📚 Cambridge KS3 Maths: Mixed Review – Page 317 Exercise 1 | 剑桥初中数学：第317页练习1 综合复习
Page 317 Exercise 1 in the Cambridge KS3 Mathematics course is a mixed question set that pulls together the main skills from Stage 9. The questions are usually short, but they expect you to show clear working and to switch confidently between number, algebra, geometry, statistics and probability. This article explains the key ideas behind the exercise and gives you a structured way to revise each topic.
剑桥初中数学第317页练习1是一套混合题，集中考查第三学段（Stage 9）的主要数学技能。题目通常不长，但要求写出清晰步骤，并能在数、代数、几何、统计和概率之间灵活切换。本文讲解这套练习背后的核心知识点，并帮助你系统地复习每个主题。
1. Overview of the Mixed Exercise | 混合练习总览
Mixed exercises such as Page 317 Exercise 1 are designed to test whether you can recognise the correct method without being told which topic is being assessed. You should read each question twice, circle key numbers, and write down the relevant formula or operation before you begin the calculation.
像第317页练习1这样的混合题，目的是考查你能否在没有提示的情况下识别正确方法。每道题应读两遍，圈出关键数字，并在开始计算前写下相关公式或运算步骤。
- Check whether the question asks for length, area, volume, average, probability or an unknown value.
- Write all working in a logical order so that the marker can follow your method.
- Always finish with the correct unit such as cm, cm², cm³, degrees or percent.
- 先判断题目要求的是长度、面积、体积、平均数、概率还是未知数。
- 所有步骤按逻辑顺序书写，让阅卷人能看懂你的方法。
- 最后一定要写上正确单位，如 cm、cm²、cm³、度或百分比。
2. Number Skills: Fractions, Decimals and Percentages | 数的技能：分数、小数和百分数
decimals and percentages. You should memorise the common equivalents such as 0.25 = ¼ = 25% and 0.75 = ¾ = 75% o compare quantities, change them all into the same form, usually decimals or percentages.
混合练习中常出现分数、小数和百分数的互化。你需要记住常见等值关系，例如 0.25 = ¼ = 25%，0.75 = ¾ = 75%。比较数量时，应把它们化成同一种形式，通常是小数或百分数。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
Percentage change is especially common. Use the following rule:
百分比变化非常常见，使用以下公式：
Percentage change = (new value − old value) ÷ old value × 100%
百分比变化 =（新值 − 旧值）÷ 旧值 × 100%
For example, if a price rises from £40 to £50, the increase is £10, so the percentage increase is 10 ÷ 40 × 100% = 25%.
例如，如果价格从 40 英镑上涨到 50 英镑，增加额为 10 英镑，因此百分比增加率为 10 ÷ 40 × 100% = 25%。
3. Algebra: Simplifying, Expanding and Solving | 代数：化简、展开与解方程
Algebra questions on a mixed page usually ask you to simplify expressions, expand brackets, or solve linear equations. For example, 3x + 5y − 2x + y simplifies to x + 6y. Expanding 3(x + 4) gives 3x + 12.
混合页上的代数题通常要求化简表达式、展开括号或解一元一次方程。例如，3x + 5y − 2x + y 化简后为 x + 6y。展开 3(x + 4) 得到 3x + 12。
x + 3 = 11, first subtract 3 from both sides to get 2x = 8, then divide by 2 to get x = 4.
解方程时，在等式两边同时进行逆运算。对于 2x + 3 = 11，先在两边减去 3，得到 2x = 8，再除以 2，得到 x = 4。
2x + 3 = 11 → 2x = 8 → x = 4
2x + 3 = 11 → 2x = 8 → x = 4
When questions involve both algebra and area, set up an equation from the words. Define the unknown clearly, build the expression, and solve with full steps.
当题目同时涉及代数和面积时，要根据题意建立方程。清楚地设出未知数，列出表达式，并完整解出结果。
4. Geometry: Area, Volume and Angles | 几何：面积、体积与角
eight, the area of a circle is πr², and the volume of a cuboid is length × width × height.
几何题通常要求选择正确的面积或体积公式。三角形面积为 ½ × 底 × 高，圆的面积为 πr²，长方体体积为长 × 宽 × 高。
Triangle: A = ½ × b × h | Circle: A = πr² | Cuboid: V = l × w × h
三角形：A = ½ × b × h | 圆：A = πr² | 长方体：V = l × w × h
8 × 5 = 20 cm². Remember that area is measured in square units and volume in cubic units.
示例：一个三角形底为 8 cm，高为 5 cm，面积为 ½ × 8 × 5 = 20 cm²。记住面积用平方单位，体积用立方单位。
- Angles on a straight line add up to 180°.
- Angles around a point add up to 360°.
- Angles in a triangle add up to 180°.
- 直线上的角之和为 180°。
- 一个点周围的角之和为 360°。
- 三角形内角之和为 180°。
5. Statistics: Charts and Averages | 统计：图表与平均数
Statistics questions ask you to calculate the mean, median, mode and range, or to read information from a bar chart, pie chart or pictogram. The mean is found by adding all values and dividing by the number of values. The median is the middle value once data is ordered, and the mode is the most frequent value.
统计题要求计算平均数、中位数、众数和极差，或从条形图、饼图、象形图中读取信息。平均数是将所有数值相加后除以数值个数。中位数是数据排序后位于中间的数，众数是出现次数最多的数。
Mean = sum of values ÷ number of values
平均数 = 数值总和 ÷ 数值个数
6 ÷ 5 = 9.2, the median is the third number 8, the mode is 8, and the range is 14 − 5 = 9.
对于数据集 5、8、8、11、14，总和为 46，平均数为 46 ÷ 5 = 9.2，中位数为第三个数字 8，众数为 8，极差为 14 − 5 = 9。
When reading charts, check the scale carefully. A bar chart may start at 0, but a pictogram may use one symbol to represent several items, so always multiply correctly.
读图时要仔细检查刻度。条形图通常从 0 开始，但象形图中一个符号可能代表多个单位，因此一定要正确乘以倍数。
6. Probability Basics | 概率基础
Probability measures how likely an event is. It is always a number between 0 and 1, where 0 means impossible and 1 means certain. The basic formula is:
概率衡量一个事件发生的可能性。它始终是 0 到 1 之间的数，0 表示不可能，1 表示必然发生。基本公式为：
Probability = number of favourable outcomes ÷ total number of outcomes
概率 = 有利结果数 ÷ 所有可能结果数
ies to 1/2. For mutually exclusive events, add the probabilities: P(A or B) = P(A) + P(B).
对于公平的六面骰子，掷出 6 的概率是 1/6。掷出偶数的概率是 3/6，化简为 1/2。对于互斥事件，概率可以相加：P(A 或 B) = P(A) + P(B)。
t happening, subtract the event probability from 1. For example, P(not 6) = 1 − 1/6 = 5/6.
如果题目问事件未发生的概率，用 1 减去该事件的概率。例如，P(不是 6) = 1 − 1/6 = 5/6。
7. Common Mistakes to Avoid | 常见错误与避免方法
In mixed exercises, students often lose marks by making small errors that are easy to fix. The most frequent mistakes include forgetting units, mixing area with perimeter, making sign errors when solving equations, using frequency instead of total outcomes in probability, and misreading chart scales.
在混合练习中，学生常因一些容易纠正的小错误而失分。最常见的错误包括忘记单位、混淆面积与周长、解方程时符号出错、在概率题中把频数当作总结果数，以及读错图表刻度。
- After finding a length, write cm or m; after an area, write cm² or m²; after a volume, write cm³ or m³.
- Check that your probability answer is between 0 and 1.
- When using the distributive law, multiply every term inside the bracket, including negative signs.
- Order the data before finding the median.
- 求长度后写上 cm 或 m；求面积后写 cm² 或 m²；求体积后写 cm³ 或 m³。
- 检查概率答案是否在 0 到 1 之间。
- 使用分配律时，括号内每一项都要乘，包括负号。
- 求中位数前先给数据排序。
8. Worked Example from a Mixed Exercise | 混合练习例题解析
Here is a typical mixed problem: a triangle has base 10 cm and height 4 cm. A rectangle has width 5 cm and unknown length x. The triangle and rectangle have the same area. Find x.
下面是一道典型的混合题：一个三角形底为 10 cm，高为 4 cm。一个矩形宽为 5 cm，长为未知数 x。三角形和矩形面积相等。求 x。
Triangle area = ½ × 10 × 4 = 20 cm²
三角形面积 = ½ × 10 × 4 = 20 cm²
algebra and geometry combine in one question, which is exactly the style used on page 317 mixed exercises.
矩形面积为 5 × x，因此方程为 5x = 20。两边除以 5，得到 x = 4 cm。这个例子展示了代数与几何如何结合在一道题中，这正是第317页混合练习的常见题型。
5x = 20 → x = 4 cm
5x = 20 → x = 4 cm
9. Exam-Style Practice Tips | 考试风格练习建议
To do well on mixed exercises, practise without looking at topic headings and decide the method yourself. Use the following routine: read the question twice, list the given information, write the formula, substitute the numbers, solve, and state the answer with units.
要在混合练习中取得好成绩，应不看主题标题独立练习，自己判断方法。使用以下流程：题目读两遍，列出已知信息，写出公式，代入数字，解出结果，并带单位写出答案。
- Set a timer for short practice sets to build speed.
- Mark your answers using the mark scheme and note where you lost method marks.
- Redo missed questions after two days to check retention.
- Create a personal error log for each topic.
- 计时完成短篇练习，提升速度。
- 对照评分标准批改答案，记下丢失步骤分的地方。
- 两天后重做错题，检验是否真正掌握。
- 为每个主题建立个人错误记录。
10. Summary and Revision Checklist | 总结与复习清单
Page 317 Exercise 1 is not about learning new content; it is about retrieving and applying what you already know. Make sure you can convert between fractions, decimals and percentages, simplify and solve expressions, use area and volume formulas, find averages, and compute basic probabilities.
第317页练习1不是学习新内容，而是提取并应用你已经掌握的知识。确保你能熟练转换分数、小数和百分数，化简并解方程，使用面积和体积公式，求平均数，并计算基本概率。
Before attempting the exercise again, use this checklist: units written, working shown, equation balanced, probability between 0 and 1, and answer checked against the question.
再次尝试该练习前，请使用以下清单：单位已写，步骤已展示，方程两边平衡，概率在 0 到 1 之间，答案已与题目核对。
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