Transcript 2-1-inductive-reasoning-edited-oct-10-2012x

Folding Paper
How many rectangles?
# of folds
1st fold
2nd fold
# of Rectangles
2
4
3rd fold
8
4th fold
16
5th fold
32
25th fold
50
Can you find
a pattern
that you can
describe
and then
predict how
many
rectangles
on the next
fold?
2.1 Inductive Reasoning
Chapter 2 Pg 82-84
Objectives:
• I CAN use patterns to make conjectures.
• I CAN disprove geometric conjectures using
counterexamples.
DoDEA Standards Addressed in this lesson:
G.6: Proof and Reasoning
Students apply geometric skills to making conjectures, using axioms and theorems,
understanding the converse and contrapositive of a statement, constructing logical
arguments, and writing geometric proofs.
G.1.1: Demonstrate understanding by identifying and giving examples of undefined
terms, axioms, theorems, and inductive and deductive reasoning
2
Patterns and Inductive Reasoning
You will learn to identify patterns and use inductive reasoning.
If you were to see dark, towering clouds
approaching, you might want to take
cover.
What would cause you to think
bad weather is on its way ?
Your past experience tells you that a
thunderstorm is likely to happen.
When you make a conclusion based on a pattern of examples or past
events, you are using inductive reasoning.
3
What teams will go to the next
Super Bowl?
How do you know?
What evidence did you use to make your
prediction?
Inductive Reasoning is used to predict a future
event based on observed patterns.
Patterns and Inductive Reasoning
You can use inductive reasoning to find the next terms in a sequence.
Find the next three terms of the sequence:
3,
6,
X2
24,
12,
X2
X2
48,
X2
96,
X2
Find the next three terms of the sequence:
7,
+1
8,
16,
11,
+3
+5
23,
+7
32
+9
5
Patterns and Inductive Reasoning
Draw the next figure in the pattern.
Lesson 2-1 Patterns and Inductive
Reasoning
6
Example #1
Describe how to sketch the
4th figure. Then sketch it.
Each circle is divided into twice as
many equal regions as the figure
number. The fourth figure should
be divided into eighths and the
section just above the horizontal
segment on the left should be
shaded.
7
Example #2
Describe the pattern. Write
the next three numbers.
7,
 21,
3
 63,
3
 189,...
3
Multiply by 3 to get the next
number in the sequence.
189  3  567
567  3  1701
1701 3  5103
8
Example 1C: Identifying a Pattern
Find the next item in the pattern.
In this pattern, the figure rotates 90° counter-clockwise each time.
The next figure is
.
Vocabulary
conjecture
inductive reasoning
counterexample
What is a conjecture?
What is inductive reasoning?
Conjecture: conclusion made based
on observation
Inductive Reasoning: conjecture
based on patterns
Proving conjectures TRUE
is very hard.
Proving conjectures FALSE
is much easier.
What is a counterexample?
Counterexample: example that
How do you disprove a conjecture?
shows a conjecture is false
What are the steps for inductive
reasoning?
How do you use inductive
reasoning?
Steps for Inductive Reasoning
1. Find pattern.
2.Make a conjecture.
3.Test your conjecture
or find a counterexample.
11
To show that a conjecture is always true,
you must prove it.
To show that a conjecture is false, you have to
find only one example in which the conjecture is
NOT true. This case is called a counterexample.
A counterexample can be a
drawing, a statement, or a number.
Steps for Inductive Reasoning
Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find
a counterexample.
Check It Out!
Show that the conjecture is false by finding a
counterexample.
Supplementary angles are always adjacent.
True or false? If false provide a counterexample.
This drawing is a
counterexample
to the statement,
making it false.
23°
157°
The supplementary angles are not adjacent, so the conjecture is
false.
Example #3
Make and test a conjecture
about the sum of any 3
consecutive numbers.
(Consecutive numbers are
numbers that follow one
after another like
3, 4, and 5.)
3  4  5  12  4  3
6  7  8  21  7  3
8  9 10  27  9  3
11 12 13  36 12  3
Conjecture:
The sum of any 3 consecutive numbers
is 3 times the middle number.
1  0  1  0  0  3
20  21  22  63  21 3
15
Example #4
Conjecture:
The sum of two numbers is
always greater than the
larger number.
True or false?
2  0  2
sum > larger number
2  0
A counterexample was found,
so the conjecture is false.
16
Hannah sells snow cones during soccer tournaments.
She records data for snow cone sales and temperature.
a. Predict the amount of snow cone sales when the temperature is 100°F.
b. Is it reasonable to use the graph to predict sales for when the temperature is
15ºF? Explain. One Possible Answer: Sales decrease as temperature drops.
Sales at 100°F is
predicted to be in
this range.
$4500 to $5000
17
Objective Practice:
Go to flippedmath.com
Select courses tab
MyGeometry
Semester 1
Unit 2
Section 2.1
Watch the Inductive Reasoning video –
This video is accessible from any internet connection.
USE IT IF YOU ARE STUCK AT HOME
Complete Packet 2.1 while listening to the video.
After video complete the Practice 2.1 exercises.
HW Complete the entire Packet 2.1
DUE FRIDAY Oct 12