1. Using Inductive Reasoning
Lesson 7
Using Inductive Reasoning

2. Find the number of dots there will be in the 5th arrangement of the series
15 dots You just used inductive reasoning Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. In other words, observing a pattern

3. More on Inductive Reasoning
Explain the pattern you saw to come up with 15 dots and use that to find how many will be in the 7th? For the nth in a series, add n to the previous number, 28 dots You just made a conjecture Conjecture is a statement that is believed to be true If a conjecture can be proven to be true, then it becomes a theorem

4. Will Inductive Reasoning always produce a true conjecture?
Consider these prime numbers:
3, 11, 17, 43, and 101
Conjecture - All prime numbers are odd
True or False? Explain
False, 2 is a prime number
2 is a counterexample because it disproves a conjecture
We will do more with counterexamples in Lessons 10 and 14
Remember, a prime number is a number with only two factors; 1 and itself
Is 1 a prime number? Explain
No, 1 only has one factor, not two
Prime numbers and square are good series to recognize
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,…
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, …

5. Validity of conjectures
Not valid
After gardening flowers for several years, Shawndell noticed that each rose he saw had thorns on their stems
He made the conjecture “All roses have thorns”
Why is this valid?
The conjecture was made after observing many examples over several years
At the zoo Shea saw a brown bear, two giraffes, 5 buffalos, an otter, a herd of deer, and several pelicans
He made the conjecture “All bears are brown”
Why is this invalid?
The conjecture was made after only one example

6. Testing a Conjecture 2 2 −1 3 =1 True 4 2 −1 3 =5 True
Once a conjecture is made, you may be asked to test the conjecture
Testing a conjecture is not the same as proving it, but it can disprove a conjecture
To prove a conjecture is true, all possible examples must be shown to be true
How can you prove a conjecture is false?
Find a counterexample
Only one counterexample is need to disprove a conjecture
The square of a an even number minus 1 is divisible by 3
Test for numbers 2, 4, and 6
2 2 −1 3 =1 True
4 2 −1 3 =5 True
6 2 −1 3 = False

7. Application in research
After weeks of taking walks around her neighborhood, Janet noticed all birds she spotted were black
She then made the conjecture “Only black birds live in her neighborhood”
Is this conjecture valid?
How can she test this conjecture?
Can it be proven true/false? How?
This is a valid conjecture because she made the statement after many examples
To test this conjecture she must continue to observe all birds in her area and their colors
This cannot be proven to be true since the population of the birds, even in her area, will continue you to change with new births and migration (not to mention Janet’s lifespan)
However, it can be proven false by just finding one counterexample
Can you think of one bird that is not black?

8. Challenge Example
Isaac draws one marble from a bag of marbles and records its color before returning it to the bag. He repeats this experiment twenty times, and has gotten a red marble all twenty times. He makes a conjecture that “all marbles in the bag are red”. Cassie observed all 20 experiments and made a conjecture of her own. “There exists a differently colored marble in the bag.”
Only one of the conjectures is true, which one do you think is more valid? Why?
Both are equally valid
20 experiments supports Isaac’s and by replacing the marble each time supports Cassie’s since the number of marbles in the bag changed never changed
What would be the best way to prove which conjecture is true?
Empty the bag to observe the color of all the marbles in the bag, since performing the experiment could not prove or disprove either conjecture

9. Questions?
It is important to be able to explain each step in an inductive proof and to make sure that the assumptions made are clearly indicated in the proof, so that people can follow the argument
The same care must take place when making a conjecture so others are able to reach the same conclusion as you or find a counterexample
Inductive reasoning is one of the first steps we take in developing our logical reasoning
Later we will learn about deductive reasoning
Both are important skills to have in developing logical arguments