Spain 2º ESO Mathematics

Chapters: 6

1. Number sense

Counting · Quantity · Meaning of operations · Number relations · Proportional reasoning · Financial education

Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.
Number Systems and Encoding – A number system is a way to write numbers using a set of digits and a base. Decimal (base 10) uses 0–9, binary (base 2) uses 0 and 1, octal (base 8) uses 0–7, and hexadecimal (base 16) uses 0–9 and A–F. To go from decimal to any base, divide repeatedly by the base and read remainders bottom to top; for fractions, multiply by the base and read the integer parts top to bottom. To go to decimal, multiply each digit by its place value and add. Binary ↔ octal uses groups of 3 bits, binary ↔ hex groups of 4. Text is stored with encoding schemes: ASCII (7-bit, 128 characters), ISCII (8-bit, Indian scripts) and Unicode (every script), stored as UTF-8 (1–4 bytes) or UTF-32 (4 bytes).
Integers and Signed Numbers – Integers are the whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …. On a number line, numbers grow to the right. Adding a positive moves right, adding a negative moves left. Subtracting a number is the same as adding its opposite. For × and ÷, same signs give a positive answer and different signs give a negative answer. The same sign rules work for signed decimals like −2.5.
Fundamental Theorem of Arithmetic – Every whole number bigger than 1 is either a prime or can be written as a product of primes in exactly one way (only the order can change). This is the Fundamental Theorem of Arithmetic. Using these prime "building blocks": HCF = product of the smallest powers of the common primes; LCM = product of the greatest powers of all primes. For two numbers, HCF × LCM = a × b.
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).
Smart Ways to Manage Your Finances – Inflation makes prices rise, so the same money buys less. Money kept in a bank earns interest: simple interest is paid only on the original amount, while compound interest also earns interest on earlier interest, so it grows faster over time. A budget plans income into needs, wants and savings. Savings are kept safe; investments can grow but carry risk, and higher possible returns mean higher risk. Insurance shares risk among many people. Income tax is paid on income above a limit, at rates that rise with income.

2. Measurement sense

Magnitude · Measuring · Estimation and relations

Measurement and Units: How We Measure Anything – To measure something is to compare it with a fixed amount called a unit. Every measurement has a number and a unit. Scientists everywhere use the SI system, with seven base units such as the metre, kilogram and second. Prefixes like kilo (×1000), centi (÷100) and milli (÷1000) make units bigger or smaller. A good measurement starts at zero, is read with the eye straight above the mark, and is only as accurate as the smallest division (least count). Rounded values hide a small range, given by upper and lower bounds.
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.
Estimation and Order of Magnitude – An estimate is a quick, sensible answer made with rounded numbers. Count a small part, multiply up, and give the answer to a sensible precision. The order of magnitude is the nearest power of ten, such as 10² for about 100.

3. Spatial sense

2D and 3D shapes · Location and coordinate systems · Movements and transformations · Geometric visualisation and modelling

Geometry Basics: 2D Shapes, 3D Solids, Nets and Scale – Geometry is the maths of shape, size and position. A 2D (flat) shape has length and width only: triangles, squares, hexagons, circles. A 3D (solid) shape also has depth: cubes, prisms, pyramids, cylinders. A solid has faces (flat sides), edges (where two faces meet) and vertices (corners). A net is a flat pattern that folds into a solid. Two shapes are congruent if they have the same shape and size, and similar if they have the same shape but a different size. A tessellation covers a floor with shapes and leaves no gaps. A scale drawing shows a real object smaller or bigger by a fixed scale factor.
The Cartesian Plane: Coordinates, Distance and Midpoint – Two number lines that cross at right angles turn a flat surface into a map where every point has an address (x, y). The axes meet at the origin O(0, 0) and cut the plane into four quadrants. Distance between two points comes from Pythagoras: square the x-gap and the y-gap, add, take the square root. The midpoint is the average of the x values and the average of the y values. Three points are collinear when the two short distances add up to the long one, and they make a right angle when the squares of the two short sides add up to the square of the long side.
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.
Geometric Modelling: Describing Real Objects with Simple Solids – Geometric modelling means replacing a real object with simple shapes (cuboid, cylinder, cone, sphere, prism) so we can calculate with it. The cycle is: look at the real object, simplify it, measure, calculate volume or surface area, then check the answer against reality and improve the model. Scale changes lengths by k, areas by k² and volumes by k³. Geometry also helps us see patterns in nature and art, like hexagons in honeycombs.

4. Algebraic sense

Patterns · Mathematical model · Variable · Equality and inequality · Relations and functions · Computational thinking

Sequences and Progressions – A sequence is a list of numbers in a fixed order. We can describe it with a recursive rule (how to get the next term from the last one) or an explicit rule (a formula for term n). In an arithmetic progression (AP) we add the same number every time. In a geometric progression (GP) we multiply by the same number every time. Fractals and the Tower of Hanoi are fun patterns that hide these rules.
Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.
Algebraic Expressions: Variables, Terms and Simplifying – An algebraic expression uses letters (variables) and numbers joined by +, −, × and ÷, like 3x + 2. A variable stands for a number that can change. An expression is made of terms (3x and 2). In 3x, 3 is the coefficient; a term with no letter, like 2, is the constant. Like terms have exactly the same letter part (3x and 5x) and can be added; unlike terms (3x and 2, or x and x²) cannot. Substitution means putting a number in place of the letter to find the value. Expanding removes brackets: a(b + c) = ab + ac. Factorising is the reverse: take out the common factor. A formula is an expression that gives one quantity from others, and an identity is true for every value of the letter.
Linear Equations in One Variable – A linear equation in one variable has one unknown (like x) with power 1, for example 2x + 3 = 11. An equation is like a balance: both sides are equal. To solve it, do the same thing to both sides (add, subtract, multiply or divide by the same non-zero number) until x is alone. Always check by putting the answer back in.
Functions: Input, Rule, Output – A function is a rule that gives exactly one output for each input. We write it as f(x). The output f(a) is called the image of a; an input that gives a certain output is a preimage. A function can be shown as words, a table, a formula or a graph. Its graph can go up (increasing), go down (decreasing) and have a highest point (maximum) or lowest point (minimum).

5. Stochastic sense

Organising and analysing data · Uncertainty · Inference

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.
Sampling: Learning About a Population from a Sample – A population is the whole group we want to know about; a sample is a smaller part we actually check. A good sample is chosen at random so that it represents the population. Simple random sampling gives everyone an equal chance; stratified sampling takes the right share from each group; systematic sampling takes every k-th item. Different samples give slightly different answers (sampling variation), but bigger samples wobble less (law of large numbers). A biased sample gives a wrong answer however big it is.

6. Socio-emotional sense

Beliefs, attitudes and emotions · Teamwork and decision-making · Inclusion, respect and diversity

Self-Management: Taking Charge of Your Feelings, Goals and Time – Self-management is the skill of guiding your own feelings, thoughts and actions so you can reach your goals. It starts with self-awareness: noticing what you feel and how strong it is. Then you pause before reacting (Stop, Think, Act) and use calming tools such as slow breathing. Your beliefs and thoughts shape your feelings and actions, so changing an unhelpful thought ('I'm useless') into a helpful one ('I can't do it yet') changes what you do. Good self-managers set SMART goals, plan their time by importance, build healthy habits and bounce back from setbacks (resilience). These skills help in school, friendships and future careers.
Groups: How They Form and How They Change Us – A group is two or more people who interact, share a goal, follow norms and feel 'we'. People join groups for security, status, self-esteem, needs and goals. Groups grow through stages: forming, storming, norming, performing, adjourning. Their structure has roles, norms, status and cohesiveness. Types: primary/secondary, formal/informal, in-group/out-group. Groups also change our behaviour: in social loafing each person tries less in a shared task; in group polarisation the group's view becomes more extreme after discussion.
History of Mathematics – Mathematics grew over thousands of years in many places. People first counted with tally marks. Egypt and Babylon used geometry for land and building, and Babylon counted in 60s. Greek thinkers such as Thales, Pythagoras, Euclid and Hypatia turned geometry into proofs. India gave place value with zero, scholars in Baghdad built algebra, and the ideas reached Europe. Women and men from every continent have shaped maths, often against unfair barriers.