Netherlands VWO 3 (onderbouw) Mathematics

1. Equations

Using equations · Harder equations and inequalities

• 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.
• 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.

2. Data and chance

Interpreting and representing data · Analysing datasets · Probability

• Statistics: Mean, Median and Mode of Grouped Data – When data is put into classes (like marks 20–30), we cannot see each value. We use the middle of each class to find the mean, the running total to find the median, and the tallest bar to find the mode. All three tell us the 'centre' of the data in different ways.
• 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.
• 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.

3. Mathematical thinking

Problem solving · Mathematical modelling · Reasoning and proof · Algorithms

• 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.
• 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.
• 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.
• 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.

4. Maths in the world

Applying maths in real situations

Coming soon