided positions.

Within G1 introductory pattern work, the useful habit is to test a proposed expression at several positions. A rule that fails on an already-given term can be rejected immediately, even if the first test happened to work.

11. The same percentage increase and decrease do not cancel

Someone might claim that raising a positive amount by 20% and then reducing it by 20% always returns to the original. Start with 100. After a 20% increase it becomes 120.

Now decrease 120 by 20%. The decrease is 24, leaving 96, not 100. This is a counterexample to the cancellation claim. The mistake was treating both percentage changes as though they had the same base.

Ask the pupil to label 100% separately at each stage. The first change uses 100 as its base; the second uses 120. This teaches percentage meaning more effectively than memorising a special exception.

12. With negative numbers, bigger-looking digits can mean a smaller value

Consider the statement ‘the number with the larger digit is always larger’. Compare −8 and −3. Eight is the larger positive digit, but −8 is less than −3 on a number line.

The number −8 lies farther to the left. The numerical value becomes greater as we move right, whether both numbers are negative or not.

The counterexample helps show why signs cannot be ignored. Ask for another pair, such as −12 and −4, then compare both their magnitudes and their actual values. Words such as larger, greater, more negative and farther from zero must be interpreted accurately.

13. A 2:3 ratio does not make the first group two-thirds of the total

If red : blue = 2 : 3, the combined group has five equal ratio parts. Red occupies two of them, so red is 2/5 of the whole.

Red is indeed 2/3 of blue when both groups are measured by the same part size. That is a different comparison. The claim ‘in a ratio 2:3 the first group is 2/3 of the combined whole’ is false.

Draw two red boxes and three blue ones. The diagram makes the denominator visible. The counterexample is not a trick: it shows that a mathematical statement must specify what its reference whole is.

14. A pictogram symbol need not mean one item

Imagine a pictogram in which one book icon stands for four real books. Three icons then represent 12 books, not three. That disproves the assumption that one picture always means one physical item.

A half-icon represents two books in that chart, but a chart with a different key could assign another value. The picture alone does not define the count.

Ask your child to read the key aloud before comparing categories. For every statistical visualisation, scales and units are conditions of the interpretation. Ignoring them can turn an accurate picture into an incorrect statement.

15. Not every claim can be settled by one example

A counterexample is a way to disprove a universal claim. It is not a shortcut for proving that a claim is true in all cases. Testing 2, 4 and 6 does not prove that every even number is positive, because −2 is an even integer too.

Also be careful with existential claims. The sentence ‘at least one even prime number exists’ is established by showing that 2 is prime and even. It is not a claim about every prime. The words every and at least one ask for different kinds of evidence.

At Secondary 1 G1 level, keep this as a clear reasoning habit. There is no need to introduce formal logic notation or advanced proof techniques. Pupils can learn plenty by reading conditions and giving a correct example or counterexample.

16. A good counterexample is small and easy to verify

To disprove ‘every rectangle is a square’, you do not need an elaborate drawing. A 2 cm by 5 cm rectangle has four right angles but unequal adjacent side lengths. It is a rectangle and not a square.

To disprove ‘every odd number is prime’, nine suffices. To disprove ‘multiplication always increases a positive number’, 1/2 × 1/2 = 1/4 suffices. Each example is simple and meets the stated conditions.

When a pupil searches for a counterexample, encourage them to choose familiar numbers first. A correct example should be easy for someone else to check. Making the case complicated rarely improves the explanation.

17. Five-minute ‘always or not?’ challenge

A. ‘Every odd number is prime.’ False: 9 is odd and composite. B. ‘Multiplication always makes a positive amount larger.’ False: 1/2 × 1/2 = 1/4.

C. ‘Rectangles with equal area always have equal perimeter.’ False: 4 × 4 and 2 × 8 both have area 16, but perimeters 16 and 20. D. ‘Every prime number is odd.’ False: 2 is a prime and even.

E. ‘For every positive whole number n, n + 1 is greater than n.’ True: adding one moves one unit to the right on the number line. Ask the child why an example supports a claim but does not by itself prove the universal rule.

18. How a G1 small-group tutor can use counterexamples

One pupil may believe every odd number is prime; another may apply a rule about positive fractions to negative numbers; a third may accept a matching first sequence term as proof. A carefully chosen counterexample reveals the precise assumption behind each mistake.

eduKateSG’s published Secondary 1 Mathematics reference describes three-pupil tutorials near Sixth Avenue MRT, with 1.5-hour weekly lessons and guided correction. A tutor can invite learners to test each other’s mathematical statements, then require individual explanations.

The learning target is healthy mathematical scepticism: check the conditions, test an unfamiliar case and be willing to revise a claim when the evidence contradicts it. That habit makes a pupil more independent in G1 Mathematics and beyond.

A brief counterexample practice routine

Practice 1: State the claim carefully

Read every word, including restrictions such as positive, nonzero, greater than or exactly. A case outside the claim’s conditions cannot disprove it.

Practice 2: Try a small case

Use a familiar number or shape first: zero, one, two, one half, nine, or a small non-square rectangle. Simplicity makes the result easy to check.

Practice 3: Show why the case qualifies

For instance, explain that nine is odd before showing that it is composite. A good counterexample meets the claim’s starting condition.

Practice 4: Name the contradiction

Finish with a complete sentence: ‘Nine is odd but not prime, so the claim that every odd number is prime is false.’ Revisit with a different claim later.

Frequently asked questions

What is a counterexample?

A case that satisfies a general statement’s conditions but makes its conclusion false.

Can one counterexample disprove a statement that says ‘every’?

Yes. If the example genuinely meets the conditions and contradicts the conclusion, the universal statement is false.

Does one correct example prove a universal rule?

No. An example supports the possibility of the rule but cannot, by itself, establish that it holds for every relevant case.

Is two a counterexample to ‘all prime numbers are odd’?

Yes. Two is prime and even.

Is nine prime because it is odd?

No. Nine has factors 1, 3 and 9, so it is composite.

Does a counterexample have to be complicated?

No. Small, easy-to-check numbers are usually better when they meet the claim’s conditions.

Does G1 Secondary One require advanced proof?

This article teaches a useful reasoning technique through familiar G1 content; it does not assert that advanced formal proof is a separately examined G1 topic.

What question should parents ask?

Ask: ‘Does that claim mean always? Can you find a qualifying case where it fails?’ Then request a clear explanation.

Secondary 1 G1 Mathematics Tutor Bukit Timah | Why One Counterexample Can Disprove a Maths Rule