Spain 4º ESO Mathematics B
Chapters: 6
1. Number sense
Quantity · Meaning of operations · Number relations · Proportional reasoning
- 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).
- 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).
2. Measurement sense
Measuring · Change
- Linear Functions – A linear function adds the same amount to y every time x goes up by 1. Its rule is y = mx + c (also written f(x) = mx + b). m is the slope: rise ÷ run, the change in y for each 1 step in x. c is the y-intercept: the value of y when x = 0. Its graph is always a straight line.
- 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).
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.
- 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.
- 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.
- 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.
- 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.
5. Stochastic sense
Organising and analysing data · Uncertainty · Inference
- Correlation: Scatter Diagram, Karl Pearson's Coefficient and Spearman's Rank Correlation – Correlation tells how two variables move together. It is positive when both rise together, negative when one rises as the other falls, and zero when there is no straight-line pattern. A scatter diagram shows it as a picture. Karl Pearson's coefficient r measures its direction and strength and always lies between −1 and +1. Spearman's rank correlation R uses ranks and works for qualities like beauty or honesty; tied ranks need a small correction.
- 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.