Always, Sometimes or Never True - Set #2 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: 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! ____________________________________________________ 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. 1. The center of a circle that circumscribes a triangle is inside the triangle. Is this always, sometimes or never true? ...................................................... Reasons or examples: 2. An altitude subdivides a triangle into two similar triangles. Is this always, sometimes or never true? ...................................................... Reasons or examples: 3. (a + b)2 = a2 + b2 Is this always, sometimes or never true? ...................................................... Reasons or examples: 4. 3x2 = (3x)2 Is this always, sometimes or never true? ...................................................... Reasons or examples: 5. A shape with a finite area has a finite perimeter. Is this always, sometimes or never true? ...................................................... Reasons or examples: 6. A shape with a finite perimeter has a finite area. Is this always, sometimes or never true? ...................................................... Reasons or examples: