



Tools  Math 'Convincing and Proving' Always, Sometimes or Never True Tasks, Set #2 (solutions)
Always, Sometimes or Never True: Set #1 (solutions)  Set #2 (solutions) Critiquing 'Proofs': Set #3 (solutions)  Set #4 (solutions)
Malcolm Swan
Jim Ridgway
Here is an example of what we mean:
The aim of this assessment is to provide the opportunity for you to:
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.
The center of the circle that circumscribes a triangle sometimes lies inside the triangle. The statement is true when the triangle is acute. In the case of a right triangle the center of the circumscribing circle lies on the triangle. In the case of an obtuse triangle, the center of the circumscribing triangle lies outside the triangle.
It is sometimes true that an altitude subdivides a triangle into similar triangle. Two interesting case are that of an altitude drawn in an isosceles triangle from the nonbase angle, and that of an altitude drawn in a right triangle from the right angle.
This is only true when a = 0 or b = 0 or a = b = 0.
Again this is true only when x = 0
If the shape has fixed area A and dimensions x and A/x. The perimeter is thus 2(x + ^{A}/_{x}). As x is varied, this can be made as large or as small as we please. So theoretically, the perimeter may become infinite  but then is an infinitely thin rectangle still a rectangle (it feels more like a line to me!)? As an aside, it may also be noted that many fractals also have infinite boundaries but enclose finite areas. Moving up a dimension, it is also interesting to note that the trumpet shape obtained by rotating the curve y = ^{1}/_{x} around the x axis for x > 1 has a finite volume, but an infinite surface area. (Does this suggest that we can fill the shape with a finite amount of paint, but that the paint will never fully coat the surface?)
It is always true that a shape with a finite perimeter has a finite area. If the dimensions of the rectangle are x and y, then if 2(x + y) is finite and x and y are both positive, both x and y must be finite, so the area must be finite.
Critiquing 'Proofs': Set #3 (solutions)  Set #4 (solutions)

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