MMW - Midterms

Inductive Reasoning

The process of reaching a general conclusion by examining specific examples.

Conjecture

Conclusion formed in inductive reasoning that may or may not be correct.

Deductive Reasoning

The process of reaching a conclusion by applying general assumptions, procedures, or principles.

Logic Puzzles

Can be solved by using deductive reasoning and a chart that enables us to display the given information in a visual manner.

George Polya (1877-1985)

He was born in Hungary and moved to the United States in 1940.

Polya's Problem Solving

Four-step strategy for solving mathematical problems.

Understand the Problem

This part of Polya's four-step strategy is often overlooked.

Understand the Problem

Can you restate the problem in your own words?

Understand the Problem

Can you determine what is known about these types of problems?

Understand the Problem

Is there an extraneous information that is not needed to solve the problem?

Understand the Problem

Is there a missing information that, if known, would allow you to solve the problem?

Understand the Problem

What is the goal?

Devise a Plan

Successful problem solvers use a variety of techniques when they attempt to solve a problem.

Devise a Plan

Make a list of the known information.

Devise a Plan

Make a list of information that is needed.

Devise a Plan

Draw a diagram.

Devise a Plan

Make an organized list that shows all the possibilities.

Devise a Plan

Make a table or a chart.

Devise a Plan

Work backwards.

Devise a Plan

Try to solve a similar but simpler problem.

Devise a Plan

Look for a pattern.

Devise a Plan

Write an equation. If necessary, define what each variable represents.

Devise a Plan

Perform an experiment.

Devise a Plan

Guess at a solution, then check your result.

Carry Out the Plan

Once you have devised a plan, you must carry it out.

Carry Out the Plan

Work carefully.

Carry Out the Plan

Keep an accurate and neat record of all your attempts.

Carry Out the Plan

Realize that some of your initial plans will not work and that you may have to devise another plan or modify your existing plan.

Review the Solution

Ensure that the solution is consistent with the facts of the problem.

Review the Solution

Interpret the solution in the context of the problem.

Review the Solution

Ask yourself if there are generalizations of the solution that could apply to other problems.

Sequence

An ordered list of numbers.

Term of Sequence

The numbers in a sequence that are separated by commas.

…

Indicate that the sequence continues.

Subscript Notation an

Designate the 𝑛𝑡ℎ term of a sequence

a1

first term of a sequence

an

represents the 𝑛𝑡ℎ term of a sequence

nth-Term Formula for a Sequence

nth term = 𝑑𝑛 + (𝑎 − 𝑑)

"nth" term

A formula with "n" in it which enables you to find any term of a sequence without having to go up from one term to the next.

Term Number (n)

Position of a term in a sequence.

First Term (a)

Initial value in a sequence.

Difference between the terms (d)

Constant value added between terms in a sequence.

Difference Table

Shows the differences between successive terms of the sequence.

First Differences

Differences between consecutive terms in a sequence.

Second Differences

Differences of the first differences.

Third Differences

Differences of the second differences.