Mathematics in the Modern World Midterm Module (Weeks 6–9)

This set of vocabulary flashcards covers core concepts of mathematical reasoning, deductive and inductive logic, Polya's problem-solving framework, and mathematical patterns as outlined in the Midterm Module.

Mathematical Argument

A structured series of statements consisting of premises (assumptions or given facts) and a conclusion derived logically from those premises.

Premise

A statement assumed or proven to be true within a mathematical argument.

Conclusion

The claim arrived at through logical inference in a mathematical argument.

Valid Argument

An argument where it is impossible for the premises to be true while the conclusion is false.

Sound Argument

A valid argument whose premises are actually true in reality.

Inductive Reasoning

The process of reaching a general conclusion by observing specific examples or repeated patterns.

Conjecture

The general conclusion reached through the process of inductive reasoning.

Counterexample

A single case that proves a conjecture false.

Deductive Reasoning

The process of reaching a specific, definitive conclusion by applying general rules, definitions, axioms, or logical principles.

Understand the Problem

The first step of Polya’s strategy: identifying unknowns, given values, conditions, and constraints, then restating the problem in one's own words.

Devise a Plan

The second step of Polya’s strategy: choosing a strategy such as looking for a pattern, drawing a diagram, making a table, or writing an equation.

Carry Out the Plan

The third step of Polya’s strategy: executing the chosen strategy step by step and keeping organized notes of calculations.

Look Back

The fourth step of Polya’s strategy: reflecting and verifying the answer in the original problem statement to ensure the result makes sense contextually.

Arithmetic Sequence

A sequence formed by adding a constant difference ($d$) to each term.

Geometric Sequence

A sequence formed by multiplying each term by a constant ratio ($r$).

Difference Tables

A technique used to find the next term of a sequence by subtracting consecutive terms until a constant row of differences is found.

Recreational Mathematics

Puzzles, logic games, and brain teasers that highlight mathematical concepts in engaging ways.

Magic Square

An $n \times n$ grid filled with distinct integers such that the sum of the numbers in each row, column, and diagonal is equal to the magic constant.

Magic Constant ($M$)

The constant sum of numbers in each row, column, and diagonal of a Magic Square.

Magic Constant Formula

The formula used to find the sum for an $n \times n$ grid using numbers $1$ to $n^2$: $M = \frac{n(n^2 + 1)}{2}$.