ACT Math High-Frequency Vocabulary Handbook | ACT数学高频词汇手册

📚 ACT Math High-Frequency Vocabulary Handbook | ACT数学高频词汇手册

Building a solid ACT Math vocabulary is essential for interpreting questions quickly and accurately. Many problems rely on precise terms like “integer,” “slope,” or “mean,” and misunderstanding them can cost valuable points. This handbook presents high-frequency words grouped by topic, with clear bilingual explanations to help you master the language of the test.

建立扎实的ACT数学词汇库是快速准确理解题目的关键。许多问题依赖 “integer”、”slope” 或 “mean” 等精确术语，误读它们会丢失宝贵的分数。本手册按主题分组整理高频词汇，配以清晰的中英双语解释，帮助您掌握考试语言。

1. Arithmetic & Number Properties | 算术与数的性质

These words describe the building blocks of numbers and basic operations. ACT questions often test your knowledge of factors, multiples, and divisibility rules.

这些词语描述了数字的基本构成和运算。ACT考题经常考查你对因数、倍数和整除规则的理解。

Integer: A whole number that is not a fraction or decimal; it can be positive, negative, or zero. The instruction “x is an integer” immediately limits the possible solutions.

整数：一个非分数、非小数的整数，可以是正数、负数或零。题目中 “x is an integer” 的表述会立即限制可能的解。

Prime number: A positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself (e.g., 2, 3, 5, 7). Recognizing primes is key to prime factorization problems.

质数：大于1的正整数，且只有两个不同的正因数：1和它本身（如2、3、5、7）。辨识质数是解决质因数分解问题的基础。

Factor: A whole number that divides another number exactly without leaving a remainder. If 12 ÷ 3 = 4 with no remainder, then 3 is a factor of 12.

因数：能整除另一个数且没有余数的整数。若12 ÷ 3 = 4 无余数，则3是12的因数。

Least common multiple (LCM): The smallest positive number that is a multiple of two or more numbers. For 4 and 6, the LCM is 12, used to find common denominators.

最小公倍数：两个或多个数共有的倍数中最小的正数。4和6的最小公倍数是12，常用于寻找公分母。

Absolute value: The distance of a number from zero on the number line, always non-negative. |−5| = 5. It appears in equations and inequalities that involve distance.

绝对值：数轴上某数到零的距离，永为非负值。|−5| = 5。它出现在涉及距离的方程和不等式中。

Consecutive integers: Integers that follow each other in order, like n, n+1, n+2. ACT may ask for three consecutive odd integers or their sum.

连续整数：依序排列的整数，如n, n+1, n+2。ACT常要求三个连续奇数或求和。

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2. Fractions, Decimals & Percentages | 分数、小数与百分比

These concepts appear in nearly every ACT Math section. Being able to convert between forms and handle percentage change is a must.

这些概念几乎出现在每套ACT数学题中。能在不同形式之间转换并处理百分比变化是必备技能。

Fraction: A number expressed as a/b, where a is the numerator and b is the denominator. Fractions represent parts of a whole and are often simplified.

分数：表示为a/b的数，其中a为分子，b为分母。分数表示整体的一部分，通常要求化简。

Decimal: A non-integer number written with a decimal point, such as 3.14. Terminating decimals end, while repeating decimals have a digit pattern that continues forever.

小数：带小数点的非整数，如3.14。有限小数会终止，而循环小数有无限重复的数字模式。

Percent: A ratio that compares a number to 100; “%” means “per hundred.” Percent change = (new − original)/original × 100%.

百分比：将数与100作比较的比例，“%”表示“每百”。百分比变化 = (新值 − 原值)/原值 × 100%。

Ratio: A comparison of two quantities, written as a:b or a/b. If a recipe uses 2 cups of flour to 3 cups of sugar, the ratio is 2:3.

比：两个量的比较，写作a:b或a/b。若配方用2杯面粉对3杯砂糖，比为2:3。

Proportion: An equation that states two ratios are equal, like a/b = c/d. Cross-multiplication is used to solve for an unknown.

比例：表示两个比相等的等式，如 a/b = c/d。可用交叉相乘法求解未知数。

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3. Exponents & Radicals | 指数与根式

Exponents and roots regularly appear in simplifying expressions and solving equations. Understanding the notation is critical.

指数和根式常用于化简表达式和求解方程。理解其符号表示至关重要。

Exponent: A small raised number that indicates how many times the base is multiplied by itself. In 5³, the exponent is 3, so 5 × 5 × 5 = 125.

指数：一个上标数字，表示底数自乘的次数。在5³中，指数为3，即5 × 5 × 5 = 125。

Base: The number that is raised to a power. In 2⁴, the base is 2. Exponents rules like aⁿ × aᵐ = aⁿ⁺ᵐ require careful base identification.

底数：被乘方运算的数字。在2⁴中，底数为2。指数法则如 aⁿ × aᵐ = aⁿ⁺ᵐ 须正确识别底数。

Square root: A value that, when multiplied by itself, gives the original number. √9 = 3 because 3 × 3 = 9. The symbol √ denotes the principal (non-negative) root.

平方根：一个数自乘后得到原数的值。√9 = 3，因为3 × 3 = 9。符号√表示主平方根（非负）。

Radical: The symbol √ used to indicate a root, along with the index for higher roots. ∛8 = 2. Simplifying radicals often involves factoring out perfect squares.

根号：表示根的符号√，并带指数表示高次根。∛8 = 2。化简根式常需提取完全平方数。

Rational exponent: An exponent written as a fraction, where x¹/² is equivalent to √x. This form bridges exponents and radicals.

有理指数：写成分数形式的指数，如 x¹/² 等价于 √x。这种形式连接了指数与根式。

Scientific notation: A compact way to write very large or small numbers as a × 10ⁿ, where 1 ≤ a < 10. 4.5 × 10³ = 4500.

科学记数法：用以简洁表示极大或极小数的方法，形式为 a × 10ⁿ，其中1 ≤ a < 10。4.5 × 10³ = 4500。

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4. Algebraic Expressions & Equations | 代数表达式与方程

Algebra forms the core of the ACT Math test. Knowing the terminology of variables, terms, and equations speeds up problem-solving.

代数是ACT数学测试的核心。熟悉变量、项、方程等术语能加快解题速度。

Variable: A symbol, usually a letter, that represents an unknown number. In 3x + 5, x is the variable whose value can change.

变量：通常用字母表示的未知数。在3x + 5中，x就是可以取不同值的变量。

Coefficient: The numerical factor multiplying a variable. In 7y, 7 is the coefficient of y. Combining like terms involves adding coefficients.

系数：乘以变量的数值因子。在7y中，7是y的系数。合并同类项即系数相加。

Expression: A combination of numbers, variables, and operations without an equals sign. 2a − 5b is an expression; it cannot be “solved,” only simplified.

表达式：数字、变量和运算的组合，无等号。2a − 5b是一个表达式，只可化简，不能“解”。

Equation: A statement that two expressions are equal, containing an equals sign. Solving 3x = 12 gives x = 4.

方程：两个表达式相等的陈述，包含等号。解方程3x = 12得 x = 4。

Linear equation: An equation whose graph is a straight line, typically in the form y = mx + b. The highest exponent of the variable is 1.

线性方程：图像为直线的方程，通常呈 y = mx + b。变量的最高指数为1。

Inequality: A mathematical sentence using <, >, ≤, or ≥ instead of = . Solving inequalities often flips the inequality sign when multiplying or dividing by a negative.

不等式：使用 <, >, ≤, ≥ 而非等号的数学语句。当乘以或除以负数时，不等号方向需反转。

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5. Functions & Graphs | 函数与图像

Function notation and graph interpretation are heavily tested. The terms below will help you analyze inputs, outputs, slopes, and key points.

函数符号与图像解读是考试重点。下列术语有助于分析输入、输出、斜率和关键点。

Function: A rule that assigns exactly one output for each input. f(x) = x² is read “f of x equals x squared.” Understanding function notation is vital.

函数：每个输入对应唯一输出的规则。f(x) = x² 读作“f of x 等于 x 平方”。掌握函数符号至关重要。

Domain: The set of all possible input values (x) for a function. For f(x) = √x, the domain is x ≥ 0.

定义域：函数中所有可能输入值（x）的集合。例如 f(x) = √x 的定义域为 x ≥ 0。

Range: The set of all possible output values (y) produced by a function. For f(x) = x², the range is y ≥ 0.

值域：函数产生的所有可能输出值（y）的集合。对于 f(x) = x²，值域为 y ≥ 0。

Slope: The measure of steepness of a line, often described as rise over run. In y = mx + b, m is the slope. A positive slope rises to the right.

斜率：直线陡峭程度的度量，常描述为“垂直变化比水平变化”。在 y = mx + b 中，m为斜率。正斜率向右上升。

Intercept: The point where a graph crosses an axis. The y-intercept is (0, b) in slope-intercept form; the x-intercept occurs where y = 0.

截距：图像与坐标轴的交点。斜截式中的 y-截距为 (0, b)；x-截距出现在 y = 0 处。

Vertex: The turning point of a parabola, either its maximum or minimum. For y = a(x − h)² + k, the vertex is (h, k).

顶点：抛物线的转折点，即最大值或最小值点。对于 y = a(x − h)² + k，顶点为 (h, k)。

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6. Coordinate Geometry | 坐标几何

These terms help you navigate the xy-plane. Distance, midpoint, and slope formulas are frequently needed in ACT geometry problems.

这些术语帮助你在xy平面上定位。距离、中点和斜率公式在ACT几何题中经常出现。

Coordinate plane: A two-dimensional surface defined by a horizontal x-axis and a vertical y-axis intersecting at the origin (0, 0).

坐标平面：由水平x轴和垂直y轴相交于原点 (0, 0) 所形成的二维平面。

Ordered pair: A pair of numbers (x, y) that give the location of a point on the plane. The first number is the x-coordinate, the second is the y-coordinate.

有序对：表示平面上点位置的一对数 (x, y)。第一个数为x坐标，第二个为y坐标。

Midpoint: The point exactly halfway between two points. Its coordinates are the averages: ((x₁+x₂)/2, (y₁+y₂)/2).

中点：两点正中间的点。坐标为平均值：((x₁+x₂)/2, (y₁+y₂)/2)。

Distance formula: Derived from the Pythagorean theorem, d = √[(x₂−x₁)² + (y₂−y₁)²] gives the length between two points.

距离公式：由勾股定理导出，d = √[(x₂−x₁)² + (y₂−y₁)²] 可计算两点间的长度。

Slope formula: slope m = (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

斜率公式：斜率 m = (y₂ − y₁)/(x₂ − x₁)。平行线斜率相等；垂直线斜率互为负倒数。

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7. Plane Geometry | 平面几何

Shapes, angles, and the Pythagorean theorem form the backbone of ACT geometry. Memorizing area and circumference formulas can save time.

图形、角度和勾股定理构成了ACT几何的基础。记忆面积和周长公式能节省时间。

Angle: The figure formed by two rays sharing a common endpoint. Measured in degrees; right angle = 90°, acute < 90°, obtuse > 90° but < 180°.

角：由两条共端点的射线构成的图形。以度计量，直角=90°，锐角<90°，钝角>90°但<180°。

Parallel lines: Lines in a plane that never intersect. They are marked with arrow symbols and have the same slope when graphed.

平行线：平面上永不相交的直线。常以箭头标记，在图像中具有相同斜率。

Perpendicular lines: Lines that intersect at a right angle (90°). Their slopes multiply to −1 if neither is vertical.

垂直线：相交成直角（90°）的直线。若非垂直，则斜率之积为−1。

Pythagorean theorem: a² + b² = c², where c is the hypotenuse of a right triangle. Applies to find side lengths in many ACT questions.

勾股定理：a² + b² = c²，其中c为直角三角形的斜边。ACT许多问题用它求边长。

Area: The amount of space inside a 2D shape. Rectangle area = length × width; triangle area = ½ × base × height; circle area = πr².

面积：二维图形内部的空间量。矩形面积=长×宽；三角形面积=½×底×高；圆面积=πr²。

Circumference: The distance around a circle, given by 2πr or πd. When the radius is known, this formula is tested with word problems.

周长：圆的周线距离，公式为2πr或πd。已知半径时，应用题常考查该公式。

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8. Solid Geometry & Measurement | 立体几何与测量

Problems about 3D shapes often ask for volume or surface area. Familiarize yourself with the standard formulas and units.

三维图形题常要求计算体积或表面积。请熟悉标准公式和单位。

Volume: The amount of space a 3D object occupies. Rectangular solid: V = lwh; cylinder: V = πr²h; cone: V = ⅓πr²h; sphere: V = ⁴⁄₃πr³.

体积：三维物体占据的空间量。长方体：V=lwh；圆柱：V=πr²h；圆锥：V=⅓πr²h；球体：V=⁴⁄₃πr³。

Surface area: The total area of all faces or surfaces of a solid. For a rectangular prism, SA = 2(lw + lh + wh).

表面积：立体所有面的总面积。长方体表面积 SA = 2(lw + lh + wh)。

Prism: A solid with two congruent parallel bases and rectangular sides. Volume = base area × height.

棱柱：有两个全等平行底面和矩形侧面的立体。体积=底面积×高。

Cylinder: A solid with two parallel circular bases. Often appears in real-world measurement questions.

圆柱：有两个平行圆形底面的立体。常出现于实际测量问题中。

Units: Square units (cm², in²) for area, cubic units (cm³, in³) for volume. Converting between units is a common trap.

单位：面积用平方单位（cm², in²），体积用立方单位（cm³, in³）。单位换算是一个常见陷阱。

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9. Trigonometry | 三角学

ACT trigonometry focuses on right triangles and basic ratios. Only a few definitions are needed to handle most questions.

ACT三角学侧重直角三角形和基本比率。掌握少数定义即可应对大多数问题。

Sine (sin): In a right triangle, sin θ = opposite / hypotenuse. Used to find unknown sides or angles.

正弦 (sin)：直角三角形中，sin θ = 对边/斜边。用于求未知边或角。

Cosine (cos): cos θ = adjacent / hypotenuse. cos 0° = 1.

余弦 (cos)：cos θ = 邻边/斜边。cos 0° = 1。

Tangent (tan): tan θ = opposite / adjacent. Also equals sin θ / cos θ.

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