1. Folding Paper How many rectangles?
# of folds
# of Rectangles
Can you find a pattern that you can describe and then predict how many rectangles on the next fold?
1st fold 2 4
2nd fold
3rd fold 8
4th fold 16
5th fold 32
25th fold 50

2. 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

3. What You'll Learn 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.

4. 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.

5. 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, ,
24, , ,
X 2
X 2
X 2
X 2
X 2
Find the next three terms of the sequence:
7, 8, ,
16,
23, 32
+ 1
+ 3
+ 5
+ 7
+ 9

6. Patterns and Inductive Reasoning
Draw the next figure in the pattern.
Lesson 2-1 Patterns and Inductive Reasoning

7. 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.

8. Example #2 Describe the pattern. Write the next three numbers.
Multiply by 3 to get the next number in the sequence.

9. 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

10. Vocabulary
conjecture
inductive reasoning
counterexample

11. Conjecture: conclusion made based on observation
What is a conjecture?
Conjecture: conclusion made based on observation
What is inductive reasoning?
Inductive Reasoning: conjecture based on patterns
Proving conjectures TRUE
is very hard.
Proving conjectures FALSE
is much easier.
What is a counterexample?
How do you disprove a conjecture?
Counterexample: example that shows a conjecture is false
What are the steps for inductive reasoning?
How do you use inductive reasoning?
Steps for Inductive Reasoning
Find pattern.
Make a conjecture.
Test your conjecture or find a counterexample.

12. A counterexample can be a drawing, a statement, or a number.
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.

13. Steps for Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a counterexample.

14. Show that the conjecture is false by finding 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.

15. The sum of any 3 consecutive numbers is 3 times the middle number.
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.)
Conjecture:
The sum of any 3 consecutive numbers is 3 times the middle number.

16. A counterexample was found, so the conjecture is false.
Example #4 Conjecture: The sum of two numbers is always greater than the larger number. True or false?
sum > larger number
A counterexample was found, so the conjecture is false.

17. 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

18. Objective Practice: DUE FRIDAY Oct 12 Go to flippedmath.com
Select courses tab MyGeometry Semester Unit 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