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Tools - Math 'Convincing and Proving' Always, Sometimes or Never True Tasks, Set #2
Always, Sometimes or Never True: Set #1 (solutions) || Set #2 (solutions)
Critiquing 'Proofs': Set #3 (solutions) || Set #4 (solutions)
Malcolm Swan
Mathematics Education
University of Nottingham
Malcolm.Swan@nottingham.ac.uk
Jim Ridgway
School of Education
University of Durham
Jim.Ridgway@durham.ac.uk
Introduction:
You will be given a number of statements. You must decide if each statement is
- always true, or
- sometimes true, or
- never true
You must provide full and convincing reasons for your decision. If you think that a statement is sometimes true, you must fully explain when it is true and when it is not true.
Here is an example of what we mean:
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Example:
 
Weaker response:
This statement is sometimes true.
It is true when both numbers are 0 and when both numbers are 2.
It is not true when one number is 2 and one number is 3.
Stronger response:
This statement is sometimes true.
Suppose one number is x and one number is y.
The statement says that: x+y = xy
This simplifies to the condition that y = x/(x-1)
A few pairs of numbers when it works are therefore:
(0, 0); (2, 2); (3, 3/2); (4, 4/3); (5, 5/4) ......
There are also other pairs which work!
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The aim of this assessment is to provide the opportunity for you to:
- test statements to see how far they are true;
- provide examples or counterexamples to support your conclusions
- provide convincing arguments or proofs to support your conclusions
For each statement, say whether it is always, sometimes or never true.
You must provide several examples or counterexamples to support your decision.
Try also to provide convincing reasons for your decision.
You may even be able to provide a proof in some cases.
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1. The center of a circle that circumscribes a triangle is inside the triangle.
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Is this always, sometimes or never true? ......................................................
Reasons or examples:
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2. An altitude subdivides a triangle into two similar triangles.
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Is this always, sometimes or never true? ......................................................
Reasons or examples:
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Is this always, sometimes or never true? ......................................................
Reasons or examples:
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5. A shape with a finite area has a finite perimeter.
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Is this always, sometimes or never true? ......................................................
Reasons or examples:
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6. A shape with a finite perimeter has a finite area.
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Is this always, sometimes or never true? ......................................................
Reasons or examples:
Always, Sometimes or Never True: Set #1 (solutions) || Set #2 (solutions)
Critiquing 'Proofs': Set #3 (solutions) || Set #4 (solutions)
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