Netherlands HAVO 4 (bovenbouw, 2e fase) Mathematics A
Chapters: 5
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
Arithmetic · Algebra · Counting problems
- Percentages – Per cent means "out of 100", so x% = x/100. To find x% of an amount, multiply by x/100. Percentage change = (change ÷ original) × 100. To increase by r%, multiply by (1 + r/100); to decrease by r%, multiply by (1 − r/100). Successive changes multiply their multipliers. For reverse percentage, divide the new value by the multiplier.
- 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)
Tables · Graphs, equations and inequalities · Formulas with one or more variables
- 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).
- 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.
4. Change
Describing change
- Rate of Change: How Fast Does It Change? – Rate of change tells you how much an output changes for each one-unit change in the input. A graph can rise (increasing), fall (decreasing), or turn at a highest or lowest point. To measure change between two inputs a and b, find the change in input, Δx = b − a, and the change in output, Δy = f(b) − f(a). The average rate of change is Δy ÷ Δx, which is the slope of the straight line (secant) through the two points. A table of differences shows if growth is steady (constant differences, linear) or speeding up (growing differences, for example exponential). If the two points come very close, the average rate becomes the slope of the graph at one point. Sequences are lists of values with a rule: a recursive formula builds each term from the one before, a direct formula gives any term at once.
5. Statistics (part 1)
Interpreting and judging data displays · Processing data · Statistics with ICT
- 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.
- 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.
- 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.