Always, Sometimes or Never True - Set #1 (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:
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.
1. The more digits a number has, then the larger is its value.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Sometimes true.
This statement is true when we are dealing with positive integers only.
For negative integers, such as -23 and -234, the more digits, the smaller the value of the number.
For decimals, the number of digits tells you nothing about the size of the number (e.g., 0.62 >
0.236 but 0.12< 0.236).
2. If you multiply 12 by a number, the answer will be greater than 12.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Sometimes true.
12x > 12 only when x > 1.
3. The square of a number is greater than that number.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Sometimes true.
x2 > x when x(x - 1) > 0. That is only when x < 0 or x > 1.
4. If two rectangles have the same perimeter, they have the same area.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Sometimes true.
This is only true when the rectangles are identical.
5. Pentagons have fewer right angles than rectangles.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Always true.
Pentagons can have at most 3 right angles. Rectangles must have four.
If one tries to draw a five-sided polygon with four or more right angles, then it either degenerates
into a rectangle, or has three parallel sides and thus cannot be closed.
6. Quadrilaterals tessellate.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Always true.
This follows from the property that the four angles of a quadrilateral total 360û.