Mathematics Year 7: Conjecture, test and justify with letters
Mathematics, Year 7 (Portugal): From a pattern to a statement: test it, tell testing from validating, and justify it with letters or knock it down with a counterexample.
Classify, spot a pattern, conjecture
Classify these numbers by the sets they belong to: 7 and 12 are natural numbers (ℕ); −4 is an integer (ℤ) but not natural; 2/5 and −1.5 are rational numbers (ℚ) but not integers. All are rational (−4 = −4/1) and the natural ones are integers too. From that regularity comes a conjecture: ℕ ⊂ ℤ ⊂ ℚ, each set inside the next. A conjecture is a statement you believe because of a pattern. You test it on cases; to validate it, you must show that it holds always.
Testing is not validating
Conjecture: “for every natural number n, n × (n + 1) + 17 is prime”. Test: n = 1 gives 19, then 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227 and, for n = 15, 257: fifteen tests, fifteen primes. Tempting to believe it. But n = 16 gives 16 × 17 + 17 = 289 = 17 × 17, which is not prime. Testing tries cases and can support a conjecture; validating shows it holds for every case. A single counterexample is enough to show a conjecture is false.
A case: three consecutive integers
Conjecture: the sum of three consecutive integers is three times the middle one. Tests: 4 + 5 + 6 = 15 = 3 × 5, and (−3) + (−2) + (−1) = −6 = 3 × (−2). Both work, but they only test. To validate, call the middle integer n: the one before is n − 1 and the one after is n + 1. Then (n − 1) + n + (n + 1) = n + n + n + (−1 + 1) = 3 × n. Nothing depends on the value of n, positive or negative, so it is true in every case. The letter carries the whole argument in one line.
“Minus a” is not always negative
Ana tested −a with a = 3, 8 and 20 and got −3, −8 and −20. Her conjecture: “−a is always a negative number”. But the letter a can hold any integer, including a negative one. With a = −4: −a = −(−4) = 4, which is positive. One counterexample and the conjecture is false. The reasonable mistake is to read the minus sign as “negative”; it means the opposite of a. To avoid it, test with different kinds of number: positive, negative and also 0, since −0 = 0.
Where you see it: five ways of justifying
Whoever has to convince someone uses one of these. Counterexample: “subtraction is commutative” fails, 5 − 3 = 2 but 3 − 5 = −2. Exhaustion: with few cases, check them all (the divisors of 12 are 1, 2, 3, 4, 6 and 12). Generic example: an argument that does not depend on the number chosen. Logical coherence: chaining facts you already know. Reductio ad absurdum: if an even number plus an odd one were even, the odd would be even − even, so even: absurd. Each fits its own situation.
Test what you learned
Sara tried “the sum of two integers is an integer” on 3 + 4 = 7, −5 + 2 = −3 and −7 + (−1) = −8. What has she done?
- a) She validated it, because three cases are enough
- b) She disproved it, because two of the sums are negative
- c) Nothing yet: tests only count after at least ten cases
- d) She tested it on three cases; to validate it she must show it holds for all integers
Correct answer: d) She tested it on three cases; to validate it she must show it holds for all integers — Right: cases can support a conjecture, but validating means showing it holds for every case (or finding one where it fails).
Conjecture: “n × (n + 1) + 17 is prime for every natural number n”. What does n = 16 give, and what does it show?
- a) 289 = 17 × 17, not prime: a counterexample, so the conjecture is false
- b) 289, which is prime because it is odd, so the conjecture still holds
- c) 289 is not prime, but the conjecture still holds, since it worked 15 times
- d) 273 = 3 × 91, not prime, so it is a counterexample
Correct answer: a) 289 = 17 × 17, not prime: a counterexample, so the conjecture is false — Right: 16 × 17 = 272 and 272 + 17 = 289 = 17 × 17. One counterexample is enough to refute it.
Ana claims: “−a is always a negative number”. Which reply is best?
- a) She is right: the minus sign always makes a number negative
- b) It depends on a: for a = 3, −a = −3, but for a = −4, −a = 4, a counterexample
- c) She is right, because she tested a = 3, 8 and 20 and all gave negatives
- d) She is wrong the other way: −a is always positive
Correct answer: b) It depends on a: for a = 3, −a = −3, but for a = −4, −a = 4, a counterexample — Right: the letter can hold a negative number, and one counterexample is enough to refute the statement.
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