📚 Understanding Fractions: Halves, Thirds, and Fourths | 理解分数：二分之一、三分之一和四分之一
Welcome, young mathematicians! Today we are going to learn about fractions. A fraction is a way to describe parts of a whole object or a set. When we share a pizza equally among friends, we are actually using fractions. Let’s explore how to name, read, and compare fractions such as one-half, one-third, and one-fourth.
欢迎，小小数学家们！今天我们一起来学习分数。分数是用来描述一个整体物体或一组物体中一部分的方法。当我们和朋友平均分披萨时，实际上就在使用分数。让我们一起来认识、读写并比较二分之一、三分之一和四分之一这样的分数吧。
1. What Is a Fraction? | 什么是分数？
A fraction is a number that describes equal parts of a whole. We use two numbers to write a fraction, with a small line between them. The number above the line is called the numerator, and the number below the line is called the denominator. The denominator tells us how many equal parts the whole has been divided into, and the numerator tells us how many of those parts we are talking about.
分数是用来描述一个整体被分成若干等份后其中几份的数。书写分数时，我们用两个数字，并在它们之间画一条小横线。横线上方的数字叫分子，横线下方的数字叫分母。分母告诉我们整体被分成了多少等份，分子告诉我们表示的是其中的几份。
For example, imagine a chocolate bar made of 4 equal squares. If you eat 3 squares, you have eaten 3 out of 4 pieces of the whole bar. We write this as ¾ and read it as “three-fourths.”
例如，想象一块由4个相等小方格组成的巧克力。如果你吃了3格，就吃掉了整条巧克力4份中的3份。我们把它写成 ¾，读作”四分之三”。
Fractions appear in our daily lives all the time. When you cut a cake, fold a paper, or divide a group of toys, you are making fractions. Understanding fractions helps you share fairly and compare sizes correctly.
分数在我们的日常生活中随处可见。当你切蛋糕、折纸或把一盒玩具平均分配时，你就在创造分数。理解分数能帮助你公平地分享物品，并正确地比较大小。
2. Equal Parts | 等分
To write a fraction, the parts of the whole must be equal. Equal parts have exactly the same size and shape. If we cut a square into four parts of different sizes, we cannot use fractions to name those parts, because they are not equal. Fractions only work when every part is exactly the same.
要写出一个分数，整体分成的各部分必须相等。等分是指大小和形状完全相同的部分。如果我们将一个正方形切成大小不同的四块，就不能用分数来命名这些部分，因为它们不相等的。分数只有在每一部分都完全相同的时候才有意义。
Let us look at some examples. A circle cut into two equal semicircles has 2 equal parts. A rectangle divided into 3 equal strips has 3 equal parts. A square divided into 4 equal small squares has 4 equal parts. In each case, the parts are identical, so we can use fractions.
我们来看几个例子。一个圆被切成两个相等的半圆，就有2个等份。一个长方形被分成3个相等的小长条，就有3个等份。一个正方形被分成4个相等的小正方形，就有4个等份。每一种情况下，各部分都是相同的，所以我们才能用分数来表示。
When the parts are equal, we can give them special names. Two equal parts make “halves,” three equal parts make “thirds,” and four equal parts make “fourths.” These names tell us how many groups the whole has been split into.
当各部分相等时，我们就能给它们特别的名称。2等份叫”二分”（两个一半），3等份叫”三分”，4等份叫”四分”。这些名称告诉我们整体被分成了几组。
3. Halves (½) | 二分之一（½）
When a whole is divided into 2 equal parts, each part is called one-half. The symbol for one-half is ½. For example, if you cut a round pancake into 2 equal pieces, each piece is ½ of the pancake. If you put the two halves back together, you have the whole pancake again.
当一个整体被分成2个相等的部分时，每一部分都叫二分之一，符号写作 ½。例如，如果你把一个圆煎饼切成2个相等的块，每一块就是这张煎饼的 ½。如果你把两个半块重新拼在一起，就又得到了整个煎饼。
Two halves always make one whole. We can write this as a simple addition sentence:
两个二分之一合起来总是等于一个整体。我们可以把它写成一道简单的加法算式：
½ + ½ = 1
Let us think about another example. Suppose you fold a rectangular paper towel in half. The folded paper has 2 identical layers. Each layer is ½ of the whole towel. When you unfold it, you see the whole rectangle again. This folding activity helps you see that two halves equal one whole.
我们再想一个例子。假设你把一张长方形的纸巾对折。折好的纸巾有2个完全相同的层。每一层就是整张纸巾的 ½。当你把它展开时，就再次看到完整的长方形。这个折纸活动能帮助你直观地理解”两个一半等于一个整体”。
4. Thirds (⅓) | 三分之一（⅓）
When a whole is divided into 3 equal parts, each part is called one-third, written as ⅓. If you take two of those three equal parts, you have two-thirds, written as ⅔. Three thirds make one whole: ⅓ + ⅓ + ⅓ = 1.
当一个整体被分成3个相等的部分时，每一部分叫三分之一，写作 ⅓。如果你从中取出两部分，就是三分之二，写作 ⅔。三个三分之一组成一个整体：⅓ + ⅓ + ⅓ = 1。
⅓ + ⅓ + ⅓ = 1
Here is a tasty example. A sandwich is cut into 3 equal pieces. If you eat one piece, you have eaten ⅓ of the sandwich. If you eat two pieces, you have eaten ⅔ of the sandwich, and only ⅓ is left for your friend.
这里有一个美味的小例子。一个三明治被切成3个相等的块。如果你吃一块，就吃掉了三明治的 ⅓。如果你吃两块，就吃掉了 ⅔，只剩下 ⅓ 留给你的朋友。
Drawing pictures can help us understand thirds. Draw a rectangle and divide it into 3 equal vertical strips. Shade one strip: that is ⅓. Shade two strips: that is ⅔. Notice that the shaded parts are always equal in size because the strips were cut equally.
画图能帮助我们理解三分之一。画一个长方形，把它分成3个相等的竖条。涂色1条，那就是 ⅓；涂色2条，那就是 ⅔。注意涂色部分的大小始终相同，因为这些竖条是被等分的。
5. Fourths (¼) | 四分之一（¼）
When a whole is divided into 4 equal parts, each part is called one-fourth, written as ¼. Three of those parts are written as ¾ and read as “three-fourths.” Four fourths make one whole: ¼ + ¼ + ¼ + ¼ = 1.
当一个整体被分成4个相等的部分时，每一部分叫四分之一，写作 ¼。其中三部分写作 ¾，读作”四分之三”。四个四分之一组成一个整体：¼ + ¼ + ¼ + ¼ = 1。
¼ + ¼ + ¼ + ¼ = 1
Let us think about a pizza cut into 4 slices. One slice is ¼ of the whole pizza. If you eat 3 slices, you have eaten ¾ of the pizza, and ¼ of the pizza is still on the plate. If all four people each take one slice, the whole pizza is gone, which equals 1 whole.
我们来想一个比萨被切成4片的情景。一片是整个比萨的 ¼。如果你吃3片，就吃掉了比萨的 ¾，盘子里还剩 ¼。如果四个人每人拿一片，整个比萨就分完了，合计正好等于1个整体。
Remember that the denominator tells us the total number of equal parts. For fourths, the denominator is 4. The numerator tells us how many parts we have chosen. So in ¾, the numerator is 3 and the denominator is 4.
记住：分母告诉我们总共有多少个等份。对于四分来说，分母是4。分子告诉我们选出了多少份。所以在 ¾ 中，分子是3，分母是4。
6. Numerator and Denominator | 分子和分母
Every fraction has a top number and a bottom number. The top number is the numerator, and the bottom number is the denominator. The denominator shows the total number of equal parts in the whole, and the numerator shows how many of those parts are being counted or shaded.
每个分数都有一个上面的数和一个下面的数。上面的数叫分子，下面的数叫分母。分母表示整体被分成的总等份数，分子表示被数出或涂色的份数。
Let us look at a table to practice reading different fractions:
我们来看一个表格，练习读一读不同的分数：
| Fraction | Numerator | Denominator | Read As |
| ½ | 1 | 2 | one-half |
| ⅓ | 1 | 3 | one-third |
| ⅔ | 2 | 3 | two-thirds |
| ¼ | 1 | 4 | one-fourth |
| ¾ | 3 | 4 | three-fourths |
Look at ½ and ¾. The bigger numerator in ¾ means more parts are chosen, but we must always look at the denominator to know the size of each part. A fraction is like a friendly pair: the denominator names the size of the pieces, and the numerator counts how many pieces we have.
观察 ½ 和 ¾。¾ 的分子更大，表示选出的份数更多，但我们总要看看分母，才能知道每一份有多大。分数就像一对好朋友：分母告诉我们每一份的大小，分子数一数我们手里有多少份。
7. Comparing Fractions | 比较分数
In Grade 2, we compare fractions by looking at the size of the parts, not just the size of the numbers. Here is an important rule: when the same whole is divided into more equal parts, each part becomes smaller. So ¼ is smaller than ½, and ⅓ is smaller than ½.
在二年级，我们通过观察部分的大小来比较分数，而不只是看数字的大小。这里有一条重要规则：同一个整体分成的份数越多，每一份就越小。所以，¼ 小于 ½，⅓ 也小于 ½。
Imagine two identical chocolate bars. The first bar is cut into 2 equal pieces. The second bar is cut into 4 equal pieces. One piece from the first bar is much larger than one piece from the second bar. Therefore, we say ½ is greater than ¼.
想象两块完全相同的巧克力。第一块被切成2个相等的块，第二块被切成4个相等的块。第一块巧克力的一块比第二块的一块大得多。因此，我们说 ½ 大于 ¼。
We use the symbols > (greater than), < (less than), and = (equal to) to compare fractions. Here are some examples:
我们用 >（大于）、<（小于）和 =（等于）来比较分数。下面是一些例子：
½ > ¼
⅓ > ¼
¾ > ½
Drawing fraction bars side by side is the best way to compare. If the bars are the same length, divide one into 2 parts and the other into 4 parts. You will clearly see that one big half is longer than one small fourth. Always draw or imagine equal wholes before comparing.
画并排的分数条是比较分数最好的方法。如果两个长条一样长，把其中一个分成2份，另一个分成4份。你会清楚地看到，大的半个长条比小的四分之一条更长。比较之前，一定要先画出或想象出相同的整体。
8. Fractions of a Group | 一组物体的分数
Fractions are not only for single objects like sandwiches or pizzas. We can also use fractions to describe part of a group of objects. For example, if there are 4 toy cars and 1 of them is red, then ¼ of the cars are red.
分数不仅可以用来描述三明治、比萨这样的单个物体，也可以用来描述一组物体中的一部分。例如，如果有4辆玩具车，其中1辆是红色，那么红色车就占全部的 ¼。
Let us try another example. There are 6 apples in a basket, and 3 of them are green. What fraction of the apples are green? Since 3 out of 6 apples are green, we can write 3/6. We can also notice that 3 is exactly half of 6, so half of the apples are green.
我们再试一个例子。篮子里有6个苹果，其中3个是绿色的。绿色苹果占几分之几？因为6个苹果中有3个是绿色，所以我们可以写作 3/6。我们还能注意到，3正好是6的一半，所以可以说一半的苹果是绿色的。
When we write fractions of a group, the denominator is the total number of objects in the group, and the numerator is the number of objects with the special feature we are describing.
写一组物体的分数时，分母是这一组物体的总数，分子是符合我们描述特征的那部分物体的数量。
9. Real-Life Fractions | 生活中的分数
Fractions are everywhere around us! When you look at a clock, the time “half past five” uses the fraction half, because 30 minutes is ½ of an hour. In the kitchen
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