Algebra Fundamentals: Concepts and Operations

Fundamental Concepts

• Numbers: Integers, natural numbers (positive integers), whole numbers (non-negative integers), rational numbers (fractions), irrational numbers (cannot be expressed as a fraction), real numbers (rational or irrational), and complex numbers (include imaginary numbers).
• Operations: Addition, subtraction, multiplication, division, exponentiation, and roots (e.g., square root). Order of operations (PEMDAS/BODMAS) dictates the sequence for evaluating expressions.
• Sets: Collections of objects. Sets are described using roster notation (listing elements) or set-builder notation (defining a rule).
• Logic: Statements can be true or false. Connectives include AND, OR, NOT.
• Variables: Symbols representing unknown quantities.
• Equations: Statements that show the equality between two expressions.
• Inequalities: Statements that show the relationship between expressions using symbols like <, >, ≤, ≥.

Algebra

• Variables and expressions: Evaluate expressions by substituting given values for the variables. Combine like terms.
• Solving equations: Isolate the variable to find its value. Strategies include addition, subtraction, multiplication, and division. Check solutions.
• Linear equations: Equations of the form ax + b = 0, where a and b are constants. Graph as a straight line. The slope and y-intercept are key features.
• Quadratic equations: Equations of the form ax² + bx + c = 0. Can be solved using factoring, the quadratic formula, or completing the square.
• Systems of equations: Two or more equations with the same variables. Methods for solving include graphing, substitution, and elimination.
• Functions: A relation where each input (x-value) has exactly one output (y-value). Notation includes f(x). Graph by plotting points and identifying key features.
• Polynomials: Expressions containing one or more variables with exponents.
• Rational expressions: Expressions containing variables in the numerator and/or denominator. Simplification involves factorisation and cancelling.

Geometry

• Shapes: Triangles, quadrilaterals (squares, rectangles, parallelograms, trapezoids), polygons (multiple sided shapes).
• Angles: Formed by two rays sharing a common endpoint. Types include acute, obtuse, right, straight, and reflex angles.
• Area and perimeter: Measure of the space enclosed by a shape and the distance around the shape respectively.
• Volume and surface area: Measure of space occupied by a 3D shape and the combined area of its faces.
• Circles: Defined by a centre point and radius. Formulas for circumference and area.
• Trigonometric ratios: Relationships between angles and sides of right angled triangles (sine, cosine, tangent).

Calculus

• Limits: The value that a function approaches as the input approaches a certain value.
• Derivatives: The rate of change of a function. Used for finding slopes, tangents, and maximum/minimum points.
• Integrals: The area under a curve. Used for finding areas, volumes, and accumulating values.

Statistics

• Data collection and organization: Gathering, classifying, and summarizing data.
• Measures of central tendency: Mean, median, mode.
• Measures of dispersion: Range, variance, standard deviation.
• Probability: The likelihood of an event occurring.

Discrete Math

• Counting principles: Fundamental counting principle, permutations, combinations.
• Logic gates: AND, OR, NOT, XOR, NAND, NOR.
• Proof techniques: Direct proof, proof by contradiction
• Graphs and trees: Visual representations of relationships between objects.

Applications

• Solving real-world problems: Mathematical concepts applied in different fields.
• Modeling: Using mathematical equations to represent real-world situations and predict outcomes.
• Problem-solving: Applying mathematical concepts to find solutions.

Problem-Solving Strategies

• Understanding the problem: Identify the given information and the goal.
• Devising a plan: Choose an appropriate strategy (e.g., setting up an equation, drawing a diagram).
• Carrying out the plan: Perform the necessary calculations and steps.
• Looking back: Check the answer and determine if it makes sense in the context of the problem.