Spain 1º Bachillerato General Mathematics
Chapters: 6
1. Number sense
Counting · Sense of operations · Relationships · 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.
- Data Analysis – Data analysis means turning raw data into answers. It follows a cycle: ask a question, collect data, clean it (remove errors, repeats and blanks), organise and transform it, analyse it with summaries such as mean, median, range and patterns, show it with a good chart, and draw a careful conclusion. Watch for outliers, small samples and bias, and remember that a correlation between two things does not prove that one causes the other. Data must also be stored safely and used with permission.
- 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
Measurement · Change
- 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.
- Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.
3. Spatial sense
Visualisation, reasoning and geometric modelling
- 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 and computational thinking
Patterns · Mathematical model · 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.
- 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.
- 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).
- 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.
5. Stochastic sense
Data organisation and analysis · Uncertainty · Probability distributions · 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.
- Conditional Probability, Multiplication Rule and Independent Events – Conditional probability is the chance of A when we already know B has happened. We throw away every outcome outside B and count again: P(A|B) = P(A ∩ B) ÷ P(B). Turned around, this gives the multiplication rule P(A ∩ B) = P(B)·P(A|B). If knowing B does not change the chance of A, the events are independent and P(A ∩ B) = P(A)·P(B).
- Probability Distributions of Discrete Random Variables – A random variable X turns each outcome of an experiment into a number. Its probability distribution lists every value x with its probability P(X = x); each P is between 0 and 1 and they add to 1. The mean E(X) = Σx·P(x) is the long-run average (balance point). The variance Var(X) = Σ(x − μ)²P(x) = E(X²) − μ² measures spread; σ = √Var. Special models: uniform, binomial B(n, p) with mean np and variance np(1 − p), and Poisson with mean = variance = λ.
- 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
- Social-Emotional Skills and Mathematical Processes – Doing maths well is not only about numbers. Social-emotional skills help you notice your feelings, calm stress, keep going after mistakes, work well with others and believe you can grow. The mathematical processes are the habits good problem solvers use: understand the problem, make a plan, carry it out and look back; reason and prove; connect ideas; choose tools; represent ideas in words, tables, graphs and models; and communicate clearly.