📚 Year 8 Essential Maths Higher Homework: Core Skills and Exam-Style Practice | Year 8 数学高阶作业：核心技能与考试题型练习
Welcome to this Year 8 Essential Maths Higher homework guide. Here you will find the core skills, common methods and exam-style practice needed to complete your homework with confidence.
欢迎阅读本 Year 8 Essential Maths Higher 作业指南。你将在这里找到完成作业所需的核心技能、常用方法和考试题型练习。
1. Understanding the Homework Brief | 理解作业要求
Before starting, read the homework sheet carefully. Identify the topics and the marks available for each question. This helps you allocate time and effort where it matters most.
开始前，仔细阅读作业单。找出每道题考查的主题和分值。这有助于你把时间和精力分配到最重要的地方。
Higher-tier questions often combine two skills, such as a fraction problem inside an area question. Highlight command words like 'show that', 'solve', 'simplify' and 'estimate' so you know what the final answer must do.
高阶题目经常把两种技能结合起来，比如在面积题里包含分数运算。圈出 "证明"、"求解"、"化简"、"估算" 等指令词，这样你就知道最终答案需要达到什么要求。
Write down what you are given and what you need to find before you calculate. This plan keeps multi-step questions organised and reduces careless mistakes.
在计算之前，写下已知条件和需要求出的量。这个计划能让多步题保持条理，减少粗心错误。
2. Number Skills: Fractions, Decimals and Percentages | 数字技能：分数、小数与百分比
vert fluently between fractions, decimals and percentages. For example, 0.375 = 3/8 = 37.5%. These conversions appear constantly in real-life and exam questions.
你必须熟练地在分数、小数和百分比之间转换。例如，0.375 = 3/8 = 37.5%。这些转换在现实生活和考试题中经常出现。
n denominator first. For 2/3 + 1/4, use 12 as the common denominator: 8/12 + 3/12 = 11/12.
分数相加或相减时，一定要先找到最小公分母。对于 2/3 + 1/4，用 12 作公分母：8/12 + 3/12 = 11/12。
For percentage increase or decrease, use the formula below. Remember: a decrease gives a negative value inside the brackets, so the final percentage change can be negative.
求百分比增加或减少时，使用下面的公式。记住：减少会在括号内产生负值，所以最终的百分比变化可以是负数。
Percent change = (new value − old value) ÷ old value × 100%
百分比变化 = (新值 − 旧值) ÷ 旧值 × 100%。
3. Ratio and Proportion | 比与比例
Simplify ratios by dividing all parts by the highest common factor. For example, 8:12 simplifies to 2:3 because both numbers divide by 4.
通过除以最大公因数来化简比。例如，8:12 化简为 2:3，因为两个数都能除以 4。
To share an amount in a ratio, first find the total number of parts. Share £45 in the ratio 2:3: total parts = 2 + 3 = 5, one part = £45 ÷ 5 = £9, so the shares are 2 × £9 = £18 and 3 × £9 = £27.
按比例分配金额时，先求总份数。按 2:3 分 £45：总份数 = 2 + 3 = 5，每份 = £45 ÷ 5 = £9，所以各部分为 2 × £9 = £18 和 3 × £9 = £27。
Always check that the final shares add back to the original total. £18 + £27 = £45, so the answer is correct.
始终检查最终各份相加是否等于原来的总数。£18 + £27 = £45，所以答案正确。
Amount per part = total ÷ sum of ratio parts
每份金额 = 总数 ÷ 比的总份数。
4. Algebraic Expressions and Simplification | 代数表达式与化简
or example, 5x + 3y − 2x + 7y = 3x + 10y. Terms with different letters cannot be combined.
合并同类项要相加系数。例如，5x + 3y − 2x + 7y = 3x + 10y。不同字母的项不能合并。
acket by the term outside. For 3(2x − 5), multiply 3 by 2x and by −5: 3(2x − 5) = 6x − 15.
展开括号就是把括号外的项乘以括号内的每一项。对于 3(2x − 5)，把 3 分别乘以 2x 和 −5：3(2x − 5) = 6x − 15。
After expanding, always collect like terms again. For 2(x + 4) + 3(x − 1), expand to get 2x + 8 + 3x − 3, then simplify to 5x + 5.
展开后，要再次合并同类项。对于 2(x + 4) + 3(x − 1)，展开得到 2x + 8 + 3x − 3，然后化简为 5x + 5。
a(b + c) = ab + ac
a(b + c) = ab + ac。
5. Solving Linear Equations | 解一元一次方程
subtract 2x from both sides: 3x − 3 = 9; add 3 to both sides: 3x = 12; divide by 3: x = 4.
解方程要对两边做相同的运算。对于 5x − 3 = 2x + 9，两边先减 2x：3x − 3 = 9；两边加 3：3x = 12；两边除以 3：x = 4。
o 5x − 3 = 2x + 9: left side 5(4) − 3 = 17, right side 2(4) + 9 = 17, so x = 4 is correct.
始终用代入法检查答案。把 x = 4 代回 5x − 3 = 2x + 9：左边 5(4) − 3 = 17，右边 2(4) + 9 = 17，所以 x = 4 正确。
When the variable appears on both sides, move the smaller variable term first. This keeps the coefficient of x positive and reduces sign errors.
当变量出现在等号两边时，先移动较小的变量项。这样可以让 x 的系数保持为正，减少符号错误。
If ax + b = c, then x = (c − b) ÷ a
如果 ax + b = c，那么 x = (c − b) ÷ a。
6. Sequences and the nth Term | 数列与第 n 项
e term-to-term rule first by subtracting consecutive terms. For 7, 10, 13, 16, the rule is 'add 3', because 10 − 7 = 3 and 13 − 10 = 3.
先通过相邻项相减找出递推规则。对于 7、10、13、16，规则是 "加 3"，因为 10 − 7 = 3，13 − 10 = 3。
of 3, the nth term is 3n + 4. Check n = 1: 3(1) + 4 = 7; n = 2: 3(2) + 4 = 10. Both match.
按位置的第 n 项公式给出了任意项。公差为 3 时，第 n 项公式为 3n + 4。检查 n = 1：3(1) + 4 = 7；n = 2：3(2) + 4 = 10。两项都吻合。
20, 16, 12, 8, the difference is −4, so the nth term is −4n + 24. Always test with n = 1.
如果数列是递减的，公差为负数。对于 20、16、12、8，公差为 −4，所以第 n 项公式为 −4n + 24。一定要用 n = 1 检验。
nth term = common difference × n + zero term
第 n 项 = 公差 × n + 零项。
7. Geometry: Angles and Polygons | 几何：角与多边形
Use basic angle facts: angles on a straight line add to 180°, angles around a point add to 360°, and vertically opposite angles are equal. These facts solve most angle diagrams.
使用基本角度事实：直线上的角之和为 180°，一个点周围的角之和为 360°，对顶角相等。这些事实能解决大多数角度图问题。
where n is the number of sides. A pentagon has n = 5, so the sum is (5 − 2) × 180° = 540°.
对于多边形，内角和为 (n − 2) × 180°，其中 n 是边数。五边形 n = 5，所以内角和为 (5 − 2) × 180° = 540°。
to find one interior angle. For a regular pentagon, one interior angle = 540° ÷ 5 = 108°.
在正多边形中，所有内角相等。用内角和除以 n 得到一个内角。对于正五边形，一个内角 = 540° ÷ 5 = 108°。
Sum of interior angles = (n − 2) × 180°
内角和 = (n − 2) × 180°，其中 n 是多边形的边数。
8. Area, Perimeter and Volume | 面积、周长与体积
nd trapeziums. The area of a triangle is ½ × base × vertical height, not the sloping side.
掌握矩形、三角形、平行四边形和梯形的面积公式。三角形面积是 ½ × 底 × 垂直高，而不是斜边。
For compound shapes, split them into rectangles and triangles. Find each area separately, then add or subtract. Always write the units
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