Spain 2º Bachillerato Mathematics II
Chapters: 6
1. Number Sense
Sense of operations · Relations
- Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
2. Measurement Sense
Measurement · Change
- Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.
- Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.
3. Spatial Sense
Two- and three-dimensional geometric shapes · Location and representation systems · Visualisation, reasoning and geometric modelling
- Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.
- 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 · 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.
- Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
- 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.
- Curve Sketching Using Derivatives – To sketch y = f(x): find the domain and intercepts, solve f'(x) = 0 for stationary points, use the sign of f'(x) to see where the curve rises or falls, use f''(x) for concavity and points of inflection, check asymptotes and end behaviour, then join everything smoothly. Example: y = x³ − 3x has a maximum at (−1, 2), a minimum at (1, −2) and an inflection at (0, 0).
5. Stochastic Sense
Uncertainty · Probability distributions
- 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.
6. Socio-affective Sense
Beliefs, attitudes and emotions · Decision-making · Inclusion, respect and diversity
- Mental Health: Feelings, Stress and Support – Mental health is how we think, feel, cope and get along with others. Like physical health, everyone has it and it moves up and down. Emotions are useful signals. Stress adds up like water in a bucket; healthy coping (sleep, movement, talking, calm breathing, planning) lets it drain. Self-esteem and good relationships protect us. If feelings stay very heavy for two weeks or more, or stop daily life, talk to a trusted adult or a professional — asking for help is a sign of strength.
- Research Skills: From a Question to a Finished Project – Research is a careful way of finding an answer. You ask a clear, focused question, plan how to answer it, find information and check that each source can be trusted, collect and analyse your own data, draw a conclusion that the evidence supports, and share it while crediting every source you used.