Secondary 3 G2 Mathematics Tutor Bukit Timah | Mathematical Reasoning and Counterexamples. Did you know? One example can support a conjecture, but one counterexample can disprove a universal claim.
This Secondary 3 G2 Mathematics tutor guide for Bukit Timah families explains mathematical reasoning and counterexamples with worked examples, checks and realistic limits. Read the quick answer first, then use the relevant section to repair a specific misunderstanding.
Quick answer
A mathematical claim states a relationship that can be tested. ‘Every even number is divisible by two’ follows from the definition of even. ‘Every prime number is odd’ is false because two is prime and even. A counterexample is one valid case that makes a universal claim false. For ‘all prime numbers are odd’, the number two is enough. A hundred confirming examples would not repair the false universal statement.
Find the exact question
What is a mathematical claim? · What is a counterexample? · A pattern is not yet a proof · A geometry counterexample · A fraction counterexample · How to justify a true identity · Test a proposed formula · Find the hidden condition · A real-world model needs assumptions · Explain your reasoning in one paragraph
What is a mathematical claim?
A mathematical claim states a relationship that can be tested. ‘Every even number is divisible by two’ follows from the definition of even. ‘Every prime number is odd’ is false because two is prime and even.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
What is a counterexample?
A counterexample is one valid case that makes a universal claim false. For ‘all prime numbers are odd’, the number two is enough. A hundred confirming examples would not repair the false universal statement.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
A pattern is not yet a proof
The numbers 2, 4, 6 and 8 are even, but seeing several examples does not prove every integer is even. Students should distinguish an observed sequence from a general conclusion.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
A geometry counterexample
The claim ‘every quadrilateral with equal opposite sides is a rectangle’ is false: a slanted non-rectangular parallelogram has equal opposite sides but need not have right angles.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
A fraction counterexample
The claim ‘multiplying a number always makes it bigger’ fails for 8×1/2=4. It also fails for multiplication by zero or negative numbers. The multiplier’s value matters.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
How to justify a true identity
For any real a and b, (a+b)²=a²+2ab+b² by expansion. A worked substitution is a check, but the algebraic expansion establishes the identity generally.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Test a proposed formula
If someone claims (x+3)²=x²+9, substitute x=2: the left side is 25 and the right side is 13. The mismatch disproves the claimed identity.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Find the hidden condition
The statement ‘dividing by x preserves an equation’ needs x≠0. If x=0, division is undefined. Recording restrictions is part of mathematical reasoning.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
A real-world model needs assumptions
If six identical workers complete a fixed job in eight hours, twelve workers might complete it in four only if their rates remain constant and work can be divided without delay. State assumptions rather than claiming a universal rule.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
Explain your reasoning in one paragraph
Write the claim, show the example or derivation, then explain what the evidence establishes. For a counterexample, state explicitly why the chosen value satisfies the original conditions but violates the conclusion.
Try a new example with changed values and explain why the method applies. Check the original conditions and units rather than relying only on a calculator result.
A seven-day revision routine
Day 1: identify the key definition and the given quantities. Day 2: reproduce a worked example with reasons. Day 3: change the numbers. Day 4: solve an unlabelled problem. Day 5: diagnose the first error. Day 6: combine the idea with another topic. Day 7: retest independently after a delay.
Frequently asked questions
How should I start? Name the unknown and what the question actually asks. How can I verify the answer? Substitute it into the original information and check that all restrictions are met. Why can a correct-looking number be wrong? The model may have used the wrong reference, denominator or condition.
Official syllabus and tutorial context
Consult the 2027 SEC G2 Mathematics K210 syllabus for the exact assessment scope. The unchanged eduKateSG three-pupil Mathematics reference describes tutorials near Sixth Avenue. Confirm the current tutor, class fit, fees, venue and timetable directly.
Further Mathematics revision
Related resources: Quadratic Graphs, Standard Form and Accuracy and Probability Rules.