Netherlands VWO 4 (bovenbouw, 2e fase) Mathematics B
Chapters: 3
1. Skills
General skills · Profile-specific skills · Mathematical skills
- 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.
- 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.
- Problem Solving in Maths – A problem is a question where you do not yet know the method. A good solver follows four steps: understand and analyse the problem, make a plan using a strategy (heuristic) such as drawing a picture, trying small cases, making a table, finding a pattern, guessing and checking, or working backwards, carry out the plan, and look back to check and reflect. The handshake problem, with n people and n(n-1)/2 handshakes, shows all the steps.
2. Functions, graphs and equations (part 1)
Formulas and functions · Standard functions · Functions and graphs
- 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).
- Graphs of Basic Functions and Solving Equations with Graphs – A graph shows every point (x, y) with y = f(x). Five shapes come up again and again: the parabola y = x², the cubic y = x³, the half-parabola y = √x, the V-shape y = |x| and the hyperbola y = k/x. Learn each one's domain, range, symmetry, where it rises or falls, and where it crosses the axes. Then use graphs to solve equations (where graphs cross) and inequalities (where one graph is above the other).
- Transformations of Functions – A transformation changes the graph of a parent function y = f(x) without changing its basic shape. Adding outside the function moves it vertically: f(x) + k moves up k. Changes inside the brackets act horizontally and the opposite way: f(x − h) moves right h. Multiplying outside, a·f(x), stretches vertically by a (and flips in the x-axis if a < 0). Multiplying inside, f(bx), squeezes horizontally by factor 1/b (and flips in the y-axis if b < 0). All together: y = a·f(b(x − h)) + k, and the point (0, 0) of the parent moves to (h, k).
3. Geometry with coordinates (part 1)
Geometric skills
- Logic and Proof: How Mathematicians Show Something Is Always True – A proof is a chain of correct reasons that shows a statement is true in every case. Statements are true or false; 'if P then Q' (P ⇒ Q) means Q must hold whenever P does, and its converse Q ⇒ P may not. P ⇔ Q means both directions hold. Main methods: direct proof (from known facts step by step, often with algebra like 2n + 1 for odd numbers); proof by exhaustion (check every possible case); disproof by counterexample (one failing case kills a claim); proof by contradiction (assume the opposite and reach something impossible). Geometric proofs use given facts, definitions and theorems (parallel lines, congruent and similar triangles), with a reason for every line. Analytic proofs use coordinates; synthetic proofs use pure geometry.