📚 Math G3 Practice Book: Core Skills and Worked Examples | 数学三年级练习册：核心技能与例题精讲
This practice book is designed for Grade 3 learners who are building a strong foundation in arithmetic, geometry, measurement, and data handling. It focuses on the core skills that appear most often in school assessments and international primary checkpoints. Each section pairs a concise explanation with worked examples, so students can practise independently or with a tutor.
本练习册专为三年级学生编写，帮助他们在算术、几何、测量和数据处理方面打下扎实基础。全书聚焦学校测评和国际小学阶段考查中最常见的核心技能。每一节都配有简明讲解和例题精解，方便学生独立练习或在老师指导下使用。
Twelve short units cover the full Grade 3 toolkit: place value, the four operations, fractions, measurement, perimeter and area, basic geometry, and simple statistics. Every unit ends with a practice tip that can be done on paper in five to ten minutes.
全书共十二个短单元，覆盖三年级完整的技能体系：数的认识、四则运算、分数、测量、周长与面积、基础几何以及简单统计。每个单元都以一条练习建议收尾，用纸笔五到十分钟即可完成。
1. Place Value up to 10,000 | 万以内数的认识
In Grade 3, students must read, write, and compare numbers up to 10,000. A digit's value depends on its place: ones, tens, hundreds, thousands. For example, in 4,728 the digit 4 stands for 4,000, the digit 7 stands for 700, the digit 2 stands for 20, and 8 stands for 8 ones.
在三年级，学生需要会读、会写并会比较 10,000 以内的数。一个数字的大小取决于它所在的数位：个位、十位、百位、千位。例如，在 4,728 中，数字 4 表示 4,000，数字 7 表示 700，数字 2 表示 20，数字 8 表示 8 个一。
| Thousands 千位 | Hundreds 百位 | Tens 十位 | Ones 个位 |
| 4 | 7 | 2 | 8 |
| 4,000 | 700 | 20 | 8 |
Common exercises ask students to write a number in expanded form. The number 3,506 is split as follows.
常见练习要求学生进行数的拆分，把 3,506 写成展开式如下。
3,506 = 3,000 + 500 + 6
When comparing numbers, always start from the highest place value: 5,210 > 4,999 because 5 thousands are greater than 4 thousands, even though the last three digits of 4,999 look larger.
比较数的大小时，一定要从最高位开始比较：5,210 > 4,999，因为 5 个千大于 4 个千，尽管 4,999 后三位看起来更大。
Practice tip: write five different four-digit numbers, order them from least to greatest, and then read each one aloud using correct place-value language.
练习建议：写出五个不同的四位数，将它们从小到大排列，然后用正确的数位名称大声读出每一个数。
2. Comparing and Ordering Numbers | 数的比较与排序
ess than, and = means equal to. The open end of the symbol always faces the larger number.
三年级学生要掌握三个比较符号：> 表示大于，< 表示小于，= 表示等于。符号开口的一侧永远朝向较大的数。
4,728 > 4,278 630 < 1,003 25 × 4 = 100
Ordering a set of numbers means arranging them either from least to greatest (ascending) or from greatest to least (descending). A reliable method is to compare the thousands digit first, then the hundreds, then the tens, then the ones.
排序就是把一组数按从小到大（升序）或从大到小（降序）排列。可靠的方法是先比千位，再比百位，再比十位，最后比个位。
- Worked example: order 3,905 3,509 3,950 3,059 from least to greatest.
例题：把 3,905、3,509、3,950、3,059 从小到大排列。 - Step 1: all four numbers have 3 thousands, so compare hundreds: 0 < 5 < 9 < 9.
第一步：四个数的千位都是 3，比较百位：0 < 5 < 9 < 9。 - Step 2: between 3,905 and 3,950, compare tens: 0 < 5, so 3,905 comes first.
第二步：3,905 与 3,950 比较十位：0 < 5，所以 3,905 在前。 - Answer: 3,059 < 3,509 < 3,905 < 3,950.
答案：3,059 < 3,509 < 3,905 < 3,950。
A very common error is to compare the first digit that looks different, ignoring place value. Always check the number of digits first: any four-digit number is greater than any three-digit number.
最常见的错误是看到某一位不同就直接下结论，而忽略了数位。先看位数：任意四位数都大于任意三位数。
3. Rounding and Number Patterns | 近似数与数列规律
Rounding to the nearest 10 or 100 is a core Grade 3 skill that supports estimation. Look at the digit to the right of the target place: if it is 5 or more, round up; if it is 4 or less, round down.
四舍五入到最近的十或百，是三年级支持估算的核心技能。看目标数位右边的那一位：大于或等于 5 就进位，小于或等于 4 就舍去。
468 → 470 (nearest 10) 468 → 500 (nearest 100)
Number patterns involve finding the rule that links consecutive terms. A sequence such as 250, 300, 350, 400 increases by 50 each time, so the next three terms are 450, 500, 550.
数列规律就是找出相邻两项之间的关系。例如 250、300、350、400 每次增加 50，因此后面三项是 450、500、550。
- Skip counting by 3: 3, 6, 9, 12, 15, 18 …
跳数 3：3、6、9、12、15、18…… - Skip counting by 4: 4, 8, 12, 16, 20, 24 …
跳数 4：4、8、12、16、20、24…… - Skip counting by 25: 25, 50, 75, 100, 125 … (useful for money and time)
跳数 25：25、50、75、100、125……（用于钱币和时间很实用）
Practice tip: write a sequence with a hidden rule, ask a partner to find the rule, then swap roles. Explaining the rule out loud is the fastest way to secure it.
练习建议：自己写一个隐藏规律的数列，请同伴找出规律，然后互换角色。把规律说出来，是掌握它最快的方法。
4. Addition and Subtraction with Regrouping | 进位加法与退位减法
Grade 3 arithmetic includes adding and subtracting numbers up to four digits with regrouping. Regrouping means exchanging 10 ones for 1 ten, or 1 hundred for 10 tens, when a column total exceeds 9.
三年级算术包括四位以内数的进位加法和退位减法。进位是指当某一数位的和超过 9 时，将 10 个一换成 1 个十，或将 1 个百换成 10 个十。
456 + 278 = 734
- Add ones: 6 + 8 = 14, write 4 and carry 1 ten.
个位相加：6 + 8 = 14，写 4，向十位进 1。 - Add tens: 5 + 7 + 1 = 13, write 3 and carry 1 hundred.
十位相加：5 + 7 + 1 = 13，写 3，向百位进 1。 - Add hundreds: 4 + 2 + 1 = 7.
百位相加：4 + 2 + 1 = 7。 - Result: 734.
结果：734。
Subtraction with regrouping is the reverse process. When the top digit is smaller than the bottom digit, borrow 1 from the next column on the left.
退位减法则是相反的过程。当被减数的某一位小于减数的对应位时，就向左边的相邻数位借 1。
700 − 268 = 432 check: 432 + 268 = 700
A useful habit is to check every subtraction by adding the answer back to the number subtracted. If the total does not match the original number, a regrouping mistake has been made.
一个好习惯是：每道减法都用加法验算，把差与减数相加。如果结果不等于原来的被减数，说明退位过程中出错了。
5. Multiplication: Meaning, Facts and Properties | 乘法：意义、口诀与性质
+ 6 = 24. The numbers being multiplied are called factors, and the answer is the product.
乘法是加法的简便运算。4 × 6 表示 4 个 6，等于 6 + 6 + 6 + 6 = 24。相乘的两个数叫因数，结果叫积。
4 × 6 = 24 6 × 4 = 24 (commutative property 交换律)
d ×10, then use doubling strategies for ×4 and ×8, and the ×5 table plus one group for ×6.
三年级学生应熟练背诵到 10 × 10 的乘法口诀。先记容易的：×1、×2、×5、×10，然后用翻倍策略记 ×4 和 ×8，用 5 的口诀加一组来记 ×6。
- 2 × 7 = 14, so 4 × 7 = 28 (double), and 8 × 7 = 56 (double again).
2 × 7 = 14，所以 4 × 7 = 28（翻倍），8 × 7 = 56（再翻倍）。 - Any number × 1 equals itself; any number × 0 equals 0.
任何数 × 1 等于它本身；任何数 × 0 都等于 0。 - 3 × 4 × 2 can be grouped as (3 × 4) × 2 = 24 or 3 × (4 × 2) = 24.
3 × 4 × 2 可以分组为 (3 × 4) × 2 = 24，也可以 3 × (4 × 2) = 24。
Practice tip: practise the facts that are still slow, not the ones already known. Five minutes a day on three tricky facts beats thirty minutes on the whole table.
练习建议：重点练还不熟的口诀，不要重复练已经会的内容。每天五分钟攻克三条难记口诀，比三十分钟通背整张表更有效。
6. Division and Remainders | 有余数的除法
7 means 42 objects can be shared into 6 groups of 7, and it is the inverse of 6 × 7 = 42.
除法是把一个总量平均分成几组，或者求能分成多少组。42 ÷ 6 = 7 表示 42 个物体可以平均分成 6 组，每组 7 个；它是 6 × 7 = 42 的逆运算。
Not every division works out exactly. When a number cannot be divided evenly, there is a remainder, and the remainder must always be smaller than the divisor.
并不是所有除法都能整除。当不能正好分完时，就会出现余数，而且余数一定小于除数。
47 ÷ 5 = 9 remainder 2 check: 5 × 9 + 2 = 47
- Divisor 除数 = the number you divide by (5).
- Quotient 商 = the result of the division (9).
- Remainder 余数 = what is left over (2), always less than the divisor.
余数是剩下的部分（2），永远小于除数。
A classic Grade 3 question: 47 pencils are packed into boxes of 5. How many full boxes are there, and how many pencils are left? The answer is 9 full boxes and 2 pencils left, and one more box would be needed to hold all 47.
一道典型三年级题目：47 支铅笔每 5 支装一盒，能装满几盒，还剩几支？答案是装满 9 盒，还剩 2 支；如果要把 47 支全部装完，则还需要再准备一个盒子。
7. Two-Step Word Problems | 两步应用题
Many Grade 3 problems need two operations before the final answer. A reliable routine is: Read the question, Underline the numbers, Choose the operations, Solve, and Check.
许多三年级应用题需要两步运算才能得到最终答案。可靠的步骤是：读题、画出关键数据、确定运算、计算、检查。
Worked example: A shop has 3 boxes of pencils with 24 pencils in each box. The shopkeeper sells 15 pencils. How many pencils are left?
例题：商店有 3 盒铅笔，每盒 24 支。店主卖出了 15 支，还剩多少支？
- Step 1 (total): 3 × 24 = 72 pencils.
第一步（求总数）：3 × 24 = 72 支。 - Step 2 (subtract): 72 − 15 = 57 pencils.
第二步（减去卖出）：72 − 15 = 57 支。 - Answer: 57 pencils remain.
答案：还剩 57 支。
Bar models help a lot here. Draw one bar for the total, cut it into 3 equal parts for the boxes, then cross out the 15 that were sold. The picture makes the two steps obvious.
线段图在这里非常有用。先画一条表示总数的线段，平均分成 3 份表示 3 盒，再划掉卖出的 15 支。图一画出来，两步运算就一目了然了。
A key reading skill is spotting the question mark: some problems ask for the total, others ask for what is left, and a few ask how many more are needed. Always reread the final sentence before writing the answer.
关键的读题技能是看清问题问的是什么：有的求总数，有的求剩余，还有的问"还差多少"。写完答案前，一定要把最后一句再读一遍。
8. Fractions as Equal Parts | 分数的初步认识
A fraction describes equal parts of a whole. The bottom number (denominator) tells how many equal parts the whole is divided into, and the top number (numerator) tells how many parts are taken.
分数表示把一个整体平均分成的若干份。下面的数（分母）表示整体被平均分成几份，上面的数（分子）表示取了多少份。
½ one half 二分之一 ¼ one quarter 四分之一 ¾ three quarters 四分之三
An important word is equal. A shape cut into four pieces of different sizes does not show quarters, because the parts are not equal.
关键词是"平均分"。一个图形被分成四块但大小不同，就不能表示四分之一，因为没有平均分。
- Compare fractions with the same denominator: 3/7 > 2/7 because 3 parts are more than 2 parts.
同分母比较：3/7 > 2/7，因为 3 份比 2 份多。 - Fraction of a set: ½ of 12 = 6, and ¼ of 12 = 3.
求一个数的几分之几：12 的 ½ 是 6，12 的 ¼ 是 3。 - Unit fractions get smaller as the denominator grows: ½ > ⅓ > ¼.
分子为 1 时，分母越大分数越小：½ > ⅓ > ¼。
label each fold. Seeing that ½ = 2/4 = 4/8 builds the idea of equivalent fractions early.
练习建议：把一条纸带折成二分之一、四分之一和八分之一，并在折痕处标出分数。看到 ½ = 2/4 = 4/8，能提前建立等值分数的概念。
9. Measurement and Time | 测量与时间
Grade 3 measurement covers length, mass, capacity and time. Students should choose sensible units and convert between them confidently.
三年级的测量内容包括长度、质量、容积和时间。学生要能选择合适的单位，并熟练进行单位换算。
| Quantity 量 | Units 单位 | Conversion 换算 |
| Length 长度 | mm, cm, m, km | 1 km = 1,000 m; 1 m = 100 cm |
| Mass 质量 | g, kg | 1 kg = 1,000 g |
| Capacity 容积 | mL, L | 1 L = 1,000 mL |
| Time 时间 | seconds, minutes, hours | 1 hour = 60 minutes; 1 minute = 60 seconds |
Reading a clock to the nearest minute is a key skill. If the minute hand points to the 7, the time is 35 minutes past the hour; the hour hand has already moved past the hour number, so 7:35, not 8:35.
认读钟表到整分钟是关键技能。如果分针指向 7，就是整点过后 35 分；此时时针已经越过了整点数字，因此是 7:35，而不是 8:35。
er and record their lengths in both centimetres and millimetres, for example 7 cm = 70 mm.
练习建议：在家用尺子量三件物品，并用厘米和毫米两种单位记录长度，例如 7 cm = 70 mm。
10. Perimeter and Area of Rectangles | 长方形的周长与面积
Perimeter is the distance around the outside of a shape, while area is the amount of surface it covers. They are measured in different units: perimeter in cm or m, area in cm² or m².
周长是图形一周的长度，面积是图形表面的大小。两者的单位不同：周长用 cm 或 m，面积用 cm² 或 m²。
Perimeter of rectangle 长方形周长: P = 2 × (l + w)
Area of rectangle 长方形面积: A = l × w
Worked example: a rectangle is 8 cm long and 3 cm wide.
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图