China 八年级（初二） Mathematics
Chapters: 12
1. Ch.13 Triangles
Concept of a triangle · Sides; medians, angle bisectors, altitudes · Interior and exterior angles · Why proofs are needed · Practice: centre of gravity of a thin plate
- Triangles: Types, Sides, Medians and Altitudes – A triangle is a closed shape with 3 sides, 3 corners (vertices) and 3 angles. The three angles always add up to 180°. By sides, a triangle is equilateral (3 equal sides), isosceles (2 equal) or scalene (none equal). By angles it is acute (all under 90°), right (one 90°) or obtuse (one over 90°). Any two sides together must be longer than the third side: this is the triangle inequality. A median goes from a corner to the middle of the opposite side, an angle bisector splits the angle in two equal parts, and an altitude drops straight down at 90°. A triangle cannot change shape when pushed, so it is the strongest shape for bridges, roofs and towers.
- Lines and Angles: Linear Pair, Vertically Opposite and Parallel Lines – An angle is the turn between two rays that start from the same point. Angles on a straight line add up to 180° (linear pair). When two lines cross, the opposite angles are equal. When a transversal cuts two parallel lines, corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side add up to 180°. The reverse is also true, and we can prove such facts by contradiction: assume the opposite and show it leads to something impossible.
- Logic and Proof: How Mathematicians Show Something Is Always True – A proof is a chain of correct reasons that shows a statement is true in every case. Statements are true or false; 'if P then Q' (P ⇒ Q) means Q must hold whenever P does, and its converse Q ⇒ P may not. P ⇔ Q means both directions hold. Main methods: direct proof (from known facts step by step, often with algebra like 2n + 1 for odd numbers); proof by exhaustion (check every possible case); disproof by counterexample (one failing case kills a claim); proof by contradiction (assume the opposite and reach something impossible). Geometric proofs use given facts, definitions and theorems (parallel lines, congruent and similar triangles), with a reason for every line. Analytic proofs use coordinates; synthetic proofs use pure geometry.
- Properties of Triangles and Their Centres – The angles of every triangle add up to 180°. A triangle has four famous centres. Medians meet at the centroid G, which cuts each median 2 : 1. Perpendicular bisectors meet at the circumcentre O, the centre of the circle through the corners. Angle bisectors meet at the incentre I, the centre of the circle inside that touches all sides. Altitudes meet at the orthocentre H. O, G and H lie on one line, the Euler line.
2. Ch.14 Congruent triangles
Congruent triangles and properties · Tests for congruence (SAS, ASA, AAS, SSS, HL) · Angle bisector properties · Axiomatic method (history)
- Congruence of Triangles – Two triangles are congruent when one can be placed exactly on top of the other: all three sides and all three angles match. You do not need to check all six parts. Any one of SAS, SSS, ASA, AAS or RHS is enough. SSA is not enough. In an isosceles triangle the angles opposite the equal sides are equal, and the converse is also true.
- Angle Bisector – An angle bisector is a ray that cuts an angle into two equal parts. Every point on it is the same distance from both arms of the angle, and every point that is the same distance from both arms lies on it. You can draw it with only a compass and a ruler.
- Euclid's Geometry: Definitions, Axioms and the Five Postulates – Geometry began as practical measuring of land and altars in Egypt, India and Mesopotamia. Indian Sulbasutras (like Baudhayana's) gave rope rules for making squares, doubling a square and the diagonal rule. Around 300 BCE, Euclid of Alexandria organised geometry as a chain of reasoning: start from a few definitions, common-sense axioms and five geometry postulates, and prove everything else. The fifth postulate is about when two lines meet, and it leads to the idea of parallel lines.
3. Ch.15 Axial symmetry
Axial symmetry; perpendicular bisector · Drawing symmetric figures · Isosceles and equilateral triangles · Sides vs angles inequality · Practice: shortest path
- 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.
- Congruence of Triangles – Two triangles are congruent when one can be placed exactly on top of the other: all three sides and all three angles match. You do not need to check all six parts. Any one of SAS, SSS, ASA, AAS or RHS is enough. SSA is not enough. In an isosceles triangle the angles opposite the equal sides are equal, and the converse is also true.
- Shortest Path by Reflection – To go from A to a straight road and then to B (both on the same side), flip B over the road to get B′. The shortest route is the straight line from A to B′; where it crosses the road is the best stop P. It works because PB = PB′ for every point P on the road, so AP + PB = AP + PB′, and a straight line is the shortest way from A to B′.
4. Ch.16 Multiplying polynomials
Laws of powers · Multiplying polynomials · Difference of squares; perfect squares · Yang Hui triangle
- Laws of Exponents – An exponent tells how many times a base is multiplied by itself. Same base: multiply → add exponents, divide → subtract, power of a power → multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root.
- Multiplying Polynomials – To multiply polynomials, multiply every term of the first by every term of the second (the distributive law), then collect like terms. Coefficients multiply; powers of the same letter add (x² · x³ = x⁵). A rectangle of area tiles shows every product: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6. To divide a polynomial by a monomial, divide each term by it.
- 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.
- Yang Hui's (Pascal's) Triangle – Yang Hui's triangle starts with 1 at the top. Every number below is the sum of the two numbers above it, and the edges are all 1. Row n lists the coefficients of (a + b)ⁿ, so row 4 (1 4 6 4 1) gives (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴. Each row is symmetric and adds up to 2ⁿ. The same triangle is called Pascal's triangle in Europe and Meru Prastara in India.
5. Ch.17 Factorisation
Common factor method · Using identities · x²+(p+q)x+pq type
- 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.
6. Ch.18 Fractional expressions
Fractional expressions and basic property · Multiplying and dividing · Adding and subtracting · Integer exponents · Fractional equations
- Laws of Exponents – An exponent tells how many times a base is multiplied by itself. Same base: multiply → add exponents, divide → subtract, power of a power → multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root.
- Rational Equations – A rational (fractional) equation has the unknown in the denominator of a fraction. First write the banned values that make any denominator zero. Then multiply every term by the lowest common denominator (LCD) to clear the fractions, and solve the equation that is left (often linear or quadratic). Finally check each root: a root that is a banned value is extraneous and is thrown out. The same method solves rate problems such as boats on rivers and people working together.
7. Ch.19 Quadratic radicals
Radicals and properties · Multiplication and division · Addition and subtraction
- 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.
8. Ch.20 Pythagorean theorem
Pythagorean theorem and uses · Converse and uses
- 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.
9. Ch.21 Quadrilaterals
Quadrilaterals and polygons (angle sums) · Parallelograms; midline theorem · Rectangles, rhombuses, squares
- Quadrilaterals (4-gons) – A quadrilateral (4-gon) has 4 sides and 4 angles that add to 360°. In a parallelogram, opposite sides are parallel and equal, opposite angles are equal, and the diagonals cut each other in half. Each of these facts also works as a test. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. The three medians of a triangle meet at one point that cuts each median in the ratio 2 : 1.
10. Ch.22 Functions
Concept of function · Representing functions
- Functions: Input, Rule, Output – A function is a rule that gives exactly one output for each input. We write it as f(x). The output f(a) is called the image of a; an input that gives a certain output is a preimage. A function can be shown as words, a table, a formula or a graph. Its graph can go up (increasing), go down (decreasing) and have a highest point (maximum) or lowest point (minimum).
11. Ch.23 Linear functions
Linear function concept · Graphs and properties · Links to equations and inequalities · Real problems · Practice: music and mathematics
- Linear Functions – A linear function adds the same amount to y every time x goes up by 1. Its rule is y = mx + c (also written f(x) = mx + b). m is the slope: rise ÷ run, the change in y for each 1 step in x. c is the y-intercept: the value of y when x = 0. Its graph is always a straight line.
- Pair of Linear Equations in Two Variables – Two equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ each make a straight line. The answer that fits both is the point where the lines meet. Lines that cross give one answer, parallel lines give none, and lines that lie on top of each other give endless answers. We can find the answer by drawing (graph), by substitution or by elimination.
- Mathematics in Music – A sound is a wave. Its frequency f (in hertz) is the pitch, and its period is T = 1 / f. Strings of length ratio 1 : 2 sound an octave apart, 2 : 3 a fifth. A piano octave has 12 equal steps, each a factor of 2^(1/12) = 1.0595. Rhythm splits a bar into fractions, for example 3 + 3 + 2 = 8 beats.
12. Ch.24 Data analysis
Central tendency (mean, weighted mean, median, mode) · Spread (variance) · Quartiles and box plots · Grouping data · Practice: student fitness survey
- Statistics: Asking Questions, Averages and Stacked Bar Graphs – Statistics starts with a question that has many possible answers. We collect data, organise it, show it in a graph, and find one number that sums it up. The mean is the fair share, the median is the middle value, and the mode is the most common value. A weighted average gives more importance to some values. Stacked and 100% stacked bar graphs show how a whole is split into parts.
- Measures of Dispersion: Range, Mean Deviation, Variance and SD – Dispersion means spread: how far the values sit from the centre. Range = largest − smallest. Mean deviation = average distance from the mean (or median). Variance = average of squared distances from the mean. Standard deviation = √variance. The same ideas work for grouped data when every term is multiplied by its frequency.
- Student Fitness Survey: A Data Project from Start to Finish – A survey project has two parts. First collect data fairly: one question, one test, same rules for everyone, results written with units. Then analyse it: sort, find the mean, median and quartiles, draw a box plot, and write a short report with a clear finding and one honest limit.