South 중학교 3학년 Mathematics 3
Chapters: 6
1. Real numbers
Square roots · Irrational numbers · Ordering real numbers · Operations with radicals
- Square Roots: Build a Square, Find Its Side – The square root of a number n is the number that, multiplied by itself, gives n. √25 = 5 because 5 × 5 = 25. Picture n tiles arranged in a square: the root is the side. Perfect squares (1, 4, 9, 16, …) have whole-number roots. Other roots are irrational and lie between two whole numbers; we estimate them or simplify them, like √50 = 5√2.
- 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.
- 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. Factorisation and quadratic equations
Expanding and factorising · Quadratic equations
- 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.
- 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.
3. Quadratic functions
Quadratic functions and graphs · Properties of parabolas
- 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).
4. Trigonometric ratios
Trigonometric ratios · Using trigonometric ratios
- 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°.
5. Properties of circles
Chords and tangents · Inscribed angles
- 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°.
6. Statistics
Median, mode, mean · Variance and standard deviation · Scatter plots and correlation
- 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.