📚 Year 9 Core Homework Compressed: Essential Skills and Worked Examples | 九年级核心作业精编：核心技能与例题解析
This revision guide compresses the Year 9 core homework programme into ten essential skill areas. It gives clear rules, worked examples and common pitfalls so that you can complete homework accurately and revise efficiently.
这份复习指南把九年级核心作业计划浓缩为十个核心技能领域。它给出清晰规则、例题和常见易错点，帮助你准确完成作业并高效复习。
1. Number and Place Value | 数与位值
In Year 9 core homework, you must be confident with positive and negative numbers, powers of 10, and standard form. For example, 3.2 × 10⁴ means 3.2 × 10,000, which equals 32,000. When multiplying by 10ⁿ, move the decimal point n places to the right; when dividing, move it n places to the left.
在九年级核心作业中，你需要熟练掌握正负数、10 的幂和标准形式。例如，3.2 × 10⁴ 表示 3.2 × 10,000，等于 32,000。乘以 10ⁿ 时，小数点向右移动 n 位；除以 10ⁿ 时，小数点向左移动 n 位。
A common error is writing 0.00056 as 5.6 × 10⁻³ instead of 5.6 × 10⁻⁴. Count the places from the first non-zero digit, not from the decimal point. With negative indices, the number is smaller than 1.
一个常见错误是把 0.00056 写成 5.6 × 10⁻³，而正确应为 5.6 × 10⁻⁴。要从第一个非零数字开始数位数，而不是从小数点开始数。负指数表示该数小于 1。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
To compare fractions, decimals and percentages, convert them to the same form. For example, ⅗ = 0.6 = 60%. To find a percentage of an amount, multiply the amount by the percentage as a decimal: 15% of £80 = 0.15 × 80 = £12.
比较分数、小数和百分比时，先把它们化成同一种形式。例如，⅗ = 0.6 = 60%。求一个数的百分比时，用该数乘以百分比化成的小数：80 英镑的 15% = 0.15 × 80 = 12 英镑。
Percentage increase and decrease often appear in homework. A jumper priced at £40 with a 25% decrease becomes £40 × (1 − 0.25) = £30. To reverse a percentage change, divide by the multiplier.
百分比的增加和减少经常出现在作业中。一件标价 40 英镑的毛衣降价 25%，变为 40 × (1 − 0.25) = 30 英镑。要逆向求原价，则除以对应的乘数。
3. Ratio and Proportion | 比与比例
A ratio compares parts of a whole. The ratio 3:2 means 3 + 2 = 5 equal parts in total. If a drink is mixed in the ratio 3:2 and the total volume is 600 ml, one part is 600 ÷ 5 = 120 ml, so the parts are 360 ml and 240 ml.
比用来比较整体中的各个部分。3:2 表示总共有 3 + 2 = 5 等份。如果一种饮料按 3:2 混合，总体积为 600 毫升，则一份是 600 ÷ 5 = 120 毫升，因此两部分分别为 360 毫升和 240 毫升。
Direct proportion means that as one quantity increases, the other increases at the same rate. If 5 pens cost £3, then 20 pens cost £3 × (20 ÷ 5) = £12 because the number of pens is multiplied by 4.
正比例意味着一个量增加时，另一个量以相同速率增加。如果 5 支笔花费 3 英镑，那么 20 支笔花费 3 × (20 ÷ 5) = 12 英镑，因为笔的数量乘了 4。
4. Algebraic Expressions | 代数表达式
Simplify expressions by collecting like terms. For example, 5a + 3b − 2a + 7b = 3a + 10b. Only terms with exactly the same letter part can be combined; a and a² are not like terms.
通过合并同类项来化简表达式。例如，5a + 3b − 2a + 7b = 3a + 10b。只有字母部分完全相同的项才能合并；a 和 a² 不是同类项。
Expanding brackets uses the distributive law: 4(2x + 3) = 8x + 12. For double brackets, (x + 3)(x + 2) = x² + 5x + 6. Remember to multiply every term in the first bracket by every term in the second.
去括号使用分配律：4(2x + 3) = 8x + 12。对于双括号，(x + 3)(x + 2) = x² + 5x + 6。记住要把第一个括号中的每一项与第二个括号中的每一项相乘。
5. Solving Linear Equations | 解一元一次方程
To solve an equation, keep it balanced by doing the same operation to both sides. For 3x + 5 = 20, subtract 5 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5.
解方程时，要对两边进行相同的运算以保持等式平衡。对于 3x + 5 = 20，两边先减 5 得到 3x = 15，然后两边除以 3 得到 x = 5。
If the unknown appears on both sides, collect the variable terms on one side. Example: 5x − 2 = 2x + 7. Subtract 2x from both sides: 3x − 2 = 7. Add 2: 3x = 9, so x = 3.
如果未知数出现在等号两边，就把含未知数的项移到一边。例如：5x − 2 = 2x + 7。两边减去 2x：3x − 2 = 7。再加 2：3x = 9，所以 x = 3。
6. Coordinates and Linear Graphs | 坐标与线性图像
The coordinate (x, y) gives the horizontal position first and the vertical position second. The midpoint of two points (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). For (2, 3) and (6, 7), the midpoint is (4, 5).
坐标 (x, y) 先给出水平位置，再给出竖直位置。两点 (x₁, y₁) 和 (x₂, y₂) 的中点坐标是 ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)。对于 (2, 3) 和 (6, 7)，中点为 (4, 5)。
A straight line has equation y = mx + c, where m is the gradient and c is the y-intercept. If y = 2x + 3, the gradient is 2 and the line crosses the y-axis at (0, 3).
直线的方程是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。如果 y = 2x + 3，则斜率为 2，图像与 y 轴交于 (0, 3)。
The gradient between two points is the change in y divided by the change in x: m = (y₂ − y₁) ÷ (x₂ − x₁). This is often remembered as rise over run.
两点间的斜率等于 y 的变化量除以 x 的变化量：m = (y₂ − y₁) ÷ (x₂ − x₁)。这通常记作铅垂升高除以水平距离。
7. Geometry: Area and Perimeter | 几何：面积与周长
Perimeter is the distance around a shape. For a rectangle with length l and width w, perimeter P = 2l + 2w. Area is the space inside: A = l × w. Always include units such as cm, m² or cm².
周长是围绕图形一周的长度。长为 l、宽为 w 的长方形，周长 P = 2l + 2w。面积是图形内部的大小：A = l × w。始终要写单位，如 cm、m² 或 cm²。
The area of a triangle is A = ½ × base × height. The height must be perpendicular to the base. For a base of 8 cm and height of 5 cm, A = ½ × 8 × 5 = 20 cm².
三角形的面积公式为 A = ½ × 底 × 高。高必须与底垂直。底为 8 厘米、高为 5 厘米时，A = ½ × 8 × 5 = 20 平方厘米。
The area of a trapezium is A = ½ × (a + b) × h, where a and b are the parallel sides and h is the perpendicular height.
梯形的面积公式为 A = ½ × (a + b) × h，其中 a 和 b 是两条平行边，h 是垂直高度。
8. Angles and Properties of Shapes | 角与图形的性质
On a straight line, angles add to 180°. Around a point, angles add to 360°. In a triangle, the interior angles add to 180°. These facts allow you to find missing angles without measuring.
直线上的角相加为 180°。一个点周围的角相加为 360°。三角形内角相加为 180°。利用这些事实，你可以不测量就求出缺失的角。
In a parallelogram, opposite angles are equal and co-interior angles add to 180°. If one angle of a parallelogram is 65°, the angle next to it is 180° − 65° = 115°.
在平行四边形中，对角相等，同旁内角相加为 180°。如果平行四边形的一个角为 65°，与它相邻的角就是 180° − 65° = 115°。
Alternate angles and corresponding angles are equal when two parallel lines are cut by a transversal. Recognising these angle pairs is useful for proving that lines are parallel.
当两条平行线被一条截线所截时，内错角相等，同位角相等。识别这些角对有助于证明两条直线平行。
9. Statistics and Probability | 统计与概率
The mean is the sum of values divided by the number of values. For 4, 7, 8, 10, 11, the mean is (4 + 7 + 8 + 10 + 11) ÷ 5 = 40 ÷ 5 = 8. The median is the middle value when the data are ordered.
平均数等于所有数值之和除以数值个数。对于 4、7、8、10、11，平均数为 (4 + 7 + 8 + 10 + 11) ÷ 5 = 40 ÷ 5 = 8。中位数是数据按顺序排列后位于中间的值。
Probability is measured from 0 to 1. P(event) = number of favourable outcomes ÷ total number of outcomes. If a bag has 3 red, 2 blue and 5 green counters, P(red) = 3 ÷ 10 = 0.3.
概率的取值范围是 0 到 1。P(事件) = 有利结果的数量 ÷ 所有结果的总数。如果一个袋子里有 3 个红球、2 个蓝球和 5 个绿球，则 P(红球) = 3 ÷ 10 = 0.3。
The probability of an event not happening is 1 − P(event). For example, if P(rain) = 0.4, then P(no rain) = 1 − 0.4 = 0.6.
某事件不发生的概率是 1 − P(事件)。例如，如果 P(下雨) = 0.4，那么 P(不下雨) = 1 − 0.4 = 0.6。
10. Problem-Solving Strategies | 解题策略
Read the problem twice and underline key numbers and units. Write down what you know, what you need to find, and a plan. For multi-step problems, break them into smaller, manageable steps.
把题目读两遍，并在关键数字和单位下画线。写下已知条件、要求解的量以及求解计划。对于多步骤问题，把它拆分成若干较小、可处理的步骤。
Estimate before calculating and check your answer makes sense. For example, if you calculate a speed of 1200 km/h for a bicycle, you have probably mixed up units or placed a decimal point incorrectly.
计算前先估算，并检查答案是否合理。例如，如果你算出一辆自行车的速度为 1200 km/h，那你很可能把单位弄混了或小数点位置写错了。
Use diagrams, tables or algebra to organise information. When stuck, try a simpler version of the problem with smaller numbers, then apply the same logic to the original question.
用图表、表格或代数来组织信息。遇到困难时，可以先用更小的数字简化问题，然后把同样的逻辑应用到原题上。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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