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Primary Mathematics: Conjecturing, Generalising and Testing a Rule | Worked Learning Guide

A conjecture is a mathematical idea brave enough to be tested. Examples can suggest it, counterexamples can break it, and a convincing general argument can explain why it must continue.

Primary learners make conjectures constantly: “the answer is always even”, “the square has the smallest perimeter”, “doubling both ratio terms keeps the comparison the same”, “every pattern that begins 2,4,6 must continue 8,10,12”. Some of these statements are true under suitable conditions. Some are false. Some are too vague to test.

The important capability is not guessing the teacher’s intended pattern. It is learning how to turn a notice into a testable statement, choose examples that matter, search for failure rather than only success, identify the conditions that control the result and then state the strongest generalisation the evidence can carry.

This guide sits deliberately beside the existing BTT articles on using examples and counterexamples, logic and truth conditions, and patterns and early algebra. Its distinct job is a Primary worked laboratory for moving from observed cases to a controlled general statement.

Build a conjecture · Choose examples · Search for counterexamples · State conditions · Generalise · 24 questions · Worked solutions

1. Turn a notice into a statement that could be wrong

Suppose the first five results of a process are 5, 9, 13, 17, 21. “I notice fours” is not yet a precise conjecture. “Each displayed term is four more than the previous term” is an observation. “The process will continue by adding four every time” is a conjecture about the unseen future.

A useful conjecture says clearly:

• which objects or numbers it concerns,
• what relationship is claimed,
• under what conditions it is supposed to hold.

Worked example: parity

Try 3+5=8, 7+9=16 and 11+1=12. A natural conjecture is:

The sum of any two odd whole numbers is even.

This is stronger than “the answers I tried were even”. The word any makes a general claim and creates a clear challenge: either explain why every allowed pair must work or find one allowed pair that fails.

2. Supporting examples are useful, but quantity alone does not prove a rule

Testing 100 pairs of odd numbers without finding a failure increases confidence that the conjecture is plausible. It does not by itself explain why the 101st, millionth or an arbitrarily large pair must also work.

Examples have several important jobs:

• discovering a possible pattern,
• checking whether a proposed rule fits known cases,
• exposing hidden conditions,
• finding counterexamples,
• making a later general explanation easier to see.

The mistake is not using examples. The mistake is asking examples to do the work of a general argument when the claim contains words such as all, any, always or never.

Worked example: three consecutive whole numbers

Take 4,5,6. Their sum is 15, divisible by 3. Try 10,11,12: sum 33. Try 21,22,23: sum 66.

Conjecture: the sum of any three consecutive whole numbers is divisible by 3.

The examples suggest it. A general explanation comes later: if the middle number is n, the three numbers are n−1, n, n+1 and their total is 3n.

3. Choose examples that stress the claim

Do not test only friendly examples. If a conjecture mentions whole numbers, include zero when allowed. If it mentions fractions, try proper and improper fractions where appropriate. If it concerns rectangles, try long thin and nearly square rectangles. If it concerns averages, try equal values and widely spread values.

Worked example: larger area means larger perimeter?

A 3×5 rectangle has area 15 and perimeter 16. A 1×12 rectangle has smaller area 12 but larger perimeter 26.

This immediately destroys the conjecture “a rectangle with larger area always has larger perimeter”. A long thin shape is a stress case because perimeter reacts differently from area.

Worked example: larger denominator means smaller fraction?

Compare 3/4 and 3/8: the larger denominator gives the smaller fraction when numerator and whole are fixed. But compare 7/8 and 3/4: denominator eight is larger, yet 7/8 is larger than 3/4.

The original statement was missing a condition. A refined conjecture is: for positive fractions with the same numerator and the same whole, the larger denominator gives the smaller fraction.

4. One valid counterexample defeats an “always” claim

A counterexample must satisfy the stated conditions and violate the stated conclusion.

Claim: “Every prime number is odd.” The number 2 is prime but even. Therefore the claim is false.

The counterexample does not mean “most primes are even”. It proves only that the universal claim, as written, cannot be correct.

Worked example: greedy choice

Coin values are 1,3 and4. A learner claims that taking the largest available coin first always uses the fewest coins.

For target 6, greedy gives 4+1+1, using three coins. But 3+3 uses two. One valid target defeats the word always. The Optimisation guide develops this example further.

5. A failed conjecture can be repaired

When a counterexample appears, do not discard the investigation immediately. Ask what makes the counterexample different.

“Every prime is odd” fails at 2. A repaired statement is: every prime number greater than 2 is odd.

Why? Every even number greater than 2 has 2 as a factor as well as 1 and itself, so it cannot be prime.

Worked example: square perimeter

Claim: “Among rectangles with a fixed area, the square always has the smallest perimeter.” This needs conditions. For whole-number rectangles of area 24, the possibilities 1×24,2×12,3×8,4×6 do not include a square because 24 is not a square number. The more accurate claim is that factor pairs closer together tend to produce smaller perimeter, and if a square with the required area is allowed, it is the most compact rectangle.

For a Primary enrichment problem restricted to whole-number sides, list the factor pairs rather than importing a continuous optimisation theorem that the task does not require.

6. Conditions are part of the theorem

Mathematical statements become safer when their domain is explicit.

Too loose	Controlled statement
Adding makes a number bigger.	Adding a positive number makes a real quantity larger.
A larger denominator means a smaller fraction.	With the same positive numerator and whole, a larger positive denominator gives a smaller fraction.
Multiplying makes bigger.	Multiplying a positive number by a factor greater than 1 makes it larger.
Every graph that rises has a positive relationship.	Within the displayed interval and scales, the plotted dependent value increases as the horizontal variable increases.

The purpose is not to bury Primary Mathematics under formal language. It is to teach that a rule owns the conditions under which it was established.

7. “If” does not automatically mean “if and only if”

Suppose: If a whole number is divisible by 4, then it is even. This is true. But the reverse statement, “if a number is even, then it is divisible by 4,” is false because 6 is even but not divisible by 4.

A correct one-way rule should not be reversed without testing.

Worked example: square and rectangle

If a quadrilateral is a square, then it is a rectangle. The reverse is false: a 3×5 rectangle is not a square.

This helps learners distinguish category inclusion from equality of categories.

8. Classify statements as always, sometimes or never

This routine is stronger than asking only “true or false” because it encourages counterexamples and conditions.

Worked example: sum of two numbers

• Odd + odd is even: always.
• Odd + even is odd: always.
• The sum of two whole numbers is even: sometimes.
• The sum of two consecutive whole numbers is even: never; one is odd and the other even, so the sum is odd.

Worked example: rectangle with equal sides

A rectangle has all four sides equal: sometimes. Such a rectangle is a square. Many rectangles do not have all four sides equal.

9. Generalisation replaces many cases with one relationship

Suppose a row of joined squares uses 4,7,10,13 matchsticks for Figures 1–4. The recursive rule is add three. A direct generalisation is 3n+1.

The symbols are useful only if their parts are explained. Each of n squares contributes three new matchsticks after a single starting side is supplied. Therefore total = 3n + 1.

Generalisation is not “using a letter”. It is expressing what remains structurally true as the case changes.

Worked example: odd numbers

Every odd whole number can be represented as 2n+1 for some whole number n. Add two odd numbers:

(2a+1)+(2b+1)=2a+2b+2=2(a+b+1).

The result is twice a whole number, so it is even. The expression explains all pairs at once.

10. Generalise from geometry by tracking what one change adds

A row of n unit squares has area n and perimeter 2n+2.

Why perimeter 2n+2? The top contributes n, the bottom n, and the two end sides contribute 1 each. Total = n+n+1+1.

This explanation is more stable than extrapolating the sequence 4,6,8,10. It reveals the geometric source of every term.

Worked example: border of a square

A filled n×n square has n² cells. Growing to (n+1)×(n+1) adds a new row of n+1 and a new column of n, because the shared corner has already been counted. New cells = 2n+1.

Thus consecutive square numbers differ by odd numbers: (n+1)²−n²=2n+1.

11. Generalise from arithmetic transformations

Take any two numbers with fixed sum S. Increase one by k and decrease the other by k. New sum = (A+k)+(B−k)=A+B=S.

This generalises compensation. The specific transformation 37+48 → 35+50 is one case with k=−2 on the first addend and +2 on the second.

Worked example: fixed difference

Add the same k to both quantities. New difference = (A+k)−(B+k)=A−B. Equal additions preserve a difference.

12. Generalise from ratios

If A:B=3:5, then A=3u and B=5u for some common unit u. Multiplying both actual quantities by the same positive factor changes u but preserves the 3:5 relationship.

This is why 3:5,6:10 and12:20 are equivalent ratios. The numbers differ, but the multiplicative comparison does not.

13. Boundary cases are powerful tests

A rule for whole-number n should be checked at the smallest allowed n when practical.

Proposed formula for the perimeter of n joined unit squares in a row: 2n+2. At n=1 it gives 4, correct. At n=2 it gives 6, also correct.

A formula that fails at n=1 cannot be repaired by working on larger cases first. Boundary tests find structural mistakes cheaply.

Worked example: zero

Claim: n(n+1) is even for every whole number n. At n=0 the product is zero, which is even. The boundary case does not prove the claim, but it verifies that zero was not accidentally excluded from the pattern.

14. Reverse a conjecture and test it separately

Original: if a number is a multiple of 10, its final digit is zero. Reverse: if a whole number’s final digit is zero, it is a multiple of 10. In ordinary decimal whole-number notation, both directions are true.

Original: if a number is a multiple of 6, it is even. Reverse: if a number is even, it is a multiple of 6. False; 8 is a counterexample.

The reverse statement is a new conjecture. It does not inherit truth from the original.

15. Distinguish “possible”, “always” and “guaranteed”

If three numbers have average 10, it is possible that all three are 10. It is not guaranteed; 5,10,15 also average 10.

If five consecutive whole numbers are listed, the middle one always equals their average. This can be explained by symmetric pairs around the middle: two numbers below and two above cancel their deviations.

Language is part of mathematical precision. “Can” and “must” describe different logical strengths.

16. Common conjecture errors

• Treating several supporting examples as proof of an “always” claim.
• Searching only friendly examples.
• Using a counterexample that violates the original conditions.
• Repairing a false conjecture by adding vague exceptions rather than identifying the controlling condition.
• Reversing a true implication without testing.
• Generalising from a picture whose construction rule is unknown.
• Using algebraic letters without explaining what they represent.
• Confusing “sometimes true” with “not useful”.

17. Practice: 24 original questions

Questions 1–8: Form and test conjectures

1. Try three pairs of odd whole numbers. State a conjecture about their sums.
2. Try three sets of three consecutive whole numbers. Conjecture a divisibility property of each sum.
3. A learner says every prime number is odd. Find a counterexample and repair the statement.
4. A learner says a rectangle with larger area always has larger perimeter. Find a counterexample using whole-number sides.
5. Test the statement “for fractions with the same positive numerator and whole, a larger denominator gives a smaller fraction” using three examples.
6. Test the reverse of “if a number is divisible by 4, then it is even”.
7. Classify “a rectangle has four equal sides” as always, sometimes or never.
8. Classify “the sum of two consecutive whole numbers is odd” as always, sometimes or never.

Questions 9–16: Generalise structure

9. Joined-square matchstick counts are 4,7,10,13,… Explain and write a direct rule for Figure n.
10. A row of n unit squares has perimeter 2n+2. Explain every term geometrically.
11. Show that the difference between (n+1)² and n² is 2n+1 by describing the added border.
12. Explain generally why odd+odd is even.
13. Explain generally why n(n+1) is even for whole-number n.
14. If A+B=S, increase A by k and decrease B by k. What happens to the sum?
15. If A−B=D, add k to both A and B. What happens to the difference?
16. A:B=3:5. Express both quantities using one common unit u and explain equivalent ratios.

Questions 17–24: Conditions, reversals and limits

17. A learner claims “multiplying makes a number bigger.” Give a counterexample and state a safer condition for positive numbers.
18. A learner says “the denominator is larger, so the fraction is smaller.” Give a pair of fractions showing why the numerator condition matters.
19. If a whole number is a multiple of 10, its last digit is 0. Test the reverse statement.
20. If a whole number is a multiple of 6, it is even. Test the reverse statement.
21. Three numbers have average 10. Is it possible, necessary or impossible that all are 10? Give another valid set.
22. Five consecutive whole numbers have middle number m. Explain why their average is m.
23. Sequence 2,4,6 is displayed with no generation rule. Explain why “the next term must be 8” is too strong.
24. Create one false “always” claim from Primary Mathematics, give a valid counterexample, and rewrite the claim with a condition that makes it true.

18. Worked solutions

Solutions 1–8

1. Conjecture: the sum of any two odd whole numbers is even. Example cases may suggest it; a general parity argument is needed to establish it for all cases.

2. Conjecture: the sum of three consecutive whole numbers is divisible by 3. Examples 4+5+6=15 and10+11+12=33 support it.

3. Counterexample 2. Repaired statement: every prime number greater than 2 is odd.

4. A 1×12 rectangle has area 12 and perimeter 26. A 3×5 rectangle has larger area 15 but smaller perimeter 16.

5. Examples: 3/4>3/5>3/8; 2/3>2/7; 5/6>5/9. The same positive numerator and same whole are essential conditions.

6. Reverse is false. Number 6 is even but not divisible by 4.

7. Sometimes. Squares are rectangles with four equal sides; non-square rectangles are not.

8. Always. Consecutive whole numbers have opposite parity, and odd+even is odd.

Solutions 9–16

9. Each new square shares one old side and contributes three new sticks. Begin with one initial side, then add three per square: 3n+1.

10. Top side contributes n, bottom n, and two ends contribute 1 each: n+n+1+1=2n+2.

11. Growing n×n to (n+1)×(n+1) adds one new row of n+1 and one additional column of n because the corner is already in the row. Total added=2n+1.

12. Write odd numbers as 2a+1 and2b+1. Their sum is 2(a+b+1), twice a whole number, hence even.

13. Consecutive whole numbers n and n+1 have opposite parity, so one is even. Their product therefore contains a factor 2 and is even.

14. The sum stays S because (A+k)+(B−k)=A+B.

15. The difference stays D because (A+k)−(B+k)=A−B.

16. Let A=3u and B=5u. Replacing u by ku multiplies both quantities by k and leaves the unit-count relationship 3:5 unchanged.

Solutions 17–24

17. Example: 8×1/2=4, which is smaller. Safer statement: multiplying a positive number by a factor greater than 1 makes it larger.

18. Example: 7/8>3/4 even though 8>4. A denominator comparison alone is not enough when numerators differ.

19. True for ordinary decimal whole numbers: a last digit 0 means the number is divisible by 10.

20. False. Number 8 is even but not a multiple of 6.

21. Possible, not necessary. 10,10,10 works; so does 5,10,15.

22. The five numbers are m−2,m−1,m,m+1,m+2. Their total is 5m because −2−1+1+2=0. Divide by 5 to get average m.

23. Many rules can agree with the first three terms. “Add 2” is one plausible continuation, but without a stated construction or rule the displayed values do not logically force 8.

24. Example false claim: “every even number is divisible by 4.” Counterexample: 6. Repaired claim: “every whole number divisible by 4 is even.”

19. Conjecture laboratory: from examples to a controlled statement

Investigate the sum of the first n odd numbers:

1=1, 1+3=4, 1+3+5=9, 1+3+5+7=16.

A natural conjecture is that the sum of the first n odd numbers is n².

Now represent the numbers as successive L-shaped borders. Begin with one unit square. Add three cells to make a 2×2 square. Add five cells to make a 3×3 square. Add seven to make a 4×4 square. At stage n, the new border contains 2n−1 cells and grows the previous (n−1)×(n−1) square into an n×n square.

The picture is not merely an illustration of the first four sums. It reveals a construction that can continue for any positive whole n. That is the movement from notice to conjecture to general explanation.

20. Parent and tutor guide

When a learner proposes a pattern, respond with three questions: “What exactly are you claiming?”, “What case would be hardest for your claim?”, and “What would convince someone who tried different numbers?” This keeps curiosity while introducing mathematical responsibility.

Do not turn every conjecture into formal algebra immediately. Younger learners can justify generally through pairing, arrays, bar models, factor structure, exhaustive cases or a construction that clearly repeats. Introduce a variable when it compresses an already understood relationship.

If a conjecture fails, treat the counterexample as information. Ask which condition was missing and whether a nearby true statement can be recovered. This develops resilience without pretending incorrect claims are “almost right” when their logic has genuinely failed.

The first weak link

A learner may struggle at different stages: noticing a pattern, stating it precisely, choosing test cases, recognising a valid counterexample, identifying conditions, or moving from examples to a general reason. Diagnose the earliest weak stage.

Continue to Explaining, Convincing and Proving when the conjecture is stable but the learner cannot justify why it must hold. Return to Exploring and Noticing when the learner cannot yet generate or organise useful cases.

21. Delayed return

Three days later, present four claims without indicating whether they are true: one parity claim, one fraction claim, one geometry claim and one pattern claim. Ask the learner to classify each as likely true or false, choose stress cases, search for counterexamples, and state the conditions before attempting a proof.

22. Mastery receipt

• I turn observations into precise testable conjectures.
• I know supporting examples do not prove every universal claim.
• I choose examples that stress the conditions.
• I can use one valid counterexample to refute an “always” statement.
• I repair false conjectures by identifying missing conditions.
• I test reverse statements separately.
• I generalise from structure rather than from notation alone.
• I distinguish possible, sometimes, always, never and guaranteed.

Sources and scope

University of Cambridge NRICH — Thinking Mathematically, Primary Students identifies conjecturing and generalising as a major strand of primary mathematical thinking. NRICH — Developing Mathematical Thinking, Primary Teachers provides teacher-facing tasks and guidance across these reasoning processes.

For official Singapore subject documents, use the MOE Primary curriculum and syllabus page alongside the learner’s current school programme. This article is an enrichment and reasoning guide, not an official examination requirement.

Continue the Working Mathematically collection

• Exploring, Noticing and Finding Structure
• Visualising, Representing and Seeing the Same Mathematics
• Explaining, Convincing and Proving
• BTT Primary Mathematics Learning Hub

The Quiet Return

Mathematics grows by making claims that can survive challenge. Notice something, state it clearly, try to break it, repair the conditions, and then explain why the strongest remaining statement deserves to be believed.