Mathematics for Critical Thinking

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Mathematics for Critical Thinking

• A study of the Beauty and Structure of
  Mathematics
• Using the book The Nature of Mathematics by
  Karl Smith

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Items Needed

• An Open Mind regarding Mathematics
• No caving in to preconceived notions such as I
  have never been good at Math. or
• Math is just not my thing.
• A scientific (not graphing) calculator.(graphing
  calculators are OK, but not reqd)
• A Computer with E-mail and Internet access.
• The book The Nature of Mathematics by Karl
  Smith.

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Chapter OneThe Nature of Problem Solving

• Problem Solving
• George Polya his 4 steps
• Problem Solving with Sets
• Logic (an introduction)
• Inductive and Deductive Reasoning
• Scientific Notation and Estimation

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Problem Solving

• George Polya -- Four Step Guideline
• Understand the Problem
• Devise a Plan
• Carry out the Plan
• Look Back
• Some Techniques for Devising Plans
• Look for a Related Problem
• Work on a Simplified version
• Work Backwards

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A
B
Find the number of possible routes from point A
to point B.
Assume no backtracking.
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This is a simplified version.
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This is an even simplier model.
A
B
Or simpler still
A
C
How many ways from A to C? From A to D? From A to
E? From A to F?
D
E
B
F
How many ways from A to B in this case?
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Another Problem
A man goes into a store and says to the clerk,
If you will give me an amount equal to what I
have in my pocket, Ill spend 10 here. The
store-keeper agrees and the man does the same
thing in two more stores. After he spends
10 in the 3rd store, he is completely broke.
How much did he start with?
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Solution Working Backwards
Store Exits w/ Spends Enters w/ 3
0 10 5.00 2 5 10 7.50 1
7.50 10 8.75
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Sets
A set is a collection of objects. It is usually
represented by a Venn Diagram.
A
4
16
1
2
8
Set A 1, 2, 4, 8, 16
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The Universal and Null Sets
The Null set or Empty set is usually represented
by or ? but not by ?.
The Universal set denoted by U contains all
objects under consideration.
U
A
B
C
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The Complement of a Set
The Complement of a set A consists of all the
elements of the universal set except those in set
A.
U
A
A
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Union and Intersection of two sets
A
B
A
B
A ? B
A ? B
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Two Operations on Three Sets
U
A
B
C
(A ? B) ? C
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The Relative Complement of two sets A - B
A B x x ? A and x ? B or x ? A ?
B
Here A B is region 1 only.
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Examples
Identify the numbers of the region(s) to which
each of the following refer
A ? (B ? C)
1,4,5,6,7
C (A ? B)
3
(A ? C) ? B
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