SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 32
Mathematics does not become certain because many examples worked. Examples can suggest a pattern. A counterexample can destroy a false universal claim. Proof explains why a claim must hold under stated conditions.
This guide develops conjectures, testing, counterexamples, necessary assumptions, deduction, direct proof, contradiction, parity arguments, algebraic generalisation, geometric reasoning and the difference between evidence and certainty. It extends the earlier Mathematical Communication, Working and Justification guide by focusing on how conviction is earned.
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1. A conjecture is a claim suggested by evidence
After checking 2+4=6, 6+8=14 and 10+12=22, a learner may conjecture that the sum of two even numbers is even.
The examples suggest the claim. They do not yet explain why it is always true.
2. Examples are useful for discovery
Trying cases can reveal patterns, exceptions and useful representations.
Examples help us ask better questions.
But discovery and proof are different jobs
A pattern seen repeatedly may still fail later.
3. One counterexample can disprove a universal claim
Claim: “Every prime number is odd.”
Counterexample: 2 is prime and even.
Therefore the universal claim is false.
Efficiency
To disprove “all”, one valid exception is enough.
4. A failed example does not automatically kill an existential claim
Claim: “There exists a prime number greater than 100.”
Testing 102 and finding it composite does not disprove the claim. The claim asks whether at least one example exists, not whether every example works.
5. Universal and existential statements require different evidence
“All squares are non-negative” is universal.
“Some integers are negative” is existential.
Universal claims require reasoning that covers every allowed case. Existential claims can be established by one valid example.
6. Definitions are proof tools
An even integer can be written as 2k for some integer k.
An odd integer can be written as 2k+1.
These definitions turn verbal claims into algebra that can be manipulated.
7. Prove the sum of two even integers is even
Let the even integers be 2a and 2b.
Their sum is 2a+2b = 2(a+b).
Because a+b is an integer, the sum has the form 2×integer.
Therefore the sum is even.
8. Prove the sum of two odd integers is even
Let the odd integers be 2a+1 and 2b+1.
Sum = 2a+1+2b+1 = 2(a+b+1).
Therefore the sum is even.
9. Odd plus even is odd
Let odd = 2a+1 and even = 2b.
Sum = 2a+1+2b = 2(a+b)+1.
This has the form of an odd integer.
10. Algebra can prove number patterns for every case
Consider three consecutive integers n, n+1 and n+2.
Their sum is 3n+3 = 3(n+1).
Therefore the sum of any three consecutive integers is divisible by 3.
This replaces testing many triples with one general argument.
11. A proof should expose the condition that makes the claim true
Claim: “Opposite angles formed by two intersecting lines are equal.”
The argument uses the fact that adjacent angles on a straight line sum to 180°.
The equality is not justified by the diagram looking symmetric; it follows from the angle constraints.
12. Diagrams support reasoning but do not replace it
A diagram can suggest that two lengths are equal. Unless equality is given or proved, measuring the drawing is not a proof.
Useful habit
Label what is given differently from what merely appears true.
13. Assumptions determine whether an argument is valid
If a geometry proof uses alternate angles, parallel lines must be established.
If a probability argument assumes equally likely outcomes, that assumption must fit the model.
Proof is conditional on its premises.
14. Necessary conditions are not always sufficient
Being divisible by 2 is necessary for an integer to be divisible by 6, but it is not sufficient.
8 is divisible by 2 but not by 6.
This distinction helps prevent overclaiming from partial conditions.
15. Converse statements can be false even when the original statement is true
True statement: if an integer is divisible by 6, then it is divisible by 3.
Converse: if an integer is divisible by 3, then it is divisible by 6.
The converse is false; 9 is a counterexample.
16. “If” and “only if” carry different logical loads
“If A, then B” means A is sufficient for B.
“A only if B” means B is necessary for A.
Formal logical notation may come later, but precise language already matters.
17. Contradiction can expose impossibility
Suppose an integer is claimed to be both even and odd.
Even means n=2a. Odd means n=2b+1.
Then 2a=2b+1, so 2(a−b)=1, impossible because the left side is even.
Therefore no integer is both even and odd.
18. Counterexamples should satisfy the original conditions
To disprove “all rectangles are squares”, a 3-by-5 rectangle works because it is genuinely a rectangle and not a square.
A triangle does not count as a counterexample because it fails the starting condition.
19. Extreme and boundary cases are good places to test conjectures
If a rule claims to work for all positive integers, test 1. If a geometry claim includes right angles, test 90°. If a percentage claim includes 0% or 100%, inspect those boundaries.
Counterexamples often hide at edges.
20. Changed-case reasoning tests whether the explanation is structural
If the sum of three consecutive integers is divisible by 3, ask what happens for four consecutive integers.
n+(n+1)+(n+2)+(n+3)=4n+6=2(2n+3), always even but not always divisible by 4.
Changing one condition reveals which conclusion depended on which structure.
21. Pattern recognition needs a stop condition
Sequence 2,4,8,16 suggests doubling, but finitely many terms can fit many different rules.
In school sequence questions, contextual conventions usually indicate the intended pattern. In proof, a few terms never establish a unique universal law.
22. Proof can be computational without being vague
To show 37²−35² is divisible by 4, factor:
37²−35²=(37−35)(37+35)=2×72=144.
The factorisation exposes structure more clearly than multiplying both squares separately.
23. Generalisation often begins by replacing examples with variables
Instead of checking 5+6=11, 8+9=17 and 20+21=41, represent consecutive integers as n and n+1.
Their sum is 2n+1, which is always odd.
The variable carries all cases at once.
24. Common reasoning errors
| Error | Why it fails | Repair prompt |
|---|---|---|
| “I checked ten examples, so it is proved” | Unseen cases remain | Can a variable cover all allowed cases? |
| Counterexample violates the hypothesis | Not a valid test | Does the example satisfy the original conditions? |
| Converse assumed automatically | Implication reversed | Can you find a case where B holds but A fails? |
| Diagram appearance used as proof | Visual scale not guaranteed | Which stated property forces the conclusion? |
| Necessary condition treated as sufficient | Partial requirement overclaimed | Is the condition enough by itself? |
25. Practice laboratory
- Give a counterexample to “all multiples of 3 are even”.
- State whether checking 100 examples proves a universal claim.
- Prove the sum of two even integers is even.
- Prove odd + even is odd.
- Prove the sum of two consecutive integers is odd.
- Prove the sum of three consecutive integers is divisible by 3.
- Disprove “every square number is even”.
- Give a valid counterexample to “all quadrilaterals are rectangles”.
- State the converse of “if a number is divisible by 10, then it is divisible by 5”.
- Is that converse true? Give evidence.
- Explain why a measured diagram does not prove two sides are equal.
- State one necessary condition for a number to be divisible by 6.
- State whether that condition alone is sufficient.
- Explain the difference between an example and a proof.
26. Explained answers
1. 3, 9 or 15.
2. No. It provides evidence, not coverage of every allowed case.
3. Let numbers be 2a and 2b. Sum=2(a+b), so even.
4. (2a+1)+2b=2(a+b)+1, so odd.
5. n+(n+1)=2n+1, so odd.
6. n+(n+1)+(n+2)=3(n+1), so divisible by 3.
7. 9 is a square number and odd.
8. Example: a kite that is not a rectangle.
9. If a number is divisible by 5, then it is divisible by 10.
10. False; 15 is divisible by 5 but not 10.
11. The drawing may not be to scale; equality needs a stated or proved property.
12. Divisibility by 2 or divisibility by 3.
13. No. For example, 8 is divisible by 2 but not 6.
14. An example shows one case; a proof explains why every case covered by the claim must work.
27. Complete mixed problem
Conjecture: the square of every odd integer is odd.
Test examples: 3²=9, 5²=25, 7²=49. The pattern survives.
Now prove it. Let an odd integer be 2k+1.
(2k+1)²=4k²+4k+1=2(2k²+2k)+1.
The result has the form 2m+1 for integer m=2k²+2k.
Therefore the square of every odd integer is odd.
The examples discovered the conjecture; algebra delivered the conviction.
28. Teaching conviction as a ladder
Use four questions in order:
- What do you notice?
- What examples support it?
- What example would destroy it?
- Why must it hold for every allowed case?
This ladder separates noticing, testing, falsifying and proving.
Changed-case test
After a proof, alter one premise and ask whether the conclusion survives. This reveals which assumption was doing the real work.
29. Questions students often ask
How many examples are enough for a proof?
No finite number of examples proves a universal claim unless the domain itself is finite and every case has been exhaustively checked.
Can one example prove something?
Yes, if the claim is existential: one valid example proves that at least one such case exists.
Why are counterexamples powerful?
Universal claims promise there are no exceptions, so one valid exception is decisive.
Is proof always algebra?
No. Proof can use geometry, cases, contradiction, invariants, counting or other valid reasoning structures.
30. Return path and sources
Reasoning and proof connect every topic in the Secondary 1 estate. Revisit Mathematical Communication, Working and Justification for presentation and Number Patterns, Sequences and nth-Term Generalisation for generalisation.
Official curriculum reference: MOE Secondary Syllabus Directory. Formal proof expectations vary by subject level and school, but reasoning, generalisation and justification remain transferable mathematical processes.
Editorial approach: Wintour House V1.0 · Rainbolt/CivDJ gap-tested · eduKate Publishing. Observe → conjecture → test → hunt for counterexamples → expose assumptions → generalise → prove → change one condition and test whether the argument survives.
