



Tools  Math 'Convincing and Proving' Critiquing 'Proofs' Tasks, Set #3 (solutions)
Always, Sometimes or Never True: Set #1 (solutions)  Set #2 (solutions) Critiquing 'Proofs': Set #3 (solutions)  Set #4 (solutions)
Malcolm Swan
Jim Ridgway
1. Consecutive Addends Here are three attempts at proving the following statement: When you add add three consecutive numbers, your answer is always a multiple of three. Look carefully at each attempt. Which is the best 'proof'? Explain your reasoning as fully as possible.
The best proof is attempt number ..........
Attempt 2 is more structural, but again is only related to a specific case. Attempt 3 is the best attempt at a proof.
Here are three attempts at proving the following statement: If you have two rectangles, the one with the greater perimeter will have the greater area. Which is the best 'proof'? Explain your reasoning as fully as possible.
The best proof is attempt number ..........
Attempt 3 gives a counterexample and therefore is the only correct solution  although the perimeter and area are not calculated. Attempt 1 is flawed in that you can increase the perimeter without increasing both of the sides of the rectangle. Indeed, you can decrease one side and still increase the perimeter. Thus we might have x > 0 and y < 0 so long as x + y > 0. Attempt 2 clearly only demonstrates the result for a few special cases.
A student is playing a coin turning game. She starts with three heads showing and then turns them over, two at a time.
After a while she makes the statement: Here are three attempts to prove this. Look carefully at each attempt. Which is the best 'proof'? Explain your reasoning as fully as possible.
The best proof is attempt number ..........
Attempt 2 is correct. Attempt 3 simply explains that it is impossible to reach three tails from a position showing two tails and one head. This is insufficient. (It does not take into account other positions from which one can reach three tails, such as HHT, for example).
Critiquing 'Proofs': Set #3 (solutions)  Set #4 (solutions)

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