At a glance. One lesson of about 60 minutes. It assumes factors and primes from Year 7, substitution, and expanding a squared bracket. Students build counterexamples that meet the conditions, see a pattern fail at its sixth term, and rescue a divisibility claim with a condition.
Calculator: GDC optional. The lesson is done by hand. The optional section shows GDC users a table of values for hunting a failing case, and says that a table is evidence and not a proof.
What students will learn
Know
- A conjecture is not yet proved; a proved conjecture is a theorem.
- Inductive reasoning gathers evidence; deductive reasoning proves.
- One counterexample disproves a general claim, and it must meet the claim's conditions.
Understand
- A pattern can fit a long run of cases and still fail, so evidence never becomes proof.
- A false claim can be repaired by a condition, and finding the condition is generalizing.
- A counterexample is a precise object: it meets every condition and breaks the conclusion.
Do
- Find the smallest counterexample to a claim and state it as a sentence with the numbers.
- Explain why a long run of successes is not a proof.
- Adjust a claim with a condition and test the new claim.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: testing values, factors and primes, and the formula for the regions count (questions 1 to 8).
- Criterion B. Investigating patterns: both figures. Students gather cases in a table, state a conjecture, and try to break it before they believe it (the circle regions and the divisibility claim).
- Criterion C. Communicating: writing a counterexample as a sentence that shows it meets the conditions and breaks the conclusion (worked examples 1 and 3, faded example).
- Criterion D. Applying mathematics in real-life contexts: judging how much evidence a claim needs, from the tile plate in the hook and the what-if.
The one thing to do with the figure
Let the plate settle the prediction. Ask for a number for 6 points on mini-whiteboards before anyone moves the slider. Most students write 32. Then step the points up one at a time as a class, reading the lines and crossings aloud, and stop at 5 before the final step: ask what everyone now believes.
In the second figure, ask students to find every pair a, b with a counterexample and to say what the pairs share. They find that a and b share a factor, which is the condition that turns the claim into a theorem.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 6 min | Hook: predict the number of regions for 6 points. Collect numbers and reasons, do not confirm. |
| 12 min | First figure; key idea; settle the hook; the warning card. |
| 10 min | Second figure; worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B or D, return to the warning card. |
| 17 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | No, 87 = 3 × 29; x > 5; n² + 2n + 1. |
|---|---|
| Hook | 31 regions: 1 + 15 + 15. |
| Worked examples | 41 × 41 at n = 40; 57 regions; 6 is a counterexample. |
| Faded example | n = 4 gives 9 = 3 × 3. |
| 1. What a counterexample is | C. A fits the claim; B and D fail the condition. |
| 2. Smallest counterexample | 11. 11² − 11 + 11 = 121. |
| 3. A counterexample with conditions | 2. The only even prime. |
| 4. Counting regions | 163. 1 + 36 + 126. |
| 5. Why the rule fails | 43. 41 × (41 + 1 + 1). |
| 6. Smallest counterexample with two conditions | 30. The lowest common multiple of 10 and 15. |
| 7. Working back from the crossings | 8. 8 × 7 × 6 × 5 = 1680. |
| 8. Conjecture or theorem | B. A treats evidence as proof; C and D overreach in opposite directions. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Offered a case that agrees with the claim, or one that fails the condition. | Ask: does it end in 2? Does 4 divide it? A counterexample has to end in 2 and not be divisible by 4. |
| 10 or 121 in question 2 | Gave the last value that works, or the value of the rule at the failing n. | Ask: what does the question ask for, n or the rule's value? |
| 1 or 4 in question 3 | Thought 1 is prime, or picked an even number that is not prime. | List the prime numbers and say which are even. |
| 256, 162 or 37 in question 4 | Used doubling, left out the 1, or left out the crossings. | Check the formula against 4 points: 1 + 6 + 1 = 8. |
| 41 or 42 in question 5 | Copied the n = 40 factor, or counted one 41 too few. | Write 41² + 41 + 41 as three terms and take 41 out of each. |
| 150 or 15 in question 6 | Gave the product, or one number that meets only one condition. | List the multiples of 15 and test each against 10. |
| 7 or 10 in question 7 | Stopped too early or too late in a trial. | Make a two-column table of n and the crossing count and scan it. |
| A, C or D in question 8 | Treated evidence as proof, called the pattern false, or extended it. | Ask: could a counterexample still exist beyond 100? What would settle it? |
Questions worth asking
- Why was 31 a surprise when 1, 2, 4, 8, 16 came first? Listen for: small cases can fit more than one rule, so a pattern needs a reason, not just a fit.
- Is a counterexample to a claim about primes allowed to be a composite number? Listen for: no, it must meet the condition (be prime) and break the conclusion.
- If a claim is false, can it still be useful? Listen for: yes, an added condition can make it a theorem, as with numbers that share no factor.
- Does finding no counterexample prove anything? Listen for: no, it is evidence; a proof must cover every case.
Differentiation
A way in for anyone stuck
- Give a table with columns n, rule value, and prime or not, and ask students to fill it row by row before they predict.
- Provide a sentence frame for the counterexample: “n = … gives …, which is not …, so the claim is false.”
- Use a physical plate: draw six points on paper circles and count regions by colouring each one.
Stretch
- Find the smallest counterexample to “a number divisible by 8 and by 12 is divisible by 96” and explain it with the lowest common multiple.
- Show that the rule n² + n + 41 gives a multiple of 41 for n = 82 as well.
Students learning in English as an additional language
“Conjecture” is a statement we think is true but have not proved. “Counterexample” is one case that does not fit. Say “a counter example” as three words if it helps: an example that counters the claim. “Theorem” is a statement that has been proved.
| Word | Plain meaning | Example |
|---|---|---|
| conjecture | a statement we think is true for every case but have not proved | every even number above 2 is the sum of two primes |
| counterexample | one case that meets the conditions of a claim and breaks its conclusion | 9 breaks the claim that every odd number is prime |
| theorem | a statement that has been proved | the angles of a triangle add to 180° |
| inductive reasoning | using cases to suggest a general rule | tracing a pattern through the first few terms |
| deductive reasoning | using established facts to prove a statement for every case | showing a rule with a letter |
| condition | something that must be true for a claim to apply | “if n is even” is a condition |
Sentence starters for explaining aloud or in writing:
- This is a counterexample because it meets the condition that … and it breaks …
- The claim is only a conjecture because …
- The claim becomes a theorem if we add the condition that …
Two things not to say
“It works for a lot of cases, so it's true.” A hundred cases are evidence; the proof is a different kind of thing.
“That counterexample isn't fair.” If it meets the conditions and breaks the conclusion it is fair; help the student check the conditions.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- CounterexampleGive a counterexample to: “every multiple of 3 is even.”
Any odd multiple of 3, such as 15. - TestIs n² + n + 41 prime for n = 5?
Yes: 25 + 5 + 41 = 71, which is prime.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- CountingHow many regions do 4 points make? How many do 5 make?
8 and 16: 1 + 6 + 1 and 1 + 10 + 5. - Earlier work (Year 9)Expand (n + 1)² and (n − 1)². What is their difference?
n² + 2n + 1 and n² − 2n + 1; the difference is 4n.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that the claim in the second figure is true when a and b share no factor, using the lowest common multiple.
The lowest common multiple of numbers with no shared factor is a × b, so every number divisible by both is a multiple of a × b. - ExtendA claim says: “if n² is divisible by 4 then n is divisible by 4.” Find a counterexample.
n = 2: 4 is divisible by 4, and 2 is not divisible by 4.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Paper plates and a ruler are enough: students mark points, draw lines and count regions. The With a GDC section is optional.