Homework: MTHT 400 Methods of Teaching Secondary Mathematics I \\

Reasoning

Due Nov. 15

The definitions of converse and contrapositive can easily be found on the net. (E.g. on Dr. Math http://mathforum.org/library/drmath/view/54198.html) where there

are a number of examples explaining the notions asked about below.

For negation see for example http://en.wikipedia.org/wiki/Negation

For symmetric, reflexive etc, see for example

http://mathworld.wolfram.com/SymmetricRelation.html

1, Write down the converse and contrapositive of each of the following implications.

a) If a and b are integers then ab is an integer.

b) If x is an even integer then x^2 is an even integer.

c) Every planar graph can be colored with at most 4 colors.

2) For each of the previous examples is the converse true? Why?

3) For each of the previous examples is the contrapositive true? Why?

4) Write down the negation of each of the following statements in clear and concise English.

a) Either x is not a real number or x>4 .

b) There exists a real number x such that n >x for every

integer n .

5) Let n be an integer; prove that n is odd if and only if

n^3 is odd.

6) What is wrong with the following argument which purports to

prove that every relation which is symmetric and transitive must

necessarily be reflexive as well.

Suppose R is a symmetric and transitive binary relation on a set

A and let a \in A . Then for any b with R( a, b)

we have also R(b,a) by symmetry. Since we now have both R( a, b)and R(b,a),

we have R(a,a) as well, by transtivity. Thus R(a,a) and R is reflexive.

7) You say, `every square is a rectangle' and a student looks

at you with complete incomprehension. What can you do? What

might the student be failing to understand?

Please go on to next page.

8) Read the article VOCABULARY CONCERNS IN THE MASTERY

OF MATHEMATICS: COLLEGE ALGEBRA ...

http:www.math-cs.cmsu.edu/~mjms/2003.1/Francis00.ps

If you have trouble reading postscript files, I have put a link to a pdf copy of this paper in the links

section of the MTHT 400 website.

Are some of his examples `distinctions without a difference'. I think at least two of the examples are terms for which you can find large numbers of mathematicians who adopt either of two conventions. (A similar example is: are the natural numbers

0,1,2,3 … or 1,2,3, … Can you think of other examples of careless language that could have been used in the pretest. See if you can locate the controversial examples. We will discuss these issues on November 16.