Math Humor

Deductive 1 Warm Up: Using Logical Reasoning The first person in each row will receive a hypothesis in the format of If there is a fire, then… Fill in a conclusion that LOGICALLY follows. Each statement should start with the word “then” Bend your paper so that only your conclusion shows and pass back to the person behind you. The second person will read your conclusion then as their hypothesis and write down a new conclusion. Keep going in this manner until the last person brings the paper back up to the first person in your row. We will share our results.

LT 2.1: Use inductive reasoning to make conjectures and find counterexamples

Example 1 Find the next item in the pattern: January, March, May July

Example 2 Find the next item in the pattern: 7, 14, 21, 28, 35

Example 3 Find the next item in the pattern:

Inductive Reasoning Inductive reasoning is the process of arriving at a conclusion based on a set of observations. In itself, it is not a valid method of proof. Just because a person observes a number of situations in which a pattern exists doesn't mean that that pattern is true for all situations. For example, after seeing many people outside walking their dogs, one may observe that every dog that is a poodle is being walked by an elderly person. The person observing this pattern could inductively reason that poodles are owned exclusively by elderly people.

Inductive Reasoning This is by no means a method of proof; in fact, in the real world it is a means by which people and things are stereotyped. This is, however, the most common way people reason throughout their life and learn about their surroundings.

Inductive Reasoning A hypothesis based on inductive reasoning, can, however, lead to a more careful study of a situation. By inductive reasoning, in the previous example, a viewer has formed a hypothesis that poodles are owned exclusively by elderly people. The observer could then conduct a more formal study based on this hypothesis and conclude that his hypothesis was either right, wrong, or only partially wrong.

Inductive reasoning is used in geometry in a similar way. For example – Seth observes 4 rectangles and in all 4 rectangles the diagonals are congruent. Seth concludes that the diagonals in a rectangle are always congruent.

Although we know this fact to be generally true, Seth hasn't proved it through his limited observations. However, he could prove his hypothesis using other means (which we'll learn later) and come out with a theorem (a proven statement).

In this case, as in many others, inductive reasoning led to a suspicion, or more specifically, a hypothesis, that ended up being true. The power of inductive reasoning, then, doesn't lie in its ability to prove mathematical statements. In fact, inductive reasoning can never be used to provide proofs. Instead, inductive reasoning is valuable because it allows us to form ideas about groups of things in real life.

Inductive Reasoning Whether we know it or not, the process of inductive reasoning almost always is the way we form ideas about things. Once those ideas form, we can systematically determine (using formal proofs) whether our initial ideas were right, wrong, or somewhere in between. Can you think of something you BELIEVE to be true based on observations you have witnessed? Examples – If it is January then it is cold. If I am late to school then I will get in trouble. If I don’t tie my shoes than I will trip in the hallway and everyone will laugh at me.

Example:

Sherlock Holmes, Silver Blaze In the short story, The Adventure of Silver Blaze, Sherlock Holmes solves a murder by establishing the premise that dogs bark at strangers. Silver Blaze is a champion race horse who disappears from the stables the night his trainer is murdered. Holmes has two premises: 1) The dogs didn’t bark the night of the murderer broke into the barn where Silver Blaze was stabled, and 2) dogs bark at strangers. These premises lead Holmes to the conclusion the dogs knew the murderer. His premises turns out to be correct and lead to an accurate conclusion; Holmes solves the mystery and the murder. One could also argue Holmes is using inductive reasoning, essentially reasoning by sign, (the clues), and perhaps by cause, (because dogs bark at strangers, therefore...)

You’re at school eating lunch You’re at school eating lunch. You ingest some air while eating, which causes you to belch. Afterward, you notice a number of students staring at you with disgust. You burp again, and looks of distaste greet your natural bodily function. You have similar experiences over the course of the next couple of days. Finally, you conclude that belching in public is social unacceptable. The process that lead you to this conclusion is called inductive reasoning.

A scientist takes a piece of salt, turns it over a Bunsen burner, and observes that it burns with a yellow flame. She does this with many other pieces of salt, finding they all burn with a yellow flame. She therefore makes the conjecture: “All salt burns with a yellow flame.”

Mythbusters about sharks

On her first road trip, Little Window Watcher Wilma observes a number of vehicles. Each one she observes has four wheels. She conjectures “All vehicles have four wheels.” What is wrong with her conjecture? What counterexample will disprove it?

Kenny makes the following conjecture about the sum of two numbers Kenny makes the following conjecture about the sum of two numbers. Find a counterexample to disprove Kenny’s conjecture. Conjecture: The sum of two numbers is always greater than the larger number.

Example 6 Show that the conjecture is false by finding a counterexample: for any real number x, x2 x ≥ How do you know which numbers to test when trying to find a counterexample?

Joe has a friend who just happens to be a Native American named Victor Joe has a friend who just happens to be a Native American named Victor. One day Victor gave Joe a CD. The next day Victor decided that he wanted the CD back, and so he confronted Joe. After reluctantly giving the CD back to his friend, Joe made the conjecture: “Victor, like all Native Americans, is an Indian Giver.” What is wrong with his conjecture? What does this example illustrate?

Challenge: Find a counterexample for the following: For every integer n, n2 + 1 is prime.