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MATH 104 Chapter 1
Reasoning
Inductive Reasoning
• Definition: Reasoning from specific to general
• Examples of Patterns
Deductive Reasoning
• Definition: Reasoning from general to specific
Use Inductive or Deductive Reasoning
• Example #1: What is the product of an odd
and an even number?
Divisible by 3
• Statement: If the sum of the digits of a
number is divisible by 3, then the number is
divisible by 3. True or false?
Number
Sum of digits
Sum div by 3? Number div
by 3?
Divisible by 4
• Statement: If the sum of the digits of a number is
divisible by 4, then the number is divisible by 4.
True or false?
Number
Sum of digits
Sum div by 4? Number div
by 4?
Example
•
•
•
•
•
•
Pick a number.
Multiply by 6.
Add 4.
Divide by 2.
Subtract 2.
What is your result?
Use inductive reasoning
1. Exponents:
Notice that 21=2, 22=4, 23=8, 24=16,
25=32. Predict what the last digit of 2100 is.
Use inductive reasoning to predict
Use inductive reasoning to predict the next three lines. Then
perform arithmetic to determine whether your conjecture
is correct:
2.
111 / 3 = 37
222 / 6 = 37
333 / 9 = 37
3.
1x8+
1=9
12 x 8 +
2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98,765
4. Calculator patterns
a) Use a calculator to find the answers to
6x6=
66x66=
666x666=
6666x6666=
b) Describe a pattern in the numbers being
multiplied and the resulting products.
c) Use the pattern to write the next three
multiplications and their products
d) Use a calculator to verify.
5. 142,857
5. Calculate the following:
142,857 x 2=
142,857 x 3=
142,857 x 4=
What do you notice that all of your answers
have in common? Do you think this will
continue indefinitely?
6.
Sections of a circle:
1
2
If we draw a circle with two
points on it and connected the
points, I end up with 2
sections of a circle.
1
2
3
1
2
2
4
When I draw a circle with 3
points and connect the points,
I get 4 sections of the circle.
When I use 4 points, I get 8
sections.
1
Sections– find a pattern and predict
Number of points
2
3
4
5
6
7
10
100
Number of sections
7. Try inductive or deductive
If you take a positive integer (1,2,3,4,5,…) that is NOT
divisible by 3, then square that integer, and then
subtract one, what happens? Is the result ALWAYS
divisible by 3?
n
n2
n2 -1
8.
Divisibility by 6:
Show that anytime you take three consecutive
positive integers and multiply them together
that the resulting number is divisible by 6.
Numbers
Product
Divisible by 6?
9.
Toothpicks:
Consider the following pattern:
1x1 square --4 toothpicks
(draw this…)
2x2 square--12 toothpicks
3x3 square
How many toothpicks are needed to draw:
a 4x4 square?
.
9. Toothpicks- data
Square
1x1
2x2
3x3
4x4
5x5
6x6
7x7
8x8
9x9
10x10
No. of
toothpicks
Square
No. of
toothpicks