小学数学五年级知识点中英文双语对照

五年级数学核心知识点精要 掌握五大模块，轻松玩转数学！统计与概率教你分析数据（平均数、可能性大小），图形与几何探索立方体与平面图形面积，方程与代数从字母表示数到解简易方程，数与运算涵盖分数、小数及数的整除概率定性描述事件发生几率，平均数反映数据集中趋势系统学习，夯实基础！

小学数学五年级知识点/ Core Math Topics for Grade 5

• 1. 数与运算/ Numbers and Operations
  • 1.1. 小数的运算/Decimal Operations
    • 1.1.1. 小数乘法/Decimal Multiplication
      • 规则/Rule：先按整数乘法算出积，再看因数中一共有几位小数，就从积的右边起数出几位，点上小数点/Multiply as whole numbers first. Then, the number of decimal places in the product is equal to the total number of decimal places in the factors.
      • 积的近似值/Approximation of Products ：用“四舍五入”法保留指定的小数位数/Use **rounding** to keep a specified number of decimal places
    • 1.1.2. 小数除法/ Decimal Division
      • 除数是整数/Divisor is a whole number：按整数除法的方法除，商的小数点要和被除数的小数点对齐/Divide as whole numbers. Align the decimal point in the quotient with the decimal point in the dividend
      • 除数是小数/ Divisor is a decimal：利用商不变的性质，把除数变成整数再计算/Use the property of constant quotients to convert the divisor to a whole number.
      • 商的近似值/ Approximation of Quotients：根据要求用“四舍五入”法保留指定的小数位数/Use **rounding** to keep a specified number of decimal places.
    • 1.1.3. 循环小数/Repeating Decimals
      • 一个数的小数部分，从某一位起，一个数字或者几个数字依次不断重复出现/A decimal in which a digit or a sequence of digits repeats indefinitely.
  • 1.2. 数的整除/Divisibility
    • 1.2.1. 因数与倍数/ Factors and Multiples
      • 如果a÷b=c（a, b, c都是自然数)，那么b和c是a的因数，a是b和c的倍数/If a ÷ b = c (a, b, c are natural numbers), then b and c are factors of a, and a is a multiple of b and c
    • 1.2.2. 2,3,5的倍数特征/Divisibility Rules
      • 2的倍数：个位是0,2,4,6,8/ Multiples of 2: last digit is 0,2,4,6,8.
      • 5的倍数：个位是0或5/Multiples of 5: last digit is 0 or 5.
      • 3的倍数：各位上数字的和是3的倍数/Multiples of 3: the sum of the digits is divisible by 3.
    • 1.2.3. 素数(质数)与合数/Prime and Composite Numbers
      • 素数/质数/Prime Number：一个大于1的自然数，除了1和它本身不再有别的因数(e.g., 2, 3, 5, 7...)
      • 合数/Composite Number：一个大于1的自然数，除了1和它本身还有别的因数(e.g., 4, 6, 8, 9...)
      • 1：既不是素数，也不是合数/1 is neither prime nor composite
    • 1.2.4. 公因数与公倍数/ Common Factors
      • 公因数/Common Factors：几个数公有的因数/Factors shared by two or more numbers.
      • 最大公因数/Greatest Common Factor (GCF)：公因数中最大的一个/The largest common factor.
      • 公倍数/Common Multiples：几个数公有的倍数/Multiples shared by two or more numbers.
      • 最小公倍数/ Least Common Multiple (LCM)：公倍数中最小的一个/The smallest common multiple.
      • 短除法/ Short Division：一种用于分解质因数和求最大公因数/最小公倍数的算法/An algorithm for prime factorization and finding GCF/LCM
  • 1.3. 分数 | Fractions
    • 1.3.1. 分数的意义/Meaning of Fractions: 把单位“1”平均分成若干份，表示这样的一份或几份的数/Represents parts of a whole, where the whole is divided into equal parts.
    • 1.3.2. 分数与除法 | Fractions and Division: 被除数 ÷ 除数 = 被除数/除数 (除数≠0) | a ÷ b = a/b (b ≠ 0)
    • 1.3.3. 分数的基本性质 | Fundamental Property of Fractions: 分数的分子和分母同时乘或除以同一个不为零的数，分数的大小不变/ Multiplying or dividing both the **numerator** and **denominator** by the same non-zero number does not change the value of the fraction.
    • 1.3.4. 约分 | Reducing/Simplifying Fractions: 利用分数的基本性质，把一个分数化成同它相等，但分子、分母都比较小的分数/ Dividing the numerator and denominator by their GCF.
    • 1.3.5. 通分 | Finding a Common Denominator: 把异分母分数分别化成和原来分数相等的同分母分数/ Converting fractions to equivalent fractions with a common denominator.
    • 1.3.6. 分数加减法 | Addition and Subtraction of Fractions:
      • 同分母 | With Common Denominators: 分母不变，分子相加减。| Keep the denominator, add/subtract the numerators.
      • 异分母 | With Different Denominators: 先通分，再按同分母分数加减法的法则计算/First, find a common denominator, then proceed as above.
• 2. 方程与代数/ Equations and Algebra
  • 2.1. 用字母表示数/Using Letters for Numbers
    • 2.1.1. 表示运算定律，如加法交换律 a + b = b + a / Commutative Property of Addition
    • 2.1.2. 表示计算公式，如长方形周长 C = 2(a + b) / Perimeter of a Rectangle
  • 2.2. 简易方程/Simple Equations
    • 2.2.1. 方程：含有未知数的等式/ Equation:A mathematical statement that shows that two expressions are equal, containing an **unknown** (variable).
    • 2.2.2. 等式的性质/Properties of Equality
      • 等式两边同时加上或减去同一个数，等式仍然成立/If you add or subtract the same number from both sides, the equality holds.
      • 等式两边同时乘或除以同一个不为零的数，等式仍然成立/If you multiply or divide both sides by the same non-zero number, the equality holds.
    • 2.2.3. 解方程：利用等式的性质求方程中未知数的值的过程/ Solving Equations:The process of finding the value of the unknown that makes the equation true.
    • 2.2.4. 列方程解应用题/Writing Equations to Solve Word Problems
      • 找出等量关系/Find the **relationship**
      • 设未知数为x/Let the unknown be x.
      • 列出方程/Write the equation
      • 解方程/Solve it
      • 检验/Check.
• 3. 图形与几何/ Shapes and Geometry
  • 3.1. 平面图形的面积/Area of Plane Shapes
    • 3.1.1. 平行四边形/Parallelogram
      • 公式：S = a × h / Formula: S = a × h
      • 解释：底 × 高 / Explanation: base × height (垂直距离 perpendicular distance)
    • 3.1.2. 三角形/ Triangle
      • 公式：S = a × h ÷ 2 / Formula: S = a × h ÷ 2
      • 解释：底 × 高 ÷ 2 / Explanation: (base × height) ÷ 2
    • 3.1.3. 梯形/Trapezoid
      • 公式：S = (a + b × h ÷ 2 / Formula: S = (a + b) × h ÷ 2
      • 解释：(上底 + 下底) × 高 ÷ 2 / Explanation: (**base1** + **base2**) × height ÷ 2
    • 3.1.4. 组合图形/Composite Figures
      • 通过“割”(分割 - **partitioning**)或“补”(补充 - **complementing**)转化为基本图形计算
  • 3.2. 立方体/Cuboids & Cubes
    • 3.2.1. 认识：面、棱、顶点/ Introduction: 面 (**-face**), 棱 (**edge**), 顶点 (**vertex**).
    • 3.2.2. 表面积：所有面的总面积/Surface Area: The total area of all faces.
      • 长方体/ Cuboid
        • 公式：面积=(长×宽 + 长×高 + 宽×高)×2 / S = 2(ab + ah + bh)
      • 正方体/ Cube
        • 公式：面积=6 × (棱长 × 棱长 )/ S = 6a²
    • 3.2.3. 体积/Volume: 物体所占空间的大小/The amount of 3D space an object occupies.
      • 长方体/Cuboid：V = a × b × h（length× width × height）
      • 正方体/Cube：V = a³(edge × edge× edge)
    • 3.2.4. 容积/Capacity: 容器所能容纳物体的体积 / The amount a container can hold.
    • 3.2.5. 单位/Unit
      • 体积单位/ Units of Volume: 立方米 (m³, cubic meter), 立方分米 (dm³, cubic decimeter), 立方厘米 (cm³, cubic centimeter)
      • 容积单位/Units of Capacity: 升 (L, Liter), 毫升 (mL, Milliliter)
      • 换算/ Conversion: 1 L = 1 dm³, 1 mL = 1 cm³, 1 L = 1000 mL
• 4. 统计与概率/ Statistics and Probability
  • 4.1. 平均数/Average (Mean)
    • 4.1.1. 计算：总数 ÷ 份数 = 平均数/Calculation: Sum of all values ÷ Number of values = Mean
    • 4.1.2. 统计意义：代表一组数据的平均水平，是数据的集中趋势的一种度量/Meaning: Represents the average value of a data set, a measure of **central tendency**.
  • 4.2. 可能性/Probability
    • 4.2.1. 确定性描述/Certainty Description
      • 一定：事件一定会发生/ Certain: The event will definitely happen. (Probability = 1)
      • 可能：事件可能发生，也可能不发生/Possible: The event might happen. (0 < Probability < 1)
      • 不可能：事件根本不可能发生/ Impossible: The event will never happen. (Probability = 0)
    • 4.2.2. 可能性大小/ Likelihood：定性描述事件发生的 chance（几率）