Proof and Reasoning

Reasoning and Proof

By Robert Heinsen

Standards

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.
MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.
MA.912.LT.1.3

Apply mathematical induction in a variety of applications.

Objectives

1. Understand the concept of Mathematical Reasoning

2. Differentiate between Inductive and Deductive Reasoning

3. Use Mathematical Induction to prove mathematical statements

Mathematical Reasoning

What are we doing here?

Mathematical reasoning is a critical skill that enables students to analyze a given hypothesis without any reference to a particular context or meaning.

We FIND the meaning through our understanding of mathematics rules and definitions.

Draw

Based on the pattern you notice, draw the next picture for each sequence.

You Induced an answer from your background in mathematics and your ability to reason

We Continue the Pattern

You recognized parts of each of the patterns that matched in each sequence.

We Notice a Pattern

How did we know this?

Types of Reasoning

We use facts, definitions, and accepted properties to form a logical argument and arrive at a conclusion.

Deductive

Using observations of patterns we form a conjecture, or guess, at what would come next in the pattern.

Inductive

Multiple Choice

Find the next two numbers in the pattern:

2, 5, 11, 23, 47, ...

• 95, 191
• 60, 120
• 50, 100
• 45, 89

Multiple Choice

Find the next two terms in the pattern:

A, Z, B, Y, C, ...

• G, U
• X, D

• F, V
• E, W

Multiple Choice

What can you conclude from this information?

"If you go swimming, then you will get wet.

You went swimming."

• You will get wet.
• You will stay dry.
• You will get cold.
• You will need a towel.

Multiple Choice

Is this inductive or deductive reasoning?

"The product of two even integers is always even. Because 92 and 14 are even numbers, the product is even."

• Inductive reasoning
• Deductive reasoning

So, what more can we do?

Mathematical Induction

It is a method for proving mathematical statements for every Natural number. ( {1, 2, 3, ....})

This is a rigorous proof of the statement since it takes into account an infinite number of options and shows that the statement is true for each of them.

1. Show it is true for a base case, n = 1.

2. Assume it is true for some integer k

3. Demonstrate it is true for k+1 and therefore true for all Natural numbers

Proof by induction

Let's look at an example for the sum of the first n integers