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Geometry Integrated wAlgebra
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Learn that inductive reasoning does not always lead to a good conclusion ... You may also test right, isosceles, and equilateral triangles, so long as you do ... – PowerPoint PPT presentation
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Title: Geometry Integrated wAlgebra
1
Geometry Integrated w/Algebra
- Section 1-5
- Inductive Reasoning
2
Focus
- Apply inductive reasoning to many situations
- Learn that inductive reasoning does not always
lead to a good conclusion
3
What is inductive reasoning?
- When you make a conjecture based on past
experience - conjecture a guess based on past experience
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Examples of Inductive Reasoning
- Medicine Rosario looked in the mirror one
morning and notices a rash on her face. It
occurred to her she might be allergic to a new
facial soap she had used the night before, but
then she noticed the rash on her legs as well.
The doctor asked Rosario some questions. Looked
at her rash, and decided that Rosario was
allergic to penicillin. The doctor made some
observations and used past experience to reach
this diagnosis.
5
More examples of inductive reasoning
- Mechanic could use inductive reasoning when
repairing cars - Store owner uses inductive reasoning when
deciding what to reorder based on what has sold
well in the past - Inductive reasoning is often used to solve
problems in mathematics
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Sample 1Use inductive reasoning
- Do you think the sum of any two odd numbers is
even or odd? Explain your reasoning.
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Sample 1 Solution
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Sample 1 Reasoning
- Even though many more pairs of numbers could be
checked (and the sum will always be even), there
will always be many times more pairs that have
not been checked. If hundreds or thousands or
millions of pairs are checked the pattern of
always getting an even sum seemsto lend
overwhelming support to the truth of the
conjecture. However, because of the unlimited
number of odd numbers, no one can say with
absolute certainty that some pair might show up
that is not even. - Thus, inductive reasoning can never establish the
truth of a conjecture that involve an infinite
set of mathematical objects such as numbers or
geometric figures.
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Talk it Over 1
- In sample 1, does inductive reasoning show that
the sum of two odd numbers is always even?
Explain. - Do you think the sum of any three odd numbers is
even or odd? Explain your reasoning.
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Talk it Over 1 Solution
- In sample 1, does inductive reasoning show that
the sum of two odd numbers is always even?
Explain. - No there are infinitely many odd numbers, and
there is no way to test every possible pair of
them to check that all the sums are even.
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Talk it Over 1 Solution
- Do you think the sum of any three odd numbers is
even or odd? Explain your reasoning. - It is odd the sum of two odd numbers is even,
and when you add one more odd number to this even
sum the result will be odd.
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Sample 2Make a conjecture
- Make a conjecture about a relationship among the
exterior angles of a polygon - An exterior angle of a polygon is an angle formed
by extending a side of the polygon - Problem Solving Strategy
- Use inductive reasoning
- Draw several polygons
- Measure the exterior angles and make a table
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Sample 2 Solution
exterior angle
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Table of Values
Make a conjecture based on your table of values
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What is your conjecture about the exterior angles
of a polygon?
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Talk it Over 3
- Is it a good idea to test only regular polygons
when you are making a conjecture? - Why or why not? (Note in a regular polygon, all
the sides are the same length and all the angles
are equal in measure.)
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Talk it Over 3 Solution
- Is it a good idea to test only regular polygons
when you are making a conjecture? - No the property you are testing might be unique
to regular polygons. Also if you test only
regular polygons, your conjecture would apply
only to them.
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Talk it Over 4
- Is the conjecture in Sample 2 true for interior
angles? Why or why not? - Conjecture The sum of the measures of the
exterior angles of a polygon is 360 - The sum of the measures of the interior angles of
a polygon is 360??
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Talk it Over 4 Solution
- Is the conjecture in Sample 2 true for interior
angles? Why or why not? - Conjecture The sum of the measures of the
interior angles of a polygon is 360 - No for example, you already know the sum of the
measures of the interior angles of a triangle is
180
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Shortcomings of Inductive Reasoning
- Movie executives sometimes use inductive
reasoning. - Based on past experience, they predict which
movies will be popular in the future. - They are not always correct.
- For example, the sequel of a hit movie is not
always as popular as the original.
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Counterexamples
- You cannot prove that a conjecture is true just
by using inductive reasoning. - For example, in samples 1 and 2 it would be
impossible to test all the possibilities. - However, you can disprove a conjecture by finding
any example that does not work, a counterexample.
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Sample 3
- Tell whether you think the inequality x x2 is
True or False. If you think it is false, give a
counterexample.
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Sample 3 Solution
- Problem Solving Strategy Use a Table.
?
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Sample 3 Solution
- Problem Solving Strategy Use a Table.
?
Watch out ! For a statement to be considered
true, it must be true in all cases.
This is a counterexample 0.5 is not less than
0.25
Conclusion x x2 is false
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Talk it over 5
- How many counterexamples are needed to disprove a
conjecture?
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Talk it over 5
- How many counterexamples are needed to disprove a
conjecture? - ONE
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Talk it Over 6
- Suppose you are using inductive reasoning to make
a conjecture about triangles. What types of
triangles should you test to check your
conjecture?
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Talk it Over 6
- Suppose you are using inductive reasoning to make
a conjecture about triangles. What types of
triangles should you test to check your
conjecture? - You should test acute, obtuse, and scalene
triangles. You may also test right, isosceles,
and equilateral triangles, so long as you do not
consider them exclusively.
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Talk it Over 7
- Tell whether you think the inequality x gt x is
True or False. If it is false, give a
counterexample.
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Talk it Over 7
- Tell whether you think the inequality x gt x is
True or False. If it is false, give a
counterexample. - The inequality is false any number x 0
provides a counterexample.
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Look Back
- Can you use inductive reasoning to show that a
conjecture is always true? Sometimes true? Never
true? Why or why not?
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Look Back
- Can you use inductive reasoning to show that a
conjecture is always true? Sometimes true? Never
true? Why or why not? - Except in finite cases, you cannot use inductive
reasoning to prove that a conjecture is always
true or never true, because you could not test
every possibility. - An example of a finite case may be, "Every
student in this school has a middle name." - You could question every student in the school to
test the conjecture and so could prove it true or
false. - You can use inductive reasoning to show that a
conjecture is sometimes true in fact, that is
the reason for using inductive reasoning.
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THE END
