China 高三 Mathematics

1. Ch.6 Counting principles

Addition and multiplication principles · Permutations and combinations · Binomial theorem

• 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.
• Binomial Theorem for Positive Integers – A binomial is a two-term expression like a + b. The binomial theorem tells us how to expand (a + b)ⁿ for any positive integer n without multiplying again and again: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + ⁿCₙbⁿ. There are n + 1 terms. The power of a goes down by 1 and the power of b goes up by 1 in each term, and the two powers always add up to n. The coefficients ⁿCᵣ are the numbers in row n of Pascal's triangle, where each number is the sum of the two above it. The (r + 1)th term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ. The pattern was known long ago in India (Pingala's Meru Prastara) and in Persia and China, and Pascal later studied the triangle in detail. The theorem is proved by mathematical induction using ⁿCᵣ₋₁ + ⁿCᵣ = ⁿ⁺¹Cᵣ.

2. Ch.7 Random variables

Conditional probability; total probability · Discrete random variables; mean and variance · Binomial and hypergeometric distributions · Normal distribution

• Conditional Probability, Multiplication Rule and Independent Events – Conditional probability is the chance of A when we already know B has happened. We throw away every outcome outside B and count again: P(A|B) = P(A ∩ B) ÷ P(B). Turned around, this gives the multiplication rule P(A ∩ B) = P(B)·P(A|B). If knowing B does not change the chance of A, the events are independent and P(A ∩ B) = P(A)·P(B).
• Random Variables and Probability Distributions – A random variable X gives a number to every outcome of a chance experiment. A discrete random variable takes separate values; its probability distribution lists each value x with its probability P(X = x), and these add up to 1. The mean (expected value) is E(X) = Σ x·P(x). The variance Var(X) = E(X²) − [E(X)]² measures spread, and the standard deviation is its square root.
• Binomial Distribution – Repeat the same yes/no trial n times, independently, with the same chance of success p each time. The number of successes X follows the binomial distribution B(n, p): P(X = k) = ⁿCₖ pᵏ qⁿ⁻ᵏ with q = 1 − p. Its mean is np and its variance is npq.
• Normal Distribution – A normal distribution is a continuous, symmetric, bell-shaped distribution fixed by its mean μ (centre) and standard deviation σ (spread): X ~ N(μ, σ²). Mean = median = mode. About 68% of values lie within 1σ of μ, 95% within 2σ and 99.7% within 3σ. Any normal value is turned into a standard score z = (x − μ)/σ, which follows N(0, 1); probabilities are areas under the curve, read from a table or calculator as Φ(z).

3. Ch.8 Paired data

Correlation coefficient · Linear regression (least squares) · Contingency tables; independence test

• Correlation: Scatter Diagram, Karl Pearson's Coefficient and Spearman's Rank Correlation – Correlation tells how two variables move together. It is positive when both rise together, negative when one rises as the other falls, and zero when there is no straight-line pattern. A scatter diagram shows it as a picture. Karl Pearson's coefficient r measures its direction and strength and always lies between −1 and +1. Spearman's rank correlation R uses ranks and works for qualities like beauty or honesty; tied ranks need a small correction.
• Linear Regression and the Least Squares Line – Linear regression finds the straight line ŷ = a + bx that best follows paired data (x, y). A residual is the gap between a real point and the line: e = y − ŷ. The least squares line makes the sum of squared residuals as small as possible. Its slope is b = Sxy ÷ Sxx and it always passes through the mean point (x̄, ȳ). We use it to predict y from x, but only inside the data range, and a strong link does not prove that x causes y.
• Chi-Square Test for Independence and Homogeneity – A chi-square (χ²) test checks if counts in a table are too far from what we would expect by chance. For a contingency table: E = row total × column total ÷ grand total, χ² = Σ (O − E)² ÷ E, df = (r − 1)(c − 1). If χ² is bigger than the critical value (or p < significance level), reject H0 of no association. For 2×2 tables, Yates' correction uses (|O − E| − 0.5)².

4. Elective A (science/engineering)

Calculus: limits, continuity, derivatives, definite integrals · Spatial vectors and algebra: 3×3 matrices, determinants, linear systems, isometries · Probability and statistics: continuous variables, estimation, hypothesis tests, regression

• Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.
• Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
• Hypothesis Testing – A hypothesis test checks a claim about a population using a sample. Start with the null hypothesis H₀ (no change, e.g. p = 0.5) and the alternative H₁ (what we suspect, e.g. p > 0.5). Choose a significance level such as 5%. Work out how likely the sample result (or more extreme) is if H₀ were true: the p-value. If the p-value is below the level, or the result falls in the critical region, reject H₀. Otherwise there is not enough evidence to reject it. Type I error = rejecting a true H₀; Type II error = not rejecting a false H₀.

5. Elective B (economics/social science)

Calculus · Spatial vectors and algebra · Applied statistics (orthogonal design, cluster analysis) · Models (linear, quadratic, exponential, trig, parametric)

• Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.
• Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
• 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.

6. Elective C (humanities)

Introductory logical reasoning · Mathematical models (finance, input-output, growth) · Social surveys and data analysis

• Logic: How Good Arguments Work – Logic studies good reasoning. An argument has premises (reasons) and a conclusion; inference is the step between them. In a deductive argument, if the premises are true the conclusion must be true; such an argument is valid, and it is sound if the premises really are true. One counterexample shows an argument form is invalid. Inductive arguments move from observed cases to a general or likely conclusion; they are strong or weak, never certain. Categorical propositions (A, E, I, O) and simple predicate logic (∀, ∃) help us write arguments precisely and spot fallacies.
• 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.
• 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.

7. Elective D (PE and arts)

Beauty and mathematics · Mathematics in music · Mathematics in fine art · Mathematics in sport

• Beauty and Mathematics – Things look beautiful when they are simple, balanced and orderly. Maths describes this order: symmetry (mirror and rotation), periodicity (a pattern that repeats after a fixed distance) and harmony (parts in simple ratios such as 2 : 3). A shape with rotation symmetry of order n turns by 360° ÷ n and looks the same.
• 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.
• Mathematics in Fine Art – Artists use three big ideas from maths. The golden ratio φ ≈ 1.618 gives balanced rectangles and a spiral. Perspective makes a flat drawing look deep: parallel lines meet at a vanishing point on the horizon and size shrinks in proportion to distance. A fractal is a shape made of smaller copies of itself, such as a branching tree.
• Mathematics in Sport – Maths helps athletes and organisers. A ball thrown at speed v and angle θ travels R = v² sin 2θ ÷ g (best at 45° without air drag). Statistics such as mean, median, batting average and strike rate summarise performance. Scheduling counts matches: a round-robin of n teams has n(n−1)/2 matches, a knockout has n−1.

8. Elective E (school-based)

Broadening, daily-life and local topics; university preparatory

Coming soon

9. 高考 review

Comprehensive 高考 review

Coming soon