Italy Secondaria di secondo grado – classe 2ª Mathematics (classico, linguistico, scienze umane, artistico, musicale)

Chapters: 5

1. Arithmetic and algebra

Integers, rationals and reals · Literal calculation

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].

Euclidean plane geometry · Cartesian coordinates

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.
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.

Functions and models

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.

Statistics and 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.

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.