📚 Unit 1 Higher Homework: Algebra, Number and Coordinate Skills | 第一单元高等作业：代数、数与坐标技能
This revision guide covers the core topics commonly tested in Unit 1 Higher Homework, including number types, surds, indices, standard form, algebraic manipulation, quadratic equations, inequalities, simultaneous equations, straight-line graphs, coordinate geometry and trigonometry. Use the paired English-Chinese notes to consolidate key definitions and worked methods before attempting practice questions.
本复习指南涵盖第一单元高等作业中常考的核心主题，包括数的类型、根式、指数、标准形式、代数变形、二次方程、不等式、联立方程、直线图、坐标几何和三角学。使用英中对照笔记巩固关键定义和解题方法，然后再尝试练习题。
1. Number Types and Surds | 数的类型与根式
A rational number can be written as a fraction a/b where a and b are integers and b ≠ 0. An irrational number cannot be written as a simple fraction; square roots of non-perfect squares such as √2 and √3 are irrational.
有理数可以写成分数 a/b，其中 a 和 b 是整数且 b ≠ 0。无理数不能写成简单分数；非完全平方数的平方根如 √2 和 √3 都是无理数。
A surd is an irrational root that cannot be simplified to remove the root, for example √5 or 3√7. To simplify surds, factor the number under the root into a square number and another integer: √50 = √25 × √2 = 5√2.
根式是无法化简去掉根号的无理根，例如 √5 或 3√7。化简根式时，把根号下的数分解为一个平方数和另一个整数：√50 = √25 × √2 = 5√2。
When adding or subtracting surds, combine like surds only: 2√3 + 5√3 = 7√3, but √2 + √3 cannot be simplified further. To rationalise a denominator such as 1/√2, multiply the numerator and denominator by √2 to obtain √2/2.
根式加减时只能合并同类根式：2√3 + 5√3 = 7√3，但 √2 + √3 不能进一步化简。要把 1/√2 这样的分母有理化，可将分子和分母同时乘以 √2，得到 √2/2。
2. Laws of Indices | 指数法则
Indices follow three essential multiplication, division and power rules. When multiplying the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. When dividing, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. When raising a power to another power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ.
指数遵循三个基本的乘、除和幂运算法则。同底数相乘时指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。同底数相除时指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。幂的乘方时指数相乘：(aᵐ)ⁿ = aᵐⁿ。
Negative indices represent reciprocals: a⁻ⁿ = 1/aⁿ. Fractional indices connect roots and powers: a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√(aᵐ). For example, 16^(1/2) = √16 = 4 and 27^(2/3) = (∛27)² = 3² = 9.
负指数表示倒数：a⁻ⁿ = 1/aⁿ。分数指数连接根式与幂：a^(1/n) = ⁿ√a，a^(m/n) = ⁿ√(aᵐ)。例如 16^(1/2) = √16 = 4，27^(2/3) = (∛27)² = 3² = 9。
Anything to the power zero is 1, provided the base is not zero: a⁰ = 1. This rule is often tested when simplifying expressions such as 5x⁰ + 2 = 5 × 1 + 2 = 7.
任何非零底数的零次幂都等于 1：a⁰ = 1。化简表达式如 5x⁰ + 2 = 5 × 1 + 2 = 7 时经常用到这一法则。
3. Standard Form and Estimation | 标准形式与估算
Standard form writes very large or very small numbers as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. For example, 0.00042 = 4.2 × 10⁻⁴ and 63,000 = 6.3 × 10⁴.
标准形式将很大或很小的数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。例如 0.00042 = 4.2 × 10⁻⁴，63,000 = 6.3 × 10⁴。
To multiply numbers in standard form, multiply the A values and add the powers of ten. To divide, divide the A values and subtract the powers. Always adjust the answer so the A value remains between 1 and 10.
用标准形式相乘时，将 A 值相乘并把 10 的幂指数相加。相除时，将 A 值相除并把 10 的幂指数相减。最后要调整答案，使 A 值保持在 1 到 10 之间。
Estimation checks whether a calculated answer is reasonable. Round each number to one significant figure, then perform the calculation mentally. For 48.7 × 2.13, estimate 50 × 2 = 100, so the exact answer 103.7 is sensible.
估算用于检查计算答案是否合理。将每个数四舍五入到一位有效数字，然后心算。例如 48.7 × 2.13，估算为 50 × 2 = 100，因此精确答案 103.7 是合理的。
4. Expanding and Factorising | 展开与因式分解
Expanding brackets means multiplying each term inside the bracket by the term outside. For a single bracket, a(b + c) = ab + ac. With two brackets, use the FOIL method: (x + a)(x + b) = x² + bx + ax + ab = x² + (a + b)x + ab.
展开括号是指用括号外的项乘以括号内的每一项。单个括号时 a(b + c) = ab + ac。两个括号时使用 FOIL 方法：(x + a)(x + b) = x² + bx + ax + ab = x² + (a + b)x + ab。
Factorising is the reverse of expanding. Always look for a common factor first: 6x² + 9x = 3x(2x + 3). For a quadratic expression x² + bx + c, find two numbers that multiply to c and add to b: x² + 7x + 12 = (x + 3)(x + 4).
因式分解是展开的逆运算。首先要寻找公因式：6x² + 9x = 3x(2x + 3)。对于二次式 x² + bx + c，找到两个数相乘得 c、相加得 b：x² + 7x + 12 = (x + 3)(x + 4)。
The difference of two squares is a special pattern: a² – b² = (a + b)(a – b). For example, 9x² – 25 = (3x + 5)(3x – 5). Recognising this pattern saves time in Higher homework questions.
平方差是一个特殊模式：a² – b² = (a + b)(a – b)。例如 9x² – 25 = (3x + 5)(3x – 5)。识别这一模式可以节省高等作业题的时间。
5. Solving Quadratic Equations | 解二次方程
A quadratic equation has the general form ax² + bx + c = 0, where a ≠ 0. There are three main methods: factorising, completing the square, and the quadratic formula.
二次方程的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。主要有三种解法：因式分解法、配方法和求根公式法。
Factorising is fastest when the quadratic factorises neatly. Solve x² – 5x + 6 = 0 by writing (x – 2)(x – 3) = 0, then setting each factor to zero: x = 2 or x = 3.
当二次式容易因式分解时，因式分解法最快。解 x² – 5x + 6 = 0 时写成 (x – 2)(x – 3) = 0，然后令每个因式为零：x = 2 或 x = 3。
The quadratic formula works for any quadratic: x = (-b ± √(b² – 4ac)) / 2a. The discriminant b² – 4ac tells you the nature of the roots: positive gives two real roots, zero gives one repeated root, and negative gives no real roots.
求根公式适用于任何二次方程：x = (-b ± √(b² – 4ac)) / 2a。判别式 b² – 4ac 可以判断根的性质：正数有两个实根，零有一个重根，负数没有实根。
6. Linear Inequalities | 一元一次不等式
A linear inequality uses <, >, ≤ or ≥ instead of an equals sign. Solve inequalities in the same way as equations, but reverse the inequality sign when multiplying or dividing by a negative number.
一元一次不等式使用 <、>、≤ 或 ≥ 代替等号。解不等式的方法与方程相同，但当乘以或除以负数时，必须反转不等式符号。
For example, -2x < 8 becomes x > -4 after dividing both sides by -2 and reversing the sign. Represent the solution on a number line with an open circle for < or > and a closed circle for ≤ or ≥.
例如 -2x < 8 两边除以 -2 并反转符号后得到 x > -4。在数轴上表示解集时，< 或 > 用空心圆，≤ 或 ≥ 用实心圆。
Compound inequalities such as 3 < x + 2 ≤ 7 are solved by subtracting 2 from all three parts: 1 < x ≤ 5. This represents all values greater than 1 and less than or equal to 5.
复合不等式如 3 < x + 2 ≤ 7 可以通过对三部分同时减去 2 来求解：1 < x ≤ 5。这表示所有大于 1 且小于等于 5 的值。
7. Simultaneous Equations | 联立方程
Simultaneous equations involve two or more unknowns linked by equations. The two common methods are elimination and substitution. Elimination works best when coefficients of one variable can be made equal.
联立方程包含两个或更多未知数，并由方程相互联系。两种常用方法是消元法和代入法。当一个变量的系数可以变得相等时，消元法最有效。
Solve 2x + y = 8 and x – y = 1 by adding the equations: 3x = 9, so x = 3. Substitute x = 3 into x – y = 1 to get 3 – y = 1, so y = 2. The solution is x = 3, y = 2.
解 2x + y = 8 和 x – y = 1，可将两式相加：3x = 9，所以 x = 3。将 x = 3 代入 x – y = 1 得 3 – y = 1，所以 y = 2。解为 x = 3，y = 2。
Substitution is useful when one equation already has a single variable as the subject, for example y = 2x + 1. Replace y in the second equation with 2x + 1, then solve for x.
当其中一个方程已经把单个变量作为主语时，代入法很有用，例如 y = 2x + 1。将第二个方程中的 y 替换为 2x + 1，然后解出 x。
8. Straight-Line Graphs | 直线图
The equation of a straight line is often written as y = mx + c, where m is the gradient and c is the y-intercept. The gradient is calculated by m = Δy / Δx = (y₂ – y₁) / (x₂ – x₁).
直线方程常写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距。斜率通过 m = Δy / Δx = (y₂ – y₁) / (x₂ – x₁) 计算。
Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to -1; if one gradient is m, the other is -1/m.
平行线具有相同的斜率。互相垂直的直线斜率乘积为 -1；如果一条斜率为 m，另一条为 -1/m。
To find the equation of a line, you need the gradient and one point, then use y – y₁ = m(x – x₁). Rearrange into the form y = mx + c for the final answer.
要求直线方程，需要斜率和一点，然后使用 y – y₁ = m(x – x₁)。重新整理为 y = mx + c 的形式作为最终答案。
9. Coordinate Geometry and Distance | 坐标几何与距离
The midpoint of a line segment joining points (x₁, y₁) and (x₂, y₂) is given by M = ((x₁ + x₂)/2, (y₁ + y₂)/2). This averages the x-coordinates and y-coordinates separately.
连接点 (x₁, y₁) 和 (x₂, y₂) 的线段中点由 M = ((x₁ + x₂)/2, (y₁ + y₂)/2) 给出。它分别对 x 坐标和 y 坐标取平均值。
The distance between two points is found using Pythagoras’ theorem: d = √((x₂ – x₁)² + (y₂ – y₁)²). This formula is essential for lengths in coordinate geometry.
两点之间的距离用勾股定理求出：d = √((x₂ – x₁)² + (y₂ – y₁)²)。该公式是坐标几何中求长度的基础。
For example, the distance between (1, 2) and (4, 6) is √((4 – 1)² + (6 – 2)²) = √(9 + 16) = √25 = 5. The midpoint is ((1 + 4)/2, (2 + 6)/2) = (2.5, 4).
例如 (1, 2) 和 (4, 6) 之间的距离为 √((4 – 1)² + (6 – 2)²) = √(9 + 16) = √25 = 5。中点为 ((1 + 4)/2, (2 + 6)/2) = (2.5, 4)。
10. Trigonometric Ratios | 三角比
In a right-angled triangle, the three basic trigonometric ratios are sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent.
在直角三角形中，三个基本三角比是 sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
Use the mnemonic SOH CAH TOA to remember these relationships. To find a missing side, choose the ratio involving the known side and the required side, then solve the equation.
使用记忆口诀 SOH CAH TOA 记住这些关系。求缺失边时，选择包含已知边和所求边的三角比，然后解方程。
To find a missing angle, use the inverse functions: θ = sin⁻¹(opposite / hypotenuse), θ = cos⁻¹(adjacent / hypotenuse), or θ = tan⁻¹(opposite / adjacent). Ensure your calculator is in degree mode.
求缺失角时，使用反函数：θ = sin⁻¹(对边 / 斜边)，θ = cos⁻¹(邻边 / 斜边)，或 θ = tan⁻¹(对边 / 邻边)。确保计算器处于角度模式。
11. Exam-Style Strategy | 考试策略
In Higher homework and exams, always show every step of working. Marks are awarded for correct methods, so even if the final answer is wrong, clear working can gain most of the available marks.
在高等作业和考试中，始终展示每一步计算过程。分数是按正确方法给的，因此即使最终答案错误，清晰的步骤也能获得大部分可得分数。
Read each question carefully to identify the topic and required form. Underline keywords such as ‘simplify’, ‘factorise’, ‘solve’ or ‘show that’. This helps avoid common errors such as solving when you should factorise.
仔细阅读每道题，确定主题和所需形式。在关键词如 ‘simplify’、’factorise’、’solve’ 或 ‘show that’ 下划线。这有助于避免诸如本应因式分解却去求解的常见错误。
Check your answers by substituting back into the original equation or by estimating. For quadratic equations, verify both roots satisfy the equation. For simultaneous equations, test both values in both equations.
通过代回原方程或进行估算来检查答案。对于二次方程，验证两个根都满足方程。对于联立方程，在两个方程中检验两个值。
12. Common Mistakes and Final Review | 常见错误与最终复习
Common mistakes include forgetting to reverse the inequality sign when dividing by a negative, mishandling negative indices, and losing one root when solving quadratics by factorising. Always double-check these areas.
常见错误包括除以负数时忘记反转不等式符号、错误处理负指数，以及在因式分解解二次方程时丢失一个根。务必仔细检查这些易错点。
When rationalising surds, multiply both the numerator and denominator by the same surd to keep the fraction equivalent. Do not change the value of the expression; only change its form.
分母有理化时，分子和分母要同时乘以同一个根式，以保持分数等价。不要改变表达式的值，只改变其形式。
Finally, create a one-page summary of formulas for this unit: index laws, quadratic formula, midpoint and distance formulas, and SOH CAH TOA. Review it regularly before your Higher homework assessment.
最后，为本单元制作一页公式总结：指数法则、求根公式、中点和距离公式，以及 SOH CAH TOA。在高等作业评估前定期复习。
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