1.1 – PATTERNS AND INDUCTIVE REASONING Chapter 1: Basics of Geometry.
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Transcript of 1.1 – PATTERNS AND INDUCTIVE REASONING Chapter 1: Basics of Geometry.
![Page 1: 1.1 – PATTERNS AND INDUCTIVE REASONING Chapter 1: Basics of Geometry.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697bf751a28abf838c8057a/html5/thumbnails/1.jpg)
1.1 – PATTERNS AND INDUCTIVE REASONING
Chapter 1: Basics of Geometry
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Where did Geometry come from anyhow?
‘geometry’ = ‘geo’, meaning earth, and ‘metria’, meaning measure.
Euclid = “Father of Geometry” 300 BCGreeks used Geometry for building Modern Geometry enables our computers to
work so fast.
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Notice the Pattern
Much of Geometry came from people recognizing and describing
patterns.
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Ex 1: Sketch the next figure in the pattern:
1 432
Visual Patterns
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Number Patterns
Ex 2: Describe the pattern in this sequence. Predict the next number.
1, 4, 16, 64, 256
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Number Patterns
Ex 3: Describe the pattern in this sequence. Predict the next number.
-5, -2, 4, 13, 25
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Number Patterns
Ex 4: Describe the pattern in this sequence. Predict the next number.
3, 7, 15, 31,63
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Using Inductive Reasoning
Inductive Reasoning is the process of arriving at a general conclusion based on observations of specific examples.
Specific General
Ex 5: You purchased notebooks for 4 classes. Each notebook costs more than $5.00Conclusion: All notebooks cost more than $5.00
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The Three Stages to Reason
1) Look for a pattern2) Make a Conjecture3) Verify the Conjecture: make sure its
ALWAYS true
What even is a Conjecture? Its an unproven statement that’s based on observations. You can discuss it and modify it if necessary. It ain’t a rule yet!
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Making a Conjecture
Ask a Question: What is the sum of the 1st n odd positive integers?
1) List some examples and look for a pattern.1) First odd positive integer 1 = 12
2) Sum of first 2 odd integers 1 + 3 = 4 = 22
3) Sum of first 3 odd integers 1 + 3 + 5 = 9 = 32
4) Sum of first 4 odd integers 1 + 3 + 5 + 7 = 16 = 42
2) Conjecture: the sum of the 1st n odd positive integers is n2
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Proving a Conjecture is TRUE
To prove true, you must prove it is true for EVERY case. (Every example must fit the conjecture)
To prove a conjecture false, you only need to provide one counter example
Ex 6: Everyone in our class has blonde hair. Counter Example: Mrs. Pfeiffer, Mr. Nguyen…
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Find the Counter Example
Ex 7: Show the conjecture is false by finding a counterexample:
Conjecture: The difference of two positive numbers is always positive.
Counter Example: 2 - 8 = -6
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Find the Counter Example
Ex 8: Find the Counter Example:
Conjecture: the square of any positive number is always greater than the number itself.
Counter Examples:
1) 12 = 12) (0.5)2 = 0.25 which is NOT greater than
0.5
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Unproven Conjecture
Some Conjectures have been around for hundreds of years and are still unknown to be true or false. Goldbach’s Conjecture: all even numbers greater than
2 can be written as the sum of 2 primes. 4 = 2 + 2 6 = 3 + 3 8 = 3 + 5, etc.
It is unknown whether this is true for ALL cases, but it has not yet been disproven. A $1,000,000 prize is offered to the person that
can crack Goldbach’s Conjecture!
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Pattern Puzzle
To get the next number in a sequence, you multiply the previous number by 2 and
subtract 1. If the fourth number is 17, what is the first number in the sequence?
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Find the Pentagon that has no twin… the one that is different from all the others.
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Homework
Page 6: # 5 – 39 Odd, 53-71 Odd