Grade 8 Math Unit 1 Answer Guide: Number and Algebra Essentials | 八年级数学第一单元答案解析：数与代数基础

📚 Grade 8 Math Unit 1 Answer Guide: Number and Algebra Essentials | 八年级数学第一单元答案解析：数与代数基础

This answer guide covers the core topics in Grade 8 Mathematics Unit 1: number operations, fractions, decimals, percentages, roots, algebraic expressions, linear equations, ratio, sequences, and coordinate graphs. Each section provides worked explanations, common errors, and exam-style tips to help you check your answers and build strong foundations.

本答案解析涵盖八年级数学第一单元的核心主题：整数运算、分数、小数、百分数、方根、代数表达式、一次方程、比、数列和坐标图像。每节提供详细解题说明、常见错误与考试技巧，帮助你核对答案并打牢基础。

1. Integers and Order of Operations | 整数与运算顺序

Integer questions in Unit 1 often test negative signs and the correct order of operations. Always follow BIDMAS: Brackets, Indices, Division and Multiplication from left to right, Addition and Subtraction from left to right.

第一单元的整数题常考负号与运算顺序。务必遵循 BIDMAS 规则：先算括号，再算指数，然后从左到右进行乘除，最后从左到右进行加减。

• Example: Evaluate -6 + 4 × (-2) ÷ 2 – 3².
• 示例：计算 -6 + 4 × (-2) ÷ 2 – 3²。

First calculate the index: 3² = 9. Then multiplication and division from left to right: 4 × (-2) = -8, then -8 ÷ 2 = -4. The expression becomes -6 + (-4) – 9 = -19.

先算指数：3² = 9。然后从左到右计算乘除：4 × (-2) = -8，再 -8 ÷ 2 = -4。表达式变为 -6 + (-4) – 9 = -19。

When subtracting a negative number, remember that two negative signs make a positive: a – (-b) = a + b. For example, 7 – (-3) = 7 + 3 = 10.

减去负数时，记住两个负号变为正号：a – (-b) = a + b。例如 7 – (-3) = 7 + 3 = 10。

2. Rational Numbers and Fraction Arithmetic | 有理数与分数运算

Rational numbers include all integers, fractions, and terminating or recurring decimals. To add or subtract fractions, first find the lowest common denominator, then combine the numerators.

有理数包括所有整数、分数以及有限小数或循环小数。要加减分数，先找最小公分母，再合并分子。

2/3 + 1/4 = 8/12 + 3/12 = 11/12

For multiplication, multiply numerators together and denominators together. For division, multiply by the reciprocal of the second fraction.

分数乘法将分子相乘、分母相乘。分数除法乘以第二个分数的倒数。

5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4

Always simplify your final answer. If the question asks for a mixed number, convert improper fractions such as 5/4 to 1 1/4.

最终答案一定要化简。如果题目要求写成带分数，将假分数如 5/4 转化为 1 1/4。

3. Decimals, Percentages and Conversions | 小数、百分数与互化

Converting between fractions, decimals, and percentages is a key Unit 1 skill. To change a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100.

分数、小数和百分数之间的互化是第一单元的重要技能。分数化小数：用分子除以分母。小数化百分数：乘以 100。

Fraction	Decimal	Percentage
1/2	0.5	50%
3/4	0.75	75%
1/5	0.2	20%

To find a percentage of a quantity, multiply the quantity by the percentage written as a decimal. For example, 15% of 240 = 0.15 × 240 = 36.

求某个量的百分数，将该量乘以百分数写成的小数。例如，240 的 15% = 0.15 × 240 = 36。

Percentage increase or decrease uses the formula: change ÷ original × 100. If a price rises from 80 to 100, the percentage increase is (100 – 80) ÷ 80 × 100 = 25%.

百分比增减使用公式：变化量 ÷ 原值 × 100。如果价格从 80 涨到 100，则增长百分比为 (100 – 80) ÷ 80 × 100 = 25%。

4. Squares, Cubes and Roots | 平方、立方与方根

A square number is the result of multiplying a number by itself, such as 4² = 16. A cube number is the result of multiplying a number by itself twice, such as 3³ = 27.

平方数是一个数乘以自身的结果，如 4² = 16。立方数是一个数乘以自身两次的结果，如 3³ = 27。

Square roots and cube roots reverse these operations. For example, √49 = 7 because 7² = 49, and ∛64 = 4 because 4³ = 64.

平方根与立方根是这些运算的逆运算。例如 √49 = 7，因为 7² = 49；∛64 = 4，因为 4³ = 64。

You should memorise square numbers up to 15² and cube numbers up to 5³. This helps you spot roots quickly and estimate values such as √50, which lies between √49 = 7 and √64 = 8.

需要熟记 15 以内的平方数和 5 以内的立方数。这有助于快速识别方根并估计数值，如 √50 介于 √49 = 7 与 √64 = 8 之间。

5. Algebraic Expressions and Simplification | 代数表达式与化简

In algebra, like terms have exactly the same variable part. Only like terms can be added or subtracted. For example, 3x + 5x = 8x, but 3x + 5y cannot be simplified further.

在代数中，同类项具有完全相同的变量部分。只有同类项才能相加或相减。例如 3x + 5x = 8x，但 3x + 5y 不能进一步化简。

When multiplying terms, multiply the numbers and the variables separately. For example, 4x × 2y = 8xy, and 3a² × 2a = 6a³.

项相乘时，分别将数字与变量相乘。例如 4x × 2y = 8xy，3a² × 2a = 6a³。

Expanding brackets uses the distributive law: a(b + c) = ab + ac. For example, 5(2x – 3) = 10x – 15.

展开括号使用分配律：a(b + c) = ab + ac。例如 5(2x – 3) = 10x – 15。

Factorising is the reverse of expanding. To factorise 6x + 9, take out the highest common factor 3, giving 3(2x + 3).

因式分解是展开的逆运算。对 6x + 9 因式分解，提取最大公因数 3，得到 3(2x + 3)。

6. Solving Linear Equations | 解一元一次方程

To solve a linear equation, perform the same operation on both sides to isolate the variable. For two-step equations, undo addition or subtraction first, then undo multiplication or division.

解一元一次方程时，对等式两边执行相同运算以分离变量。对于两步方程，先消去加减，再消去乘除。

Solve: 5x + 3 = 23

Subtract 3 from both sides: 5x = 20. Then divide both sides by 5: x = 4.

两边同时减 3：5x = 20。然后两边同时除以 5：x = 4。

Equations with variables on both sides require collecting like terms first. For example, 7x – 4 = 3x + 8 becomes 4x = 12 after subtracting 3x and adding 4, so x = 3.

变量在等式两边的方程需要先合并同类项。例如 7x – 4 = 3x + 8，先减 3x 再加 4，得到 4x = 12，所以 x = 3。

Always check your solution by substituting it back into the original equation. A correct answer makes both sides equal.

始终将解代入原方程检验。正确答案应使等式两边相等。

7. Ratio and Proportion | 比与比例

Ratios compare two or more quantities in the same unit. To simplify a ratio, divide all parts by their greatest common factor. For example, 12:18 simplifies to 2:3.

比用于比较两个或更多同单位的量。化简比时，将所有部分除以最大公因数。例如 12:18 化简为 2:3。

To share a quantity in a given ratio, find the total number of parts first, then divide the quantity by that total. For example, share £120 in the ratio 3:5.

按给定比分配数量时，先求总份数，再用数量除以总份数。例如，按 3:5 分配 120 英镑。

Total parts = 3 + 5 = 8. One part = 120 ÷ 8 = 15. The shares are 3 × 15 = 45 and 5 × 15 = 75.

总份数 = 3 + 5 = 8。一份 = 120 ÷ 8 = 15。分配结果为 3 × 15 = 45 和 5 × 15 = 75。

Direct proportion means two quantities increase or decrease at the same rate. If 4 pens cost £6, then 10 pens cost 10/4 × 6 = £15.

正比例意味着两个量以相同速率增加或减少。如果 4 支笔花费 6 英镑，那么 10 支笔花费 10/4 × 6 = 15 英镑。

8. Number Patterns and Sequences | 数字规律与数列

A sequence is a list of numbers formed according to a rule. Arithmetic sequences have a constant difference between consecutive terms, called the common difference.

数列是按一定规则排列的一组数。等差数列的相邻项之间差恒定，称为公差。

For example, in the sequence 4, 9, 14, 19, the common difference is 5. The next term is 19 + 5 = 24.

例如，数列 4, 9, 14, 19 的公差为 5。下一项为 19 + 5 = 24。

The nth term of an arithmetic sequence can be written as a + (n – 1)d, where a is the first term and d is the common difference. For the sequence above, the nth term is 4 + (n – 1) × 5 = 5n – 1.

等差数列的第 n 项可表示为 a + (n – 1)d，其中 a 为首项，d 为公差。上述数列的第 n 项为 4 + (n – 1) × 5 = 5n – 1。

Linear sequences have an nth term in the form an + b. To find it, use the common difference as the coefficient of n, then adjust the constant to match the first term.

线性数列的第 n 项形式为 an + b。求法：将公差作为 n 的系数，再调整常数项使首项匹配。

9. Coordinate Plane and Linear Graphs | 坐标平面与线性图像

The coordinate plane uses two perpendicular axes: the x-axis horizontal and the y-axis vertical. A point is written as (x, y), where x is the horizontal distance and y is the vertical distance from the origin.

坐标平面使用两条垂直的数轴：x 轴水平，y 轴垂直。点的坐标写作 (x, y)，其中 x 表示到原点的水平距离，y 表示垂直距离。

Linear graphs represent equations of the form y = mx + c, where m is the gradient and c is the y-intercept. The gradient measures steepness: rise over run.

线性图像表示形如 y = mx + c 的方程，其中 m 为斜率，c 为 y 轴截距。斜率衡量倾斜程度：纵向变化量 ÷ 横向变化量。

To plot y = 2x + 1, create a table of values for x from -2 to 2, calculate y, plot the points, then draw a straight line through them.

绘制 y = 2x + 1 时，为 x 从 -2 到 2 建立数值表，计算 y，描点后用直线连接。

The gradient between two points (x₁, y₁) and (x₂, y₂) is (y₂ – y₁) ÷ (x₂ – x₁). A positive gradient slopes upward, a negative gradient slopes downward.

两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率为 (y₂ – y₁) ÷ (x₂ – x₁)。正斜率向上倾斜，负斜率向下倾斜。

10. Common Errors and Exam Tips | 常见错误与应试技巧

Many marks are lost through small mistakes. Watch out for sign errors when subtracting negatives, and always write each step clearly.

许多分数因小错误而丢失。减去负数时要当心符号错误，并始终清晰地写出每一步。

• Do not confuse -3² with (-3)². -3² = -9, but (-3)² = 9.
• 不要混淆 -3² 与 (-3)²。-3² = -9，但 (-3)² = 9。
• When simplifying fractions, divide by the greatest common factor, not just any common factor.
• 化简分数时，除以最大公因数，而不仅仅是任意公因数。
• In ratio problems, always find the total number of parts before dividing.
• 在比的问题中，一定要先求总份数再分配。
• Check algebraic solutions by substituting back into the original equation.
• 通过将解代回原方程来检验代数答案。

In the exam, show all working. Even if the final answer is wrong, method marks can be awarded for correct steps. Time management is also crucial: spend about one minute per mark.

考试时要展示所有解题过程。即使最终答案错误，正确步骤也可获得方法分。时间管理也很关键：大约每分得分点用一分钟。

Regular practice with past papers and answer guides like this one will improve accuracy and speed. Review your mistakes and keep a list of errors to avoid.

使用真题和类似本答案解析的材料进行定期练习，可提高准确性和速度。复习错题并列出需要避免的错误清单。

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