1. The document discusses inductive reasoning and making conjectures based on patterns. It defines inductive reasoning as looking for patterns and making conjectures or unproven statements based on observations. 2. It outlines a 3 stage process for inductive reasoning: looking for a pattern in examples, making a conjecture, and verifying the conjecture is true for all cases through deductive reasoning. 3. A counterexample can disprove a conjecture by providing one example where it is false. While proving a conjecture true requires showing it is true for all cases.

1. 1.1 Patterns and Inductive Reasoning
Goal 1: Finding and Describing Patterns
You will be able to find and describe _____ and ________
patterns.
visual numerical

2. Can you sketch the next figure in the pattern?
?
1 2 3 4 5
5

3. Can you predict the next number in the pattern?
0, 2, 4, 6, ?
Answer: 8
3, 1, 4, 1, 5,
Answer: 2 (These are the digits of pi.)
9, ?

4. 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?

5. 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.

6. Definition
• Inductive Reasoning is the process of looking for _______
and making __________.
• 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?
patterns
conjectures

7. 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.

8. What is the next element in the pattern?

9. 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

10. 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?

11. Goldbach’s Conjecture:
Every even number greater than 2 can be written as the
sum of two primes.
Select an even number from 50 to 100 and show that
Goldbach’s conjecture is true for that number.
Think and Discuss
Have you proven Goldbach’s conjecture?
No. You’ve shown it is true only for the number you
selected.

12. 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?