Netherlands VWO 4 (bovenbouw, 2e fase) Mathematics A
Chapters: 4
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. Algebra and counting
Algebra · Counting problems
- 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.
- 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.
3. Relationships (part 1)
Standard functions
- Functions: Composite, Inverse and Standard Graphs – A function is a rule that gives exactly one output for each allowed input. The allowed inputs are the domain; the outputs are the range. Two functions can be joined: g(f(x)) means do f first, then g. An inverse function f⁻¹ undoes f, and its graph is the mirror image of the graph of f in the line y = x. Only one-to-one functions have an inverse.
4. Statistics and probability (part 1)
Research question and design · Visualising data · Summarising data · Statistics with ICT
- The Statistical Inquiry Cycle – A statistical inquiry answers a question with data in five steps: Problem (ask a clear question), Plan (decide who to ask, which variable, how), Data (collect and record), Analysis (tables, graphs, averages) and Conclusion (answer the question, state limits, ask new questions). Then the cycle can start again.
- Data Representation: How Computers Store Numbers, Text, Images and Sound – Data is raw facts; information is data given meaning; knowledge is information we can use. A computer stores all data as bits (0 or 1). 8 bits make a byte, and 1 kB = 1000 bytes, 1 MB = 1000 kB, 1 GB = 1000 MB, 1 TB = 1000 GB. Numbers are stored in binary, where place values double: 1, 2, 4, 8 and so on. Text uses a character set: in ASCII 'A' is 65; Unicode covers every script. A bitmap image is a grid of pixels; size = width × height × colour depth. Sound is sampled: size = sample rate × bit depth × seconds. Vector images store shapes instead of pixels. Compression makes files smaller: lossless keeps every bit, lossy throws some detail away.