Italy Scuola secondaria di primo grado – classe 3ª Mathematics
Chapters: 4
1. Numbers
Number sets and operations · Ratios, fractions and percentages · Divisibility · Powers and roots
- 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).
- Percentages – Per cent means "out of 100", so x% = x/100. To find x% of an amount, multiply by x/100. Percentage change = (change ÷ original) × 100. To increase by r%, multiply by (1 + r/100); to decrease by r%, multiply by (1 − r/100). Successive changes multiply their multipliers. For reverse percentage, divide the new value by the multiplier.
- Divisibility: Multiples, Divisors, GCD and LCM – a is divisible by b when a = b × k for a whole number k: b is a divisor (factor) of a and a is a multiple of b. Every whole number a can be written as a = b × q + r with 0 ≤ r < b (division with remainder); b divides a exactly when r = 0. Quick tests tell divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 11 from the digits. The GCD is the greatest common divisor, found by prime factors or by Euclid's algorithm; LCM is the least common multiple, and GCD × LCM = a × b. Two numbers are coprime when their GCD is 1.
- Laws of Exponents – An exponent tells how many times a base is multiplied by itself. Same base: multiply → add exponents, divide → subtract, power of a power → multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root.
2. Space and figures
Plane figures and constructions · Similarity and Pythagoras · Area, circle and pi · Transformations and solids
- Geometric Constructions: Drawing Exactly with Compass and Straightedge – A geometric construction draws a figure exactly using only a compass and a straightedge (a ruler used for straight lines). Key constructions: the perpendicular bisector of a segment, the bisector of an angle, a perpendicular from a point to a line, angles of 60°, 30°, 90° and 45°, a triangle from three sides (SSS), two sides and the included angle (SAS) or two angles and a side (ASA), and regular polygons such as the hexagon. Each works because equal compass arcs make equal lengths, which give congruent triangles.
- 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.
- Circle: Circumference, Pi and Area – The distance around a circle is its circumference: C = π × d = 2πr. The space inside is its area: A = πr². Pi (π) is the number you get when you divide the circumference by the diameter. It is about 3.14 for every circle, big or small.
- Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.
3. Relations and functions
Formulas, proportionality and graphs · Linear equations
- Ratio and Proportion – A ratio a : b compares two amounts of the same kind. Divide both parts by the same number to simplify it. To share an amount in a ratio, add the parts, find one part, then multiply. Two quantities are in direct proportion when y = k·x (double one, double the other) and in inverse proportion when x·y = k (double one, halve the other).
- 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.
4. Data and prediction
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.
- Introduction to Probability: Scale, Experiments, Sample Spaces and Trees – Probability is a number from 0 to 1 that tells how likely something is. 0 means it can never happen, 1 means it will surely happen. We can find it by doing an experiment many times (empirical probability), or by listing every possible result (the sample space) and counting the ones we want. Tree diagrams and tables help us list results when two things happen together.