KS2 Year 4 Fractions: Equivalent Fractions, Simplification and Ordering — KS2 四年级分数：等值分数、化简与比较排序

一、分数的基本概念：分子、分母与整体 | Basic Concepts of Fractions: Numerator, Denominator, and the Whole

分数是数学中最基础也最重要的概念之一。一个分数表示整体被平均分成若干份后，取其中若干份的数量。它由两部分组成：位于上方的分子（numerator）和位于下方的分母（denominator）。分子告诉我们”取了多少份”，分母告诉我们”整体被分成了多少等份”。例如，在分数 3/4 中，分母 4 表示整体被分成 4 等份，分子 3 表示我们取了其中的 3 份。

A fraction is one of the most fundamental and important concepts in mathematics. A fraction represents how many parts of a whole we have when the whole is divided into equal parts. It consists of two parts: the numerator on top and the denominator on the bottom. The numerator tells us “how many parts we have taken”, and the denominator tells us “how many equal parts the whole is divided into”. For example, in the fraction 3/4, the denominator 4 tells us the whole is divided into 4 equal parts, and the numerator 3 tells us we have taken 3 of those parts.

理解”整体”的概念至关重要。整体可以是一个披萨、一条巧克力棒、一组物品，甚至是一个数字。当我们说 1/2 时，关键在于这个”一半”是相对于什么整体而言的。半个大披萨和半个小披萨的量是不同的，尽管它们都表示为 1/2。这就是为什么在解答分数问题时，首先要明确”整体是什么”。

Understanding the concept of “the whole” is crucial. The whole can be a pizza, a chocolate bar, a set of objects, or even a number. When we say 1/2, the key is what the “half” is relative to. Half a large pizza and half a small pizza are different amounts, even though both are expressed as 1/2. That is why, when solving fraction problems, the first step is always to identify “what is the whole”.

在 KS2 四年级阶段，学生需要能够用图形直观地表示分数。最常见的方法是使用分数条（fraction bar）或面积模型（area model）。例如，画一个长方形并将其平均分成 5 份，然后将其中 2 份涂色，就直观地表示了 2/5。这种视觉化方法帮助学生建立分数概念的直觉理解，为后续学习等值分数和分数运算打下坚实基础。

At the KS2 Year 4 level, students need to be able to represent fractions visually. The most common methods are using fraction bars or area models. For example, drawing a rectangle, dividing it into 5 equal parts, and shading 2 of them visually represents 2/5. This visual approach helps students build an intuitive understanding of fraction concepts, laying a solid foundation for later learning about equivalent fractions and fraction operations.

二、等值分数的定义：不同写法，相同大小 | Definition of Equivalent Fractions: Different Notation, Same Value

等值分数（equivalent fractions）是指写法不同但数值大小完全相同的分数。例如，1/2、2/4、3/6 和 4/8 都是等值分数 – 它们在数轴上占据同一个位置，代表完全相同的量。理解等值分数的核心在于：当你将分子和分母同时乘以或除以同一个非零整数时，分数的值保持不变。

Equivalent fractions are fractions that look different but represent exactly the same value. For example, 1/2, 2/4, 3/6, and 4/8 are all equivalent fractions – they occupy the same position on a number line and represent the exact same quantity. The core understanding is: when you multiply or divide both the numerator and the denominator by the same non-zero integer, the value of the fraction remains unchanged.

这个性质可以从分数的基本定义来证明。以 1/2 和 2/4 为例：如果把一个整体分成 2 等份取 1 份，与把同一个整体分成 4 等份取 2 份，在量上是完全相同的。用面积模型可以更直观地展示：画出两个同样大小的长方形，第一个平分成 2 列并涂色 1 列，第二个平分成 4 列并涂色 2 列 – 涂色面积完全一样。

This property can be proven from the basic definition of fractions. Take 1/2 and 2/4 as an example: dividing a whole into 2 equal parts and taking 1, versus dividing the same whole into 4 equal parts and taking 2, are completely identical in quantity. An area model demonstrates this even more intuitively: draw two rectangles of the same size, divide the first into 2 columns and shade 1 column, divide the second into 4 columns and shade 2 columns – the shaded area is exactly the same.

在四年级的数学课程中，学生通常通过“乘法规则”来生成等值分数：将分子和分母同时乘以 2、3、4 等整数。例如从 2/3 出发，乘以 2 得到 4/6，乘以 3 得到 6/9，乘以 4 得到 8/12 – 这些全部等值。同样地，“除法规则”用于简化分数（将在后面章节详细讨论）。掌握等值分数是分数加减运算的前提条件，因为不同分母的分数需要先通分才能相加。

In Year 4 mathematics, students typically learn to generate equivalent fractions using the “multiplication rule”: multiply both the numerator and denominator by the same integer such as 2, 3, or 4. For example, starting from 2/3, multiply by 2 to get 4/6, by 3 to get 6/9, by 4 to get 8/12 – all of these are equivalent. Similarly, the “division rule” is used to simplify fractions (discussed in detail later). Mastering equivalent fractions is a prerequisite for adding and subtracting fractions, because fractions with different denominators need to be converted to a common denominator first.

三、利用分数墙（Fraction Wall）直观比较等值分数 | Using a Fraction Wall to Visually Compare Equivalent Fractions

分数墙（Fraction Wall）是 KS2 数学教学中极为有效的视觉工具。它由多条平行的横条组成，每条横条被等分成不同的份数：第一条保持完整（1 等份，代表 1），第二条分成 2 等份，第三条分成 3 等份，以此类推。通过在分数墙上观察对齐的垂直线，学生可以一目了然地发现等值分数。

The Fraction Wall is an extremely effective visual tool in KS2 mathematics teaching. It consists of multiple parallel horizontal bars, each divided into a different number of equal parts: the first bar stays whole (1 part, representing 1), the second is divided into 2 equal parts, the third into 3 equal parts, and so on. By observing the vertical alignment lines on the fraction wall, students can discover equivalent fractions at a glance.

例如，在一条从 1 到 12 等份的分数墙上，可以清楚地看到：1/2 的边界线与 2/4、3/6、4/8、5/10、6/12 的边界线完全对齐。同样，1/3 与 2/6、3/9、4/12 对齐；2/3 与 4/6、6/9、8/12 对齐。这种视觉对齐不需要任何计算，直观地证明了等值分数的存在和规律。

For example, on a fraction wall with bars divided from 1 to 12 equal parts, you can clearly see: the boundary line of 1/2 aligns perfectly with those of 2/4, 3/6, 4/8, 5/10, and 6/12. Similarly, 1/3 aligns with 2/6, 3/9, and 4/12; 2/3 aligns with 4/6, 6/9, and 8/12. This visual alignment requires no calculation and intuitively proves the existence and pattern of equivalent fractions.

构建分数墙也是一种优秀的课堂活动。学生可以自己动手画出分数墙，用彩色笔标出不同的等值分数列。这不仅加深了对等值分数的理解，也强化了对”整体被分成越多份，每份越小”这一核心概念的认知 – 在分数墙上可以直观地看到，1/12 的每一份远小于 1/2 的每一份。

Building a fraction wall is also an excellent classroom activity. Students can draw their own fraction wall and use colored pencils to mark different equivalent fraction columns. This not only deepens their understanding of equivalent fractions but also reinforces the core concept that “the more parts a whole is divided into, the smaller each part is” – on the fraction wall, you can visually see that each part of 1/12 is much smaller than each part of 1/2.

四、最简分数与化简：用最大公因数”约分” | Simplest Form and Simplification: Reducing Fractions Using the Greatest Common Factor

一个分数如果分子和分母没有大于 1 的公因数，就称为最简分数（simplest form）。例如，3/4 是最简分数（3 和 4 的最大公因数是 1），而 6/8 不是最简分数（6 和 8 的最大公因数是 2，可以化简为 3/4）。将分数化为最简形式的过程叫做约分（simplification）。

A fraction is in its simplest form if the numerator and denominator have no common factor greater than 1. For example, 3/4 is in simplest form (the greatest common factor of 3 and 4 is 1), while 6/8 is not (the greatest common factor of 6 and 8 is 2, so it can be simplified to 3/4). The process of reducing a fraction to its simplest form is called simplification.

约分的方法非常直接：找到分子和分母的最大公因数（GCF, Greatest Common Factor），然后用分子和分母同时除以这个数。例如，化简 12/18：12 和 18 的最大公因数是 6，所以 12÷6=2，18÷6=3，得到 2/3。对于 KS2 四年级的学生，他们通常通过试除较小的公因数（如 2、3、5）来逐步化简，而不是一次找到最大公因数。

The method for simplification is straightforward: find the greatest common factor (GCF) of the numerator and denominator, then divide both by this number. For example, to simplify 12/18: the GCF of 12 and 18 is 6, so 12÷6=2, 18÷6=3, giving 2/3. For KS2 Year 4 students, they typically simplify step by step using smaller common factors (such as 2, 3, 5) rather than finding the GCF in one step.

分步约分法示例：化简 24/36。首先发现 24 和 36 都是偶数，可以同时除以 2，得到 12/18。然后发现 12 和 18 也都是偶数，再除以 2，得到 6/9。最后发现 6 和 9 都可以被 3 整除，除以 3 得到 2/3。2/3 的分子分母没有大于 1 的公因数，所以是最简分数。这种方法虽然步骤多一些，但逻辑清晰，更适合初学者。

Step-by-step simplification example: simplify 24/36. First, notice both 24 and 36 are even, so divide both by 2 to get 12/18. Then notice 12 and 18 are also even, divide by 2 again to get 6/9. Finally, notice both 6 and 9 are divisible by 3, divide by 3 to get 2/3. The numerator and denominator of 2/3 have no common factor greater than 1, so it is in simplest form. Although this method involves more steps, the logic is clear and it is more suitable for beginners.

五、分数比较的三个层次：同分母、同分子、不同分母分子 | Three Levels of Fraction Comparison: Same Denominator, Same Numerator, Different Both

比较分数的大小是 KS2 四年级数学中的重要技能。根据分数类型的不同，比较策略分为三个层次。第一层次 – 同分母分数：分母相同时，分子越大的分数越大。例如，比较 3/7 和 5/7，因为 5>3，所以 5/7 > 3/7。这个规则非常直观：整体被分成了相同的份数，取的份数越多，分数就越大。

Comparing the size of fractions is an important skill in KS2 Year 4 mathematics. Depending on the type of fractions, comparison strategies fall into three levels. Level One – same denominator fractions: when denominators are the same, the larger the numerator, the larger the fraction. For example, comparing 3/7 and 5/7: since 5>3, 5/7 > 3/7. This rule is very intuitive: the whole is divided into the same number of parts, and the more parts you take, the larger the fraction.

第二层次 – 同分子分数：分子相同时，分母越小的分数越大。例如，比较 2/5 和 2/7，虽然分子都是 2，但 2/5 的每份比 2/7 的每份更大（因为整体被分成的份数更少），所以 2/5 > 2/7。这里的关键洞察是：分母越大，每份越小。许多初学者容易搞错这个规则，误以为分母大的分数就大，需要特别注意。

Level Two – same numerator fractions: when numerators are the same, the smaller the denominator, the larger the fraction. For example, comparing 2/5 and 2/7: although both have numerator 2, each part of 2/5 is larger than each part of 2/7 (because the whole is divided into fewer parts), so 2/5 > 2/7. The key insight here is: the larger the denominator, the smaller each part. Many beginners get this rule wrong, mistakenly thinking that a larger denominator means a larger fraction – this requires special attention.

第三层次 – 分子和分母都不同：这种情况需要将分数转换为等值分数，使它们具有相同的分母（通分），然后比较分子。例如，比较 3/4 和 5/6：找到 4 和 6 的最小公倍数 12，将 3/4 转换为 9/12，将 5/6 转换为 10/12，显然 10/12 > 9/12，因此 5/6 > 3/4。通分是比较不同分母分数的通用方法，也是后续分数加减运算的基础。

Level Three – different numerators and denominators: in this case, you need to convert the fractions to equivalent fractions with a common denominator (finding a common denominator), then compare the numerators. For example, comparing 3/4 and 5/6: find the least common multiple of 4 and 6, which is 12. Convert 3/4 to 9/12 and 5/6 to 10/12. Clearly, 10/12 > 9/12, so 5/6 > 3/4. Finding a common denominator is the universal method for comparing fractions with different denominators, and is also the foundation for later fraction addition and subtraction.

六、在数轴上排列分数：从小到大建立数感 | Ordering Fractions on a Number Line: Building Number Sense from Smallest to Largest

将分数放置在数轴上是培养数感（number sense）的绝佳方法。数轴提供了一个线性的、可视化的框架，帮助学生理解分数在整体数量体系中的位置。在 KS2 四年级，学生需要能够将一组分数按照从小到大的顺序排列在数轴上。

Placing fractions on a number line is an excellent way to develop number sense. A number line provides a linear, visual framework that helps students understand where fractions sit within the overall number system. At KS2 Year 4, students need to be able to order a set of fractions from smallest to largest on a number line.

排列分数的标准步骤是：首先，将所有分数通分为同分母分数。然后，比较分子的大小：分子越小，分数越靠近数轴的左端（0）；分子越大，分数越靠近右端。例如，将 1/2、2/3、3/4、1/3 和 5/6 从小到大排列：通分到分母 12，得到 6/12(1/2)、8/12(2/3)、9/12(3/4)、4/12(1/3) 和 10/12(5/6)，按分子从小到大排列为：4/12、6/12、8/12、9/12、10/12，即：1/3 < 1/2 < 2/3 < 3/4 < 5/6。

The standard procedure for ordering fractions is: first, convert all fractions to equivalent fractions with a common denominator. Then, compare the numerators: the smaller the numerator, the closer the fraction is to the left end of the number line (0); the larger the numerator, the closer to the right. For example, ordering 1/2, 2/3, 3/4, 1/3, and 5/6 from smallest to largest: convert to denominator 12, giving 6/12(1/2), 8/12(2/3), 9/12(3/4), 4/12(1/3), and 10/12(5/6). Ordering numerators from smallest to largest: 4/12, 6/12, 8/12, 9/12, 10/12, i.e.: 1/3 < 1/2 < 2/3 < 3/4 < 5/6.

一个常用的技巧是利用基准分数（benchmark fractions）来快速判断分数的相对大小。最常见的基准分数是 1/2。判断一个分数是大于、等于还是小于 1/2，可以帮助快速排序。例如，3/8 小于 1/2（因为 3/8 = 0.375，1/2 = 0.5），而 5/8 大于 1/2。另一个有用的基准是 1/4 和 3/4。这种基准比较法在实际问题中非常实用。

A useful technique is to use benchmark fractions to quickly judge the relative size of fractions. The most common benchmark fraction is 1/2. Determining whether a fraction is greater than, equal to, or less than 1/2 helps with quick ordering. For example, 3/8 is less than 1/2 (because 3/8 = 0.375, 1/2 = 0.5), while 5/8 is greater than 1/2. Other useful benchmarks are 1/4 and 3/4. This benchmark comparison method is very practical in real problems.

七、单位分数与非单位分数：理解”一份”与”多份” | Unit Fractions and Non-Unit Fractions: Understanding “One Part” vs “Multiple Parts”

分数可以分为两大类型：单位分数（unit fractions）和非单位分数（non-unit fractions）。单位分数是分子为 1 的分数，如 1/2、1/3、1/4、1/5 等，它们代表整体的”一份”。非单位分数是分子大于 1 的分数，如 2/3、3/4、5/8 等，它们由多个单位分数组成。

Fractions can be divided into two main types: unit fractions and non-unit fractions. A unit fraction is a fraction with numerator 1, such as 1/2, 1/3, 1/4, 1/5, etc., representing “one part” of the whole. A non-unit fraction is a fraction with a numerator greater than 1, such as 2/3, 3/4, 5/8, etc., composed of multiple unit fractions.

理解非单位分数与单位分数的关系是四年级数学的关键概念。例如，3/4 可以理解为 3 个 1/4，即 1/4 + 1/4 + 1/4。这种理解自然地引出分数的累加性质，也为后续学习带分数（如 1 1/4 = 5/4）和假分数铺平了道路。在 KS2 课程中，学生需要能够将非单位分数分解为若干个单位分数之和，也要能从若干个单位分数组合成一个非单位分数。

Understanding the relationship between non-unit fractions and unit fractions is a key concept in Year 4 mathematics. For example, 3/4 can be understood as three 1/4’s, i.e., 1/4 + 1/4 + 1/4. This understanding naturally leads to the additive property of fractions and paves the way for later learning about mixed numbers (such as 1 1/4 = 5/4) and improper fractions. In the KS2 curriculum, students need to be able to decompose a non-unit fraction into the sum of several unit fractions, and also to combine several unit fractions into a non-unit fraction.

下面的练习模式在 KS2 考试中非常常见：在数轴上，从 0 开始，每次跳 1/5，跳 4 次到达什么位置？答案是 4/5。反过来说，4/5 就是 4 个 1/5 的累积。这种”跳跃计数”的方法将分数与整数计数联系起来，帮助学生将已有的整数知识迁移到分数领域。

The following exercise pattern is very common in KS2 exams: on a number line, starting from 0, jumping 1/5 each time, where do you land after 4 jumps? The answer is 4/5. Conversely, 4/5 is the accumulation of four 1/5’s. This “counting by jumps” method connects fractions to whole number counting, helping students transfer their existing whole number knowledge to the domain of fractions.

八、分数应用题：披萨、巧克力与日常生活中的等值分数 | Fraction Word Problems: Pizza, Chocolate, and Equivalent Fractions in Daily Life

分数在日常生活中的应用无处不在。最常见的例子是分享食物：如果一个披萨被切成 8 片，你吃了 2 片，那么你吃了 2/8，也就是 1/4 个披萨。如果一个巧克力棒有 12 小块，妹妹吃了 3 小块，她吃了 3/12，也就是 1/4。虽然一个用八分制、一个用十二分制，但都是 1/4 – 这就是等值分数在现实世界中的体现。

Fractions appear everywhere in daily life. The most common example is sharing food: if a pizza is cut into 8 slices and you eat 2 slices, you have eaten 2/8, which is 1/4 of the pizza. If a chocolate bar has 12 small pieces and your sister eats 3 pieces, she has eaten 3/12, which is also 1/4. Although one uses eighths and the other uses twelfths, both are 1/4 – this is equivalent fractions at work in the real world.

应用题是 KS2 四年级考试的重要题型。典型题目如：”萨姆和艾米各有一条同样长的巧克力棒。萨姆把自己的巧克力分成 4 等份，吃了 3 份。艾米把自己的巧克力分成 8 等份，吃了 6 份。谁吃得更多？”解答：萨姆吃了 3/4，艾米吃了 6/8。因为 3/4 和 6/8 是等值分数（分子分母同时乘以 2），所以两人吃得一样多。这类题目考察学生对等值分数的理解和应用能力。

Word problems are an important question type in KS2 Year 4 exams. A typical problem: “Sam and Amy each have a chocolate bar of the same size. Sam divides his chocolate into 4 equal parts and eats 3 parts. Amy divides her chocolate into 8 equal parts and eats 6 parts. Who eats more?” Solution: Sam eats 3/4, Amy eats 6/8. Since 3/4 and 6/8 are equivalent fractions (numerator and denominator both multiplied by 2), they eat the same amount. This type of problem tests students’ understanding and application of equivalent fractions.

更复杂的应用题可能涉及比较不同基准的分数。例如：”一个水壶装了 5/6 的水，另一个同样大小的水壶装了 3/4 的水。哪个水壶装的水更多？”通过通分（分母 12），5/6 = 10/12，3/4 = 9/12，所以 5/6 > 3/4。在解答这类题目时，画图辅助思考总是个好习惯 – 画出两个矩形，分别分成 6 份涂 5 份和分成 4 份涂 3 份，可以直观验证答案。

More complex word problems may involve comparing fractions with different benchmarks. For example: “Jug A is 5/6 full of water, Jug B of the same size is 3/4 full. Which jug has more water?” By finding a common denominator (12): 5/6 = 10/12, 3/4 = 9/12, so 5/6 > 3/4. When solving such problems, drawing a diagram to aid thinking is always a good habit – draw two rectangles, divide one into 6 parts and shade 5, divide the other into 4 parts and shade 3, to visually verify the answer.

九、家长辅导指南：在家轻松教孩子掌握等值分数 | Parent’s Guide: Teaching Your Child Equivalent Fractions at Home with Ease

作为家长，你不必是数学专家也能有效帮助孩子掌握分数概念。以下是一些经过验证的家庭辅导策略。首先，使用实物操作：切水果、折纸、分饼干都是极好的分数教学活动。将一个苹果切成 4 块，问孩子”你吃了 1 块，是几分之几？如果切成 8 块吃了 2 块呢？”让孩子亲手操作、亲眼观察，抽象概念就变得具体可感。

As a parent, you don’t need to be a math expert to effectively help your child master fraction concepts. Here are some proven home-tutoring strategies. First, use physical objects: cutting fruit, folding paper, and dividing cookies are all excellent fraction teaching activities. Cut an apple into 4 pieces and ask your child, “You ate 1 piece – what fraction is that? What if I cut it into 8 pieces and you ate 2?” Letting children manipulate objects and observe with their own eyes turns abstract concepts into concrete, tangible experiences.

其次，利用在线互动工具。有许多免费的数学网站提供虚拟分数墙和分数条，孩子可以拖拽操作，直观探索等值分数。第三，将分数融入日常对话：”我们已经走了路程的 1/3″、”这个蛋糕还剩下 2/5″、”你的作业完成了 3/4″。这些随口的表述让分数成为孩子日常生活的一部分，而不是只在数学课本上出现的”难题”。

Second, use online interactive tools. Many free math websites offer virtual fraction walls and fraction bars that children can drag and manipulate, intuitively exploring equivalent fractions. Third, weave fractions into everyday conversation: “We have completed 1/3 of the journey”, “There is 2/5 of the cake left”, “You have finished 3/4 of your homework”. These casual mentions make fractions part of your child’s everyday life, rather than “difficult problems” that only appear in maths textbooks.

最后，保持耐心和积极的态度。分数是许多孩子遇到的第一个真正抽象化的数学概念，需要时间来内化。如果孩子犯错了 – 比如认为 1/3 大于 1/2（因为 3>2） – 不要直接说”错了”，而是引导他们画图或使用实物来自己发现规律。错误是学习的机会，通过亲手验证来纠正误解，比简单记住”分母越大分数越小”要有效得多。

Finally, maintain patience and a positive attitude. Fractions are often the first truly abstract mathematical concept children encounter, and they need time to internalize. If your child makes a mistake – such as thinking 1/3 is larger than 1/2 (because 3>2) – don’t simply say “wrong”. Instead, guide them to draw a diagram or use physical objects to discover the pattern themselves. Mistakes are learning opportunities; correcting misconceptions through hands-on verification is far more effective than simply memorizing “the larger the denominator, the smaller the fraction”.

十、典型例题精讲：从基础到进阶的分步解析 | Worked Examples: Step-by-Step Analysis from Basic to Advanced

例题 1（基础）：写出 3/5 的两个等值分数。解答：将分子和分母同时乘以 2：3×2=6，5×2=10，得到 6/10。同乘以 3：3×3=9，5×3=15，得到 9/15。验证：3/5 = 6/10 = 9/15 = 0.6。

Example 1 (Basic): Write two equivalent fractions for 3/5. Solution: Multiply numerator and denominator by 2: 3×2=6, 5×2=10, giving 6/10. Multiply by 3: 3×3=9, 5×3=15, giving 9/15. Verification: 3/5 = 6/10 = 9/15 = 0.6.

例题 2（基础）：将 18/24 化简为最简分数。解答：找到 18 和 24 的公因数。18 的因数：1, 2, 3, 6, 9, 18。24 的因数：1, 2, 3, 4, 6, 8, 12, 24。最大公因数（GCF）是 6。18÷6=3，24÷6=4，所以最简分数是 3/4。

Example 2 (Basic): Simplify 18/24 to its simplest form. Solution: Find the common factors of 18 and 24. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The greatest common factor (GCF) is 6. 18÷6=3, 24÷6=4, so the simplest form is 3/4.

例题 3（中等）：将下列分数从小到大排列：2/5、3/10、4/5、1/2、7/10。解答：通分到分母 10（因为 5、10 和 2 的最小公倍数是 10）。2/5 = 4/10，3/10 = 3/10，4/5 = 8/10，1/2 = 5/10，7/10 = 7/10。比较分子：3 < 4 < 5 < 7 < 8。所以：3/10 < 2/5 < 1/2 < 7/10 < 4/5。

Example 3 (Medium): Order the following fractions from smallest to largest: 2/5, 3/10, 4/5, 1/2, 7/10. Solution: Convert to denominator 10 (the LCM of 5, 10, and 2 is 10). 2/5 = 4/10, 3/10 = 3/10, 4/5 = 8/10, 1/2 = 5/10, 7/10 = 7/10. Compare numerators: 3 < 4 < 5 < 7 < 8. Therefore: 3/10 < 2/5 < 1/2 < 7/10 < 4/5.

例题 4（进阶）：比较 5/8 和 7/12 的大小。解答：通分。8 和 12 的最小公倍数（LCM）是 24。5/8 = (5×3)/(8×3) = 15/24。7/12 = (7×2)/(12×2) = 14/24。因为 15/24 > 14/24，所以 5/8 > 7/12。也可以使用交叉乘法：5×12=60，7×8=56，60>56，所以 5/8 > 7/12。两者结果一致。

Example 4 (Advanced): Compare 5/8 and 7/12. Solution: Find a common denominator. The LCM of 8 and 12 is 24. 5/8 = (5×3)/(8×3) = 15/24. 7/12 = (7×2)/(12×2) = 14/24. Since 15/24 > 14/24, 5/8 > 7/12. Alternatively, use cross-multiplication: 5×12=60, 7×8=56, 60>56, so 5/8 > 7/12. Both methods give the same result.

Summary | 总结

分数是 KS2 四年级数学的重要基石。本文系统介绍了分数的基本概念 – 分子和分母的含义、等值分数的生成原理（分子分母同乘同除非零整数）、分数墙的直观应用、最简分数的化简方法（利用最大公因数约分）、以及分数比较的三个层次（同分母比较分子、同分子比较分母、不同分母通分后比较）。我们还探讨了单位分数与非单位分数的关系、分数在数轴上的排列技巧、以及日常生活中常见的分数应用题。掌握这些核心技能，学生不仅能够轻松应对 KS2 考试中的分数问题，更将为后续学习分数运算、小数、百分数和比例推理奠定坚实的数学基础。

Fractions are a crucial cornerstone of KS2 Year 4 mathematics. This article has systematically introduced the basic concepts of fractions – the meaning of numerator and denominator, the principle of generating equivalent fractions (multiplying or dividing both numerator and denominator by the same non-zero integer), the intuitive use of fraction walls, the method of simplifying fractions to their simplest form (using the greatest common factor to reduce), and the three levels of fraction comparison (same denominator: compare numerators; same numerator: compare denominators; different both: find a common denominator then compare). We have also explored the relationship between unit and non-unit fractions, techniques for ordering fractions on a number line, and common fraction word problems in daily life. By mastering these core skills, students will not only be able to confidently handle fraction problems in KS2 exams but will also build a solid mathematical foundation for subsequent learning in fraction operations, decimals, percentages, and proportional reasoning.

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