Mastering Directed Numbers | 掌握有向数

📚 Mastering Directed Numbers | 掌握有向数

Directed numbers, also known as integers or signed numbers, are numbers that carry a positive or negative sign. They appear everywhere in real life – from temperatures below zero to bank account overdrafts. At KS3, understanding how to add, subtract, multiply, and divide these numbers is a foundational skill for algebra and beyond.

有向数，也称为带符号数或正负数，是带有正号或负号的数字。它们在现实生活中随处可见——从零下的温度到银行账户透支。在 KS3 阶段，理解如何加减乘除这些数字是学习代数及其他内容的基础技能。

1. What Are Directed Numbers? | 什么是有向数？

Directed numbers are numbers with a sign (+ or –) in front of them. Positive numbers are greater than zero, while negative numbers are less than zero. Zero itself is neither positive nor negative. The sign tells us the direction on the number line: positive to the right, negative to the left.

有向数是前面带有正号或负号的数。正数大于零，负数小于零。零本身既不是正数也不是负数。符号告诉我们在数轴上的方向：正数向右，负数向左。

For example, +5 (or simply 5) means 5 steps right from 0, while –3 means 3 steps left from 0. The larger the negative number, the smaller its value. Thus, –10 is less than –2.

例如，+5（或简写为 5）表示从 0 向右走 5 步，而 –3 表示从 0 向左走 3 步。负数越大，其值越小。因此，–10 比 –2 小。

On a number line, every directed number has an opposite. The opposite of +4 is –4, and their sum is always zero. This concept is key when solving equations later.

在数轴上，每个有向数都有一个相反数。+4 的相反数是 –4，它们的和始终为零。这个观念对于以后解方程至关重要。

2. Adding Directed Numbers | 有向数的加法

To add directed numbers, think of moving along the number line. If you add a positive number, move to the right. If you add a negative number, move to the left.

要对有向数做加法，可以想象在数轴上移动。如果加上一个正数，向右移动；如果加上一个负数，向左移动。

For instance, 5 + (–3) means start at 5, then move 3 steps left, arriving at 2. That is the same as 5 – 3. When adding two negatives, such as (–2) + (–5), start at –2 and move 5 steps left, ending at –7.

例如，5 + (–3) 表示从 5 开始，向左移动 3 步，到达 2。这与 5 – 3 相同。当两个负数相加时，例如 (–2) + (–5)，从 –2 开始向左移动 5 步，到达 –7。

A useful pattern: adding a negative always results in a smaller number, while adding a positive results in a larger number.

一个有用的规律：加上一个负数总是得到更小的数，而加上一个正数得到更大的数。

Quick rules:

Positive + Positive = Positive (normal addition)
Negative + Negative = Negative (add the number parts, keep the negative sign)
Positive + Negative or Negative + Positive (subtract the smaller number part from the larger, keep the sign of the larger number part)

快速规则：

正 + 正 = 正（普通加法）
负 + 负 = 负（将数字部分相加，保留负号）
正 + 负或负 + 正（用较大的数字部分减去较小的数字部分，保留较大数字部分的符号）

3. Subtracting Directed Numbers | 有向数的减法

Subtracting a directed number can be turned into an addition problem: subtracting a number is the same as adding its opposite. This rule simplifies everything.

减去一个有向数可以转化为加法问题：减去一个数等于加上它的相反数。这条规则能简化一切。

For example, 4 – (–2) means 4 + (+2) = 6. Subtracting a negative makes the value go up. Another example: –3 – (+5) means –3 + (–5) = –8.

例如，4 – (–2) 等于 4 + (+2) = 6。减去一个负数会使数值增加。另一个例子：–3 – (+5) 等于 –3 + (–5) = –8。

So whenever you see two signs next to each other, replace them with a single sign using these rules:

+ (+) becomes +
+ (–) becomes –
– (+) becomes –
– (–) becomes +

因此，当你看到两个符号连在一起时，用以下规则将它们替换为一个符号：

+ (+) 变成 +
+ (–) 变成 –
– (+) 变成 –
– (–) 变成 +

These shortcuts are often remembered as: two like signs make a plus, two unlike signs make a minus.

这些速记常被记作：同号得正，异号得负。

4. Multiplication and Division Patterns | 乘法和除法的模式

Multiplying and dividing directed numbers follows a sign pattern. The rule is simple: if the signs are the same, the answer is positive; if the signs are different, the answer is negative.

有向数的乘法和除法遵循一个符号模式。规则很简单：同号得正，异号得负。

Examples:

(+3) × (+4) = +12 (positive × positive = positive)
(–3) × (–4) = +12 (negative × negative = positive)
(+3) × (–4) = –12 (positive × negative = negative)
(–3) × (+4) = –12 (negative × positive = negative)

例如：

(+3) × (+4) = +12（正 × 正 = 正）
(–3) × (–4) = +12（负 × 负 = 正）
(+3) × (–4) = –12（正 × 负 = 负）
(–3) × (+4) = –12（负 × 正 = 负）

The same rule applies to division: 12 ÷ (–3) = –4, and (–12) ÷ (–3) = +4. This pattern holds because division is the inverse of multiplication.

同样的规则适用于除法：12 ÷ (–3) = –4，而 (–12) ÷ (–3) = +4。这个模式成立是因为除法是乘法的逆运算。

Many students find it helpful to ignore the signs first, perform the multiplication or division with the number parts, then determine the sign of the result.

许多学生发现，先忽略符号，用数字部分进行乘除，再确定结果的符号会很有帮助。

5. Order of Operations with Directed Numbers | 含有向数的运算顺序

When directed numbers appear in longer expressions, you must follow the order of operations (BIDMAS/BODMAS): Brackets, Indices (powers), Division/Multiplication (left to right), Addition/Subtraction (left to right).

当有向数出现在较长的表达式中时，你必须遵循运算顺序（BIDMAS/BODMAS）：先括号，再指数（幂），然后乘除（从左到右），最后加减（从左到右）。

Consider the expression: –3 + 4 × (–2). First, multiply: 4 × (–2) = –8. Then add: –3 + (–8) = –11. If you ignore order, you might incorrectly add first and get a wrong answer.

考虑表达式：–3 + 4 × (–2)。首先计算乘法：4 × (–2) = –8。然后加法：–3 + (–8) = –11。如果忽略了顺序，先做加法，就会得到错误答案。

Brackets can change the meaning. For example, (–3 + 4) × (–2) gives (+1) × (–2) = –2. Always work inside brackets first.

括号可以改变含义。例如，(–3 + 4) × (–2) 得 (+1) × (–2) = –2。始终先计算括号内的部分。

Special care is needed with powers: (–2)² means (–2) × (–2) = +4, while –2² means –(2²) = –4. The brackets matter greatly.

需要特别注意幂运算：(–2)² 表示 (–2) × (–2) = +4，而 –2² 表示 –(2²) = –4。括号至关重要。

6. The Number Line and Absolute Value | 数轴与绝对值

The number line is the best visual tool for understanding directed numbers. Zero sits in the middle, positive numbers extend to the right, and negative numbers to the left. Distances on the number line represent absolute value.

数轴是理解有向数的最佳可视化工具。零位于中间，正数向右延伸，负数向左延伸。数轴上的距离代表绝对值。

The absolute value of a number is its distance from zero, always a non-negative quantity. It is written with vertical bars: |–7| = 7, |+4| = 4. Absolute value ignores the sign.

一个数的绝对值是它到零的距离，始终是一个非负数。用竖线表示：|–7| = 7，|+4| = 4。绝对值忽略符号。

Understanding absolute value helps with comparing negative numbers. For example, |–12| > |–5| means –12 is farther from zero, so –12 < –5.

理解绝对值有助于比较负数。例如，|–12| > |–5| 表示 –12 离零更远，所以 –12 < –5。

7. Real-World Applications: Temperature and Finance | 实际应用：温度和财务

Directed numbers model many everyday situations. Temperature changes: if it is –4°C and rises by 10°C, the new temperature is –4 + 10 = +6°C. A drop from 3°C by 5°C gives –2°C.

有向数为许多日常情景建模。温度变化：如果当前温度为 –4°C，升高 10°C，则新温度为 –4 + 10 = +6°C。从 3°C 下降 5°C 变为 –2°C。

In banking, a positive balance means you have money; a negative balance is an overdraft. If your account is at –£25 and you deposit £40, your new balance is –25 + 40 = +£15.

在银行业务中，正余额表示你有存款；负余额表示透支。如果你的账户是 –£25，你存入 £40，则新余额为 –25 + 40 = +£15。

Elevation above and below sea level is another example. A diver at –30 m ascending 10 m goes to –20 m.

海平面以上的海拔和海平面以下的深度是另一个例子。一名位于 –30 米的潜水员上升 10 米，会到达 –20 米。

These contexts make it easier to see why subtracting a negative yields an increase – paying off a debt increases your net worth.

这些情境使我们更容易理解为什么减去一个负数会导致增加——还清债务会增加你的净值。

8. Combining Addition and Subtraction without a Calculator | 无计算器下的加减混合运算

When you have a string of additions and subtractions of directed numbers, the best approach is to convert all subtractions into additions of the opposite. Then group all positives together and all negatives together, and finally combine.

当你面对一连串有向数的加减运算时，最好的方法是将所有减法转化为加相反数。然后分别将正数归为一组，负数归为一组，最后合并。

Example: (–6) + (+3) – (–2) – (+4) + (–1). First rewrite: –6 + 3 + 2 – 4 – 1. Now group positives: 3 + 2 = 5. Group negatives: –6 – 4 – 1 = –11. Final answer: 5 + (–11) = –6.

示例：(–6) + (+3) – (–2) – (+4) + (–1)。首先改写为：–6 + 3 + 2 – 4 – 1。然后正数归组：3 + 2 = 5；负数归组：–6 – 4 – 1 = –11。最终答案：5 + (–11) = –6。

This method reduces sign errors. It also reinforces the idea that the expression is just a sum of signed terms.

这个方法能减少符号错误。它也强化了表达式就是若干带符号项之和的概念。

9. Multiplying and Dividing by Negative One | 乘以负一和除以负一

Multiplying any number by –1 simply reverses its sign. So, 7 × (–1) = –7, and –7 × (–1) = +7. Division by –1 has the same effect: 5 ÷ (–1) = –5.

任何数乘以 –1 就改变其符号。因此，7 × (–1) = –7，而 –7 × (–1) = +7。除以 –1 效果相同：5 ÷ (–1) = –5。

This is useful when simplifying expressions or solving equations. For example, –x = 4 means x = –4, which is multiplying (or dividing) both sides by –1.

这在化简表达式或解方程时很有用。例如，–x = 4 意味着 x = –4，相当于两边同时乘以（或除以）–1。

The double negative rule in multiplication: –(–a) = a. So, the opposite of negative is positive.

乘法中的双重否定规则：–(–a) = a。因此，负数的相反数是正数。

10. Common Misconceptions and Pitfalls | 常见误解与易错点

Many learners think that adding two negatives makes a positive, confusing it with multiplication. Adding (–3) + (–4) is –7, not +7. Only in multiplication does negative × negative = positive.

许多学习者认为两个负数相加会得正数，与乘法混淆。 (–3) + (–4) 等于 –7，不是 +7。只有乘法中负负得正。

Another common error: –5² is often interpreted as (–5)² = 25, but unless brackets are present, –5² = –25. The exponent applies only to the 5, not the minus sign.

另一个常见错误：–5² 常被当作 (–5)² = 25，但除非有括号，否则 –5² = –25。指数仅作用于 5，不作用于负号。

When handling a long expression like –4 + (–3) × 2, students may add first. Always apply BIDMAS. The correct method: multiply first: –3 × 2 = –6, then –4 + (–6) = –10.

在处理诸如 –4 + (–3) × 2 这样的长表达式时，学生可能会先做加法。始终遵循 BIDMAS 规则。正确方法是先乘：–3 × 2 = –6，然后 –4 + (–6) = –10。

Misreading the subtraction of a negative is also frequent. –5 – (–3) looks complicated, but changing to –5 + 3 = –2 makes it simple.

误读减去一个负数也很常见。–5 – (–3) 看起来很复杂，但改成 –5 + 3 = –2 就简单了。

11. Practice Questions with Worked Solutions | 练习题与详细解答

Test your understanding with these examples:

用以下例子检验你的理解：

Simplify: (–8) + (+5) – (–3) – (+2).
Rewrite: –8 + 5 + 3 – 2 = (–8 – 2) + (5 + 3) = –10 + 8 = –2.
Calculate: (–3) × (–7) ÷ (–1).
Multiply first: (–3) × (–7) = +21. Then 21 ÷ (–1) = –21.
What is the value of (–4)² – (–3)³?
(–4)² = 16. (–3)³ = –27. So 16 – (–27) = 16 + 27 = 43.
If the temperature drops from 6°C by 11°C, what is the new temperature?
6 – 11 = –5°C.

化简：(–8) + (+5) – (–3) – (+2)。
改写为：–8 + 5 + 3 – 2 = (–8 – 2) + (5 + 3) = –10 + 8 = –2。
计算：(–3) × (–7) ÷ (–1)。
先乘：(–3) × (–7) = +21，然后 21 ÷ (–1) = –21。
求 (–4)² – (–3)³ 的值。
(–4)² = 16，(–3)³ = –27，所以 16 – (–27) = 16 + 27 = 43。
如果温度从 6°C 下降 11°C，新温度是多少？
6 – 11 = –5°C。

Always double-check your sign rules and operation order.

始终仔细核对你的符号规则和运算顺序。

12. Summary and Key Takeaways | 总结与关键要点

Directed numbers are a vital part of the KS3 mathematics curriculum. Mastering them requires confidence in the four operations and the ability to switch between visual, numerical, and real-world approaches.

有向数是 KS3 数学课程的重要组成部分。掌握它们需要对四则运算有信心，并能够在视觉、数字和现实世界的方法之间灵活切换。

Key takeaways:

Two signs become one: like signs give +, unlike signs give –.
Adding a negative means moving left; subtracting a negative means moving right.
Multiplication and division: same signs = positive, different signs = negative.
Always follow BIDMAS, especially with powers and brackets.
Use number lines and real-life contexts to build intuition.

关键要点：

双符号化简：同号得 +，异号得 –。
加上负数表示向左移动；减去负数表示向右移动。
乘法和除法：同号得正，异号得负。
始终遵循 BIDMAS 运算顺序，尤其注意幂和括号。
利用数轴和实际生活情境培养直觉。

With consistent practice, directed numbers will become second nature, paving the way for algebra, graphs, and advanced calculations.

通过持续练习，有向数将成为你的第二本能，为代数、图像和高级计算铺平道路。

Published by TutorHao | Cambridge KS3 Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mastering Directed Numbers | 掌握有向数