Italy Secondaria di secondo grado – classe 2ª Mathematics (liceo scientifico)
Chapters: 5
1. Arithmetic and algebra
Integers, rationals and reals · Polynomials · Equations, inequalities, systems · Vectors and matrices
- Rational Numbers: Number Line, Density and Decimals – A rational number is any number that can be written as p/q, where p and q are integers and q is not 0. Examples: 3/4, −5/3, 7 (= 7/1), 0.25 (= 1/4). To place p/q on the number line, cut each unit into q equal parts and hop p parts from 0. Between any two rational numbers there is always another one (their average), so there are endlessly many between them. The decimal of a rational number either ends (terminating) or repeats a block for ever (non-terminating repeating).
- Polynomials: Zeroes and Coefficients – A polynomial like ax² + bx + c has a degree (highest power). A zero is a value of x that makes it 0. On a graph, zeroes are the x-coordinates where the curve y = p(x) meets the x-axis. A polynomial of degree n has at most n zeroes. For ax² + bx + c with zeroes α, β: α + β = −b/a and αβ = c/a. A quadratic with given zeroes is k[x² − (α+β)x + αβ].
- Pair of Linear Equations in Two Variables – Two equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ each make a straight line. The answer that fits both is the point where the lines meet. Lines that cross give one answer, parallel lines give none, and lines that lie on top of each other give endless answers. We can find the answer by drawing (graph), by substitution or by elimination.
- Scalars and Vectors for Motion in a Plane (Class 11) – A scalar has only size; a vector has size and direction. Vectors are equal if their size and direction match. Multiplying by a number changes the length (a negative number flips it). Vectors add tail-to-head (triangle or parallelogram law); A − B = A + (−B). Any vector in a plane is A = Ax î + Ay ĵ with Ax = A cos θ, Ay = A sin θ. A·B = AB cos θ is a scalar; A×B has size AB sin θ and is perpendicular to both.
2. Geometry
Euclidean plane geometry · Pythagoras theorem · Transformations and circle · Cartesian coordinates and conics · Circular functions and triangles
- Euclid's Geometry: Definitions, Axioms and the Five Postulates – Geometry began as practical measuring of land and altars in Egypt, India and Mesopotamia. Indian Sulbasutras (like Baudhayana's) gave rope rules for making squares, doubling a square and the diagonal rule. Around 300 BCE, Euclid of Alexandria organised geometry as a chain of reasoning: start from a few definitions, common-sense axioms and five geometry postulates, and prove everything else. The fifth postulate is about when two lines meet, and it leads to the idea of parallel lines.
- Pythagoras Theorem – In any right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides: a² + b² = c².
- Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.
- Coordinate Geometry: Distance and Section Formula – Every point on a flat plane has an address (x, y). The distance between A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]; it is just Pythagoras on the x-gap and the y-gap. A point P that cuts AB inside in the ratio m : n is P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). When m = n, P is the midpoint.
- Trigonometric Ratios (sin, cos, tan) – In a right triangle, pick one sharp (acute) angle θ. The side facing θ is the opposite, the side touching θ (not the longest) is the adjacent, and the longest side is the hypotenuse. sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. cosec, sec and cot are their flips. The ratios depend only on the angle, not on the size of the triangle. Learn the table for 0°, 30°, 45°, 60°, 90°.
3. Relations and functions
Sets and functions · Linear and quadratic functions · Other elementary functions
- Relations and Functions: Cartesian Product, Domain, Range, Graphs – An ordered pair (a, b) has a first and a second place, so (1, 2) ≠ (2, 1). The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B, and n(A × B) = n(A) × n(B). A relation from A to B is any subset of A × B; its domain is the set of first elements and its range the set of second elements, while B is the co-domain. A function is a special relation where every element of A has exactly one image. We study constant, identity, polynomial, rational, modulus, signum, exponential, log and greatest integer functions with their graphs, and add, subtract, multiply and divide functions point by point.
- Quadratic Equations – A quadratic equation has x² as its highest power: ax² + bx + c = 0 with a ≠ 0. It has at most two roots. Find them by factorising (split the middle term, then set each bracket to zero) or by the formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells the nature of roots: D > 0 two different real roots, D = 0 two equal roots, D < 0 no real roots.
4. Data and prediction
Descriptive statistics · Probability
- Statistics: Asking Questions, Averages and Stacked Bar Graphs – Statistics starts with a question that has many possible answers. We collect data, organise it, show it in a graph, and find one number that sums it up. The mean is the fair share, the median is the middle value, and the mode is the most common value. A weighted average gives more importance to some values. Stacked and 100% stacked bar graphs show how a whole is split into parts.
- Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.
5. Elements of computer science
Algorithms
- Introduction to Problem Solving – Problem solving on a computer has stages: analyse the problem (inputs, outputs, rules), develop an algorithm (a finite, clear, ordered set of steps), code it in a programming language, test it with different inputs, and debug (find and remove errors). An algorithm can be shown as a flowchart (oval = start/stop, parallelogram = input/output, rectangle = process, diamond = decision, arrows = flow) or as pseudocode (structured plain English). Decomposition breaks a big problem into smaller sub-problems that are solved separately and then joined.