📚 The 6 ACT Math Topics: A Complete Summary | ACT数学六大考点归纳
The ACT Math test covers a broad range of mathematical skills, but all 60 questions are drawn from six major content areas. Understanding what these topics are, how often they appear, and which key formulas you need to know is the first step toward a high score. This article breaks down the six core categories, gives examples of the concepts in each, and offers practical tips for efficient preparation.
ACT数学考试虽然涵盖面广，但全部60道题都来自六个主要知识板块。弄清楚这些考点有哪些、出现频率如何、必须掌握哪些核心公式，是拿到高分的第一步。本文逐一拆解这六大类别，给出每类包含的具体概念示例，并提供高效备考的实用建议。
1. Overview of the Six ACT Math Categories | ACT数学六大类别概览
ACT Math is a 60-minute, 60-question multiple-choice section. The questions are not ordered by topic, but they do roughly increase in difficulty. The six content areas and their approximate percentages are: Pre-Algebra (20–25%), Elementary Algebra (15–20%), Intermediate Algebra (15–20%), Coordinate Geometry (15–20%), Plane Geometry (20–25%), and Trigonometry (5–10%). Knowing this distribution helps you allocate your study time wisely.
ACT数学部分共60道选择题，限时60分钟。题目并非按知识点排序，但难度大致逐渐上升。六大类别及其大致比例是：算术与预备代数(20–25%)、初级代数(15–20%)、中级代数(15–20%)、坐标几何(15–20%)、平面几何(20–25%)、三角学(5–10%)。了解这样的分值分布，可以帮助你合理分配备考时间。
2. Pre-Algebra (20–25%) | 算术与预备代数
This category tests your grasp of basic number operations, fractions, decimals, factors, ratios, proportions, percentages, and simple descriptive statistics. You will also face questions on absolute value, exponents and square roots, scientific notation, and elementary probability. Though the arithmetic is straightforward, careful reading is essential to avoid overlooking a unit conversion or a multi-step percent problem.
这一类别考察基础运算、分数、小数、因数、比和比例、百分比以及简单的描述统计。你还会遇到绝对值、指数与平方根、科学记数法和初等概率问题。虽然计算本身不复杂，但必须仔细审题，避免在单位换算或多步百分比问题上疏忽大意。
- Key topics: factors and multiples, prime numbers, order of operations, mean/median/mode, data interpretation from charts.
- 重点知识：因数与倍数、质数、运算顺序、平均数/中位数/众数、从图表中解读数据。
Example: A shirt originally costs $40. It is first discounted by 25%, and then an additional 10% off is applied to the reduced price. What is the final price? Focus on taking percentages sequentially rather than adding percentages.
示例：一件衬衫原价40美元，先打七五折，再在折后价上打九折，最终价格是多少？注意要逐步求百分比，不要简单地把折扣比例相加。
3. Elementary Algebra (15–20%) | 初级代数
Elementary Algebra covers solving linear equations and inequalities, evaluating algebraic expressions, simplifying polynomials, working with integer exponents, and understanding basic function notation. Word problems that translate into one or two linear equations are also common. The key is to build comfort with symbolic manipulation and to check your solutions efficiently.
初级代数包括求解一元一次方程与不等式、求代数式的值、化简多项式、整数指数运算，以及理解基本的函数记号。由文字题转化为一两个线性方程的题目也很常见。关键在于熟练掌握符号运算，并能高效验算。
- Must-know skills: multiplying binomials using FOIL, factoring common factors, solving |x + 2| = 5, graphing simple inequalities on a number line.
- 必备技能：用FOIL法则展开二项式、提取公因式、求解绝对值方程|x+2|=5、在数轴上表示一元一次不等式。
For example, if 3(x − 4) + 2x = 7, first expand, then collect like terms, and solve for x. Always double-check by substituting the value back into the original equation.
例如，3(x−4)+2x=7，先去括号，再合并同类项，求出x。最好将解代回原方程验证。
4. Intermediate Algebra (15–20%) | 中级代数
This section extends algebra to quadratic equations, systems of equations, functions, rational and radical expressions, sequences, and logarithms. You need to be comfortable with factoring quadratics, using the quadratic formula, understanding domain and range, and simplifying expressions with square roots or rational exponents. Matrices and complex numbers may appear but are less frequent.
中级代数将代数知识扩展到二次方程、方程组、函数、有理式与根式、数列和对数。你需要熟练掌握二次三项式的因式分解、使用求根公式、理解定义域与值域，以及化简含有平方根或有理指数的式子。矩阵和复数偶尔出现，但频率较低。
The quadratic formula is written as: x = [−b ± √(b² − 4ac)] / 2a. Remember to check the discriminant b² − 4ac to determine the nature of the roots.
二次方程求根公式为：x = [−b ± √(b² − 4ac)] / 2a。记住通过判别式b² − 4ac的正负来判断根的性质。
Systems of equations can be solved by substitution, elimination, or by graphing. On the ACT, the numbers are usually designed to work out neatly, so choose the method that minimises arithmetic mistakes.
解方程组可以用代入法、消元法或图解法。ACT的题目通常数据设计得比较工整，选择自己最不容易算错的方法即可。
5. Coordinate Geometry (15–20%) | 坐标几何
Coordinate geometry focuses on the graphical representation of lines, parabolas, circles, and other conic sections. You will need to find slope, midpoint, distance, equations of lines, and intercepts. Understanding how transformations shift graphs is also tested. Many questions can be solved by plugging points into equations or by visualising the graph mentally.
坐标几何侧重直线、抛物线、圆以及其他圆锥曲线的图像。你需要会求斜率、中点、距离、直线方程和截距。图像平移变换也是考点。很多题目可以通过将点的坐标代入方程或在大脑中想象图像来解决。
- Line formulas: slope m = (y₂ − y₁)/(x₂ − x₁), distance d = √[(x₂ − x₁)² + (y₂ − y₁)²], midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2).
- 直线相关公式：斜率 m = (y₂ − y₁)/(x₂ − x₁)，距离 d = √[(x₂ − x₁)² + (y₂ − y₁)²]，中点 M = ((x₁+x₂)/2, (y₁+y₂)/2)。
Given the equation y = 2x² − 4x + 1, you may be asked to identify the vertex. Use the vertex formula x = −b/(2a) to get x = 1, then substitute to find y = −1. Recognising standard forms saves precious time.
已知方程 y = 2x² − 4x + 1，可能需要找出顶点。用顶点公式 x = −b/(2a) 得到 x = 1，再代入求得 y = −1。熟悉标准式能省下大量时间。
6. Plane Geometry (20–25%) | 平面几何
Plane geometry is the largest category and tests properties of triangles, quadrilaterals, circles, angles, parallel lines, area, perimeter, and volume of simple solids. You need to memorise formulas for the area of a triangle, trapezoid, and circle, as well as the Pythagorean theorem and the relationships in special right triangles (45°-45°-90° and 30°-60°-90°). You should also know the sum of interior angles of a polygon.
平面几何是占比最大的类别，考察三角形、四边形、圆、角、平行线的性质，以及面积、周长和简单立体体积。需要熟记三角形、梯形和圆的面积公式，毕达哥拉斯定理，以及特殊直角三角形的边比关系(45°-45°-90°和30°-60°-90°)。多边形的内角和公式也要掌握。
For a regular polygon with n sides, the sum of interior angles is (n − 2) × 180°, and each interior angle measures [(n − 2) × 180°]/n. Many problems combine several shapes, so draw additional lines to break figures into simpler pieces.
正n边形的内角和为 (n − 2) × 180°，每个内角的度数为 [(n − 2) × 180°]/n。许多题目把多个图形复合在一起，添加辅助线将其拆分成简单图形往往很有效。
Volume formulas for cubes, rectangular prisms, and cylinders are frequently used. Also remember that the ratio of areas of similar figures equals the square of the similarity ratio.
正方体、长方体和圆柱的体积公式经常用到。还要记住，相似图形的面积比等于相似比的平方。
7. Trigonometry (5–10%) | 三角学
Though only a handful of questions appear, trigonometry can be an easy source of points if you know the basics. The ACT primarily tests right-triangle trigonometry: sine, cosine, tangent, and their reciprocal functions. You should also understand the unit circle, radian measure, and basic trigonometric identities such as sin²θ + cos²θ = 1.
三角学虽然只占少数几道题，但如果掌握了基础，就是很容易的得分点。ACT主要考查直角三角形的三角学：正弦、余弦、正切及其倒数函数。你还需要理解单位圆、弧度制，以及简单的三角恒等式，比如 sin²θ + cos²θ = 1。
SOHCAHTOA is your ally: Sin θ = opposite/hypotenuse, Cos θ = adjacent/hypotenuse, Tan θ = opposite/adjacent. Use these to find missing sides and angles. For non-right triangles, the law of sines and law of cosines may be tested, but less often.
SOHCAHTOA记忆法很有用：正弦 = 对边/斜边，余弦 = 邻边/斜边，正切 = 对边/邻边。利用这些关系求缺失的边长和角度。非直角三角形中，正弦定理和余弦定理也可能考到，但频率较低。
Using radian mode when needed: 180° = π radians. Converting from degrees to radians involves multiplying by π/180°, and the reverse by 180°/π.
注意弧度制：180° = π 弧度。角度转弧度乘以 π/180°，弧度转角度乘以 180°/π。
8. Strategy for Using This Topic Summary | 如何利用考点归纳高效备考
Now that you know the six content domains, prioritise your study by focusing first on areas where you can gain the most points quickly. For many students, Pre-Algebra and Plane Geometry offer the greatest room for improvement because of their large share. Dedicate consistent time to practice each category, using official ACT practice tests to identify your weak spots. When reviewing mistakes, ask yourself: Did I miss a formula? Did I misread the question? Did I rush a calculation?
知道了这六个知识领域，你可以先重点攻克最容易快速提分的板块。对多数学生来说，算术与预备代数、平面几何占比高，提分空间最大。坚持分配时间进行每个类别的专项练习，利用官方ACT真题找出薄弱点。复盘错题时，问自己：是公式没记住？审题出错？还是计算太急？
Keep a formula sheet for the ones you tend to forget, and review it briefly the night before the test. Remember that the ACT provides some basic trigonometric identities and a few geometry formulas in the test booklet, but you should still memorise them to save time. Finally, practice mental math and estimation to avoid spending too long on a single question. Time management is key to finishing all 60 questions.
把自己容易忘的公式整理成一张清单，考前晚上快速过一遍。记住虽然考卷上会给出一些基本的三角恒等式和几何公式，但为了省时间，还是应该背下来。最后，多练习心算和估算，避免在某一题上耗时过多。时间管理是完成全部60题的关键。
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