Thirty-six further investigations: distinguish a valid rule from a lucky example

The first twenty-four checks showed how to test numerical, algebraic and geometric claims. The next thirty-six extend the habit into powers, factors, coordinate patterns, inequality notation, scale and physical units. The objective is not to memorise sixty answers; it is to learn which evidence would make a claim reliable. Some proofs, formal inequalities and transformations are later-stage extensions depending on the actual school Mathematics level.

When a statement has an ‘always’, ‘every’ or ‘never’, one valid counterexample can disprove it. When a statement concerns one proposed value, substitution may be enough to check that value. When it claims two expressions are identical for all permitted inputs, show the underlying algebraic property or a general construction. A few matching numerical trials are not a general proof.

Additional reasoning example 25. Exponent as product

Statement. The statement 2³=2×3 is true.

Decision. False.

Justification. The exponent counts three copies of two.

Independent check. 2×2×2=8, not six.

Additional reasoning example 26. Negative square

Statement. (−3)² is negative nine.

Decision. False.

Justification. The complete negative three is squared, creating two negative factors.

Independent check. (−3)(−3)=+9.

Additional reasoning example 27. Opposite square answers

Statement. If x²=25 then x must equal five.

Decision. False over real numbers.

Justification. Negative five also has square twenty-five.

Independent check. Both +5 and −5 satisfy the equation.

Additional reasoning example 28. Principal root

Statement. √25=−5.

Decision. False.

Justification. The principal square root is nonnegative.

Independent check. √25=5; −5 is another solution to x²=25.

Additional reasoning example 29. Operation precedence

Statement. 24÷6×2=2 under conventional left-to-right equal precedence.

Decision. False.

Justification. Division and multiplication share precedence, processed left to right.

Independent check. (24÷6)×2=8.

Additional reasoning example 30. Brackets

Statement. 24÷(6×2)=2.

Decision. True.

Justification. The denominator is the grouped product twelve.

Independent check. 24÷12=2.

Additional reasoning example 31. Side doubled

Statement. If a square’s side doubles, its area doubles.

Decision. False.

Justification. Both perpendicular dimensions double, so area quadruples.

Independent check. Side 3 area9; side6 area36.

Additional reasoning example 32. Cube edge doubled

Statement. Doubling a cube edge makes its volume eight times larger.

Decision. True.

Justification. Three independent dimensions each double.

Independent check. 2³=8; edge3 volume27 and edge6 volume216.

Additional reasoning example 33. Same volume shape

Statement. If two cuboids have equal volume, they must have the same edge lengths.

Decision. False.

Justification. Different factors can produce the same product.

Independent check. 2×3×4=24 and 1×4×6=24.

Additional reasoning example 34. Same perimeter area

Statement. Rectangles of equal perimeter always have equal area.

Decision. False.

Justification. Perimeter sum does not determine side product.

Independent check. 10×5 and 9×6 both have perimeter 30, but areas50 and54.

Additional reasoning example 35. Parallelogram symmetry

Statement. A quadrilateral can have 180° rotational symmetry with no reflection axis.

Decision. True.

Justification. A general oblique parallelogram is restored by a half-turn about its diagonal intersection.

Independent check. Mirror folding usually fails for the generic oblique case.

Additional reasoning example 36. Every quadrilateral

Statement. Opposite angles of every quadrilateral are equal.

Decision. False.

Justification. That property requires additional shape conditions.

Independent check. An arbitrary trapezium need not have matching opposite angles.

Additional reasoning example 37. Multiples

Statement. Every positive multiple of six is also a multiple of three.

Decision. True.

Justification. 6n=3(2n) for positive whole n.

Independent check. Twelve and eighteen both divide by three.

Additional reasoning example 38. LCM candidate

Statement. Any common multiple of two numbers is automatically their LCM.

Decision. False.

Justification. A common multiple need not be the smallest positive one.

Independent check. For 4 and 6, twelve is LCM but twenty-four is also common.

Additional reasoning example 39. Rotation order

Statement. A scalene triangle’s rotational symmetry order is zero.

Decision. False under the full-turn convention.

Justification. The trivial 360° turn always restores a shape.

Independent check. A scalene triangle has order one.

Additional reasoning example 40. An nth-term shortcut

Statement. A sequence increasing by three always has nth term 3n.

Decision. False.

Justification. The starting term fixes a constant offset.

Independent check. 5,8,11,… has rule 3n+2.

Additional reasoning example 41. Term number

Statement. In a pattern, n always means number of objects.

Decision. False.

Justification. If n is a stage index, expression may determine the number of objects.

Independent check. For 3n+2 at stage 4, the value is fourteen objects.

Additional reasoning example 42. Same at one value

Statement. If 3x and 5x are equal at x=0, they are identical for all x.

Decision. False.

Justification. Agreement at one point does not prove an identity.

Independent check. At x=1, the expressions are three and five.

At this checkpoint, contrast testing a proposed number with proving a general identity. Checking x=0 in an equation can show that zero satisfies the equation, but cannot prove that two expressions are the same for every x. To establish a general statement, explain the algebraic relationship, such as why 2(x+3)=2x+6 follows from distributing two to both terms. The quality of the reason matters more than the number of numerical trials.

Additional reasoning example 43. Equal multiples

Statement. If 2x=2y for real x and y, then x=y.

Decision. True.

Justification. Divide both sides by the same nonzero factor two.

Independent check. Substitution preserves the equality.

Additional reasoning example 44. Equal squares

Statement. If x²=y² for real x,y, then x=y.

Decision. False.

Justification. The numbers can be opposites.

Independent check. x=3,y=−3 have equal squares but different values.

Additional reasoning example 45. Division by zero

Statement. An equation remains equivalent after dividing both sides by zero.

Decision. False.

Justification. Division by zero is undefined.

Independent check. There is no legal reverse multiplication check.

Additional reasoning example 46. Inclusive inequality

Statement. The condition x≥3 allows x=3.

Decision. True.

Justification. The equality bar includes the endpoint.

Independent check. 3≥3 holds.

Additional reasoning example 47. Strict inequality

Statement. The condition x>3 allows x=3.

Decision. False.

Justification. Strict greater-than excludes equality.

Independent check. 3>3 is false.

Additional reasoning example 48. Change of sign in inequality

Statement. Multiplying both sides of a true inequality by a negative preserves the inequality direction.

Decision. False.

Justification. Multiplying by a negative reverses order; this is a later algebraic extension.

Independent check. 2<5 but −2>−5.

Additional reasoning example 49. Sample inference

Statement. Eight fictional pupils’ responses prove the preference of every resident in Punggol.

Decision. False.

Justification. A small selected sample may not represent a larger population.

Independent check. The claim is only about the eight respondents.

Additional reasoning example 50. Straight line proportional

Statement. Every straight-line graph of y against x represents direct proportion.

Decision. False.

Justification. A nonzero intercept prevents the relation y=kx.

Independent check. y=2x+3 gives y=3 at x=0.

Additional reasoning example 51. Zero coefficient

Statement. 5x+0=5x for every real x.

Decision. True.

Justification. Adding zero changes nothing.

Independent check. At any input both forms have identical value.

Additional reasoning example 52. Graph scale

Statement. Every grid square on any graph represents one numerical unit.

Decision. False.

Justification. A scale is chosen and labelled independently for each axis.

Independent check. One tick may represent five units.

Additional reasoning example 53. Signed multiplication

Statement. A product of two negative real numbers is positive.

Decision. True when both are strictly negative.

Justification. Multiplication sign rules yield a positive product.

Independent check. (−3)(−4)=12.

Additional reasoning example 54. Combine mean values

Statement. The combined mean of two groups always equals the midpoint of their reported means.

Decision. False.

Justification. Different group sizes give different weights.

Independent check. Three values mean10 and one value30 combine to mean15, not20.

Additional reasoning example 55. Fraction division

Statement. Dividing a positive amount by a proper fraction less than one must produce a smaller numerical quotient.

Decision. False.

Justification. Smaller divisor means more divisor-size groups fit.

Independent check. (3/4)÷(1/8)=6, greater than 3/4.

Additional reasoning example 56. Prime one

Statement. One is a prime number.

Decision. False.

Justification. A prime has exactly two distinct positive factors, while one has one.

Independent check. Positive factors of one are only {1}.

Additional reasoning example 57. Approximate pi

Statement. π equals exactly 3.14.

Decision. False.

Justification. 3.14 is a decimal approximation, not the full value of π.

Independent check. The circumference formula uses π unless the question specifies an approximation.

Additional reasoning example 58. Perfect square root

Statement. Every positive integer has an integer square root.

Decision. False.

Justification. Only perfect squares have whole-number principal square roots.

Independent check. √2 is not a whole number.

Additional reasoning example 59. Three-four-five triangle (extension)

Statement. A triangle with sides 3,4,5 is a right triangle.

Decision. True.

Justification. The largest side squared equals the sum of squares of other two.

Independent check. 3²+4²=25=5².

Additional reasoning example 60. Continuous variable

Statement. Connecting all categorical bar-chart points by a smooth line always gives a meaningful interpretation.

Decision. False.

Justification. Categories may not have meaningful intermediate positions.

Independent check. Reading and cycling as labels are not a continuous numerical scale.

Counterexamples are precise: they must satisfy the stated conditions

A valid counterexample must lie within the claim’s domain. To disprove ‘every odd positive integer greater than one is prime’, nine works because it is odd, greater than one and composite. A negative even number would not work because it does not meet the stated conditions. This distinction teaches pupils to read a universal statement carefully before trying to refute it.

Likewise, to disprove ‘every rectangle is a square’, a 6-by-4 rectangle works because it has all four required right angles and unequal adjacent sides. A slanted parallelogram would not be a valid counterexample because it is not necessarily a rectangle in the first place. The wrong counterexample can demonstrate a shape misunderstanding, even when the conclusion of the student’s argument happens to be true.

A useful teaching challenge asks a pupil to propose a false claim and then construct one valid counterexample. For example: ‘Every number ending in five is prime.’ The number fifteen disproves it because it ends in five and equals three times five. The student should explain both why the proposed example belongs to the class and why it violates the conclusion.

A clear mathematical explanation has three parts

Part	What to include	Example
Given conditions	State the values, units, diagram facts or permitted inputs	‘The figure is a rectangle with sides 6 cm and 4 cm’
Relevant mathematical property	Name the rule that genuinely applies	‘Area equals length times width’
Consequence and check	Calculate, compare or exhibit a counterexample	‘Area=24 cm², so a claimed area of 20 is false’

Students do not need to write these three headings in every answer. A single clear line can contain all of them: ‘Since the triangle’s interior angles total 180°, the third angle is 180−45−65=70°.’ The structure is useful when a child has been writing numerical steps without showing why a particular operation belongs. A teacher can ask the student to speak the missing property before adding unnecessary lines of working.

Another example concerns a discounted fictional price. ‘The $80 sale price is 80% of the original because a 20% discount was applied; therefore original=$80÷0.8=$100.’ This sentence identifies the reference whole, the relationship and the calculation. If the pupil instead writes 80×1.2=96, the error arises in the model of the percentage base, not in multiplication fluency.

Using an error log to separate false assumptions from slips

A student may use a true formula on the wrong figure. The circumference of a circle is 2πr, but the complete perimeter of a semicircle includes its exposed straight diameter as well. The formula itself is valid; its application to the wrong boundary is not. A useful error log records ‘arc length mistaken for total perimeter’ instead of the vague label ‘forgot formula’.

Another student may choose the correct equation 4x+7=55 and then misdivide forty-eight by four. The representation is right and the arithmetic is wrong. Their next practice should emphasise division accuracy and inverse multiplication, whereas the first pupil needs a diagram and boundary-tracing exercise. Effective small-group teaching does not prescribe the same corrections to every wrong final number.

Parents can keep a short record of the first incorrect line, the corrected reason and one new question attempted after a delay. The delayed problem should change the numbers, wording or shape. Otherwise the student may only have memorised the worked correction. A concept has been meaningfully repaired when a child can use it on an unfamiliar case without being told which rule to apply.

A practical ten-minute reasoning diagnostic for parents

Begin with three statements: ‘Every square is a rectangle’, ‘Every rectangle is a square’, and ‘Every odd number is prime’. Ask the pupil to classify each and justify it. The correct responses are true, false and false. For the two false statements, require a specific valid counterexample: a non-square rectangle, and nine or fifteen for the prime claim. For the true statement, require the general defining property of a square having four right angles.

Then give 3(x+2)=3x+6 and ask why it holds for any number x. A useful explanation is that each of three copies of the complete group contains x and two, producing three x’s and six units. A child who merely tries x=1 and says ‘it worked’ has supporting examples but not yet the general reason. Ask what would change if x were negative or a fraction.

Finally show a hypothetical bar chart with labelled values ninety-five and one hundred but a baseline starting at ninety. Ask whether one hundred is twice ninety-five. The student should use the actual numbers, not the apparent bar heights, and explain the limit of the visual. This trains evidence-based interpretation across data as well as algebra.

A six-week optional reasoning route

Week	Concept	Independent check
1	True versus false numerical statements	Uses substitution or exact arithmetic appropriately
2	Universal words and valid counterexamples	Chooses examples within the stated domain
3	Explain a general property	Uses equal groups, distributive law or shape definitions
4	Check equations, ratios, rates and units	Reconstructs original condition
5	Evaluate graphs, diagrams and assumptions	Rejects unsupported claims with reasons
6	Mixed unfamiliar statements after delayed practice	Selects a justification method independently

The six weeks describe a possible learning progression, not a fixed 2028 school examination schedule. The G1, G2 and G3 subject levels vary in the mathematical forms they expect, and formal proofs may come later. A pupil still unsure of whole-number comparisons should begin with simpler statements; a pupil secure in basic reasoning may explore algebraic identities or carefully chosen geometry arguments.

What a three-pupil small-group discussion can contribute

Imagine three learners asked whether x+x=x² is always true. One agrees because both expressions contain two x’s; a second rejects it by testing x=3; a third can explain generally that addition of two equal terms gives 2x while multiplication gives x². Each response reveals a different level of understanding. A small-group tutor can make those differences visible through discussion and assign different next questions.

A purposeful 90-minute session can start with a short retrieval, move to two shared claims and ask each pupil to defend a position. The tutor should model respectful mathematical disagreement: challenge the calculation or assumption, not the person. After discussion, each student should complete a new claim independently and explain why their decision is supported by sufficient evidence.

Parents selecting Punggol Mathematics tuition can ask how the tutor distinguishes a correct answer by guesswork from a correct answer with understanding, and how corrections are tested after a delay. Class size is an opportunity for more feedback, not proof of guaranteed grades. Sound mathematical confidence comes from reasons a learner can reproduce without the tutor present.

Frequently asked questions

Can one counterexample disprove a claim about every case?

Yes, provided the example meets all the stated conditions and violates the conclusion. One odd composite number such as nine disproves the statement ‘every odd integer greater than one is prime’.

Can three examples prove a universal mathematical statement?

Not by themselves. A general explanation must establish why the property holds across the whole permitted domain. Checking several numerical values may suggest the pattern and detect errors but does not replace the needed reasoning.

Is checking a solution the same as solving an equation?

No. Substitution can verify that a proposed value satisfies the original equation. Solving is the process of finding all values under its stated conditions. A good solution often includes both.

Does every Primary-to-Secondary learner need formal proof notation immediately?

No. Clear reasons, valid counterexamples and independent checks can be taught with familiar numbers and shapes. Formal notation and advanced proof structures should follow the relevant school Mathematics syllabus and readiness.

Can a pupil have a right number but a wrong explanation?

Yes. A guessed or memorised value may work in one question but fail after the context changes. Ask which property connects the given quantities to the answer, and test it in a changed example.

Why teach counterexamples to students who struggle with arithmetic?

Very simple counterexamples can strengthen numerical understanding: nine is odd but composite, and a 6-by-4 rectangle is not a square. Use appropriate difficulty so the reasoning clarifies rather than overwhelms.

Official guidance and related eduKate learning

• MOE G1 Mathematics syllabus — mathematical processes and appropriate numerical scope.
• MOE G2/G3 Mathematics syllabuses — reasoning and subject-level examples.
• Word-problem decoding, models and units — selecting a valid relationship.
• Linear equations and answer checks — equivalence and substitution.
• Geometry and shape properties — choosing justified theorems.
• Punggol Secondary 1 Mathematics 2028 learning hub — the full transition route.
• eduKate Secondary 1 Mathematics tutorial standard — concept explanation and independent transfer.

A justified answer is one whose reasoning survives a change of numbers or diagram. A learner who can state a claim precisely, apply a valid general rule, test candidates and construct counterexamples when needed will be better prepared for unfamiliar Mathematics than one who only remembers the final number.

Final investigation: why a correct example does not prove a rule

Consider the statement ‘for every real number x, 3x equals 5x.’ When x=0, both expressions equal zero. A student who tests only this one value could conclude that the statement is true. But x=1 produces three and five, so the universal claim is false. The example at zero was a special case where the two formulas happened to agree, not a general algebraic identity.

A parallel mistake occurs in geometry. A pupil may draw a rectangle that happens to be a square and observe that its diagonals are lines of symmetry. From that one example, it would be invalid to claim that every rectangle has diagonal symmetry. A non-square rectangle, such as a six-by-four one, disproves the claim. A square inherits rectangle properties and has additional special properties; one unusually symmetric member of a class need not represent the whole class.

Parents can ask the child to finish three short sentences: ‘The claim says …’, ‘My example satisfies the conditions because …’, and ‘It proves or disproves the claim because …’. This language makes the reasoning visible. A child who correctly chooses nine to disprove an odd-prime claim but cannot explain that nine is odd and composite has not completed the argument. The example must both belong to the stated category and violate the claimed property.

For a true universal claim, look for the mathematical structure that applies to every permitted case. The identity 2(x+3)=2x+6 holds because doubling the complete group x+3 creates two x units and six constant units, whatever real value x has. The distributive law gives the general explanation; testing a few values afterward provides additional checks. This distinction between a proof and an example prepares pupils for more advanced reasoning while remaining accessible with familiar algebra.

A final two-minute exit check

Give the learner two statements: ‘Every square is a rectangle’ and ‘Every rectangle is a square.’ The first is true because a square has all four right angles required of rectangles. The second is false, and a 6-by-4 rectangle is a counterexample. The student should provide the definition for the true claim and the counterexample for the false one rather than use the same kind of evidence for both.

Then show x²=16 and ask whether x=4 is a correct solution. It is, but it is not the only real solution: x=−4 also works. Checking one candidate is different from finding the complete solution set. A pupil who can make these distinctions without a prompt has learned a valuable reasoning habit. That matters more than memorising the answers to a long sequence of true-or-false questions.