Transcript Document

2-1 Inductive Reasoning &
Conjecture
INDUCTIVE REASONING is reasoning that
uses a number of specific examples
to arrive at a conclusion
When you assume an observed pattern
will continue, you are using
INDUCTIVE REASONING.
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2-1 Inductive Reasoning &
Conjecture
A CONCLUSION reached using
INDUCTIVE REASONING is called a
CONJECTURE.
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2-1 Inductive Reasoning &
Conjecture
Example 1
Write a conjecture that describes the
pattern in each sequence.
Use your conjecture to find the next term
in the sequence.
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2-1 Inductive Reasoning &
Conjecture
Example 1a
What 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 = 48.
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2-1 Inductive Reasoning &
Conjecture
Example 1b
What is the next term?
2, 4, 12, 48, 240
Conjecture:
To get a new term, multiply the
previous number by the position of
the new number.
The next term is240•6 =1440.
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2-1 Inductive Reasoning &
Conjecture
Example 1c
?
1
4
9
Conjecture: The number of small triangles
is the perfect squares.
1 , 2 , 3 1, 4, 9
2
2
2
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2-1 Inductive Reasoning &
Conjecture
The next big triangle should have _____
little triangles.
9
4
1
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2-1 Inductive Reasoning &
Conjecture
EX 2
Make a conjecture about each value or
geometric relationship. List or draw some
examples that support your conjecture.
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2-1 Inductive Reasoning &
Conjecture
EX 2a
The sum of an odd number and an even
number is __________.
Conjecture: The sum of an odd number
an odd number
and an even number is ______________.
Example: 1 +4 = 5
Example: 26 +47 = 73
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2-1 Inductive Reasoning &
Conjecture
EX 2b
For points L, M, & N, LM = 20, MN = 6,
AND LN = 14.
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L
M
N
14
6
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2-1 Inductive Reasoning &
Conjecture
L
M
N
Conjecture: N is between L and M.
OR
L, M, and N are collinear.
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2-1 Inductive Reasoning &
Conjecture
Counterexample:
An example that proves a statement false
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2-1 Inductive Reasoning &
Conjecture
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!!!
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2-1 Inductive Reasoning &
Conjecture
Assignment:
p.93 – 96
(#14 – 30 evens, 40 – 44, 64 – 66 all)
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2-3 Conditional Statements
CONDITIONAL STATEMENT
A 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.
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2-3 Conditional Statements
HYPOTHESIS
the part of a conditional statement
immediately following the word IF
CONCLUSION
the part of a conditional statement
immediately following the word THEN
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2-3 Conditional Statements
Example 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 sides
Conclusion: it is a hexagon
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2-3 Conditional Statements
Example 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
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2-3 Conditional Statements
Example 2
Write the statement in if-then form, Then
identify the hypothesis and conclusion of
each conditional statement.
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2-3 Conditional Statements
a.) 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 dog
Conclusion:
it is Mrs. Lochmondy’s favorite animal
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2-3 Conditional Statements
b.) 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
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2-3 Conditional Statements
CONVERSE:
The statement formed by exchanging the
hypothesis and conclusion
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2-3 Conditional Statements
Example 3
Write 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.
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2-3 Conditional Statements
Assignment:
p.109-111(#18 – 30, evens 50, 52)
For #50 & 52, only write the conditional &
converse.
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2-4 Deductive Reasoning
Recall from section 2-1…..
When you assume an observed pattern
will continue, you are using
INDUCTIVE REASONING.
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2-4 Deductive Reasoning
Deductive reasoning uses
facts, rules, definitions, or properties to
reach logical conclusions from given
statements.
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2-4 Deductive Reasoning
EX 1
Determine whether each conclusion is
based on inductive reasoning (a pattern)
or deductive reasoning (facts).
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2-4 Deductive Reasoning
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.
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2-4 Deductive Reasoning
b.) 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.
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2-4 Deductive Reasoning
c.) 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.
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2-4 Deductive Reasoning
d.) 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.
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2-5 Postulates and Paragraph Proofs
Postulate or axiom –
a statement that is accepted as true
without proof
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2-5 Postulates and Paragraph Proofs
Theorem ‒
A statement in geometry that has
been proven
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2-5 Postulates and Paragraph Proofs
Proof –
a logical argument in which each
statement you make is supported by a
statement accepted as true
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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.
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2-5 Postulates and Paragraph Proofs
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.
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2-5 Postulates and Paragraph Proofs
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.
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2-5 Postulates and Paragraph Proofs
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.
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2-5 Postulates and Paragraph Proofs
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
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plane.
2-5 Postulates and Paragraph Proofs
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.
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2-5 Postulates and Paragraph Proofs
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.
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2-5 Postulates and Paragraph Proofs
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.
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2-5 Postulates and Paragraph Proofs
Midpoint Theorem:
If M is the midpoint of AB
then AM  MB .
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2-5 Postulates and Paragraph Proofs
Assignment:
p.120(#10-15 all)
p.129-130(#16-28 even, 34-40 even)
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2-6 Algebraic Proof
Day 1
See p.134 for the book’s way of explaining this.
PROPERTY
DESCRIPTION OF
PROPERTY
Addition Property of
Equality
Add the same value to
both sides of an equation.
Subtract the same value
Subtraction Property of
from both sides of an
Equality
equation.
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2-6 Algebraic Proof
Day 1
DESCRIPTION OF
PROPERTY
PROPERTY
Multiply both sides of an
Multiplication
equation by the same
Property of Equality
nonzero value.
Divide both sides of an
Division Property of
equation by the same
Equality
nonzero value.
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2-6 Algebraic Proof
Day 1
PROPERTY
DESCRIPTION OF
PROPERTY
Reflexive Property of
Equality
a=a
Symmetric Property
of Equality
If a =b,
then b = a.
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2-6 Algebraic Proof
Day 1
PROPERTY
Transitive Property of
Equality
DESCRIPTION OF
PROPERTY
If a = b and b = c,
then a = c.
If a = b, then a can be
Substitution Property substituted for b in any
equation or expression.
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2-6 Algebraic Proof
Day 1
PROPERTY
DESCRIPTION OF
PROPERTY
Distributive Property
a(b + c) = ab + ac
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2-6 Algebraic Proof
Day 1
EXAMPLES
State the property that justifies each
statement.
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2-6 Algebraic Proof
Day 1
EX 1
If 3x = 15, then x = 5.
Answer:
Division Property of Equality
OR
Multiplication Property of Equality
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2-6 Algebraic Proof
Day 1
EX 2
If 4x + 2 = 22, then 4x = 20.
Answer:
Subtraction Property of Equality
OR
Addition Property of Equality
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2-6 Algebraic Proof
Day 1
EX 3
If 7a = 5y and 5y = 3p, then 7a = 3p.
Answer:
Transitive Property of Equality
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2-6 Algebraic Proof
Day 1
EX 4
If
AB  PQ
and
PQ  MN
, then
AB  PQ .
Answer:
Transitive Property of Equality
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2-6 Algebraic Proof
Day 1
EX 5
DE  DE
Answer:
Reflexive Property of Equality
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2-6 Algebraic Proof
Day 1
EX 6
If 3 = AB, then AB = 3.
Answer:
Symmetric Property of Equality
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2-6 Algebraic Proof
Day 1
EX 7
If AB = 7.5 and CD = 7.5, then AB = CD.
Answer:
Substitution Property of Equality
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2-6 Algebraic Proof
Day 1
Assignment:
p.137-141(#1-4,9-16,46-48,56-58)
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2-6 Algebraic Proof
Day 2
EX 8
Write a justification for each step.
Prove:
If 2(5 – 3a) – 4(a + 7) = 92, then a = ‒11.
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
2(5 – 3a) – 4(a + 7) = 92
Given
10 – 6a – 4a – 28 = 92
Distributive Prop
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
–10a – 18 = 92
Substitution Prop
–10a – 18 = 92
+18 +18
Addition Prop
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2-6 Algebraic Proof
Day 2
STATEMENTS
–10a = 110
10 a  110

10
10
a = –11
REASONS
Substitution Prop
Division Prop
Substitution Prop
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2-6 Algebraic Proof
Day 2
EX 9
Write a two-column proof to verify
the conjecture.
If 4(3x + 5) – 4x = 2(3x + 5) – 8,
then x = –9.
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
4(3x + 5) – 4x = 2(3x + 5) – 8
Given
12x + 20 – 4x = 6x + 10 – 8
Distributive
Prop
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
8x + 20 = 6x + 2
Substitution Prop
8x + 20 = 6x + 2
–6x
–6x
Subtraction Prop
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
2x + 20 = 2
Substitution Prop
2x + 20 = 2
–20 –20
Subtraction Prop
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
2x = –18
Substitution Prop
2 x  18

2
2
Division Prop
x = –9
Substitution Prop
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2-6 Algebraic Proof
Day 2
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
d 5
t
.
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Write a two column proof
to verify this conjecture.
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
d = 20t + 5
Given
d – 5 = 20t
Subtraction Prop
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
d 5
t
20
Division Property
d 5
t
20
Symmetric Prop
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2-6 Algebraic Proof
Day 2
EX 11
Write a two-column proof to prove
the conjecture.
If A  B, mB = 2mC,
and mC = 45, then mA = 90.
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
A  B
mB = 2mC
mC = 45
mA = mB
Given
Defn of ≅ s
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2-6 Algebraic Proof
Day 2
STATEMENTS
REASONS
mA = 2mC
Substitution Prop
mA = 2 (45)
Substitution Prop
mA = 90
Substitution Prop
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2-6 Algebraic Proof
Day 2
Assignment:
p.137 – 139(#5 – 7,17 – 19)
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2-6 Algebraic Proof
Day 2
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2-6 Algebraic Proof
Day 2
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2-6 Algebraic Proof
Day 2
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2-6 Algebraic Proof
Day 2
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