1 2-1 inductive reasoning & conjecture inductive reasoning is reasoning that uses a number of...

1

2-1 Inductive Reasoning & Conjecture

INDUCTIVE REASONING is reasoning thatuses a number of specific examples to arrive at a conclusion

When you assume an observed pattern will continue, you are using

INDUCTIVE REASONING.

1

2


A CONCLUSION reached usingINDUCTIVE REASONING is called aCONJECTURE.

2

3


Example 1Write a conjecture that describes the pattern in each sequence.

Use your conjecture to find the next term in the sequence.

3

4


Example 1aWhat is the next term?3, 6, 12, 24,

Conjecture: Multiply each term by 2 to get the next term.

The next term is 24 •2 = 4

48.

5


Example 1bWhat is the next term?

2, 4, 12, 48, 240Conjecture:

To get a new term, multiply the previous number by the position of the new number.The next term is

5

240•6 =1440.

6


Example 1c

Conjecture: The number of small trianglesis the perfect squares.

69,4,13,2,1 222

?1 4 9

What is the next

shape?

7


The next big triangle should have _____ little triangles.

7

94

1

8


EX 2Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture.

8

9


EX 2aThe sum of an odd number and an even number is __________.

Conjecture: The sum of an odd number and an even number is ______________.

9

an odd numberExample: 1 +4 = 5Example: 26 +47 = 73

10


EX 2bFor points L, M, & N, LM = 20, MN = 6, AND LN = 14.

10

L MN

14 6

20

11


Conjecture: N is between L and M. OR

L, M, and N are collinear.

11

L MN

12


Counterexample:An example that proves a statement false

12

13


Give a counterexample to prove the statement false.

If n is an integer, then 2n > n.One possible counterexample:

n = – 8 because…2(– 8) = – 16 and – 16 > – 8 is false!!!

13

14


Assignment:p.93 – 96 (#14 – 30 evens, 40 – 44, 64 – 66 all)

14

15

2-3 Conditional Statements

CONDITIONAL STATEMENTA statement that can be written in if-then form

An example of a conditional statement: IF Portage wins the game tonight, THEN we’ll be sectional champs.

15

16


HYPOTHESISthe part of a conditional statement immediately following the word IF

CONCLUSIONthe part of a conditional statement immediately following the word THEN

16

17

2-3 Conditional StatementsExample 1

Identify the hypothesis and conclusion of the conditional statement.

a.) If a polygon has 6 sides, then it is a hexagon.

Hypothesis: a polygon has 6 sidesConclusion: it is a hexagon

17

18

2-3 Conditional StatementsExample 1 (continued)

b.) Joe will advance to the next round if he completes the maze in his computer game.

Hypothesis: Joe completes the maze in his computer game

Conclusion: he will advance to the next round

18

19


19

Example 2Write the statement in if-then form, Then identify the hypothesis and conclusion of each conditional statement.

20

2-3 Conditional Statementsa.) A dog is Mrs. Lochmondy’s favorite animal.If-then form:

If it is a dog, then it is Mrs. Lochmondy’s favorite animal.Hypothesis:

it is a dogConclusion:

it is Mrs. Lochmondy’s favorite animal

20

21

2-3 Conditional Statementsb.) A 5-sided polygon is a pentagon.If-then form:

If it is a 5-sided polygon then it is a pentagon.

Hypothesis:it is a 5-sided polygon

Conclusion:it is a pentagon

21

22


CONVERSE:The statement formed by exchanging thehypothesis and conclusion

22

23

2-3 Conditional StatementsExample 3Write the conditional and converse of the statement.

Bats are mammals that can fly.Conditional:

If it is bat, then it is a mammal that can fly.Converse:

If it is a mammal that can fly, then it is a bat.

23

24


Assignment:p.109-111(#18 – 30, evens 50, 52)For #50 & 52, only write the conditional & converse.

24

25

2-4 Deductive Reasoning

Recall from section 2-1…..When you assume an observed pattern

will continue, you are usingINDUCTIVE REASONING.

25

26


Deductive reasoning usesfacts, rules, definitions, or properties toreach logical conclusions from given statements.

26

27


EX 1

Determine whether each conclusion is based on inductive reasoning (a pattern) or deductive reasoning (facts).

27

28


a.) In a small town where Tom lives, the month of April has had more rain than any other month for the past 4 years. Tom thinks that April will have the most rain this year. Answer: inductive reasoning –Tom is basing his reasoning on the pattern of the past 4 years.

28

29

2-4 Deductive Reasoningb.) Sondra learned from a metoerologist that if it is cloudy at night, it will not be as cold in the morning as if there were no clouds at night. Sondra knows it will be cloudy tonight, so she believes it will not be cold tomorrow morning. Answer: deductive reasoning ‒Sondra is using facts that she has learned about clouds and temperature.

29

30

2-4 Deductive Reasoningc.) Susan works in the attendance office and knows that all of the students in a certain geometry class are male. Terry Smith is in that geometry class, so Susan knows that Terry Smith is male.Answer: deductive reasoning –Susan’s reasoning is based on a fact she acquired while working in the attendance office.

30

31

2-4 Deductive Reasoningd.) After seeing many people outside walking their dogs, Joe observed that every poodle was being walked by an elderly person. Joe reasoned that poodles are owned exclusively by elderly people.

Answer: inductive reasoning –Joe made his conclusion based on a pattern he saw.

32

2-5 Postulates and Paragraph Proofs

Postulate or axiom –

a statement that is accepted as true without proof

33


Theorem ‒

A statement in geometry that has been proven

34


Proof –

a logical argument in which each statement you make is supported by a statement accepted as true

35

2-5 Postulates and Paragraph Proofs Examples of Postulates

KNOW THESE POSTULATES!!! 2.1 Through any 2 points there is

exactly one line.

2.2 Through any 3 noncollinear points there is exactly one plane.

36


2.3 A line contains at least 2 points.

2.4 A plane contains at least 3 noncollinear points.

2.5 If 2 points lie in a plane, then the entire line containing those 2 points lies in that plane.

37


2.6 If 2 lines intersect, then they intersect in exactly one point.

2.7 If 2 planes intersect, then they intersect in a line.

38


Example 2

Use the diagram on page 126 for these next examples.

Explain how the picture illustrates that each statement is true. Then state the postulate that can be used to show each statement is true.

39


a.) Points F and G lie in plane Q and on line m. Therefore, line m lies entirely in plane Q.

Answer:

Points F and G lie on line m, and the line lies in plane Q.

(2.5) If 2 points lie in a plane, then the line containing the 2 points lies in the plane.

40


b.) Points A and C determine a line.

Answer:

Points A and C lie along an edge, the line that they determine.

(2.1) Through any 2 points there is exactly one line.

41


c.) Lines s and t intersect at point D.

Answer:

Lines s and t of this lattice intersect at only one location point D.

(2.6) If 2 lines intersect, then their intersection is exactly one point.

42


d.) Line m contains points F and G. Point E can also be on line m.

Answer:

The edge of the building is a straight line m. Points E, F, and G lie along this edge, so they line along line m.

(2.3) A line contains at least 2 points.

43


Midpoint Theorem:

If M is the midpoint of

then .

AB

MBAM

44


Assignment:p.120(#10-15 all)

p.129-130(#16-28 even, 34-40 even)

2-6 Algebraic ProofDay 1

See p.134 for the book’s way of explaining this.

PROPERTYDESCRIPTION OF

PROPERTY

Addition Property of Equality

Subtraction Property of Equality

45

Add the same value toboth sides of an equation.

Subtract the same valuefrom both sides of an equation.



PROPERTY

Multiplication Property of Equality

Division Property of Equality

46

Multiply both sides of an equation by the same

nonzero value.

Divide both sides of an equation by the same

nonzero value.



PROPERTY

Reflexive Property of Equality

Symmetric Property of Equality

47

a = a

If a =b, then b = a.



PROPERTY

Transitive Property of Equality

Substitution Property

48

If a = b and b = c, then a = c.

If a = b, then a can be substituted for b in any equation or expression.



PROPERTY

Distributive Property

49

a(b + c) = ab + ac


EXAMPLES

State the property that justifies each statement.

50


51

EX 1If 3x = 15, then x = 5.

Answer:Division Property of Equality

ORMultiplication Property of Equality


52

Answer: Subtraction Property of Equality

ORAddition Property of Equality

EX 2

If 4x + 2 = 22, then 4x = 20.


EX 3

If 7a = 5y and 5y = 3p, then 7a = 3p.

53

Answer: Transitive Property of Equality


EX 4

If and , then

54

PQAB MNPQ

.PQAB

Answer:Transitive Property of Equality


EX 5

55

DEDE Answer:

Reflexive Property of Equality


EX 6

If 3 = AB, then AB = 3.

56

Answer:Symmetric Property of Equality


EX 7

If AB = 7.5 and CD = 7.5, then AB = CD.

57

Answer: Substitution Property of Equality


58

Assignment:p.137-141(#1-4,9-16,46-48,56-58)


EX 8

Write a justification for each step.

Prove:

If 2(5 – 3a) – 4(a + 7) = 92, then a = ‒11.

59


STATEMENTS REASONS

2(5 – 3a) – 4(a + 7) = 92

10 – 6a – 4a – 28 = 92

60

Given

Distributive Prop


STATEMENTS REASONS

–10a – 18 = 92

–10a – 18 = 92 +18 +18

61

Substitution Prop

Addition Prop


STATEMENTS REASONS

–10a = 110

a = –11

62

10

110

10

10

a

Substitution Prop

Division Prop

Substitution Prop


EX 9

Write a two-column proof to verify the conjecture.

If 4(3x + 5) – 4x = 2(3x + 5) – 8, then x = –9.

63


STATEMENTS REASONS

4(3x + 5) – 4x = 2(3x + 5) – 8

12x + 20 – 4x = 6x + 10 – 8

64

Given

DistributiveProp


STATEMENTS REASONS

8x + 20 = 6x + 2

8x + 20 = 6x + 2 –6x –6x

65

Substitution Prop

Subtraction Prop


STATEMENTS REASONS

2x + 20 = 2

2x + 20 = 2 –20 –20

66

Substitution Prop

Subtraction Prop


STATEMENTS REASONS

2x = –18

x = –9

67

2

18

2

2

x

Substitution Prop

Division Prop

Substitution Prop


EX 10

If the distance d an object travels is given by d = 20t + 5, the time t that the object travels is given by

Write a two column proof to verify this conjecture.

68

.20

5

dt


STATEMENTS REASONS

d = 20t + 5

d – 5 = 20t

69

Given

Subtraction Prop


STATEMENTS REASONS

70

td

20

5

20

5

dt

Division Property

Symmetric Prop


EX 11

Write a two-column proof to prove the conjecture.

If , mB = 2mC, and mC = 45, then mA = 90.

71

BA


STATEMENTS REASONS

mB = 2mC mC = 45

mA = mB

72

BA Given

Defn of ≅ s


STATEMENTS REASONS

mA = 2mC

mA = 2 (45)

mA = 90

73

Substitution Prop

Substitution Prop

Substitution Prop


Assignment:

p.137 – 139(#5 – 7,17 – 19)

74


75


76


77


78

1 2-1 inductive reasoning & conjecture inductive reasoning is reasoning that uses a number of...

Documents