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1/7/2009
Section 1.2
Reasoning Mathematically
a) Add the following numbers.
1+3
3+7
35 + 31
b) Does every squared number equal itself?
0² =
1² =
2² =
c) The product of two odd numbers is what type
of number?
5x9=
3x7=
11 x 13 =
1) Inductive Reasoning
` involves the use of information from specific
examples to draw a general conclusion
(generalization). A pattern or relationship is
detected through observation.
` is a central process in elementary school
math since children form most of their
general mathematics ideas by seeing pattern
in examples
`
`
A counterexample is an example that shows a
generalization to be false.
What was the counterexample used to
disprove the generalization in part b of the
previous example?
d) The three times I went on a picnic it rained.
Sequences: patterns involving ordered arrangements of
numbers, geometric figures, letters, etc. The numbers,
figures, etc. are called the terms of the sequence.
◦ Arithmetic sequence: numeric sequence in which
each term is obtained from the previous term by
adding
ddi
a fi
fixed
d number
b ((common diff
difference).
)
2, 4, 6, 8, 10, 12, …
◦ Geometric sequence: numeric sequence in which
each term is obtained from the previous term by
multiplying a fixed number (common ratio).
1, 3, 9, 27, 81, 243, …
◦ Fibonacci sequence: each term is the
sum of the two preceding terms
1, 1, 2, 3, 5, 8, 13, 21, …
◦ Column extension patterns
◦ Row relationship patterns
efficient)
(usually more
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`
x
64, 32, 16, 8
x
16, 17, 19, 23, 31
x
1, 2, 5, 14
x
1 4
1,
4, 9
9, 16
x
2, 5, 10, 17
x
1,8, 27, 64
x
2, 10, 42, 170
`
`
`
`
Counting numbers
`
Even numbers
`
Odd numbers
`
2, 6, 10, 14, …
Consider 4, 8, 12, 16, 20, …
nth term =
Find the 7th term.
`
`
The rules for some sequences can be stated
algebraically.
Finding the rule can sometimes be simple,
sometimes challenging.
1, 2, 3, 4, …
`
What is the first term? The next three terms?
In many everyday situations a rule relates two sets of
numbers. This relationship can often be represented
with a formula.
Consider the following.
A 10-inch candle burns according to the information
in the table below.
Time
0 1
(min.)
Height
10 8
(in.)
2, 4, 6, 8, …
1, 3, 5, 7, …
`
`
`
2) Deductive Reasoning
` the process of reaching a necessary
conclusion from given facts or hypothesis
` involves drawing conclusions from given true
statements by using rules of logic
` is used by detectives solving crimes and by
philosophers thinking logically
Suppose the rule for the nth term of a
sequence is 2n+1.
`
`
`
`
`
`
`
2 3 4
5
6
What is the height of the candle after burning 3
minutes
After 5 minutes?
Write a formula that represents the relationship
between the height of the candle and the time spent
burning.
The painting “By Numbers” was stolen from Jane
Dough’s Illinois home last night. Today the painting
was found in an alley. Who was the thief?
The only suspects are Jane Dough (the owner of the house),
Jeeves (the butler), Sharky (the pool man), and Fluffy (the
pet Doberman).
Doberman) The police have gathered the following
evidence.
The painting was stolen between 8 p.m. and 9 p.m.
Fluffy can’t carry a painting.
Jeeves has been in Liverpool all week.
Sharky’s and Jane’s fingerprints were on the canvas.
Jane was with three friends at the movie from 7:30 p.m. to
10:00 p.m. last night.
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`
Understanding if-then statements and deciding
when they are true.
`
`
`
If-then statements are called conditionals.
If p, then q.
p: hypothesis
`
`
`
`
q: conclusion
`
Consider the following statement made by your basketball
coach: “If you play well in practice, then you will start in
tomorrow’s game.”
In which of the following cases would you feel that you were
being treated unfairly and the coach didn’t tell the truth?
Case 1: You play well in practice.
You start the game.
(Hypothesis true)
(Conclusion true)
Case 2: You play well in practice.
You do not start the game
game.
(Hypothesis true)
(Conclusion false)
Case 3: You do not play well in practice. (Hypothesis false)
You start the game.
(Conclusion true)
Case 4: You do not play well in practice. (Hypothesis false)
You do not start the game.
(Conclusion false)
`
Using rules of logic -- Determine whether each of
the following is TRUE or FALSE.
` If you live in Tuscaloosa, then you live in
Alabama.
`
`
Given: You live in Tuscaloosa.
C
Conclusion:
l i
You
Y
li
live iin Al
Alabama.
b
This is called affirming the hypothesis.
Given: You do not live in Alabama.
Conclusion: You do not live in Tuscaloosa.
This is called denying the conclusion.
3) Proportional Reasoning
` involves drawing conclusions or solving
problems with either the formal or informal use
of proportions (statements which assert that
two ratios are equal).
` A dog
d
owner gives
i
each
h off her
h three
th
dogs
d
a
puppy biscuit twice a day. How many puppy
biscuits does the dog owner give out in one
week?
Given: You live in Alabama
Conclusion: You live in Tuscaloosa.
This is called assuming the converse.
If you live in Alabama, then you live in
Tuscaloosa.
`
Given: You do not live in Tuscaloosa.
Conclusion: You do not live in Alabama.
This is called assuming the inverse.
If you do not live in Tuscaloosa, then you
do not live in Alabama.
4) Spatial Reasoning
` involves visualizing 3-D geometric figures or
objects to draw conclusions about properties or
relationships involving these figures or objects.
` A Rubik’s Cube™ is a 3x3x3 cube that is
comprised of 27 smaller cubes. Each of the six
faces of the large cube is painted a different
color (red,
(red blue,
blue green,
green yellow,
yellow orange,
orange and
white).
` How many of the smaller cubes are painted on
only one face?
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