Key Areas in Mathematics

Arithmetic
- Basic operations: addition, subtraction, multiplication, division.
- Properties: commutative, associative, distributive.
- Order of operations: PEMDAS/BODMAS.
Algebra
- Variables and constants: representation of numbers.
- Expressions and equations: solving for unknowns.
- Functions: linear, quadratic, polynomial, rational.
Geometry
- Shapes: triangles, quadrilaterals, circles.
- Properties: area, perimeter, volume.
- Theorems: Pythagorean theorem, properties of angles.
Trigonometry
- Trigonometric ratios: sine, cosine, tangent.
- Unit circle: understanding angles in radians and degrees.
- Applications: solving triangles, wave functions.
Calculus
- Differentiation: rates of change, slopes of curves.
- Integration: area under curves, accumulation of quantities.
- Fundamental theorem of calculus: connects differentiation and integration.
Statistics
- Data collection: surveys, experiments, observational studies.
- Descriptive statistics: mean, median, mode, standard deviation.
- Inferential statistics: hypothesis testing, confidence intervals.
Probability
- Basic concepts: sample space, events, outcomes.
- Probability rules: addition and multiplication rules.
- Distributions: normal distribution, binomial distribution.
Discrete Mathematics
- Set theory: concepts of sets, subsets, unions, intersections.
- Graph theory: vertices, edges, paths, cycles.
- Combinatorics: counting principles, permutations, combinations.

Number Theory
- Prime numbers, divisibility, greatest common divisor, least common multiple.
Matrices
- Operations: addition, multiplication, determinants, inverses.
- Applications: solving systems of equations, transformations.
Logic
- Statements, truth values, logical operators.
- Proof techniques: direct, indirect, contradiction, contrapositive.
Mathematical Modeling
- Representation of real-world scenarios using mathematical expressions.
- Applications in economics, biology, engineering.

Basic operations include addition, subtraction, multiplication, and division.
Properties include commutative (a + b = b + a), associative ((a + b) + c = a + (b + c)), and distributive (a(b + c) = ab + ac).
Order of operations follows PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) or BODMAS.

Variables represent unknown numbers, while constants are fixed values.
Expressions combine numbers and variables, while equations include an equality sign. Solving for unknowns involves isolating variables.
Functions relate inputs to outputs; common types include linear (straight-line equations), quadratic (parabolas), polynomial (multi-term), and rational (ratios of polynomials).

Fundamental shapes include triangles, quadrilaterals, and circles, each with unique properties.
Area, perimeter, and volume are key measurements for shapes: area measures the extent of a surface, perimeter measures the distance around a shape, and volume measures space within a 3D object.
Theorems like the Pythagorean theorem (a² + b² = c² in a right triangle) provide relationships between side lengths.

Key trigonometric ratios include sine, cosine, and tangent, relating angles to opposite and adjacent sides in right triangles.
The unit circle is essential for understanding angles in both radians and degrees, connecting circular movement with trigonometric functions.
Applications include solving triangles for unknown sides and angles and modeling periodic phenomena like waves.

Differentiation focuses on calculating rates of change and finding slopes of curves at given points.
Integration helps determine areas under curves and accumulates quantities over intervals.
The fundamental theorem of calculus establishes a connection between differentiation and integration, stating that integration can reverse differentiation.

Data collection methods include surveys, experiments, and observational studies to gather information effectively.
Descriptive statistics summarize data through measures like mean (average), median (middle value), mode (most frequent value), and standard deviation (spread of data).
Inferential statistics involve hypothesis testing to make predictions or conclusions about a population from sample data, using confidence intervals for uncertainty estimation.

Fundamental concepts include sample space (all possible outcomes), events (specific outcomes), and outcomes (results of random processes).
Key probability rules include addition (finding the probability of either of two events) and multiplication (finding the probability of two independent events happening together).
Common distributions include normal distribution (bell curve representation of data) and binomial distribution (representing the number of successes in a series of trials).

Set theory involves understanding sets (collections of elements), subsets, unions (combining sets), and intersections (common elements).
Graph theory studies structures made of vertices (points) and edges (connections), exploring paths and cycles.
Combinatorics focuses on counting principles, including permutations (ordered arrangements) and combinations (unordered selections).

Prime numbers are greater than one and have no divisors other than one and themselves.
Key concepts also include divisibility rules and calculations of the greatest common divisor and least common multiple.

Matrix operations include addition, multiplication, calculating determinants, and finding inverses.
Matrices can solve systems of equations and represent transformations in geometry.

Statements have truth values (True/False) and involve logical operators like AND, OR, and NOT.
Proof techniques include direct proof, indirect proof, proof by contradiction, and contrapositive proof.

Mathematical modeling represents real-world scenarios using equations and expressions, allowing for analysis and predictions.
Applications span various fields, including economics (market trends), biology (population dynamics), and engineering (stress analysis).

Regular practice of problem-solving enhances understanding and retention of concepts.
Utilize visual aids such as graphs and charts to facilitate comprehension of complex ideas.
Review and condense each topic regularly to reinforce foundational knowledge.
Leverage online resources and exercises for additional practice and exposure to diverse problems.