Honors Geometry Section 1.0 Patterns and Inductive Reasoning
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Transcript of Honors Geometry Section 1.0 Patterns and Inductive Reasoning
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Honors Geometry Section 1.0
Patterns and Inductive Reasoning
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Geometry, like much of mathematics and science,
developed when people began recognizing and describing
patterns. Much of the reasoning in geometry consists of three
steps.
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1.
2.
3.
Recognize a pattern.
Make a conjecture about the pattern.
A conjecture is an educated guess based on past observations.
Prove the conjecture.
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Example 1: Give the next two terms in each sequence of numbers and describe the pattern in words.
2, 6, 18, 54… 1 1 1
1, , , ,...2 4 8
486 ,162
3by Multiply 32
1- ,161
21-by Multiply
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Example 1: Give the next two terms in each sequence of numbers and describe the pattern in words.
1, 3, 5, 7, 9…
1, 1.1, 21.1, 21.12, 321.12, …
13 ,11 2 Add
4321.123 ,123.321rightfar then theandleft far
theinteger tolarger next theAdd
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A sequence can be specified by an equation or “rule”. For the first
example (2,6,18,54,…), the sequence can be specified by the rule
where n = 1,2,3, etc. corresponding to the 1st term, 2nd term, 3rd term ,
etc.
12 3nna
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Example 2: Write a rule for the nth term for the 2nd and 3rd sequences in example 1.
1, 3, 5, 7, 9
1 1 11, , , ,...
2 4 8 1
211
n
na
)1(21 nan
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Reasoning based on past observations is called inductive reasoning.
Keep in mind that inductive reasoning does not guarantee a correct conclusion.
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Later in the course, we will prove a conjecture is true using deductive reasoning. To prove a conjecture is
false, you need to show a single example where the conjecture is
false. This single example is called acounterexample.
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Example 2: Show the conjecture is false.
The product of two positive numbers is always greater than the larger number.
If m is an integer*, then m2 > 0.
771 5.372
1
002
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You should know the following sets of numbers:
whole numbers:
integers:
rational numbers:
irrational numbers:
,...3,2,1,0
,...2,1,0,1,2...,
integers are b and a and b
a as written becan
rationalnot is