Geometry: Week 6 - Faculty Perry,...

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Perry High School

Kevin M. Bond, PHD

Geometry: Week 6 Monday: 2.3 – Deductive Reasoning

Tuesday: 2.3 – Work Day

Wednesday: 2.4 – Reasoning with Properties from Algebra

Thursday: 2.5 – Proving Statements about Segments

Friday: 2.5 – Work Day

Next Week: 2.6, Review, Exam 2

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Perry High School

Kevin M. Bond, PHD

Monday: Mindfulness Training

This week: Loving Kindness Meditation

http://marc.ucla.edu/mpeg/05_Loving_Kindness_M

editation.mp3

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Kevin M. Bond, PHD

Debrief 2.2

Mindfulness Exercise

Questions on 2.2?

Mixed Review, p. 85: Evens

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Perry High School

Kevin M. Bond, PHD

Reasoning

Inductive Reasoning: Uses patterns to

arrive at a conclusion (conjecture).

Deductive Reasoning: Uses facts, rules,

definitions or properties to arrive at a

conclusion.

In either case, easiest to use symbols.

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Kevin M. Bond, PHD

Symbolism in Logic

Turn statements into letters.

Examples

John went to the movies = J

Sally likes to eat at Macarena = S

It rains = R

The Grass is Wet = G

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Perry High School

Kevin M. Bond, PHD

Conditional Statements Symbolism

Conditional Statements:

If hypothesis, then conclusion

If p, then q: pq

Verbally:

“If P then Q”

“P implies Q”

“P only if Q”

PQ,

Inverse?

Converse?

Contrapositive?

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Kevin M. Bond, PHD

Example 1


If it rains, then the grass is wet.

Key: p = “it rains”; q = “the grass is wet”

Translate into conditional: p q

Converse:

Inverse:

Contrapositive:

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Kevin M. Bond, PHD

Example 2


If I have a square, then each angle is 90°.

Key: p = I have a square; q = each angle is 90°

Translate into conditional: p q

Converse: q p

Inverse: ~p ~q

Contrapositive: ~q ~p

Is this a true biconditional?

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Perry High School

Kevin M. Bond, PHD

Inductive Reasoning

Inductive Reasoning – uses patterns to arrive at

a conclusion (conjecture)

• Establishes or increases the probability of a

conclusion, i.e., the conjecture is possible or

likely, but may still be false.

• E.g., Every time I run I get a cramp in my leg.

Therefore, if I run tonight, I will get a cramp in

my leg.

• Strong – Probably true

• Weak – Probably false

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Kevin M. Bond, PHD

Deductive Reasoning

Deductive Reasoning – uses facts, rules,

definitions, or properties to arrive at a conclusion.

• A “chain” of reasoning that either holds together

or falls apart.

• E.g., All Cretans are liars. Ramsy is a Cretans.

Therefore, Ramsy is a liar.

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Kevin M. Bond, PHD

Deductive Reasoning

Deductive Reasoning – uses facts, rules,

definitions, or properties to arrive at a conclusion.

• Valid – assuming the premises are true, the

conclusion necessarily is also true.

• Invalid – assuming the premises are true, it is

still possible for the conclusion to be false.

• Sound – Valid & the premises are actually true.

• Unsound – either not valid or at least one

premise is false.

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Kevin M. Bond, PHD

Deductive Reasoning

Some patterns of arguments are always valid.

• disjunctive syllogism, hypothetical syllogism

• modus ponens, modus tollens

• constructive dilemma, destructive dilemma

Some patterns of arguments are always invalid.

• affirming the consequent

• denying the antecedent,

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Kevin M. Bond, PHD

Modus Ponens

Law of Detachment

1. If it is raining, then

the street is wet.

2. It is raining.

3. Therefore, ____?

Rewrite symbolically

and solve.

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Kevin M. Bond, PHD

Modus Ponens

Law of Detachment


the street is wet.

2. It is raining.

3. Therefore, ____?

Key:

p=it is raining

q=the street is wet

p --> q

p

----------

q

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Kevin M. Bond, PHD

MP: Law of Detachment

1. All men are mortal.

2. Socrates is a man.

3. Therefore,____?

Rewrite in standard

form

Map symbolically

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Perry High School

Kevin M. Bond, PHD

MP: Law of Detachment

1. All men are mortal.

2. Socrates is a man.

3. Therefore,____?

Key:

p = someone is a man

q = that someone is a

mortal

p --> q

p

----------

q

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Kevin M. Bond, PHD

Hypothetical Syllogism

Law of Syllogism


the street is wet.

2. If the street is wet,

then the street is

slippery.

3. Therefore, if it is

raining,_____?

Hypothetical Syllogism

PQ

QR

Therefore, PR

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Kevin M. Bond, PHD

Syllogism Example

• If the circumference of a

circle is 8pi, then the

diameter is 8 inches.

• If the diameter of a circle

is 8 inches, then its radius

is 4 inches.

• If the radius of a circle is

4 inches, then its area is

16pi square inches.

• The circumference of

circle O is 8pi.

What can we conclude

about circle O?

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Kevin M. Bond, PHD

Syllogism Example

• If the circumference of a

circle is 8, then the

diameter is 8 inches.

• If the diameter of a circle

is 8 inches, then its radius

is 4 inches.

• If the radius of a circle is

4 inches, then its area is

16pi square inches.

• The circumference of

circle O is 8.

Key:

p=the circumference of

a circle is 8

q=the diameter of a

circle is 8 in.

r=the radius of a circle

is 4 in.

s=the area of a circle is

16.

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Kevin M. Bond, PHD

Syllogism Example

Symbolically:

p q

q r

r s

-------

???

Key:

p=the circumference of

a circle is 8

q=the diameter of a

circle is 8 in.

r=the radius of a circle

is 4 in.

s=the area of a circle is

16.

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Kevin M. Bond, PHD

Syllogism Example

Symbolically:

p q

q r

r s

-------

???

What if I know p is true?

p q

q r

r s

p

-------

???

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Kevin M. Bond, PHD

Another Example

If quadrilateral ABCD is

a square, then the

opposite sides of

ABCD are parallel.

If the opposite sides of

a quadrilateral are

parallel, then the

quadrilateral is a

parallelogram.

ABCD is a square.

Rewrite in symbols.

Show argument with a

conclusion regarding

ABCD being a

square.

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Kevin M. Bond, PHD

Another Example

If quadrilateral ABCD is

a square, then the

opposite sides of

ABCD are parallel.

If the opposite sides of

a quadrilateral are

parallel, then the

quadrilateral is a

parallelogram.

ABCD is a square.

Key:

p=quadrilateral ABCD is

a square

q=the opposite sides of

ABCD are parallel

r=the quadrilateral is a

parallelogram

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Kevin M. Bond, PHD

Another Example

Symbolically:

p q

q r

p

--------

???

Key:

p=quadrilateral ABCD is

a square

q=the opposite sides of

ABCD are parallel

r=the quadrilateral is a

parallelogram

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Perry High School

Kevin M. Bond, PHD

2.3 Practice

Day 1 (Today)

• 2.3, page 91

• 1-7

• 56-63

Day 2 (Tomorrow)

8–42 all

45–48 all

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Kevin M. Bond, PHD

Tuesday: Mindfulness Training



editation.mp3

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Perry High School

Kevin M. Bond, PHD

2.3 Work Day


Review

Day 1 (Yesterday)

• 2.3, page 91

• 1-7

• 56-63

Due by end of class

8–42 all

45–48 all

Show me when

completed

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Perry High School

Kevin M. Bond, PHD

Wednesday:

Mindfulness Training



editation.mp3

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Kevin M. Bond, PHD

Debrief 2.3



Questions on 2.3?

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Kevin M. Bond, PHD

Review

1. Rewrite, symbolically,

the biconditional as a

conditional and its

converse: We will go to

the beach if and only if it

is sunny.

2. Give a counterexample

to: (a) If a polygon has

four equal sides, then it

is a square. (b) if a

vehicle has wheels, then

it is a car.

3. Determine whether

the statement can be

combined with its

converse to form a

true biconditional: If

2x>8, then x=5.

4. What is the law of

detachment?

5. What is the law of

syllogism?

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Perry High School

Kevin M. Bond, PHD

2.4: Reasoning with Properties from Algebra

Solve each equation, give a reason for what you

do:

1. 3x=27

2. X+6=-17

3. X-9=18

4. (2/3)x=6

5. -x=4

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Kevin M. Bond, PHD

Algebraic Properties of Equality

• 2.4, Page 96

• Two Line proofs, i.e., justification of steps.

• Ex. 1

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Kevin M. Bond, PHD

Geometric Properties

Algebra Properties often have Geometric

Equivalents.

Page 98, Properties

Example #5

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2.4 Work

Due when class starts

2.4, Page 99

1-9 (together?)

10-15 all

16-26 evens

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35-50, all

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Perry High School

Kevin M. Bond, PHD

Thursday:

Mindfulness Training



editation.mp3

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Perry High School

Kevin M. Bond, PHD

Debrief 2.4


Questions on

Practice and Applications?

Mixed Review p. 101: Evens

I’ll spot check these while you work…

Start Reading 2.5

– Definitions, formulas, examples, etc.

– If finish, start guided practice and problems.

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Kevin M. Bond, PHD

2.5 Proving Statements

• Theorems

• Two-Column Proof

• Paragraph Proof

• Constructions

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Definitions

• Theorem – a true

statement that follows

from other true

statements

• Two-Column Proof –

a numbered proof that

has statements and

reasons showing a

logical order of an

argument.

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Kevin M. Bond, PHD

Let’s look at the text

Page 102

• Example 1

• Example 2

• Example 3

Two Column Proofs

Statement | Reason

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Construction Review

Constructions:

Recall How To…

• Copy a Segment

• Copy an Angle

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2.5 Practice

Day One (today)

Guided Practice

2.5, page 104+

1-5

29-39, odd

Day Two (tomorrow)

2.5, page 105+

6–11 all

12–18 all

21

22

28–40 even

Show when completed

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Perry High School

Kevin M. Bond, PHD

Friday: Mindfulness Training



editation.mp3

43

Perry High School

Kevin M. Bond, PHD

2.5 Practice

Day One (yesterday)

Guided Practice

2.5, page 104+

1-5

29-39, odd

Day Two (today)

2.5, page 105+

6–11 all

12–18 all

21

22

28–40 even

Show when completed Next Week: 2.6, Review, Exam on Friday

Geometry: Week 6 - Faculty Perry,...

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