Chapter 1: Viewing Mathematics
1.2 Mathematics as Reasoning
1.2.1. Types of Reasoning
1.2.1.1. Several different types of reasoning
1.2.1.1.1. inductive
1.2.1.1.2. deductive
1.2.1.1.3. proportional
1.2.1.1.4. spatial
1.2.1.2. Sometimes used individually
1.2.1.3. Sometimes used in combination
1.2.2. Inductive Reasoning and Patterns
1.2.2.1. Description of inductive reasoning: Inductive reasoning involves the use of
information from specific examples to draw a general conclusion. The general
conclusion drawn is called a generalization.
1.2.2.2. An observed pattern from a finite number of trials is concluded to always work
that way
1.2.2.3. Procedure for using the inductive reasoning process (p. 16)
1.2.2.3.1. Check several examples of a possible relationship
1.2.2.3.2. Observe that the relationship is true for every example you checked
1.2.2.3.3. Conclude that the relationship is probably true for all other examples and
state a generalization
1.2.2.4. Description of counterexample: A counter example is an example that shows a
generalization to be false.
1.2.2.5. While an INFINITE number of examples CANNOT PROVE a generalization is
true, ONE counterexample CAN PROVE a generalization is false.
1.2.2.6. Your Turn p. 17
1.2.2.7. Patterns
1.2.2.7.1. Mathematics sometimes defined as study of patterns
1.2.2.7.2. sometimes order not important to see pattern
1.2.2.7.3. sometimes order is critical to finding pattern
1.2.2.8. Sequences
1.2.2.8.1. Sequence: a pattern involving an ordered arrangement of numbers,
geometric figures, letters, or other entities
1.2.2.8.2. Terms of a sequence: the individual numbers, geometric figures, or letters
that make up the sequence. Terms are usually separated by a comma
1.2.2.8.3. Numerical sequences are classified by the relationship between consecutive
terms
1.2.2.8.4. Arithmetic sequence: a numerical sequence where each successive term is
obtained by adding a fixed number to the preceding term
1.2.2.8.5. Common difference: the fixed number used to find terms in an arithmetic
sequence
1.2.2.8.6. Geometric sequence: a numerical sequence where each successive term is
obtained by multiplying a fixed number by the preceding term
1.2.2.8.7. Common ratio: the fixed number used to find terms in a geometric sequence
1.2.2.8.8. There are many sequences that are not arithmetic nor geometric
1.2.2.8.8.1.
1, 3, 6, 10, … (triangular numbers)
1.2.2.8.8.2.
2, 6, 12, 20, … (rectangular numbers)
1.2.2.8.8.3.
1, 4, 9, 16, … (square numbers)
1.2.2.8.9. Your Turn p. 19
1.2.2.8.10. Generalizing an arithmetic sequence
1.2.2.8.10.1.
a, a + d, a + 2d, a + 3d, …
Term #
1
2
3
4
5
6
7
…
n
Term
a
a+d
a + 2d
a + 3d
a + 4d
a + 5d
a + 6d
…
a + (n - 1)d
Term #
1
2
3
4
5
6
7
…
n
Term
a
ar
ar^2
ar^3
ar^4
ar^5
ar^6
…
ar^(n-1)
1.2.2.8.10.2.
1.2.2.8.10.3.
The nth term of an arithmetic sequence is a + (n – 1)d
1.2.2.8.11. Generalizing a geometric sequence
1.2.2.8.11.1.
a, ar, ar2, ar3, …
1.2.2.8.11.2.
1.2.2.8.11.3.
The nth term of an geometric sequence is ar(n – 1)
1.2.2.9. Patterns in real-world data tables
1.2.2.9.1. Frequently recorded in a table of values similar to the ones used above
1.2.2.9.2. Two types of patterns often occur
1.2.2.9.2.1.
column extension patterns: pattern found in the numbers in the
column
1.2.2.9.2.2.
row relationship patterns: pattern found comparing consecutive
rows
1.2.2.9.2.3.
Column extensions are useful, but row relationship patterns are
more efficient
1.2.2.9.3. Your Turn p. 23
1.2.2.10. Patterns in sequences of number sentences
1.2.2.10.1.
Term #
1
2
3
4
5
6
7
…
n
Term
1+2
1+2+3
1+2+3+4
1+2+3+4+5
1+2+3+4+5+6
1+2+3+4+5+6+7
1+2+3+4+5+6+7+8
…
1+2+3+4+5+6+7+8+…+n
= Term
(2 x 3)/2
(3 x 4)/2
(4 x 5)/2
(5 x 6)/2
(6 x 7)/2
(7 x 8)/2
(8 x 9)/2
…
?
Result
3
6
10
15
21
28
36
…
?
1.2.3. Deductive Reasoning
1.2.3.1. used for drawing logical conclusions
1.2.3.2. presenting convincing arguments
1.2.3.3. presenting proofs
1.2.3.4. If-then statements
1.2.3.4.1. Called conditional statements
1.2.3.4.2. If part is called the hypothesis or antecedent
1.2.3.4.3. Then part is called the conclusion or the consequent
1.2.3.4.4. Your Turn p. 25
1.2.3.4.5. Procedure for deciding when a conditional statement is true or false
1.2.3.4.5.1.
First decide whether the hypothesis and the conclusion are true or
false
1.2.3.4.5.2.
Then use the following to decide if the statement is true or false
1.2.3.4.5.2.1. When both the hypothesis and the conclusion are true, the
conditional statement is true
1.2.3.4.5.2.2. When both the hypothesis and the conclusion are false, the
conditional statement is true
1.2.3.4.5.2.3. When the hypothesis is true and the conclusion is false, the
conditional statement is false
1.2.3.4.5.2.4. When the hypothesis is false and the conclusion is true, the
conditional statement is true
1.2.3.5. Truth table for conditional statements
1.2.3.5.1. summarizes the information for determining the truth of a conditional
statement
1.2.3.5.2. use p for the antecedent
1.2.3.5.3. use q for the consequent
1.2.3.5.4. p  q read “If p, then q.” or “p implies q”
p
T
T
F
F
q
T
F
T
F
pq
T
F
T
T
1.2.3.5.5.
1.2.3.5.6. p  q is only false when the antecedent is true and the consequent is false;
otherwise it is true
1.2.3.6. Rules of logic
1.2.3.6.1. Rule A: used when a conditional and its antecedent are both true p. 26
1.2.3.6.2. Rule B: used when a conditional is true and its consequent is false p. 26
1.2.3.6.3. Your Turn p. 27
1.2.3.6.4. Logic Rule A – affirming the hypothesis or affirming the antecedent
1.2.3.6.5. Logic Rule B – denying the conclusion or denying the consequent
1.2.3.6.6. other rules of logic are also used
1.2.3.6.7. Description of deductive reasoning: Deductive reasoning involves drawing
conclusions from given true statements using rules of logic
1.2.3.6.8. Procedure for using the deductive reasoning process
1.2.3.6.8.1.
Start with a true statement, often in if-then form
1.2.3.6.8.2.
Note given information about the truth or falsity of the hypothesis or
the conclusion (antecedent or consequent)
1.2.3.6.8.3.
Use a rule of logic to determine the truth or falsity of the hypothesis
or the conclusion (antecedent or consequent)
1.2.3.6.9. Sometimes rules are used in sequence
1.2.3.6.10. Your Turn p. 29
1.2.4. Proportional Reasoning
1.2.4.1. A statement asserting two ratios are equal is a proportion
1.2.4.2. Description of proportional reasoning: Proportional reasoning involves
drawing conclusions or solving problems with either the formal or informal use of
proportions
1.2.4.3. useful way to solve problems and draw conclusions about data
1.2.4.4. can be helpful in making everyday decisions
1.2.4.5. Your Turn p. 30
1.2.5. Spatial Reasoning
1.2.5.1. used in everyday situations and mathematics
1.2.5.2. helps make decisions about 2-D objects
1.2.5.3. helps make decisions about 3-D objects
1.2.5.4. optical illusions – when we see contradictory visual information
1.2.5.5. Description of spatial reasoning: Spatial reasoning involves visualizing 2-D
and 3-D geometric figures or objects to draw conclusions about properties or
relationships involving those figures or objects
1.2.5.6. Spatial sense is a big key to students being able to READ
1.2.5.7. Your Turn p. 32
1.2.6. Importance of Reasoning Process
1.2.6.1. Inductive reasoning – generalizing from an observed pattern seen in several
examples
1.2.6.2. Deductive reasoning – based on using rules of logic to draw conclusions from
given information
1.2.6.3. Proportional reasoning – based on using ratios and proportions
1.2.6.4. Spatial reasoning – based on visualizing properties and relationships in 2-D and
3-D geometric figures or objects (There are 7 subcategories of spatial sense)
1.2.7. Problems and Exercises p. 33
1.2.7.1. Home work: 4-7, 9, 11-13, 18, 21, 25, 27, 29-31, 35-37