South 중학교 2학년 Mathematics 2
Chapters: 7
1. Rational numbers and decimals
Repeating decimals and rational numbers
- Rational Numbers: Number Line, Density and Decimals – A rational number is any number that can be written as p/q, where p and q are integers and q is not 0. Examples: 3/4, −5/3, 7 (= 7/1), 0.25 (= 1/4). To place p/q on the number line, cut each unit into q equal parts and hop p parts from 0. Between any two rational numbers there is always another one (their average), so there are endlessly many between them. The decimal of a rational number either ends (terminating) or repeats a block for ever (non-terminating repeating).
2. Expressions
Laws of exponents · Adding and subtracting polynomials · Multiplying and dividing by monomials
- 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.
- Introduction to Polynomials – An algebraic expression is made of terms like 3x², −5x and 7. It is a polynomial when every power of the variable is a whole number (0, 1, 2, …). The degree is the biggest power. Degree 1 polynomials, y = ax + b, are called linear. They model things that grow or shrink by the same amount each step. a is the slope (change per step) and b is the y-intercept (starting value).
3. Inequalities and simultaneous equations
Inequalities and their properties · Solving linear inequalities · Simultaneous linear equations in two unknowns
- Linear Inequalities in One Variable – An inequality compares two expressions with <, >, ≤ or ≥. A linear inequality in one variable looks like ax + b < c. Its answer is usually a whole range of numbers, not one number. We solve it like an equation: we may add or subtract the same number on both sides, and multiply or divide by the same positive number. If we multiply or divide by a negative number, the sign must flip. We show the answer on a number line: a hollow dot for < or > (end not included) and a filled dot for ≤ or ≥ (end included), with the shaded part showing all solutions. Double inequalities like −1 ≤ x < 3 give a piece of the line. If x must be a natural number or an integer, only the whole numbers in that range count.
- 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.
4. Linear functions
Idea of a function · Linear functions and their graphs · Slope and intercepts · Linear functions and two-variable equations · Graphs and simultaneous equations
- Functions: Composite, Inverse and Standard Graphs – A function is a rule that gives exactly one output for each allowed input. The allowed inputs are the domain; the outputs are the range. Two functions can be joined: g(f(x)) means do f first, then g. An inverse function f⁻¹ undoes f, and its graph is the mirror image of the graph of f in the line y = x. Only one-to-one functions have an inverse.
- 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.
5. Properties of triangles and quadrilaterals
Isosceles triangles · Circumcentre and incentre · Properties of quadrilaterals
- 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.
- 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.
- 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.
6. Similarity and Pythagoras
Similar figures and ratio · Similar triangle conditions · Parallel lines and segment ratios · 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.
- 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.
7. Probability
Counting outcomes · Probability and its basic properties
- 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.