Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Is 51 a prime number? Give a reason.
Answer
No: 51 = 3 × 17, so it has a factor other than 1 and itself. - What are the next two terms of 2, 5, 8, 11, …?
Answer
14 and 17: the rule is add 3. - Factorize 4x + 6.
Answer
2(2x + 3).
Words used in this lesson
- conjecture
- a statement that seems true but is not yet proved. For example: n² + n + 41 is always prime.
- counter-example
- one case that shows a statement is false. For example: 2 is a prime that is not odd.
- axiom
- a fact accepted without proof. For example: two points lie on exactly one straight line.
- theorem
- a statement that has been proved. For example: the sum of two odd numbers is even.
- proof
- reasoning that shows a statement is true for every case. For example: 2m + 1 and 2n + 1 add to 2(m + n + 1).
How many tests are enough?
A student tests the expression n² + n + 41 for n = 0, 1, 2 and 3. She gets 41, 43, 47 and 53, and all four are prime. “It always makes a prime!” she says. Is four tests enough? Is forty? Decide how far you think the pattern will go before you touch the figure.
A conjecture is a statement you believe is true for every case but have not yet shown. A theorem is a statement that has been proved, step by step, from axioms (facts accepted without proof) and from theorems already proved.
Testing cases can make a conjecture more likely, but when there are infinitely many cases it can never prove it. One counter-example is enough to disprove it. A proof is reasoning that covers every case at once.
Move n along the row. Each dot is one test.
n = 0 gives 41, which is prime. 1 test, no failures.
Show me the steps
- n = 0: 0 + 0 + 41 = 41, prime. n = 3: 9 + 3 + 41 = 53, prime.
- n = 10: 100 + 10 + 41 = 151, prime. Ten tests and no failures yet.
- n = 39: 1521 + 39 + 41 = 1601, prime. Forty tests in a row, all prime.
- Move one more step and read the caption. Does the pattern survive?
Two slips to watch for. “It worked every time I tried” is evidence, not a reason that it must work next time. And a counter-example needs only one case, so look in awkward places: 0, 1, a fraction, a very large number.
Worked examples
Conjecture: “squaring a whole number always gives a bigger number.”
- Test an awkward case: n = 1.
- 1² = 1, which is not bigger than 1. This single case is a counter-example, so the conjecture is false.
Conjecture: “the sum of two odd numbers is even.” Prove it.
- Any odd number is one more than a multiple of 2, so write two odd numbers as 2m + 1 and 2n + 1, where m and n are whole numbers.
- Add them: (2m + 1) + (2n + 1) = 2m + 2n + 2.
- Factorize: 2m + 2n + 2 = 2(m + n + 1). That is 2 times a whole number, so it is even.
- m and n could be any whole numbers, so this covers every pair of odd numbers. It is a proof, and the statement is now a theorem.
Here is the same proof as a picture. Every odd number is some pairs of dots and one spare. Move m and n and watch the two spares join to make one more pair.
5 + 9 = 14 = 2 × 7: all pairs, so even. The two spare dots make one more pair.
Show me the steps
- m = 0, n = 0: 1 + 1 = 2. Each number is one spare dot, and the two spares make one pair: 2 = 2 × 1.
- m = 2, n = 4: 5 + 9 = 14. That is 2 + 4 = 6 pairs, plus one pair made from the two spares: 7 pairs, 14 = 2 × 7.
- m = 6, n = 1: 13 + 3 = 16 = 2 × 8. Whatever m and n are, the sum is 2(m + n + 1), so it is always even.
Your go. The first step is given. Prove that the sum of three consecutive whole numbers is always a multiple of 3. Call the three numbers n, n + 1 and n + 2.
- Their sum is n + (n + 1) + (n + 2).
- Simplify, then factorize, then say why this covers every case. Then check.
Check the answer
n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). That is 3 times a whole number, and n can be any whole number, so the sum is always a multiple of 3.
Check before you go on
1. Which of these is a proof that the sum of two even numbers is even?
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. Work out n² + n + 41 when n = 7.
3. Work out n² + n + 41 when n = 12.
Level 2 · Building
4. Which single number is a counter-example to “every prime number is odd”?
5. Conjecture: “n² + 1 is even for every positive whole number n.” What is the smallest positive whole number n that is a counter-example?
6. Three consecutive numbers starting at n add up to 3(n + 1). Use this to find their sum when n = 14.
Level 3 · Stretch
7. A student claims that n² − n + 11 is prime for every whole number n. Find the smallest whole number n for which it is not prime.
8. Two odd numbers are 2m + 1 and 2n + 1, so their product is (2m + 1)(2n + 1) = 2(2mn + m + n) + 1. For the odd numbers 7 and 9, find 2mn + m + n.
Something to think about
A student says: "I have checked 1000 cases and the conjecture was true every time, so I have proved it." What would you say? What would a proof need that 1000 cases do not give?
One way to think about it
A proof must cover every case, and there are infinitely many whole numbers, so no number of checked cases can finish the job. A proof uses reasoning that works for any n at once, such as writing an odd number as 2m + 1. A single counter-example, on the other hand, can end the argument at once.
Exit ticket
9. Which one fact is enough to show that “every square number is even” is false?