At a glance. One lesson of about 60 minutes. Conjecture, counter-example, axiom and theorem; a short algebraic proof built from 2m + 1.
Calculator: no calculator. No calculator: the arithmetic is small and the point is the logic.
What students will learn
Know
- A conjecture is a statement believed true but not yet proved.
- One counter-example disproves a statement.
- A theorem is a statement that has been proved.
Understand
- Checking cases can support a statement but, with infinitely many cases, cannot prove it.
- A proof uses letters for any case, so the reasoning holds for all of them at once.
Do
- Test a formula by substitution.
- Find a counter-example, including the smallest one.
- Follow and complete a short algebraic proof.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: substitution and the vocabulary of proof (questions 2, 3, 4, 9).
- Criterion B. Investigating patterns: students predict how far the pattern runs, test it, and write a general statement about when it fails.
- Criterion C. Communicating: complete reasoning with a stated conclusion, such as “2 times a whole number, so even”.
- Criterion D. Applying mathematics in real-life contexts: judging a claim made from data in the “Something to think about” task (“I checked 1000 cases”).
The one thing to do with the figure
Before anyone touches the slider, ask for a vote: how far will the pattern go? Most say “forever” or “a long way”. Move n to 39 slowly, then ask for another vote before moving one more step.
The surprise at n = 40 is the lesson: forty successes proved nothing. Do not tell students that this formula is famous or special; let them decide how many tests they would trust in future.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 10 min | Hook and vote; the figure with a second vote at n = 39. |
| 10 min | Vocabulary: conjecture, counter-example, axiom, theorem; the key card in their own words. |
| 10 min | Worked examples; the faded proof in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | No (51 = 3 × 17); 14 and 17; 2(2x + 3). |
|---|---|
| Worked examples | n = 1 is a counter-example to the squaring claim; the odd-plus-odd proof gives 2(m + n + 1). |
| Faded example | 3n + 3 = 3(n + 1), a multiple of 3 for every whole number n. |
| 1. Proof or evidence | B. A and C are evidence from cases; only B uses letters for any even numbers. |
| 2. Testing a formula | 97 = 49 + 7 + 41. |
| 3. Testing a formula | 197 = 144 + 12 + 41. |
| 4. Counter-example | B, 2. C is even but not prime, so it is not a counter-example. |
| 5. Smallest counter-example | 2: 2² + 1 = 5 is odd. |
| 6. Using a proved rule | 45 = 3 × 15. |
| 7. Hunting a counter-example | 11: 121 − 11 + 11 = 121 = 11 × 11. |
| 8. Reading a proof | 31: m = 3, n = 4, so 2mn + m + n = 24 + 3 + 4. |
| 9. Disproving | B. C is the “more examples” slip; D reverses proving and disproving. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or C in question 1 | Treats examples as a proof. | Ask: could a case exist that you have not tried? What would make the argument cover all of them? |
| 62 or 77 in questions 2 and 3 | Squares as doubling. | Write 12 × 12 and ask where the 24 came from. |
| 1 in question 5 | Picks a case that fits the statement. | Ask: does your case break the statement or agree with it? |
| 43 or 42 in question 6 | Drops or misplaces the +1 in the bracket. | Substitute n = 14 into n, n + 1, n + 2 and add them as a check. |
| 0, 1 or 10 in question 7 | Stops at a value whose result is still prime. | Ask: is 101 prime? Go one value at a time and check each result for a factor. |
| 12 in question 7 | Finds a counter-example but not the smallest. | Ask what happens at n = 11 and why 11 is special. |
| 24, 7 or 63 in question 8 | Reads only part of the bracket, or the product itself. | Find m and n from 7 = 2m + 1 and 9 = 2n + 1 first, then substitute into 2mn + m + n. |
Questions worth asking
- If a formula works for n = 1 to 1000, have we proved it? No. Listen for “there are infinitely many cases”.
- Why can one counter-example do what a thousand examples cannot? Listen for “the statement says every case”.
- Why do we write odd numbers as 2m + 1? Listen for “so it is any odd number, not one in particular”.
Differentiation
A way in for anyone stuck
- Give a table with n from 0 to 5 already filled for the formula before the figure.
- For the proof, let students check 2m + 1 with m = 0, 1, 2, 3 first.
- Colour the +1 in 2m + 1 and 2n + 1 so that it can be tracked into the factorized step.
Stretch
- Prove that the product of an odd number and an even number is even.
- Prove that the sum of any two even numbers plus 1 is odd, and say why 2m + 2n + 1 cannot be written as 2 times a whole number.
Students learning in English as an additional language
Pair each term with a short example in the student's home language: counter-example and conjecture have close cognates in many languages.
| Word | Plain meaning | Example |
|---|---|---|
| conjecture | a statement that seems true but is not yet proved | n² + n + 41 is always prime |
| counter-example | one case that shows a statement is false | 2 is a prime that is not odd |
| axiom | a fact accepted without proof | two points lie on exactly one straight line |
| theorem | a statement that has been proved | the sum of two odd numbers is even |
| proof | reasoning that shows a statement is true for every case | 2m + 1 and 2n + 1 add to 2(m + n + 1) |
Sentence starters for explaining aloud or in writing:
- My conjecture is that … and I tested it for …
- A counter-example is … because …
- This is a proof, not a test, because the letters stand for …
Two things not to say
“Test a few numbers and you are sure.” That is the slip the figure is built to break.
“A proof is a long explanation.” A proof is reasoning that holds for every case, not a longer description of examples.
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.
- SubstitutionWork out n² + n + 41 when n = 5.
71. - Counter-exampleFind a counter-example to “every prime number is odd”.
2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- DisprovingClaim: “n + 1 is a multiple of 2 for every whole number n.” Give a counter-example.
n = 2 (or any even n): 3 is not a multiple of 2. - ProvingProve that the sum of two even numbers is even.
2m + 2n = 2(m + n), 2 times a whole number.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningProve that the sum of any two consecutive whole numbers is odd.
n + (n + 1) = 2n + 1, which is one more than a multiple of 2. - From Year 8Write the next two terms of 3, 7, 11, 15, … and give the rule.
19 and 23; add 4 each time.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard and by touch; “Show me the steps” gives four cases in words.
On paper: a table of n from 0 to 10 with the values filled in is enough for the first part, though it cannot reach n = 40.