China 九年级（初三） Mathematics
Chapters: 10
1. Ch.25 Quadratic equations in one variable
Concept · Solving: square roots, completing the square, formula, factorising · Discriminant; roots and coefficients · Real problems
- 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. Ch.26 Quadratic functions
Concept · Graphs and properties; vertex form · Quadratic functions and equations (approximate roots) · Maximum/minimum problems
- 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).
3. Ch.27 Inverse proportion functions
Concept · Graphs and properties · Real problems · Practice: high-speed rail quantities
- Inverse Variation (Inverse Proportion) – Two quantities vary inversely when their product stays the same: x × y = k, so y = k/x. If x is doubled, y is halved. The graph of y = k/x (k > 0) is a curve called a hyperbola that comes close to both axes but never touches them.
- Speed, Distance and Time (with Time and Work) – Speed = distance ÷ time. To change km/h into m/s multiply by 5/18. Average speed = total distance ÷ total time. Two bodies moving towards each other close the gap at the sum of their speeds; in the same direction, at the difference. A train must cover its own length (plus the platform or other train). Downstream speed = boat + stream; upstream = boat − stream. Work and pipes use the same idea: add rates (per hour), subtract leaks.
4. Ch.28 Rotation
Rotation of figures · Central symmetry
- Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.
- Symmetry: Mirror Lines, Centres and Bisectors – A figure is symmetric when one half is a perfect copy of the other. In line (axial) symmetry the copy is a mirror image across a line. In point (central) symmetry the copy is the figure turned half a circle (180°) about a centre. The perpendicular bisector of a segment and the bisector of an angle are lines of symmetry, so every point on them is the same distance from two things.
5. Ch.29 Circles
Circle concepts · Chords, arcs, central and inscribed angles; perpendicular diameter; cyclic quadrilateral · Arc length and sector area
- 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°.
- Area of Sector and Segment of a Circle – A sector is a pizza slice of a circle: its area is θ/360 × πr². Its crust is an arc of length θ/360 × 2πr. A segment is the slice minus the triangle inside it: segment = sector − triangle.
6. Ch.30 Lines and circles
Line–circle positions; tangents; tangent-length theorem · Incircle and circumcircle · Regular polygons and circles · Practice: optimisation in daily life
- Tangent to a Circle – A tangent is a line that touches a circle at exactly one point. At that point it makes a 90° angle with the radius, and two tangents drawn from one outside point are always equal in length.
- The Circle: Chords, Tangents and Triangle Circles – A circle is all points at one fixed distance (the radius) from a centre. A chord joins two points on it; the longest chord is the diameter. A line can cut a circle at two points (secant), touch it at one (tangent) or miss it. A tangent is perpendicular to the radius. Every triangle has an incircle inside and a circumcircle around it.
- Regular Polygons and Their Circles – A regular polygon has all sides equal and all angles equal. Every regular polygon has a centre O, a circumscribed circle through all its vertices (radius R) and an inscribed circle touching every side (radius r, the apothem). For n sides of length a: central angle = 360°/n, interior angle = (n − 2)·180°/n, R = a / (2 sin(180°/n)), r = a / (2 tan(180°/n)), and area = ½ · perimeter · r. Special cases: triangle R = a/√3, r = a/(2√3); square R = a/√2, r = a/2; hexagon R = a, r = a√3/2.
- Optimisation: Finding the Best Value with Quadratics – Optimisation means finding the best value: the biggest area, the largest profit or the smallest cost. We write the quantity as a quadratic function, then the vertex of its parabola gives the best value. For y = ax² + bx + c the vertex is at x = −b/(2a); if a < 0 it is a maximum, if a > 0 it is a minimum.
7. Similarity (下册)
Similar figures; ratio, golden section · Similar triangles: tests and properties · Dilation (homothety) incl. coordinates
- 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.
- Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.
8. Acute-angle trigonometry (下册)
sin, cos, tan; values of 30°, 45°, 60° · Solving right triangles; applications
- Trigonometric Ratios (sin, cos, tan) – In a right triangle, pick one sharp (acute) angle θ. The side facing θ is the opposite, the side touching θ (not the longest) is the adjacent, and the longest side is the hypotenuse. sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. cosec, sec and cot are their flips. The ratios depend only on the angle, not on the size of the triangle. Learn the table for 0°, 30°, 45°, 60°, 90°.
- Heights and Distances – The line of sight joins your eye to the object. If the object is above you, the angle between the line of sight and the horizontal is the angle of elevation. If it is below you, that angle is the angle of depression. The object, the ground and the line of sight make a right triangle, so tan θ = height ÷ distance. The angle of depression from the top equals the angle of elevation from the bottom. Hard questions use two right triangles that share one side.
9. Projection and views (下册)
Central and parallel projection · Three views; nets of prisms and cones
- Projections: Central, Parallel and Orthogonal – A projection draws a 3D object on a flat surface using straight rays. If all rays start at one point (a lamp, your eye) it is a central projection: images change size with distance and give perspective. If all rays are parallel (sunlight) it is a parallel projection: sizes do not depend on distance. If the parallel rays are at 90° to the screen it is an orthogonal projection: faces parallel to the screen show their true size. Engineers use orthogonal views, artists use perspective.
- Orthographic Views of Solids – An orthographic view shows a solid as seen straight on from one direction, with no perspective. The three main views are the front view (front elevation), the top view (plan) and the side view (side or end elevation). Each view loses one dimension, so we usually need all three to fix the shape. Views are arranged in a standard layout: first-angle projection (plan below the front view; used in India, Europe and ISO standards) or third-angle projection (plan above; used in the USA). Visible edges are thick continuous lines, hidden edges are dashed, and centre lines are chain lines. A net is the flat pattern that folds into a solid, and a cross-section is the shape you see when you cut through it.
10. Probability and statistics (standard, placement varies)
Listing outcomes with tables and tree diagrams · Estimating probability by frequency · Simple random sampling; pie charts; frequency histograms
- 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.
- Introduction to Probability: Scale, Experiments, Sample Spaces and Trees – Probability is a number from 0 to 1 that tells how likely something is. 0 means it can never happen, 1 means it will surely happen. We can find it by doing an experiment many times (empirical probability), or by listing every possible result (the sample space) and counting the ones we want. Tree diagrams and tables help us list results when two things happen together.
- 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.