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1.1 Inductive and Deductive Reasoning

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1.1 Inductive and Deductive Reasoning

Inductive reasoning is the process of arriving at a general conclusion based on observations of specific examples.

A scientist takes a piece of salt, turns it over a burning candle, and observes that it burns with a yellow flame. She does this with other pieces of salt, finding they all burn with a yellow flame. She therefore makes the following conjecture: “All salt burns with a yellow flame.”

Caveperson Stony Grok picks up a rock, drops it into a lake, and watches it sink. He picks up a second rock, drops it into the lake and it also sinks. He does this five more times, and each time the rock heads straight to the bottom of the lake. Stony conjectures: :Ura nok seblu,” which translates as: ___________________

A mathematician lands at the airport of the kingdom of Moravia. He desperately needs to use the bathroom, but he is very shy, and social customs of the kingdom would not permit him to use the wrong bathroom. He locates the doors to what appear to be two bathrooms. He observes men enter the door marked “Warvan” and women enter the door marked “Cupore.” He is finally ready to make his conjecture. How does he spell relief?

The sum of two two-digit numbers is a three-digit number.

A counterexample will show this conjecture to be false.

Find the next number:

3, 7, 11, 15, 19, ___

2, 6, 18, 54, ____

1, 4, 9, 16, 25, 36, ___

What is the maximum number of regions formed when points are connected by line segments on a circle?

10

Points Regions

11

9

1

1

1

12

17

18

13

19

2

2

20

4

3

14

16

4

3

22

24

21

23

4

8

7

25

26

5

5

16

6

30

15

29

27

28

31

6

31

2

8

What is the maximum number of regions formed when chords are constructed on a circle?

5

Chords Regions

1

0

1

4

1

2

3

6

4

2

8

11

9

3

7

2

10

4

11

7

5

16

1: A, E, F, H, I, K, L, M, N, T, V, W,

2: B, C, D, G, J, O, P, Q, R, S, U, Where should X, Y, and Z be placed?

Deductive reasoning is the process of proving a specific conclusion from one or more general statements. A conclusion that is proved true by deductive reasoning is called a theorem.

Select a number. Multiply the number by 6. Add 15 to the product. Divide the sum by 3. Subtract 5 from the quotient. Divide the difference by 2.

Conjecture:

Examples will not prove the conjecture. We need to use deductive reasoning.

Select a number. Multiply the number by 6. Add 15 to the product. Divide the sum by 3. Subtract 5 from the quotient. Divide the difference by 2.

Call the selected number n.

The first operation will give us 6n.

The second operation yields 6n+15

The third operation, 2n+5

The fourth operation, 2n

Finally n