China 高一 Mathematics
Chapters: 11
1. Ch.1 Sets and logic
Sets, relations, operations · Necessary and sufficient conditions · Universal and existential quantifiers; negation
- Sets: Representation, Types, Subsets, Venn Diagrams and Operations – A set is a well-defined collection of different objects. We write it in roster form {2, 4, 6} or set-builder form {x : x is even}. Sets can be empty, finite, infinite or equal. If every element of B is in A, B is a subset of A (B ⊂ A); a set with n elements has 2ⁿ subsets. Intervals like (a, b) and [a, b] are subsets of real numbers. The universal set U holds everything under study. With Venn diagrams we see union A ∪ B, intersection A ∩ B, difference A − B and complement A′ = U − A.
- Propositions and Conditions: The Logic Behind Maths – A proposition is a sentence that is either true or false. We join propositions with NOT, AND, OR, IF…THEN and IF AND ONLY IF. 'If p then q' is false only when p is true and q is false. Its contrapositive 'if not q then not p' always has the same truth value. When p ⇒ q, p is sufficient for q and q is necessary for p. A predicate like 'x > 3' becomes a proposition when we fix x or add 'for all' / 'there exists'.
2. Ch.2 Quadratic functions, equations, inequalities
Properties of equalities and inequalities · AM–GM inequality · Quadratic inequalities
- Inequalities: Rules, Intervals and Solving Them – An inequality says one amount is bigger or smaller than another, using <, >, ≤ or ≥. On a number line, the smaller number is on the left. You may add or subtract the same number on both sides, and multiply or divide by the same positive number, and the sign stays. If you multiply or divide by a negative number, the sign flips. The answer is usually a whole set of numbers, written as an interval such as (−∞, 4]. A quadratic inequality is solved from its roots and the shape of its graph. |x| < a means −a < x < a. Some inequalities are true for every number, like x² ≥ 0 and the AM–GM inequality.
- AM–GM Inequality – For two positive numbers a and b, the arithmetic mean (a+b)/2 is never smaller than the geometric mean √(ab). They are equal only when a = b. This gives the biggest product when the sum is fixed, and the smallest sum when the product is fixed.
- Quadratic Inequalities – A quadratic inequality asks where ax² + bx + c is above zero (> 0) or below zero (< 0). Find the roots, picture the parabola, and read the answer from the graph. When a > 0 the curve is below zero between the roots and above zero outside them. If there are no real roots (D < 0), the curve is always on one side of the x-axis.
3. Ch.3 Functions
Definition and representation · Monotonicity, extrema, parity · Power functions · Applications
- Relations and Functions: Cartesian Product, Domain, Range, Graphs – An ordered pair (a, b) has a first and a second place, so (1, 2) ≠ (2, 1). The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B, and n(A × B) = n(A) × n(B). A relation from A to B is any subset of A × B; its domain is the set of first elements and its range the set of second elements, while B is the co-domain. A function is a special relation where every element of A has exactly one image. We study constant, identity, polynomial, rational, modulus, signum, exponential, log and greatest integer functions with their graphs, and add, subtract, multiply and divide functions point by point.
- Properties of Functions: Reading a Graph Like a Story – A function gives exactly one output f(x) for each input x. From its graph we read the domain (allowed x), the range (y values reached), the zeros (where f(x) = 0), the intervals where f is positive or negative, where it increases or decreases, its maximum and minimum, and whether it is even (mirror in the y-axis) or odd (half-turn about the origin).
- Power Functions: y = a·xⁿ – A power function has the form y = a·xⁿ, where a is a number and n is a fixed exponent. Even whole powers (x², x⁴) make U shapes that are symmetric about the y-axis. Odd whole powers (x³, x⁵) make S shapes that are symmetric about the origin. All y = xⁿ with n > 0 pass through (0, 0) and (1, 1). Negative powers (x⁻¹ = 1/x) have asymptotes and are not defined at x = 0. Fractional powers are roots: x^(1/2) = √x, x^(1/3) = ∛x. The root function is the inverse of the matching power.
- 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.
4. Ch.4 Exponential and logarithmic functions
Rational and real exponents · Exponential functions · Logarithms; logarithmic functions · Zeros; bisection; function models
- 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.
- Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
- Logarithms – A logarithm tells you the power you need. log_b x = y means b^y = x. So log₂ 8 = 3 because 2³ = 8. Logs turn multiplying into adding: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a. Base 10 is the common log, base e is the natural log (ln). Any base can be changed: log_b x = log x ÷ log b.
- 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.
5. Ch.5 Trigonometric functions
Any angle; radians · Trig functions of any angle · Reduction formulae · Graphs and properties of sin, cos, tan · Sum, difference, double-angle identities · y = A sin(ωx+φ); applications
- Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.
- The Sine Wave y = A sin(ωx + φ) – The graph of y = A sin(ωx + φ) is a smooth wave. A is the amplitude (height), the period is T = 2π/ω, and the wave is shifted sideways by φ/ω (to the left if φ is positive). It models anything that repeats: a Ferris wheel, tides, sound and AC current.
6. Ch.6 Plane vectors
Vectors and linear operations · Dot product · Basis theorem and coordinates · Law of cosines and sines
- Scalars and Vectors for Motion in a Plane (Class 11) – A scalar has only size; a vector has size and direction. Vectors are equal if their size and direction match. Multiplying by a number changes the length (a negative number flips it). Vectors add tail-to-head (triangle or parallelogram law); A − B = A + (−B). Any vector in a plane is A = Ax î + Ay ĵ with Ax = A cos θ, Ay = A sin θ. A·B = AB cos θ is a scalar; A×B has size AB sin θ and is perpendicular to both.
- Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.
- Law of Sines (Sine Rule) – In any triangle, each side divided by the sine of the angle facing it gives the same number: a/sinA = b/sinB = c/sinC = 2R, where R is the radius of the circle through the three corners. Use it when you know two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA). SSA can give two triangles, one or none.
7. Ch.7 Complex numbers
Extending the number system · Arithmetic and geometric meaning · Trigonometric form (optional)
- Complex Numbers and Quadratic Equations – Some equations like x² + 1 = 0 have no real answer, because no real number has a negative square. So we add a new number i with i² = −1. A complex number is z = a + ib: a is the real part and b is the imaginary part. We add, subtract and multiply complex numbers like algebra, and replace i² by −1. To divide, we multiply top and bottom by the conjugate. On the Argand plane, z = a + ib is the point (a, b). The conjugate a − ib is its mirror image in the real axis, and the modulus √(a² + b²) is its distance from the origin. Every quadratic ax² + bx + c = 0 with D = b² − 4ac < 0 has two complex roots that are conjugates of each other.
- De Moivre's Theorem and Roots of Unity – A complex number can be written by its length r and angle θ: z = r(cos θ + i sin θ) = re^{iθ}. When you multiply, the lengths multiply and the angles add. So zⁿ = rⁿ(cos nθ + i sin nθ): this is De Moivre's theorem. It gives quick powers, formulas for cos nθ and sin nθ, and the n roots of any number. The n roots of 1 sit evenly on the unit circle like the corners of a regular polygon.
8. Ch.8 Solid geometry
Basic solids; intuitive drawings · Surface area and volume · Points, lines, planes · Parallelism · Perpendicularity
- Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.
- Surface Areas and Volumes: Combined Solids – Many real objects are two simple solids stuck together, like a cone on a cylinder. The surface area is only the outside skin you can touch, so the hidden joint is left out. The volume is the space inside, so you simply add the volumes of the parts.
- Solid Geometry: Points, Lines and Planes in Space – Solid geometry studies figures in three dimensions. Three points not on one line fix a plane. Two lines in space can be parallel, intersecting or skew (not in one plane). A line can lie in a plane, cut it, or be parallel to it; it is perpendicular to a plane if it is perpendicular to two intersecting lines of that plane. Angles in space are found by projecting onto a plane and using right triangles.
9. Ch.9 Statistics
Random sampling (simple, stratified) · Estimating the population: percentiles, central tendency, spread · Case study
- 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.
- Statistics: Mean, Median and Mode of Grouped Data – When data is put into classes (like marks 20–30), we cannot see each value. We use the middle of each class to find the mean, the running total to find the median, and the tallest bar to find the mode. All three tell us the 'centre' of the data in different ways.
10. Ch.10 Probability
Sample space; events; classical probability · Independent events · Frequency and probability; simulation
- 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.
- 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).
- 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.
11. Modelling and inquiry
Mathematical modelling project
- 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.