At a glance. One lesson of about 60 minutes. It assumes primes and factors from Year 7 and substitution into expressions. Students test a conjecture, find the smallest counterexample, count the cases that hold and predict from a pattern, and say what evidence does and does not show.
Calculator: GDC optional. The lesson is done by hand. The optional section shows GDC users a table of values for testing cases quickly, and says that a table is evidence and not a proof.
What students will learn
Know
- A conjecture is a statement not yet proved; once it is proved it is a theorem.
- One counterexample disproves a general statement.
- A counterexample must satisfy the conditions of the claim.
Understand
- Evidence from many cases makes a conjecture more convincing but does not prove it.
- A pattern can hold for a long run and then fail, and can fail and then hold again.
- Generalizing from a pattern is a prediction, and it needs checking.
Do
- Test a conjecture with a table of cases and look for a counterexample.
- State a counterexample clearly in a sentence with the numbers.
- Predict the next terms of a pattern and describe the rule.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: testing values and recognizing primes and factors (questions 1 to 7).
- Criterion B. Investigating patterns: the figure and question 7. Students collect cases in a table, describe the pattern and form a general rule, then say what would be needed to prove it.
- Criterion C. Communicating: stating a counterexample as a sentence with the substituted values (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: deciding how much evidence a claim in everyday life needs (hook and the what-if).
The one thing to do with the figure
Let students find the failure themselves. The strip only shows values up to the current n, so the first counterexample cannot be seen from the opening state. Ask for a prediction on a mini-whiteboard before any sliding, then step n up one at a time as a class. Move to the second expression and ask for a new prediction: most students shorten it.
Ask: “If I had stopped at n = 10, what would I have believed?” That is the point of the lesson.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: how many values in a row? Collect predictions on the board. |
| 12 min | Figure: step n up with the class, expression 1 then 2; key idea; warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, return to the key card. |
| 20 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 (91 = 7 × 13); 42; 9 (or any odd square). |
|---|---|
| Hook | Expression 1, n² + n + 17, gives primes for n = 1 to 15, a run of 15; n = 16 gives 289 = 17 × 17. Expression 2, 2n² + 11, gives primes for n = 1 to 10; n = 11 gives 253 = 11 × 23. |
| Worked examples | Four cases agree but are evidence only; n = 1 is a counterexample; 16. |
| Faded example | n = 3: 4 × 3 + 3 = 15 = 3 × 5, which is not prime. |
| 1. What a counterexample is | C. A is a case that fits; B is supporting evidence; D is not a case at all. |
| 2. Finding a counterexample | 4. 6 × 4 + 1 = 25. |
| 3. A counterexample to an inequality | 1. 1 is less than 2. |
| 4. Smallest counterexample | 10. 100 + 10 + 11 = 121 = 11 × 11. |
| 5. Counting cases that hold | 12. 15 cases, 3 not prime (121 = 11 × 11, 143 = 11 × 13, 221 = 13 × 17). |
| 6. A conjecture about primes | 11. 2¹¹ − 1 = 2 047 = 23 × 89. |
| 7. Predicting from a pattern | 110. 10 × 11. |
| 8. Evidence is not proof | B. A treats evidence as proof; C treats lack of testing as falsity; D invents a failure. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Chose a case that fits, supporting evidence, or an argument about untested cases. | Ask: what would a case have to do to break the claim? Then find a number ending in 3 that is not prime. |
| 25 or 3 in question 2 | Gave the value instead of n, or the last case that worked. | Ask: the question asks for n. Which n made 6n + 1 stop being prime? |
| 0 or 2 in question 3 | Forgot that ≥ includes equal. | Make a three-column table: n, n², 2n, and a tick or cross for each row. |
| 121 or 11 in question 4 | Gave the value of the expression, or a later failure. | Ask: is there a smaller n that fails? Check in order from n = 1. |
| 10, 15 or 3 in question 5 | Stopped at the first failure, assumed all hold, or counted the failures. | Mark each n in a table with a tick or cross, then count the ticks. |
| 23 or 4 in question 6 | Gave a factor of the value, or used a non-prime n. | Underline the condition in the claim: n must be prime. List the primes: 2, 3, 5, 7, 11. |
| 100 or 60 in question 7 | Used the odd-number pattern, or doubled. | Write the sums as products 1 × 2, 2 × 3, 3 × 4 and find how the two numbers depend on n. |
| A, C or D in question 8 | Treated evidence as proof, or lack of testing as falsity, or invented a failure. | Ask: what would a proof have to cover that testing cannot? |
Questions worth asking
- How many cases would be enough to be sure? Listen for: no finite number is enough; you need an argument for a general n.
- Does a counterexample have to be a small number? No: it can be large. Small cases are just cheaper to try first.
- Why does a counterexample to “n² ≥ 2n” have to use a whole number from 0 to 5? The claim is only about those values; a case outside the conditions does not test it.
- If a statement has no counterexample yet, is it true? Not necessarily: it is an open conjecture until it is proved.
Differentiation
A way in for anyone stuck
- Give a ready-made table with columns n, the expression and prime or not prime, and a list of primes up to 100.
- Start with the claim about numbers ending in 3 so that the idea of a counterexample is clear before algebra.
- Work with one expression at a time and stop for a prediction before each new row.
Stretch
- Find a value of n at which 2n² + 29 is not prime. (Think about when 29 could be a factor.) (n = 29: 2 × 29² + 29 = 29 × 59 = 1711.)
- Without working out the value, explain why n² + n + 17 is not prime when n = 17. (17² + 17 + 17 = 17 × (17 + 1 + 1) = 17 × 19: every term has the factor 17.)
Students learning in English as an additional language
“Conjecture” is a statement you think is true but cannot yet prove. “Counterexample” means an example that goes against a claim. Say “holds” for “is true in this case” and “fails” for “is false in this case”.
| Word | Plain meaning | Example |
|---|---|---|
| conjecture | a statement thought to be true but not yet proved | every n works |
| counterexample | one case where a general statement is false | 21 breaks “every odd number above 1 is prime” |
| evidence | cases that support a claim without proving it | ten cases hold |
| proof | an argument that covers every case | uses a letter |
| prime | a whole number greater than 1 with exactly two factors | 17 |
| generalization | a statement about every case, formed from specific cases | the rule |
Sentence starters for explaining aloud or in writing:
- A counterexample to this claim is … because …
- I tested … cases and they all …, so I conjecture that …
- This is evidence but not a proof because …
Two things not to say
“It works, so it is true.” Say instead: it works for the cases tested.
“Try a few more to be sure.” Say instead: a case that could fail, or an argument for all n.
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.
- CounterexampleShow that “n + 4 is prime for every whole number n ≥ 1” is false.
n = 2 gives 2 + 4 = 6, which is not prime, so n = 2 is a counterexample (n = 1 gives 5, which is prime). - SubstituteFind 3n + 2 for n = 1, 2, 3, 4.
5, 8, 11, 14.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA baker says: “every loaf I bake on a Monday sells out.” She tracks 12 Mondays and all sell out. What does this show, and what would break the claim?
It supports the claim but does not prove it for every Monday; one Monday with unsold loaves would disprove it. - Earlier work (Year 9)Expand and simplify (n + 1)² − n².
2n + 1.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningZara says n² − n + 5 is prime for n = 1 to 4. Find the first n for which it is not prime.
n = 5: 25 − 5 + 5 = 25 = 5 × 5. - Working backwardsA rule gives 3, 8, 15, 24 for n = 1, 2, 3, 4. Find the rule and the value for n = 10.
n² + 2n, which gives 120.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works 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? A table drawn on paper works the same way: the strip is just a row of ticks and crosses.