Core Knowledge Review for Year 8 SQA Advanced Mathematics | Year 8 SQA 进阶数学：核心知识点梳理

📚 Core Knowledge Review for Year 8 SQA Advanced Mathematics | Year 8 SQA 进阶数学：核心知识点梳理

This comprehensive guide summarises the essential topics in Year 8 Advanced Mathematics under the Scottish SQA curriculum. Mastering these fundamentals will build a strong foundation for National 5 and beyond.

本指南总结了苏格兰 SQA 课程下 Year 8 进阶数学的核心主题。掌握这些基础知识将为 National 5 及更高级别的学习奠定坚实基础。

1. Integers, Powers and Roots | 整数、幂与根

Negative numbers represent quantities below zero. When adding a negative, subtract its positive value; when subtracting a negative, add its positive value. For example, 5 + (-3) = 5 – 3 = 2, and 5 – (-3) = 5 + 3 = 8.

负数表示零以下的量。加上一个负数相当于减去它的绝对值；减去一个负数相当于加上它的绝对值。例如，5 + (-3) = 5 – 3 = 2，5 – (-3) = 5 + 3 = 8。

The order of operations (BIDMAS/BODMAS: Brackets, Indices, Division and Multiplication, Addition and Subtraction) must be followed to avoid mistakes. Evaluate exponents before multiplication, and perform left‑to‑right for division and multiplication.

计算时必须遵守运算顺序（括号、指数、乘除、加减），先算括号和指数，再算乘除（从左到右），最后算加减。这样可以避免错误。

Powers express repeated multiplication: 4³ = 4 × 4 × 4 = 64. The square root √ reverses squaring: √81 = 9 because 9² = 81. Cube roots follow the same principle. Any non‑zero number raised to 0 equals 1.

幂表示重复相乘：4³ = 4 × 4 × 4 = 64。平方根 √ 是平方的逆运算：√81 = 9，因为 9² = 81。立方根同理。任何非零数的 0 次幂都等于 1。

2. Fractions, Decimals and Percentages | 分数、小数与百分比

Fractions, decimals and percentages are interchangeable. The fraction ½ equals 0.5 and 50%. To convert a fraction to a decimal, divide the numerator by the denominator; to turn a decimal to a percentage, multiply by 100.

分数、小数和百分比可以互化。½ = 0.5 = 50%。分数化小数用分子除以分母；小数化百分数乘以 100。

To add or subtract fractions, write them with a common denominator. Multiply fractions by multiplying numerators together and denominators together. To divide by a fraction, multiply by its reciprocal. Example: ¾ ÷ ½ = ¾ × 2 = 1½.

加减分数先通分，分母相同后再加减分子。分数相乘时分子乘分子、分母乘分母。除以一个分数等于乘以它的倒数，如 ¾ ÷ ½ = ¾ × 2 = 1½。

Finding a percentage of a quantity: multiply by the decimal form. To increase 80 by 15%, multiply 80 by 1.15. Percentage change = (change ÷ original) × 100%.

求一个数的百分比：用该数乘以对应的小数。80 增加 15% 就是 80 × 1.15。百分比变化 = (变化量 ÷ 原量) × 100%。

3. Algebraic Expressions and Simplification | 代数表达式与化简

Simplify by collecting like terms – terms with exactly the same variable part. For instance, 5a + 3b – 2a + b = 3a + 4b. Constants are like terms with each other.

通过合并同类项化简——变量部分完全相同的项。例如 5a + 3b – 2a + b = 3a + 4b。常数之间也是同类项。

To expand brackets, multiply the term outside by each term inside: 3(x + 4) = 3x + 12, and -2(y – 5) = -2y + 10. Double brackets such as (x + 2)(x + 3) expand to x² + 5x + 6.

展开括号时，用外面的项乘以里面的每一项：3(x + 4) = 3x + 12；-2(y – 5) = -2y + 10。双重括号如 (x + 2)(x + 3) 展开得 x² + 5x + 6。

Factorising is the reverse: take out the highest common factor. 6y + 9 = 3(2y + 3). Always check by expanding back.

因式分解是展开的逆运算：提取最大公因子。6y + 9 = 3(2y + 3)。展开检验即可确认正确。

4. Solving Linear Equations and Inequalities | 解一元一次方程与不等式

Use inverse operations to isolate the variable. For 3x + 2 = 11, subtract 2 from both sides to get 3x = 9, then divide by 3 to find x = 3. Always perform the same operation on both sides.

利用逆运算使变量单独出现。解 3x + 2 = 11 时，两边同减 2 得 3x = 9，再除以 3 得 x = 3。每一步都要在等式两边做相同运算。

Inequalities (<, >, ≤, ≥) are solved similarly, but if you multiply or divide by a negative number, reverse the inequality sign. E.g., -2x < 6 becomes x > -3.

解不等式（<, >, ≤, ≥）方法类似，但若乘或除以负数，必须调转不等号。例如 -2x < 6 变为 x > -3。

Solutions can be shown on a number line: open circles for strict inequalities, closed circles for ≤

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