📚 Mastering Number Foundations: Place Value and Addition up to 100 for IGCSE | IGCSE 数学基础：100 以内的位值与加法
Number sense is the foundation of every topic in IGCSE Mathematics. Before extending arithmetic to fractions, decimals, percentages and algebra, students must be confident with how two-digit numbers are built from tens and ones. This unit reviews place value, counting strategies, mental addition and subtraction within 100, and simple word problems that appear in early IGCSE number work and calculator-free questions.
数感是 IGCSE 数学所有主题的基础。在把算术拓展到分数、小数、百分数和代数之前，学生必须熟悉两位数是如何由十位和个位组成的。本单元复习位值、计数策略、100 以内的口算加减法，以及 IGCSE 早期数字运算和非计算器题中常见的简单应用题。
1. Why Place Value Matters in IGCSE Mathematics | 为什么位值在 IGCSE 数学中重要
Place value is the rule that the position of a digit controls its value. In the number 36, the digit 3 is not just 3; it stands for 3 tens, or 30. The digit 6 stands for 6 ones, or 6. If you confuse these positions, every later calculation becomes risky, especially when regrouping in addition and subtraction.
位值是指数字所在的位置决定它的数值。在数字 36 中，数字 3 不只是 3，它代表 3 个十，即 30；数字 6 代表 6 个一，即 6。如果混淆这些位置，后面每一种计算都会变得危险，尤其是在加法和减法中需要进位或退位时。
IGCSE questions often ask students to identify place value, write numbers in expanded form, or compare two numbers using <, > and =. A firm understanding of tens and ones therefore supports both mental arithmetic and written methods across the Number strand.
IGCSE 考题经常要求学生辨认位值、用展开式写数字，或者用 <、> 和 = 比较两个数。因此，牢固掌握十位和个位能够支持数字运算部分中的口算和笔算方法。
2. Reading and Writing Two-Digit Numbers | 读写两位数
A two-digit number is made of a tens digit and a ones digit. For example, 58 is read as “fifty-eight”, not “five eight”. The digit 5 contributes 50, and the digit 8 contributes 8. Reading numbers in this grouped way reinforces the meaning of place value.
两位数由一个十位数字和一个各位数字组成。例如，58 读作 “fifty-eight”，而不是 “five eight”。数字 5 表示 50，数字 8 表示 8。以这种分组方式读数字可以强化位值的含义。
When writing numbers from words, listen carefully for the tens word and the ones word. “Sixty-three” means 60 + 3, so it is written as 63. “Seventy” means 7 tens and no extra ones, so it is written as 70. Numbers from 11 to 19 are exceptions because their words do not follow the usual tens-and-ones pattern.
根据英文单词写数字时，要仔细听十位词和个位词。”Sixty-three” 表示 60 + 3，所以写作 63。”Seventy” 表示 7 个十且没有额外的个位，所以写作 70。11 到 19 这些数字是例外，因为它们的英文单词不遵循通常的十位加个位规律。
3. Tens and Ones: The Core of Place Value | 十位与个位：位值的核心
Each two-digit number can be split into groups of ten and single ones. The number 47 means 4 groups of 10 plus 7 individual ones. Mathematically, this is written as:
每个两位数都可以拆成整十组和单个的个位。数字 47 表示 4 组十加上 7 个一。数学上可以写为：
47 = 4 × 10 + 7 × 1
This statement is useful because it connects the written digit to its value. The digit on the left is always multiplied by 10, while the digit on the right is multiplied by 1. It is the reason why 47 is greater than 7 even though 4 and 7 appear together.
这个等式很有用，因为它把写出的数字和它的值联系起来。左边的数字总是乘以 10，右边的数字总是乘以 1。这就是为什么 47 比 7 大，尽管 4 和 7 写在一起。
Base-ten blocks are often used to model this idea. One long block represents 10, and one small cube represents 1. For 47, you would draw or imagine 4 longs and 7 cubes. This visual model helps students avoid the common error of writing 607 when they mean 60 + 7 = 67.
十进制积木常用来模拟这个概念。一个长条表示 10，一个小立方体表示 1。对于 47，你会画出或想象 4 个长条和 7 个立方体。这种直观模型有助于学生避免常见的错误：把 60 + 7 = 67 误写成 607。
| Number | Tens | Ones |
|---|---|---|
| 34 | 3 | 4 |
| 80 | 8 | 0 |
| 16 | 1 | 6 |
4. Expanded Form and Standard Form | 展开式与标准式
Expanded form shows a number as the sum of the values of its digits. The standard form is the usual compact way of writing the number. For 72, the expanded form is 70 + 2. For 90 + 5, the standard form is 95.
展开式把一个数字表示为各位数值之和。标准式就是我们通常写数字的紧凑形式。对于 72，展开式是 70 + 2；对于 90 + 5，标准式是 95。
Being able to move between expanded form and standard form is an essential IGCSE skill. It is especially helpful when adding two two-digit numbers mentally, because you can add the tens and ones separately before combining them again.
能够在展开式和标准式之间转换是一项重要的 IGCSE 技能。在心算两个两位数相加时尤其有帮助，因为你可以先分别加上十位和个位，然后再合并起来。
Here is a table showing the same numbers in three ways:
下表以三种方式展示相同的数字：
| Standard form | Expanded form | Tens and ones |
|---|---|---|
| 46 | 40 + 6 | 4 tens, 6 ones |
| 93 | 90 + 3 | 9 tens, 3 ones |
| 25 | 20 + 5 | 2 tens, 5 ones |
5. Comparing Numbers Using <, > and = | 使用 <、> 和 = 比较数字
To compare two numbers, first compare the tens digits. If the tens digits are different, the number with the larger tens digit is larger. For example, 64 > 37 because 6 tens are more than 3 tens, even though 7 ones in 37 are more than 4 ones in 64.
比较两个数字时，先比较十位数字。如果十位数字不同，十位数字较大的那个数就较大。例如，64 > 37，因为 6 个十多于 3 个十，即使 37 中的 7 个一多于 64 中的 4 个一。
If the tens digits are equal, then compare the ones digits. For example, 58 < 59 because both have 5 tens, but 8 ones are fewer than 9 ones. The symbols are read as "less than", "greater than" and "equal to".
如果十位数字相同，就比较个位数字。例如，58 < 59，因为两者都有 5 个十，但 8 个一少于 9 个一。这些符号分别读作 "小于"、"大于" 和 "等于"。
45 < 54, 72 > 27, 60 = 60
On a number line or hundred chart, numbers increase as you move right and increase by 10 as you move down one row. This spatial pattern makes comparison easier to visualise.
在数轴或百数表上，数字向右移动时增大，向下移动一行时增大 10。这种空间模式使比较更容易可视化。
6. Counting On and Back in Steps | 按步长正数和倒数
Counting on means adding to a starting number. Counting back means subtracting from a starting number. In IGCSE foundation work, students are expected to count on and back by 1s, 2s, 5s and 10s from any number under 100.
正数是在起始数字上累加；倒数是从起始数字中递减。在 IGCSE 基础阶段，学生应能从 100 以内的任意数字开始，以 1、2、5、10 为步长正数和倒数。
For example, counting on by 5s from 35 gives 40, 45, 50, 55, 60. Counting back by 10s from 83 gives 73, 63, 53, 43, 33. Notice that when counting back by 10s from 83, the ones digit stays the same because only the number of tens changes.
例如，从 35 开始以 5 为步长正数得到 40、45、50、55、60。从 83 开始以 10 为步长倒数得到 73、63、53、43、33。注意，从 83 开始以 10 为步长倒数时，个位数字保持不变，因为只有十位在变化。
This pattern is used in mental arithmetic. Adding 10 to any two-digit number is the same as increasing its tens digit by 1, provided the sums do not exceed 99. Subtracting 10 is the same as decreasing its tens digit by 1.
这一规律用于口算。给任意两位数加 10，相当于把它的十位数字加 1，前提是和不超过 99。减 10 相当于把它的十位数字减 1。
7. Adding Tens and Ones Mentally | 心算加上整十数和个位数
To add a two-digit number to a one-digit number or a multiple of 10, split the calculation into tens and ones. For example, 52 + 30 is 5 tens plus 3 tens, giving 8 tens, and the 2 ones stay the same. Therefore 52 + 30 = 82.
把一个两位数与一个一位数或整十数相加时，可以把计算拆成十位和个位。例如，52 + 30 是 5 个十加 3 个十，得到 8 个十，2 个一保持不变。因此 52 + 30 = 82。
For adding a one-digit number, only the ones digit changes unless the sum of the ones crosses 10. For example, 52 + 4 = 56 because 2 ones + 4 ones = 6 ones. But 58 + 4 = 62 because 8 ones + 4 ones = 12 ones, which makes one extra ten and 2 ones.
加一位数时，只有个位数字改变，除非个位之和超过 10。例如，52 + 4 = 56，因为 2 个一加 4 个一等于 6 个一。但 58 + 4 = 62，因为 8 个一加 4 个一等于 12 个一，即多出一个十和 2 个一。
This crossing-ten skill is important for two-digit addition. An IGCSE non-calculator question may ask you to add 47 + 25 mentally. You can first add 40 + 20 = 60, then 7 + 5 = 12, and finally 60 + 12 = 72.
这种跨十的技能对两位数加法很重要。IGCSE 非计算器题可能会要求你心算 47 + 25。你可以先计算 40 + 20 = 60，再计算 7 + 5 = 12，最后计算 60 + 12 = 72。
47 + 25 = 40 + 20 + 7 + 5 = 60 + 12 = 72
8. Subtracting Tens and Ones Mentally | 心算减去整十数和个位数
To subtract a multiple of 10, only the tens digit changes. For example, 67 − 20 = 47 because 6 tens minus 2 tens equals 4 tens, and the 7 ones remain. Subtracting a one-digit number usually changes only the ones digit, unless you need to cross ten.
减去整十数时，只有十位数字改变。例如，67 − 20 = 47，因为 6 个十减 2 个十等于 4 个十，7 个一保持不变。减一位数通常只改变个位数字，除非需要跨十退位。
For example, 67 − 5 = 62 because 7 ones minus 5 ones leaves 2 ones. However, 62 − 5 requires regrouping: 2 ones are not enough to take away 5 ones, so you split one ten into 10 ones. Then 12 ones − 5 ones = 7 ones, and the remaining tens are 5, giving 57.
例如，67 − 5 = 62，因为 7 个一减 5 个一剩下 2 个一。然而，62 − 5 需要重新组合：2 个一不够减 5 个一，所以把一个十拆成 10 个一。然后 12 个一减 5 个一等于 7 个一，剩下的十位是 5，因此结果为 57。
A reliable mental method is to subtract the tens first, then the ones. For 73 − 28, think 73 − 20 = 53, then 53 − 8 = 45. This approach avoids confusion and shows each intermediate step clearly.
一种可靠的口算方法是先减十位，再减个位。对于 73 − 28，先想 73 − 20 = 53，再想 53 − 8 = 45。这种思路避免了混淆，并清晰地展示每一步中间过程。
73 − 28 = 73 − 20 − 8 = 53 − 8 = 45
9. Using a Hundred Chart for Addition and Subtraction | 使用百数表进行加减
A hundred chart organises the numbers 1 to 100 in rows of ten. Moving one square to the right adds 1. Moving one square to the left subtracts 1. Moving one square down adds 10. Moving one square up subtracts 10.
百数表把 1 到 100 的数字排成每行十个。向右移动一格加 1，向左移动一格减 1，向下移动一格加 10，向上移动一格减 10。
For example, start at 34. Move down one row to add 10, reaching 44. Move right three spaces to add 3, reaching 47. So 34 + 13 = 47. This visual movement mirrors the mental strategy of adding tens and adding ones separately.
例如，从 34 开始，向下移动一行加 10，到达 44；向右移动三格加 3，到达 47。因此 34 + 13 = 47。这种图表移动反映了先加十位再加个位的口算策略。
Similarly, start at 72. Move up one row to subtract 10, reaching 62. Move left two spaces to subtract 2, reaching 60. So 72 − 12 = 60. The hundred chart is a useful tool in early IGCSE practice because it links arithmetic to number patterns.
同样，从 72 开始，向上移动一行减 10，到达 62；向左移动两格减 2，到达 60。因此 72 − 12 = 60。百数表是 IGCSE 早期练习中有用的工具，因为它把算术与数字规律联系起来。
10. Solving One- and Two-Step Word Problems | 解一步和两步应用题
Word problems ask you to choose the correct operation and interpret the result. For a one-step problem, the language “altogether”, “total” and “in all” often signals addition, while “left”, “difference” and “how many more” often signals subtraction.
应用题要求你选择正确的运算并解释结果。对于一步问题，”altogether”、”total” 和 “in all” 等词语通常表示加法，而 “left”、”difference” 和 “how many more” 通常表示减法。
Example: A shop has 48 apples. It sells 23 apples. How many apples are left? The correct calculation is 48 − 23 = 25. Since 48 = 40 + 8 and 23 = 20 + 3, subtract the tens and ones separately: 40 − 20 = 20 and 8 − 3 = 5, giving 25.
例题：商店有 48 个苹果，卖出了 23 个，还剩多少个苹果？正确的计算是 48 − 23 = 25。因为 48 = 40 + 8，23 = 20 + 3，可以分别减去十位和个位：40 − 20 = 20，8 − 3 = 5，结果为 25。
A two-step problem requires two operations. Example: Rina has 45 stickers. She gives 20 to a friend and then buys 8 more. How many stickers does she have now? First subtract: 45 − 20 = 25. Then add: 25 + 8 = 33. Rina has 33 stickers.
两步问题需要两种运算。例题：Rina 有 45 张贴纸。她给了朋友 20 张，然后又买了 8 张。她现在有多少张贴纸？先减：45 − 20 = 25。再加：25 + 8 = 33。Rina 有 33 张贴纸。
45 − 20 = 25, then 25 + 8 = 33
11. Common Misconceptions and How to Avoid Them | 常见误区及如何避免
One common error is writing the expanded form of 60 + 7 as 607 instead of 67. This shows a misunderstanding of place value: the tens digit and ones digit must be written side by side without an extra zero.
一个常见错误是把 60 + 7 的展开式写成 607 而不是 67。这表明对位值的误解：十位数字和个位数字必须并排书写，不能额外加零。
Another misconception is reversing the digits when comparing numbers. Some students think 37 > 64 because they compare 7 and 4. The correct approach is to compare tens digits first: 3 tens is less than 6 tens, so 37 < 64.
另一个误区是在比较数字时颠倒数字。有些学生认为 37 > 64，因为他们只比较了 7 和 4。正确的方法是先比较十位数字：3 个十小于 6 个十，所以 37 < 64。
In subtraction, students sometimes subtract the smaller digit from the larger digit without considering position. For 43 − 18, a wrong approach gives
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