Mathematics: Algebra, Functions, Geometry, and Data
Mathematics in the Chiba Entrance Science paper tests fluent algebraic manipulation, function reasoning, plane and coordinate geometry, probability, and interpretation of quantitative data. Questions reward a clear setup and exact calculation rather than memorized templates.
Core rules
- Preserve equivalence: apply the same operation to both sides, and state restrictions before dividing by an expression that could be zero.
- For functions, track domain and range; a graph-reading answer should identify intercepts, slopes, and turning points.
- In geometry, label corresponding sides and angles, and keep lengths, areas, and volumes in consistent units.
- Probability requires a defined sample space; count equally likely outcomes before forming a ratio.
- In data questions, distinguish a rate from a total and a percentage change from a percentage of the original.
Worked example
For f(x) = x^2 - 4x + 3, find the roots, vertex, and minimum value. Complete the square: x^2 - 4x + 3 = (x - 2)^2 - 1. The roots satisfy (x - 2)^2 = 1, so x = 1 or x = 3. The vertex is (2, -1). Because the coefficient of x^2 is positive, the parabola opens upward, so the minimum value is -1 at x = 2. Substituting f(2) = 4 - 8 + 3 = -1 checks the result.
Traps that cost marks
- Sign slips when expanding or moving terms; substitute the answer back into the original equation.
- Squaring both sides can introduce extraneous roots, so check every candidate in the original equation.
- Mixing slope and intercept, or reading a graph's vertical axis as time when it is distance.
- Giving a root when the question asks for a sum, product, or interval.
- Rounding too early in multi-step calculations; keep exact values until the final line.