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.

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

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