PROBLEM SOLVING AND REASONING

Inductive Reasoning

The process of forming a general conclusion by examining specific examples or patterns.

Conjecture

A conclusion formed through inductive reasoning that may or may not be true.

Pattern Recognition

The process of identifying regularities or repeated relationships in data.

Deductive Reasoning

The process of reaching a specific conclusion by applying general rules principles or assumptions.

Logical Certainty

A conclusion that must be true if the reasoning and premises are correct.

Argument

A statement or set of statements (premises) that lead to a conclusion.

Premise

A statement assumed to be true and used as the basis for reasoning.

Conclusion

The final statement derived from premises using logical reasoning.

George Polya

A mathematician who developed a systematic four-step strategy for solving problems.

Polya’s Four-Step Strategy

A structured method for solving problems: Understand Devise Carry Out Review.

Understand the Problem

The first step in problem solving where the problem is carefully analyzed to determine what is given and what is asked.

Devise a Plan

The step where a strategy or method is selected to solve the problem.

Carry Out the Plan

The execution of the chosen strategy carefully and systematically.

Review the Solution

The step where the answer is checked for accuracy and reasonableness.

Organized List

A systematic listing of all possible outcomes without duplication.

Work Backwards

A problem-solving technique that starts from the desired result and reverses the steps.

Guess and Check

A strategy where a possible solution is tested and adjusted if necessary.

Logic Puzzle

A problem that requires deductive reasoning and elimination to determine correct answers.

Elimination Method

A reasoning process where impossible options are ruled out to narrow down correct answers.

Chart/Table Method

A visual tool used in logic puzzles to organize information and eliminate possibilities.

Sequence

An ordered list of numbers that follow a specific rule or pattern.

Term of a Sequence

Each individual number in a sequence.

Subscript Notation (aₙ)

A notation used to represent the nth term of a sequence.

First Term (a₁)

The first number in a sequence.

nth Term

A general formula used to determine any term in a sequence without listing previous terms.

Arithmetic Sequence

A sequence where the difference between consecutive terms is constant.

Common Difference (d)

The fixed number added or subtracted between consecutive terms in an arithmetic sequence.

nth-Term Formula (Arithmetic)

A formula used to find any term in an arithmetic sequence: an = dn + (a − d).

Summation Formula for First n Natural Numbers

A formula used to find the sum of the first n natural numbers: n(n + 1) / 2.

Difference Table

A table used to analyze patterns in sequences by examining differences between consecutive terms.

First Differences

The differences between consecutive terms in a sequence.

Second Differences

The differences between consecutive first differences.

Constant First Difference

Indicates a linear or arithmetic pattern.

Constant Second Difference

Indicates a quadratic pattern.

Period of a Pendulum

The time required for a pendulum to complete one full swing.

Square Root Relationship

A pattern where one quantity is the square root of another.

Generalization

A broad statement formed after observing patterns or repeated results.