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 = 37333 / 9 = 37

3. 1 x 8 + 1 = 9

12 x 8 + 2 = 98123 x 8 + 3 = 9871234 x 8 + 4 = 987612345 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:

If we draw a circle with two points on it and connected the points, I end up with 2 sections of a circle.

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.

2

1

2

1

4

3

2

1

1

2

Sections– find a pattern and predict

Number of points Number of sections

2

3

4

5

6

7

10

100

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 n 2 n 2 - 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 2x2 square--12 toothpicks 3x3 square (draw

this…)

How many toothpicks are needed to draw: a 4x4 square?.

9. Toothpicks- dataSquare No. of

toothpicks1x1

2x23x3

4x4

5x5

6x6

7x7

8x8

9x9

10x10

Square No. of toothpicks