Transcript Chapter 1 Tools of Geometry Section 1.1 Patterns and Inductive

CHAPTER 1
TOOLS OF GEOMETRY
Section 1.1 Patterns and Inductive Reasoning
INDUCTIVE REASONING:
REASONING BASED
ON PATTERNS YOU OBSERVE
Finding and Using a Pattern
 Examples:
 Find the next two terms of the sequence

•
3, 6, 12, 24,…
Each term is being multiplied by 2
 Next two terms: 48, 96

•
1, 2, 4, 7, 11, 16, 22, …
Adding another 1 to the previous
 Next two terms: 29, 37

EXAMPLES (CONT.)

Monday, Tuesday,
Wednesday…

Thursday, Friday
CONJECTURE:
A CONCLUSION YOU REACH
USING INDUCTIVE REASONING

Make a conjecture about the sum of the first 30
odd numbers
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•
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•
•
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1
1+3
1+3+5
1+3+5+7
1+3+5+7+9
1 + 3 + 5 + 7 + 9 + 11
=1
=4
=9
= 16
= 25
= 36
= 12
.
= 22
.
= 32
2
=4
.
= 52
.
= 62
Notice the pattern created
 (number of odd numbers, squared)
2
 The sum of the first 30 odd numbers would be 30

EXAMPLES (CONT.)

Make a conjecture about the sum of the first 35
odd numbers
2
Since the sum of the first 30 odd numbers is 30 , then
the sum of the first 35 odd numbers would be:
2
35
2
35 = 35 * 35, so:
•
TRUE OR FALSE?

If a statement is true,
we can prove it is true
for all cases


If a statement is false,
we need to provide
ONE counterexample.
A counterexample is a
specific example for
which the conjecture
is false
EXAMPLES: FIND A COUNTEREXAMPLE
FOR EACH CONJECTURE

The square of any
number is greater
than the original
number.



Counterexample:
Is
You can connect any
three points to form a
triangle
Counterexample: a
straight line
EXAMPLES (CONT)



ANY number and its
absolute value are
opposites.
Counterexample: A
positive number and
its absolute value
ie:
and

Alana makes a
conjecture about
slicing pizza. She
says that if you use
only straight cuts, the
number of pieces will
be twice the number
of cuts. Draw an
example that shows
you can make 7 pieces
using 3 cuts.
Classwork: pg. 6 # 1 – 6, 8, 11, 12, 19 –
22, 25 - 29
Homework: WS 1-1 #1-6, 13, 14