Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 7: is 87 a prime number?
Answer
No. 87 = 3 × 29, so it has a factor other than 1 and itself. - From Year 8: solve 3x − 5 > 10.
Answer
x > 5. Add 5 to both sides, then divide by 3. - From Year 9: expand (n + 1)².
Answer
n² + 2n + 1.
Words used in this lesson
- conjecture
- a statement we think is true for every case but have not proved. For example: 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. For example: 9 breaks the claim that every odd number is prime.
- theorem
- a statement that has been proved. For example: the angles of a triangle add to 180°.
- inductive reasoning
- using cases to suggest a general rule. For example: tracing a pattern through the first few terms.
- deductive reasoning
- using established facts to prove a statement for every case. For example: showing a rule with a letter.
- condition
- something that must be true for a claim to apply. For example: “if n is even” is a condition.
A pattern that is too good to be true
A tile maker in Lisbon marks n points round the edge of a circular plate and joins every pair with a straight line. No three lines meet at the same spot. The lines cut the plate into regions. One point gives 1 region, two points give 2, three give 4, four give 8 and five give 16.
Predict first: how many regions will six points give? Write down a number and the reason you chose it. Then set the first figure to 6 points and compare.
3 points: 3 lines and 0 crossings, so 1 + 3 + 0 = 4 regions. A doubling pattern predicts 4, so far so good.
Show me the steps
- 1 point: no lines, so 1 region. 2 points: 1 line, so 1 + 1 = 2 regions.
- 3 points: 3 lines and no crossing, 1 + 3 = 4 regions. 4 points: 6 lines and 1 crossing, 1 + 6 + 1 = 8 regions.
- 5 points: 10 lines and 5 crossings, 1 + 10 + 5 = 16 regions. The counts 1, 2, 4, 8, 16 look like doubling.
- 6 points: 15 lines and 15 crossings, 1 + 15 + 15 = 31 regions. Doubling predicted 32. The pattern was a coincidence of small numbers.
A conjecture is a statement about every case that has not been proved. Testing cases is inductive reasoning: it can make a conjecture more likely, never certain. A proof is deductive: it starts from facts already established and covers every case. A proved conjecture is a theorem.
One counterexample disproves a conjecture, but it must meet every condition of the claim and still break its conclusion. And a false claim can often be rescued by an extra condition: finding that condition is part of generalizing.
The second figure tests a different kind of claim, one about divisibility. It says: a whole number divisible by a and by b is divisible by a × b. Move a and b and look for a number that breaks it.
a = 3 and b = 5: the first number divisible by both is 15, which is 3 × 5. Every number divisible by both is a multiple of 15, so the claim has no counterexample.
Show me the steps
- a = 3, b = 5: the first number divisible by both is 15 = 3 × 5, so every number divisible by both is a multiple of 15. No counterexample.
- a = 4, b = 6: 12 is divisible by 4 and by 6 but not by 24. The smallest counterexample is 12.
- a = 6, b = 9: 18 is divisible by 6 and by 9 but not by 54. The smallest counterexample is 18.
- a = 7, b = 8: the first number divisible by both is 56 = 7 × 8. No counterexample.
- The claim fails exactly when a and b share a factor larger than 1. Add the condition that a and b share no factor larger than 1, and the claim becomes a theorem.
Two common mistakes. Treating a run of successes as a proof, however long the run. And offering a case that does not meet the conditions: for the claim “every multiple of 7 is odd”, 10 is not a counterexample because it is not a multiple of 7, but 14 is.
Worked examples
1. Disprove: “n² + n + 41 is prime for every whole number n.”
- The rule gives primes for a long run: n = 0 gives 41, n = 2 gives 47, n = 3 gives 53, and on. The cause of the failure is the 41: when n is a multiple of 41 or one less than one, n² + n = n(n + 1) is a multiple of 41.
- Try n = 40: 40² + 40 + 41 = 40 × 41 + 41 = 41 × (40 + 1) = 41 × 41. The value is not prime, so n = 40 is a counterexample.
2. Use the regions count: how many regions do 7 points make?
- Lines: 7 × 6 ÷ 2 = 21. Crossings: each crossing comes from 4 of the 7 points, and there are 35 ways to choose them.
- Regions = 1 + 21 + 35 = 57. Doubling would have said 64.
3. Disprove: “every multiple of 6 is a multiple of 4.”
- A counterexample must be a multiple of 6 that is not a multiple of 4. 8 is not a multiple of 6, so it does not count.
- 6 is a multiple of 6, and 6 ÷ 4 is not a whole number, so 6 is a counterexample. (12 is a multiple of both, so it is not.)
Your go. The first step is given. Disprove: “n² − 3n + 5 is prime for every whole number n.” Try n = 0, 1, 2 in order.
- n = 0 gives 5, n = 1 gives 1 − 3 + 5 = 3, n = 2 gives 4 − 6 + 5 = 3. All prime.
- Keep going: n = 3, 4, and so on, until a value is not prime. Write the counterexample as a sentence with the numbers.
Check the answer
n = 3 gives 9 − 9 + 5 = 5, which is prime. n = 4 gives 16 − 12 + 5 = 9 = 3 × 3, so n = 4 is a counterexample and the statement is false.
With a GDC (optional)
A GDC can list a rule's values in a table of values: enter the rule in the function menu, then set the table to start at 0 with a step of 1. Scrolling the table is a fast way to hunt for a case that fails.
A table is evidence, not proof: a table that stops at 30 says nothing about 31. Use it to find a counterexample, and write the counterexample as a sentence with the numbers. Everything on this page can be done by hand.
Check before you go on
1. A claim says: “every whole number that ends in 2 is divisible by 4.” Which number shows the claim is false?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. The rule n² − n + 11 gives a prime for n = 0, 1, 2 and on. For the smallest whole number n where it does not give a prime, what is n?
3. The claim “every prime number is odd” has exactly one counterexample. What is it?
Level 2 · Building
4. A plate has 9 points on its edge, every pair joined by a line, and no three lines meet at one spot. How many regions is the plate cut into? Use 1 + (lines) + (crossings); each crossing comes from 4 of the 9 points, and there are 126 ways to choose them.
5. For n = 40, n² + n + 41 = 41 × 41. For n = 41, n² + n + 41 = 41 × ?. What number goes in the place of the question mark?
Level 3 · Stretch
6. Claim: “a whole number divisible by 10 and by 15 is divisible by 150.” What is the smallest positive whole number that is a counterexample?
7. A plate has n points joined in pairs with no three lines meeting. There are 70 crossings. The crossings come from choosing 4 of the n points, which is n(n − 1)(n − 2)(n − 3) ÷ 24. What is n?
Something to think about
In the second figure the claim fails whenever a and b share a factor. Can you adjust the claim so that it is true for every a and b, and still says something useful? Test your version with a = 6 and b = 4.
One way to think about it
Yes. A number divisible by a and by b is divisible by the lowest common multiple of a and b. For 6 and 4 the lowest common multiple is 12, and 12, 24, 36 and so on are exactly the numbers divisible by both. The original claim is the special case where the lowest common multiple is a × b, which happens when a and b share no factor.
Exit ticket
8. A pattern has been checked for every whole number from 1 to 100 and fits all of them. Which statement is the most accurate?