Japan 中学3年 Mathematics
Chapters: 4
1. Numbers and expressions
Square roots · Polynomials: expansion and factorisation · Quadratic equations
- Irrational Numbers: √2, √3, the Square Root Spiral and Real Numbers – An irrational number cannot be written as p/q. Its decimal never ends and never repeats, like √2 = 1.41421356… or π. We can still mark √2 exactly on the number line: it is the diagonal of a square of side 1. We prove √2 and √3 are irrational by assuming they are fractions in lowest terms and reaching a contradiction. The square root spiral builds √2, √3, √4, √5 … one right triangle at a time. Rational and irrational numbers together form the real numbers; every repeating decimal can be turned back into p/q.
- Exploring Algebraic Identities – An identity is an equation that is true for every value of the letters. Pictures prove them: a square of side (a + b) splits into a², ab, ab and b², so (a + b)² = a² + 2ab + b². In the same way we get (a − b)², a² − b² = (a + b)(a − b), (x + a)(x + b), (a + b + c)² and (a + b)³. Read backwards, identities help us factorise, calculate fast and simplify rational expressions.
- 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. Geometry
Similarity · Inscribed and central angles · Pythagorean theorem
- Similar Triangles – Two figures are similar when they have the same shape but maybe a different size. Two triangles are similar if their matching angles are equal and their matching sides are in the same ratio. Basic Proportionality Theorem (BPT): a line parallel to one side of a triangle cuts the other two sides in the same ratio; its converse is also true. You can prove similarity with AA, SSS or SAS.
- Circles: Chords and Angles – A chord joins two points on a circle. Longer chords make bigger angles at the centre, and equal chords make equal angles. The perpendicular from the centre to a chord cuts it in half, and equal chords are the same distance from the centre. The angle an arc makes at the centre is double the angle it makes anywhere on the rest of the circle, so the angle in a semicircle is 90°. In a cyclic 4-gon, opposite angles add to 180°.
- Pythagoras Theorem – In any right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides: a² + b² = c².
3. Functions
Function y = ax²
- Quadratic Functions and Their Graphs – A quadratic function is y = ax² + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola. If a > 0 it opens up and has a lowest point; if a < 0 it opens down and has a highest point. That turning point is the vertex, at x = −b/2a. In vertex form y = a(x − h)² + k the vertex is (h, k).
4. Using data
Sampling surveys
- Sampling: Learning About a Population from a Sample – A population is the whole group we want to know about; a sample is a smaller part we actually check. A good sample is chosen at random so that it represents the population. Simple random sampling gives everyone an equal chance; stratified sampling takes the right share from each group; systematic sampling takes every k-th item. Different samples give slightly different answers (sampling variation), but bigger samples wobble less (law of large numbers). A biased sample gives a wrong answer however big it is.