📚 Year 7 Cambridge Mathematics: Core Knowledge Points Review | Year 7 Cambridge 数学：核心知识点梳理
This article provides a comprehensive overview of the essential topics covered in Year 7 Cambridge Lower Secondary Mathematics. It is designed to help students consolidate their understanding, identify key connections between concepts, and build confidence for assessments. Each section pairs English explanations with Chinese translations so that learners can access the material in both languages while reinforcing subject terminology.
本文全面梳理了 Year 7 剑桥初中数学课程的核心知识点，旨在帮助学生巩固理解、把握概念间的联系并为评估建立信心。每个小节都采用中英文配对讲解，方便双语学习者同时掌握学科术语与内容。
1. Integers and Operations | 整数与运算
Integers are whole numbers that can be positive, negative or zero. On a number line, moving right increases the value and moving left decreases it. Understanding how to add, subtract, multiply and divide with negative numbers is a fundamental skill in Year 7.
整数是正整数、负整数和零。在数轴上向右移动数值增大，向左移动数值减小。掌握正负数的加减乘除运算是七年级的基础技能。
+ (−b) = a − b. When subtracting a negative number, it becomes addition: a − (−b) = a + b.
加上一个负数等同于减去其相反数：a + (−b) = a − b。减去一个负数则转变为加法：a − (−b) = a + b。
= negative. The same signs give a positive answer, different signs give a negative answer.
负数乘除法的规则：正 × 正 = 正；负 × 负 = 正；正 × 负 = 负。同号得正，异号得负。
tion). For example, in 3 + 4 × 5, multiplication comes before addition, giving 23, not 35.
运算顺序遵循 BIDMAS 规则（括号、指数、乘除、加减）。例如 3 + 4 × 5 应先做乘法再加法，结果为 23 而非 35。
Prime numbers have exactly two factors: 1 and the number itself. The first few primes are 2, 3, 5, 7, 11, 13. A factor is a whole number that divides another number exactly, while a multiple is the product of a number and an integer.
质数只有两个因数：1 和它本身。最小的几个质数是 2, 3, 5, 7, 11, 13。因数是能整除某数的整数，倍数则是某数与整数的乘积。
The highest common factor (HCF) of two numbers is the largest factor they share; the lowest common multiple (LCM) is the smallest multiple they share. These are useful when simplifying fractions or solving problems involving repeated patterns.
两个数的最大公因数 (HCF) 是它们共有的最大因数；最小公倍数 (LCM) 是它们共有的最小倍数。这些概念在化简分数或解决重复模式问题时非常有用。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
me amount, such as 1/2 = 2/4 = 3/6. To find an equivalent fraction, multiply or divide both parts by the same non-zero number.
分数表示整体的一部分，写作分子/分母。等价分数指数值相同但形式不同的分数，如 1/2 = 2/4 = 3/6。求等价分数时可将分子分母同时乘或除以同一个非零数。
Proper fractions have a numerator smaller than the denominator; improper fractions have a numerator larger than the denominator. Mixed numbers combine a whole number and a proper fraction, such as 2½.
真分数的分子比分母小；假分数的分子比分母大。带分数由一个整数和一个真分数组合而成，例如 2½。
To add or subtract fractions, they must have a common denominator. Once the denominators are the same, simply add or subtract the numerators and keep the denominator unchanged.
分数加减时必须先化成同分母。分母相同后，只需将分子相加或相减，分母不变。
Multiplying fractions is straightforward: multiply the numerators together and the denominators together. For division, flip the second fraction (find the reciprocal) and then multiply.
分数乘法直接将分子相乘、分母相乘。分数除法则是将第二个分数取倒数后转化为乘法。
r by the denominator. Recurring decimals have a repeating pattern, such as 0.333... = 1/3.
小数是分母为 10 的幂的分数的一种书写形式。将分数化为小数可用分子除以分母。循环小数具有重复的数字模式，例如 0.333... = 1/3。
'Percent' means 'out of 100'. To convert a decimal to a percentage, multiply by 100 and add the % symbol; to convert a percentage to a fraction, write it over 100 and simplify. A strong link between these three forms helps in solving real-world problems like discounts and interest.
“百分数”表示“百分之几”。小数化为百分数需乘以 100 并加上百分号；百分数化为分数则先写成分母 100 的分数再约简。熟练掌握这三种形式之间的转换有助于解决折扣、利息等实际问题。
3. Algebra: Expressions and Equations | 代数：表达式与方程
In algebra, letters (variables) stand for unknown numbers. An expression is a combination of numbers, variables and operations, such as 3x + 2y. It does not have an equals sign. A term is a single part of an expression, for instance 3x is a term.
代数中字母（变量）代表未知数。表达式是由数字、变量和运算组成的式子，如 3x + 2y，它不含等号。项是表达式中的单个部分，例如 3x 是一项。
Like terms have exactly the same variable part; only these can be combined. For example, 5a + 3a simplifies to 8a, but 5a + 3b cannot be simplified further. This process is called collecting like terms.
同类项具有完全相同的变量部分，只有同类项才能合并。例如 5a + 3a 可化简为 8a，但 5a + 3b 不能进一步化简。这一过程叫作合并同类项。
An equation states that two expressions are equal. Solving an equation means finding the value of the variable that makes the statement true. The key principle is to keep the equation balanced: whatever you do to one side, you must do to the other.
方程表示两个表达式相等。解方程就是找出使等式成立的变量的值。基本准则是保持方程平衡：对等式一边做的任何操作，在另一边也要做同样的操作。
ke 2x − 3 = 11, first add 3 to both sides, giving 2x = 14, then divide by 2 to find x = 7.
对于一步方程如 x + 7 = 12，两边同时减去 7 得到 x = 5。对于两步方程如 2x − 3 = 11，先两边加 3 得 2x = 14，再两边除以 2 得到 x = 7。
ns 3 × 4 + 2 = 14. This skill is essential for checking solutions and evaluating formulae.
代入法指将已知数值代入变量并计算结果。若 x = 4，则 3x + 2 即为 3 × 4 + 2 = 14。这项技能对于检验解和计算公式至关重要。
4. Sequences and Patterns | 序列与规律
A sequence is a list of numbers arranged in order according to a rule. Each number in the sequence is called a term. The rule that generates the sequence is often described by looking at the difference between consecutive terms.
序列是按照某种规则有序排列的一列数。序列中的每个数称为一项。生成序列的规则常常通过观察相邻项的差来描述。
In an arithmetic sequence, the difference between successive terms is constant. For example, in 3, 7, 11, 15, ... the common difference is +4. The term-to-term rule is 'add 4 to the previous term'.
在等差数列中，相邻项的差是一个常数。例如序列 3, 7, 11, 15, ... 的公差为 +4。项到项的规则是“前一项加 4”。
Other sequences may follow a doubling pattern or use a multiplication rule. Recognising a pattern allows you to predict the next terms or find missing terms.
其他序列可能遵循翻倍或乘法规则。识别规律可以预测后续项或找出缺失的项。
The position-to-term rule expresses the relationship between the position number n and the term value. For the sequence of even numbers 2, 4, 6, 8, ..., the nth term is given by 2n. This allows you to calculate any term directly without listing all previous terms.
位置—项规则表达了位置编号 n 与项值之间的关系。对于偶数序列 2, 4, 6, 8, ...，第 n 项为 2n。这样不必列出前面所有项就能直接计算任意项。
Square numbers (1, 4, 9, 16...) follow the rule n², cube numbers (1, 8, 27, 64...) follow n³. Triangular numbers (1, 3, 6, 10...) follow the rule n(n+1)/2. Year 7 students learn to recognise these special sequences.
平方数（1, 4, 9, 16...）遵循 n² 规则，立方数（1, 8, 27, 64...）遵循 n³。三角形数（1, 3, 6, 10...）遵循 n(n+1)/2。七年级学生要学会识别这些特殊序列。
5. Geometry: Angles and Shapes | 几何：角度与图形
An angle measures the amount of turn between two lines that meet at a vertex. Angles are measured in degrees (°). Basic angle facts include: angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal.
角用于衡量两条相交直线之间的旋转量，单位为度（°）。基本角度事实：直线上的角之和为 180°；围绕一点的角之和为 360°；对顶角相等。
Acute angles are less than 90°, right angles equal 90°, obtuse angles are between 90° and 180°, and reflex angles are greater than 180° but less than 360°. Using a protractor correctly is an important practical skill.
锐角小于 90°，直角等于 90°，钝角在 90° 至 180° 之间，优角大于 180° 且小于 360°。正确使用量角器是一项重要的操作技能。
Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). The sum of the interior angles of any triangle is 180°. This property is used to find missing angles.
三角形按边可分为等边、等腰和不等边三角形，按角可分为锐角、直角和钝角三角形。任何三角形的内角和都是 180°，这一性质常用于求未知角。
Quadrilaterals are four-sided polygons. Special quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapeziums and kites. The sum of interior angles of any quadrilateral is 360°.
四边形是有四条边的多边形。特殊的四边形包括正方形、矩形、平行四边形、菱形、梯形和筝形。任何四边形的内角和为 360°。
Parallel lines are lines that never meet. When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°. Recognising these angle relationships is central to geometric reasoning at this stage.
平行线是永不相交的直线。当一条横截线穿过平行线时，同位角相等、内错角相等、同旁内角之和为 180°。识别这些角度关系是现阶段几何推理的核心。
6. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the total distance around the outside of a 2D shape. For polygons, simply sum the lengths of all sides. For a rectangle, perimeter can be calculated as 2(length + width).
周长是二维图形外边界的总长度。对于多边形，只需将所有边长相加。矩形的周长可用 2(长 + 宽) 计算。
rea of a compound shape can be found by splitting it into simpler rectangles or triangles.
面积衡量图形内部的区域大小。矩形的面积 = 长 × 宽。三角形的面积 = ½ × 底 × 高。组合图形的面积可通过拆分为简单的矩形或三角形来求解。
re the lengths of the parallel sides. Using the correct perpendicular height is essential.
平行四边形的面积 = 底 × 垂直高；梯形的面积 = ½ (a + b) × h，其中 a 和 b 为平行边的长度。务必使用垂直高度。
ns are in cm, the volume is expressed in cm³. Common units of volume include cubic centimetres (cm³) and cubic metres (m³).
体积衡量立体所占空间的大小。长方体的体积 = 长 × 宽 × 高。若所有尺寸以厘米计，体积则以 cm³ 表示。常用的体积单位有立方厘米 (cm³) 和立方米 (m³)。
Surface area is the total area of all faces of a 3D solid. For a cuboid, calculate the area of each face and add them together. Understanding the connection between nets and surface area helps with visualisation.
表面积是立体所有面的面积总和。计算长方体的表面积时，可分别求出每个面的面积再加总。理解展开图与表面积的关系有助于空间想象。
7. Ratio and Proportion | 比和比例
A ratio compares the sizes of two or more quantities. If there are 3 apples and 5 oranges, the ratio of apples to oranges is written as 3 : 5. Ratios can be simplified in the same way as fractions by dividing each part by a common factor.
比用来比较两个或多个数量的大小。若有 3 个苹果和 5 个橘子，苹果与橘子的比记为 3 : 5。比可以像分数一样通过除以公因数进行化简。
Equivalent ratios are found by multiplying or dividing all parts by the same number. For example, 2 : 3 is equivalent to 4 : 6 and 10 : 15. A ratio in its simplest form has whole numbers with no common factor other than 1.
等价比可通过将所有部分同乘或同除一个数得到。例如 2 : 3 等价于 4 : 6 和 10 : 15。最简形式的比中各项为互质的整数。
Proportion describes a part of a whole compared with the whole. It can be expressed as a fraction, decimal or percentage. For instance, if the ratio of boys to girls is 3 : 5, the proportion of boys is 3/8 of the total.
比例描述部分与整体的关系，可以用分数、小数或百分数表示。例如，若男生与女生的比为 3 : 5，则男生占整体的比例为 3/8。
Direct proportion means that two quantities increase or decrease at the same rate. If 4 pencils cost £1.20, then 1 pencil costs £0.30, and 10 pencils cost £3.00. The unitary method is a powerful tool for solving proportion problems.
正比例指两个量以相同速率增加或减少。若 4 支铅笔售价 £1.20，则 1 支铅笔 £0.30，10 支铅笔 £3.00。单位量法是解决比例问题的有力工具。
Scaling recipes and maps makes frequent use of ratio and proportion. If a map scale is 1 : 50 000, every 1 cm on the map represents 50 000 cm (or 500 m) on the ground.
食谱配比的缩放和地图应用广泛涉及比和比例。若地图比例尺为 1 : 50 000，地图上 1 cm 代表实地 50 000 cm（即 500 m）。
8. Coordinates and Graphs | 坐标与图表
A coordinate grid uses two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where they cross is the origin (0,0). A point is located using its ordered pair (x, y).
坐标网格由两条相互垂直的轴构成：x 轴（水平）和 y 轴（垂直）。它们的交点为原点 (0,0)。一个点用有序数对 (x, y) 来定位。
Reading a coordinate always moves along the x-direction first, then the y-direction. 'Along the corridor, then up or down the stairs' is a helpful reminder.
读取坐标时总是先沿 x 方向移动，再沿 y 方向，即“先横走，后竖走”。
Points can be plotted in all four quadrants. In the first quadrant both coordinates are positive; in the second, x is negative and y positive; in the third both are negative; and in the fourth, x is positive and y negative.
点可以被绘制在全部四个象限中。第一象限两个坐标均为正；第二象限 x 为负、y 为正；第三象限两者均为负；第四象限 x 为正、y 为负。
A line graph can be drawn by plotting points from a set of data and joining them with straight lines. Conversion graphs, such as those between miles and kilometres, are a practical use of straight-line graphs in Year 7.
折线图可根据一组数据描点并用直线连接而成。转换图（如英里与公里之间的转换）是七年级直线图的实际应用。
2x + 1, you can create a table of values by choosing x and calculating y, then plot the points. The gradient describes how steep the line is.
对于直线方程如 y = 2x + 1，可先通过选择 x 值计算出对应的 y 值，得到数值表，再描点连线。斜率描述直线的倾斜程度。
9. Statistics: Averages and Charts | 统计：平均数和图表
The three main averages are the mean, median and mode. The mean is found by adding all values and dividing by the number of values. The median is the middle value when the data is ordered. The mode is the most frequently occurring value.
三个主要平均数是平均数、中位数和众数。平均数是将所有数据相加再除以数据个数。中位数是将数据排序后位于中间的值。众数是出现次数最多的值。
The range measures the spread of data: largest value minus smallest value. Along with an average, the range gives a simple summary of the data set.
极差（范围）度量数据的分散程度：最大值减最小值。结合起来看，平均数和极差能对数据集给出简要概括。
Data can be organised into frequency tables. A tally chart is used to record data as it is collected. Grouped frequency tables are useful when dealing with a large spread of continuous data, such as heights or test scores.
数据可整理为频数表。计数符号表用于边收集边记录数据。在处理如身高或测验分数等大范围连续数据时，分组频数表非常有用。
Bar charts display categorical data with rectangular bars of equal width; the height of each bar represents the frequency. Pictograms use symbols or pictures to represent a certain number of items, with a key to show the scale.
条形图用宽度相等的长方形条来展示分类数据，每根条的高度代表频数。象形图使用符号或图像表示一定数量的项目，并附有图例说明比例。
Pie charts represent data as sectors of a circle. The size of each sector is proportional to the frequency. In Year 7, students interpret given pie charts and relate angles to fractions, for example 90° corresponds to ¼ of the data.
饼图以圆的扇形表示数据。每个扇区的大小与频数成正比。七年级学生需要解读给定的饼图，并将角度与分数联系起来，例如 90° 对应数据的 ¼。
10. Probability | 概率
Probability measures how likely an event is to happen. It can be expressed as a fraction, decimal or percentage. The probability scale runs from 0 (impossible) to 1 (certain). An even chance has a probability of 0.5.
概率衡量事件发生的可能性大小，可用分数、小数或百分数表示。概率标度从 0（不可能）到 1（一定发生）。机会均等的概率为 0.5。
outcomes / total number of possible outcomes, provided all outcomes are equally likely. This is the basic theoretical probability formula.
若所有结果发生的可能性相等，事件 A 的概率计算公式为：P(A) = 有利结果数 / 所有可能结果总数。这是基本的理论概率公式。
When flipping a fair coin, the probability of heads is ½. The probability of rolling a 6 on a fair dice is 1/6. The sum of the probabilities of all possible outcomes of an experiment is always 1.
抛一枚公平硬币，正面朝上的概率为 ½。掷一个均匀骰子得到 6 的概率为 1/6。一项试验所有可能结果的概率之和总是 1。
P(A). This is useful when it is easier to find the probability of the complementary event.
互补规则：某事件不发生的概率等于 1 减去它发生的概率，即 P(非 A) = 1 − P(A)。当互补事件的概率更容易求出时，这一规则非常有用。
Experimental probability is based on conducting trials or experiments, while theoretical probability is based on equally likely outcomes. Real-life experiments help students see that with more trials, the experimental probability tends to approach the theoretical value.
实验概率基于实际的试验或实验，而理论概率基于等可能性结果。现实生活中的实验能帮助学生理解，随着试验次数增加，实验概率会趋近理论值。
Sample space diagrams list all possible outcomes of two combined events, such as the sum of two dice. These diagrams are a systematic way to ensure no outcome is missed when calculating probabilities.
样本空间图列出两个组合事件的所有可能结果，例如两个骰子的和。这类图表是一种系统性方法，确保在计算概率时不会遗漏任何结果。
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