Grade 7 Math Theme Series - 200 | 7年级数学主题系列-200

📚 Grade 7 Math Theme Series – 200 | 7年级数学主题系列-200

Welcome to the Grade 7 Math Theme Series. This article focuses on 200 essential practice points that help you consolidate core skills: integers, fractions, decimals, percentages, ratios, algebra, geometry, and statistics. Each theme is presented as a short, exam-ready review.

欢迎来到7年级数学主题系列。本文聚焦200个核心练习要点，帮助你巩固整数、分数、小数、百分数、比、代数、几何与统计等关键技能。每个主题都以简短的、面向考试的形式呈现。

1. Integers and the Number Line | 整数与数轴

Integers include positive numbers, negative numbers, and zero. On a number line, numbers to the right are greater than numbers to the left. For example, −3 is less than −1 because −3 lies further left.

整数包括正数、负数和零。在数轴上，右边的数大于左边的数。例如，−3 小于 −1，因为 −3 位于更左边。

−5 < −2 < 0 < 3 < 7

When adding a negative number, move left on the number line. When subtracting a negative number, move right. Multiplication of two negatives gives a positive product.

加负数时，在数轴上向左移动；减负数时，向右移动。两个负数相乘得到正数。

Rule: same signs multiply to positive; different signs multiply to negative.

规则：同号相乘得正；异号相乘得负。
Example: (−4) × (−6) = 24

示例：(−4) × (−6) = 24

2. Order of Operations | 运算顺序

Always follow BIDMAS / PEMDAS: Brackets, Indices, Division and Multiplication, Addition and Subtraction. Work from left to right for operations of equal priority.

始终遵循括号、指数、乘除、加减的顺序。同级运算从左到右依次进行。

2 + 3 × 4² = 2 + 3 × 16 = 2 + 48 = 50

Notice that if you simply work left to right without following the order, you get a different wrong answer: (2 + 3) × 4² = 5 × 16 = 80.

注意，如果不按顺序而直接从左到右计算，会得到错误答案：(2+3)×4² = 5×16 = 80。

Inside brackets, apply the same rules.

括号内同样遵循此顺序。
Example: (6 + 2) ÷ (4 − 2) = 8 ÷ 2 = 4

示例：(6+2)÷(4−2) = 8÷2 = 4

3. Fractions: Equivalent and Simplified | 分数：等价与最简

Two fractions are equivalent if they represent the same part of a whole. To simplify a fraction, divide both numerator and denominator by their greatest common factor (GCF).

两个分数如果表示同一个整体的相同部分，则它们等价。化简分数时，用分子和分母的最大公因数 (GCF) 同时除以它们。

18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4

To add or subtract fractions, first rewrite them with a common denominator.

分数加减时，先通分，化为同分母。

Example: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2

示例：1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2
To multiply fractions, multiply numerators and denominators directly.

分数乘法：分子乘分子，分母乘分母。

4. Decimals and Place Value | 小数与位值

Every digit in a decimal has a place value: tenths, hundredths, thousandths, and so on. The first digit after the decimal point is tenths, the second is hundredths.

小数中的每个数字都有位值：十分位、百分位、千分位等。小数点后第一位是十分位，第二位是百分位。

0.375 = 3/10 + 7/100 + 5/1000 = 375/1000

When multiplying decimals, count the total number of decimal places in both factors and place the decimal point accordingly.

小数相乘时，先数两个因数中小数位数的总和，再在结果中确定小数点位置。

Example: 0.4 × 0.2 = 0.08

示例：0.4 × 0.2 = 0.08
Dividing by 10, 100, or 1000 moves the decimal point left by 1, 2, or 3 places.

除以10、100或1000时，小数点分别向左移动1、2或3位。

5. Percentages and Proportion | 百分数与比例

A percentage is a fraction out of 100. To find a percentage of a quantity, multiply the quantity by the decimal equivalent of the percentage.

百分数是分母为100的分数。求一个量的百分数时，用该量乘以百分数对应的小数。

15% of 80 = 0.15 × 80 = 12

To express one quantity as a percentage of another, divide the first by the second, then multiply by 100.

将一个量表示为另一个量的百分数时，用第一个量除以第二个量，再乘以100。

Example: 18 out of 50 = 18 ÷ 50 × 100% = 36%

示例：18÷50 = 18÷50×100% = 36%
Percentage increase or decrease = (change ÷ original) × 100%.

百分数增长或减少 = (变化量 ÷ 原量) × 100%。

6. Ratio and Direct Proportion | 比与正比例

A ratio compares two or more quantities of the same kind. Simplify a ratio by dividing all parts by their common factor.

比用来比较同类的两个或多个量。用公因数除比的每一项来化简比。

24 : 36 = 2 : 3

In direct proportion, two quantities increase or decrease at the same rate. If y is directly proportional to x, then y = kx for some constant k.

在正比例关系中，两个量以相同的速率同时增加或减少。若 y 与 x 成正比，则 y = kx，其中 k 是常数。

Example: If 3 pens cost $6, then 5 pens cost $10.

示例：3支笔6元，则5支笔10元。
Use the unitary method: find the value of one item first.

使用归一法：先求一个物品的价值。

7. Algebraic Expressions and Like Terms | 代数式与同类项

A variable is a symbol that represents an unknown number. Algebraic expressions combine numbers and variables using operations. Like terms have exactly the same variable part.

变量是表示未知数的符号。代数式通过运算结合数字与变量。同类项具有完全相同的变量部分。

3x + 2y − x + 5y = (3x − x) + (2y + 5y) = 2x + 7y

To evaluate an expression, substitute the given value for the variable.

求代数式的值时，用给定数值替换变量。

Example: evaluate 4a − 3b when a = 2 and b = 1 gives 8 − 3 = 5.

示例：当 a=2, b=1 时，4a−3b = 8−3 = 5。
Never combine different variables as one term.

不要将不同变量合并为一项。

8. Solving Linear Equations | 解线性方程

To solve an equation, do the same operation on both sides until the variable is alone. Always check your solution by substituting it back.

解方程时，两边进行相同运算，直到变量单独在一边。最后将解代回原方程检验。

2x + 5 = 17 → 2x = 12 → x = 6

If the variable appears on both sides, first gather the variable terms on one side and constants on the other.

若两边都有变量，先将变量项移到一边，常数项移到另一边。

Example: 3x − 2 = x + 6 → 2x = 8 → x = 4

示例：3x−2 = x+6 → 2x = 8 → x = 4
Check: 3(4) − 2 = 10, 4 + 6 = 10 ✓

检验：3(4)−2 = 10，4+6 = 10 ✓

9. Geometry: Area and Perimeter | 几何：面积与周长

The perimeter is the total distance around a shape. The area is the amount of space inside a shape, measured in square units.

周长是图形外边界总长度。面积是图形内部空间的大小，用平方单位表示。

Rectangle	Area = length × width
Triangle	Area = ½ × base × height
Circle	Area = πr²

Perimeter of a rectangle = 2 × (length + width)

For a circle, circumference = 2πr. Use π ≈ 3.14 or 22/7 unless specified otherwise.

圆的周长 = 2πr。若无特别说明，取 π ≈ 3.14 或 22/7。

Example: rectangle 6 cm by 4 cm has area 24 cm² and perimeter 20 cm.

示例：长为6 cm、宽为4 cm的长方形面积为24 cm²，周长为20 cm。

10. Statistics: Mean, Median, Mode, Range | 统计：平均数、中位数、众数、极差

For a data set, the mean is the sum of values divided by the number of values. The median is the middle value when data are sorted. The mode is the most frequent value. The range is the difference between the largest and smallest values.

对于一组数据，平均数是数据总和除以数据个数。中位数是排序后的中间值。众数是出现最多的值。极差是最大值与最小值之差。

Data: 4, 7, 7, 9, 13 → Mean = 40 ÷ 5 = 8; Median = 7; Mode = 7; Range = 9

If there is an even number of values, the median is the average of the two middle values.

若数据有偶数个，中位数取中间两个数的平均值。

Example: 2, 5, 8, 9 → median = (5+8) ÷ 2 = 6.5

示例：2, 5, 8, 9 的中位数为 (5+8)÷2 = 6.5
Mean is affected by outliers; median is more stable.

平均数受异常值影响，中位数更稳定。

11. Angles and Parallel Lines | 角与平行线

When a transversal intersects two parallel lines, corresponding angles are equal, alternate angles are equal, and interior angles on the same side sum to 180°.

一条直线与两条平行线相交时，同位角相等，内错角相等，同旁内角互补，和为180°。

Angles on a straight line sum to 180°; angles around a point sum to 360°

A triangle has an interior angle sum of 180°. A quadrilateral has an interior angle sum of 360°.

三角形内角和为180°，四边形内角和为360°。

Example: if two angles of a triangle are 50° and 70°, the third is 60°.

示例：三角形两个角为50°和70°，则第三个角为60°。
Vertically opposite angles are equal.

对顶角相等。

12. Probability Basics | 概率基础

Probability measures how likely an event is to happen. It always lies between 0 and 1 inclusive. A probability of 0 means impossible, and 1 means certain.

概率衡量事件发生的可能性。它的值总在0到1之间（包含0和1）。概率为0表示不可能发生，为1表示必然发生。

P(event) = number of favourable outcomes ÷ total number of equally likely outcomes

If all outcomes are equally likely, the sum of the probabilities of all outcomes is 1.

若所有结果等可能，则所有结果概率之和为1。

Example: rolling a fair dice, P(odd number) = 3/6 = 1/2.

示例：掷一颗公平骰子，掷出奇数的概率为 3/6 = 1/2。
P(complement of A) = 1 − P(A).

事件A的对立事件概率为 1 − P(A)。

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Grade 7 Math Theme Series – 200 | 7年级数学主题系列-200