Transcript 1.1 Patterns and Inductive Reasoning

Explore:
The figure shows a pattern of squares made
from toothpicks. Use the figure to complete
the following.
1 1
2 2
3 3
Record your answers.
Size of
Square
Toothpicks
1 1 2  2 3  3 4  4 5  5
4 4
...
n n
1.1 Patterns and Inductive Reasoning
Goal 1: Finding and Describing Patterns
visual and numerical
You will be able to find and describe _____
________
patterns.
Can you sketch the next figure in the pattern?
?
1
2
3
4
5
5
Can you predict the next number in the pattern?
0, 2, 4, 6, ?
Answer: 8
3, 1, 4, 1, 5, 9, ?
Answer: 2 (These are the digits of pi.)
Think and Discuss
What are some other patterns you see around you?
What are some numerical patterns you use in your daily life?
What are other types of patterns (besides visual and
numerical) you can identify?
1.1 Patterns and Inductive Reasoning
Goal 2: Using Inductive Reasoning
You will learn to use inductive reasoning to predict future
elements in a pattern.
Definition
• Inductive Reasoning is the process of looking for _______
patterns
and making __________.
conjectures
• So what’s a conjecture?
A conjecture is an unproven statement that is based on
observations, e.g.,
The sun is going to come up tomorrow.
• What are some other conjectures you make?
3 Stages of Inductive Reasoning
• Look for a Pattern
– Look at as many examples as necessary to make a conclusion.
Be sure the conjecture fits all the examples you have.
– Tables, diagrams, lists, etc. can help you find a pattern.
• Make a Conjecture
– Share it with others. Make changes as needed.
• Verify the Conjecture
– Use logical reasoning to verify it is true in all cases. (This is
called deductive reasoning, and we’ll learn more about it in
Chapter 2.
What is the next element in the pattern?
Proving a Conjecture is True or False
To prove a conjecture is true, you must prove it is true in all
cases.
To prove a conjecture is false, you only need to give one
example showing it is false. This example is called a
______________.
counterexample
Give a counterexample to the following conjecture:
The difference of two positive numbers is always a positive
number.
Sample answer: 4 – 10 = -6.
Remember: You only need one counterexample to prove a
conjecture is false. If you want to show a conjecture is
true, you must show it is true in every case.
What do you think? Is it easier to prove a conjecture is true
or that it is false? Why?
Real-life application
At a grocery store, Carlos bought 2 cans of soda for $1, Tina
bought 4 cans for $2, and Pat bought 3 cans for $1.50.
Make a conjecture about the price of a can of soda.
Think and Discuss
Why might your conjecture be false?
Can your conjecture be reasonable for the information you
have and still be false?
Can you think of some conjectures which seemed reasonable
at first but were later shown to be false?