Download 1.1 Patterns and Inductive Reasoning

Miss Stanley
The Green School
Geometry Fall 2011
Name _____________________________
Date ______________________________
Lesson 1.1 Patterns and Inductive Reasoning
Finding and using a pattern.
3, 6, 12, 24…
Triangle, Rectangle, Pentagon…
What is the pattern?
What is the pattern?
What are the next three numbers in the sequence?
What are the next three figures in the sequence?
A conjecture is a conclusion you reach using inductive reasoning.
Use the pattern to find the sum of consecutive odd numbers.
1
1+3
1+3+5
1+3+5+7
1+3+5+7+9
=1
=4
=9
= 16
=
= 12
= 22
= 32
= 42
=
Now use this same pattern to find the sum of the first 30 odd numbers.
1 + 3 + 5 + 7 + 9…
First 30 odd numbers.
=
=
A counterexample is an example where the conjecture is incorrect. Find a counterexample:
The square of any number is
greater than the original number.
You can connect any three points
to form a triangle.
Any number and its absolute
value are opposites.
Classwork: Complete the following problems by the end of the class period.
Find the next three numbers in the sequence.
3, 33, 333, 3333, …
81, 27, 9, 3, …
1 1 1 1
1, , , , ...
4 9 16 25
1, 1, 2, 3, 5, 8, 13, …
Use the table to find the following sums.
2
2+4
2+4+6
2+4+6+8
2 + 4 + 6 + 8 + 10
=1•2
=2•3
=3•4
=4•5
=5•6
=2
=6
= 12
= 20
= 30
Find the sum of the first 6 positive even numbers.
Find the sum of the first 30 positive even numbers.
What conjecture can you make about the sum of the first 100 even numbers? In other words, what rule
are you following?
Find one counterexample to show that each conjecture is false?
The sum of two numbers is greater than either
number.
The product of two positive numbers is greater than
either number.
The difference of two integers is less than either
integer.
The quotient of two proper fractions is a proper
fraction.