📚 Year 10 OCR Mathematics: Full Syllabus Breakdown | Year 10 OCR 数学：课程大纲全面解析
Year 10 is the foundation year for the OCR GCSE (9-1) Mathematics course, where students build the core skills that will carry them through to the final examinations. In this article, we walk you through every major topic area you will encounter, explain the key concepts, and show how they connect to exam success. Whether you are aiming for Grades 1 to 5 on the Foundation tier or stretching towards the top grades on the Higher tier, a solid grasp of the Year 10 syllabus is essential.
十年级是 OCR GCSE（9-1）数学课程的基础学年，学生将在这一年建立贯穿最终考试的核心技能。本文将带你全面梳理每一个主要专题，解释关键概念，并说明它们如何与考试成功挂钩。不论你的目标是在基础层取得 1 至 5 级，还是在高层冲击最高等级，扎实掌握十年级课程大纲都至关重要。
1. Number Basics and Place Value | 数字基础与位值
A firm understanding of the decimal number system and place value is the starting point for all number work. You need to be comfortable reading, writing, ordering and rounding whole numbers, as well as understanding the effect of multiplying and dividing by powers of 10.
牢固掌握十进制数系统和位值是所有数字运算的起点。你需要能够熟练地读、写、排序和四舍五入整数，并理解乘以或除以 10 的幂所产生的影响。
For negative numbers, the four operations must be applied confidently in contexts such as temperature changes or bank balances. OCR exam questions often test these skills within problem-solving scenarios, asking you to interpret a real-life situation before performing the calculation.
对于负数，必须在温度变化或银行余额等情境中自信地运用四则运算。OCR 考题经常在解决问题的场景中检验这些技能，要求你先解读实际情境再进行计算。
- Key skill: Writing 5.2 million as 5,200,000 and knowing that the digit 2 has a value of 200,000.
- 关键技能： 将 5.2 百万写成 5,200,000，并知道数字 2 的值为 200,000。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
In Year 10 you will revisit equivalent fractions, improper fractions and mixed numbers, and learn to add, subtract, multiply and divide fractions fluently. Always simplify your answer to its lowest terms unless the question specifies otherwise.
在十年级，你将复习等值分数、假分数和带分数，并学会流畅地进行分数的加减乘除。除非题目另有规定，否则始终将答案化简至最简形式。
Converting between fractions, decimals and percentages is a fundamental skill that appears across almost every topic. For instance, to turn a fraction into a decimal, divide the numerator by the denominator; to turn a decimal into a percentage, multiply by 100. OCR papers regularly include ‘list and order’ style questions where you must place a mixture of fractions, decimals and percentages in ascending or descending order.
在分数、小数和百分数之间相互转换是一项几乎贯穿所有专题的基础技能。例如，要将分数化为小数，用分子除以分母；要将小数化为百分数，乘以 100。OCR 试卷中经常出现“列出并排序”类题目，要求你将混合的分数、小数和百分数按升序或降序排列。
Fraction to decimal: 3/8 = 3 ÷ 8 = 0.375
Decimal to percentage: 0.375 × 100 = 37.5%
3. Powers, Roots and Standard Form | 幂、根与标准形式
Powers (indices) and roots are introduced early in the OCR course. You must know that a³ means a × a × a and that √a is the positive square root of a. The laws of indices for multiplication and division — aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ — are tested frequently, often alongside algebraic manipulation.
幂（指数）和根在 OCR 课程中较早引入。你必须知道 a³ 表示 a × a × a，而 √a 是 a 的正平方根。指数的乘法和除法法则——aᵐ × aⁿ = aᵐ⁺ⁿ 以及 aᵐ ÷ aⁿ = aᵐ⁻ⁿ——经常与代数操作一起被考查。
Standard form (scientific notation) is used to represent very large or very small numbers. A number in standard form is written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer. For example, the speed of light is approximately 3.00 × 10⁸ m/s. You should be able to convert between ordinary numbers and standard form, and perform calculations using a calculator.
标准形式（科学记数法）用来表示非常大或非常小的数字。标准形式的数写作 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。例如，光速约为 3.00 × 10⁸ 米/秒。你应当能够在普通数字和标准形式之间转换，并使用计算器进行计算。
4. Algebraic Expressions and Simplifying | 代数表达式与化简
Algebra makes up a large part of the Year 10 syllabus. You begin by learning to collect like terms: 3x + 2y + 5x − y simplifies to 8x + y. Expanding single brackets, such as 3(2x − 5) = 6x − 15, and factorising by taking out the highest common factor, for example 4x + 8 = 4(x + 2), are essential techniques.
代数在十年级课程大纲中占据很大比重。你首先学习合并同类项：3x + 2y + 5x − y 化简为 8x + y。展开单项括号，如 3(2x − 5) = 6x − 15，以及通过提取最大公因式进行因式分解，例如 4x + 8 = 4(x + 2)，都是基本技巧。
Substitution means replacing letters with given values. If a = 3 and b = −2, then 2a² − b = 2(3)² − (−2) = 18 + 2 = 20. OCR questions often embed substitution within a formula from geometry or science, so accurate use of negative numbers and brackets is crucial.
代入法意味着用给定数值替换字母。如果 a = 3, b = −2，则 2a² − b = 2(3)² − (−2) = 18 + 2 = 20。OCR 题目常将代入法嵌入几何或科学公式中，因此准确使用负数和括号至关重要。
5. Solving Linear Equations and Inequalities | 解线性方程与不等式
By the end of Year 10, you should be able to solve linear equations with unknowns on both sides, such as 5x − 7 = 3x + 9, giving x = 8. Rearranging formulae to change the subject, for instance making h the subject of A = ½bh, requires similar balancing skills.
在十年级结束时，你应当能够解未知数在两侧的线性方程，例如 5x − 7 = 3x + 9，得出 x = 8。变换公式使其成为某字母的函数，比如将 A = ½bh 化为以 h 为主语，需要类似的平衡技能。
Solving linear inequalities follows the same rules, but remember to reverse the inequality sign when you multiply or divide by a negative number. Representing the solution on a number line — using open or closed circles — is a typical OCR requirement.
解线性不等式遵循相同的规则，但要记住当乘或除以一个负数时，需将不等号方向反转。在数轴上表示解集——使用空心或实心圆圈——是 OCR 的典型要求。
If −2x < 6, then x > −3 (inequality sign reversed).
6. Linear Graphs and Coordinates | 直线图与坐标
The coordinate plane is used to plot straight-line graphs of the form y = mx + c. You need to identify the gradient m and the y-intercept c from an equation or a graph. Parallel lines have the same gradient, a fact frequently examined in multi-step geometry questions.
坐标平面被用来绘制 y = mx + c 形式的直线图形。你需要能从一个方程或图形中识别斜率 m 和 y 轴截距 c。平行线具有相同的斜率，这一事实经常在多步骤几何题中被考查。
Drawing a table of values, plotting points accurately and drawing the line are core competencies. OCR may ask you to find the equation of a line given two points, using the formula for gradient: m = (y₂ − y₁)/(x₂ − x₁). Later, you will use linear graphs to solve simultaneous equations graphically.
列出数值表、准确描点并画出直线是核心能力。OCR 可能会要求根据两点求直线方程，这时要用到斜率公式：m = (y₂ − y₁)/(x₂ − x₁)。往后，你将使用直线图通过图解方式解联立方程。
7. Sequences and nth Term | 序列与第 n 项
Arithmetic sequences increase or decrease by a constant difference. The nth term of an arithmetic sequence is given by an + b where a is the common difference. For the sequence 5, 8, 11, 14, …, the nth term is 3n + 2.
算术序列以一个恒定的差值递增或递减。算术序列的第 n 项由 an + b 给出，其中 a 是公差。对于序列 5, 8, 11, 14, …，第 n 项为 3n + 2。
You should also be able to recognise special sequences such as square numbers (1, 4, 9, 16, …), cube numbers and triangular numbers. OCR problems often ask whether a particular number is in the sequence, which can be checked by setting the nth term equal to that number and solving for n — if n is a positive integer, the number belongs.
你还应能够识别特殊序列，如平方数（1, 4, 9, 16, …）、立方数和三角数。OCR 问题经常询问某个特定数字是否在序列中，这可以通过令第 n 项等于该数字并解出 n 来检验——若 n 为正整数，则该数字属于该序列。
8. Ratio, Proportion and Direct Proportion | 比例、比率与正比例
Ratio notation is used to compare quantities. Simplifying a ratio 24:36 to 2:3 by dividing by the highest common factor is an everyday skill. Division in a given ratio, such as sharing £50 in the ratio 2:3, means one person gets 2/5 × £50 = £20 and the other gets £30.
比号用来比较数量。将比例 24:36 除以最大公因数简化为 2:3 是日常技能。按给定比例分配，例如将 50 英镑按 2:3 分配，意味着一人得到 2/5 × 50 = 20 英镑，另一人得到 30 英镑。
Direct proportion means two quantities increase at the same rate. If 5 pens cost £3.50, then 1 pen costs £0.70, and 8 pens cost £5.60. You will also see proportion used in scale diagrams, conversion graphs and recipes — all typical OCR contexts.
正比例意味着两个量以相同速率增加。若 5 支笔花费 3.50 英镑，则 1 支 0.70 英镑，8 支 5.60 英镑。比例还会出现在比例图、换算图和食谱中——这些都是 OCR 的典型情境。
9. Perimeter, Area and Volume | 周长、面积与体积
First, you must be able to calculate the perimeter of any rectilinear shape and the circumference of a circle (C = πd or 2πr). The area of a triangle is ½ × base × height; a parallelogram is base × perpendicular height; and a trapezium is ½(a + b)h, where a and b are the parallel sides.
首先，你必须能计算任何直线组合图形的周长以及圆的周长（C = πd 或 2πr）。三角形面积为 ½ × 底 × 高；平行四边形为底 × 垂直高度；梯形面积为 ½(a + b)h，其中 a 和 b 是平行边。
For volume, a prism’s volume is area of cross-section × length. OCR often includes compound shapes made from rectangles and triangles, requiring you to split the shape into simpler parts. Remember to state units for area (cm², m²) and volume (cm³, m³).
在体积方面，棱柱的体积为横截面积 × 长。OCR 经常给出由矩形和三角形组成的复合图形，要求你将图形拆分成更简单的部分。记住要标注面积的单位（cm², m²）和体积的单位（cm³, m³）。
Circle area: A = πr² Circumference: C = πd
10. Angles and Polygons | 角与多边形
Angle facts are the building blocks: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal. In parallel lines, alternate angles are equal, corresponding angles are equal, and interior angles sum to 180°.
角度知识是基石：直线上角的和为 180°，一点周围角的和为 360°，对顶角相等。在平行线中，内错角相等，同位角相等，同旁内角和为 180°。
For polygons, the sum of interior angles of an n-sided polygon is (n − 2) × 180°. In a regular polygon, each interior angle is this sum divided by n. Exterior angles always sum to 360°, so for a regular polygon each exterior angle is 360°/n. OCR exam questions often combine these angle rules with algebraic equations to find unknown sides or angles.
对于多边形，n 边形的内角和为 (n − 2) × 180°。在正多边形中，每个内角等于内角和除以 n。外角和始终为 360°，因此对于正多边形，每个外角为 360°/n。OCR 考题常将这些角度规则与代数方程结合，以求解未知边或角。
11. Pythagoras’ Theorem and Basic Trigonometry | 毕达哥拉斯定理与基础三角学
Pythagoras’ theorem states that in a right-angled triangle, a² + b² = c², where c is the hypotenuse. You will use it to find a missing side length, often in 2D or 3D contexts. A common OCR problem involves a ladder leaning against a wall, where the ladder is the hypotenuse.
毕达哥拉斯定理表明，在直角三角形中，a² + b² = c²，其中 c 为斜边。你将用它来求缺失的边长，常见于二维或三维情境。OCR 常见的问题涉及梯子靠墙，其中梯子为斜边。
Basic trigonometry introduces the sine, cosine and tangent ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. In Year 10, you will use these to find unknown sides or angles in right-angled triangles. Remember to set your calculator to degree mode!
基础三角学引入正弦、余弦和正切比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。在十年级，你将使用这些比值来求直角三角形中的未知边或角。记得将计算器设置为度数模式！
SOH CAH TOA: sin θ = O/H, cos θ = A/H, tan θ = O/A
12. Probability and Statistical Diagrams | 概率与统计图表
Probability is measured on a scale from 0 (impossible) to 1 (certain). The probability of an event not happening is 1 minus the probability that it does. You will construct sample space diagrams for two events and use them to find the probability of combined outcomes.
概率按从 0（不可能）到 1（确定）的尺度衡量。事件不发生的概率等于 1 减去它发生的概率。你将为一个试验的两个事件构建样本空间图，并用其求组合结果的概率。
Statistical diagrams include bar charts, pie charts, scatter graphs, and frequency tables. For grouped data, you will calculate an estimate of the mean using the midpoints of intervals. OCR places strong emphasis on interpreting diagrams — you might be given a dual bar chart and asked to compare distributions using range and median.
统计图表包括条形图、饼图、散点图和频数表。对于分组数据，你将用组中点计算平均数的估计值。OCR 非常重视解读图表——你可能会得到一个双条形图，并被要求使用极差和中位数来比较分布。
- Mean from a frequency table: Sum of (frequency × midpoint) ÷ total frequency.
- 从频数表求平均数： （频数 × 组中点）之和 ÷ 总频数。
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