📚 Flow Charts | 流程图
Flow charts are a fundamental tool in algorithm design and process mapping, widely examined in Edexcel A‑Level Decision Mathematics. They provide a visual way to represent sequences of steps, decisions and flows, making complex logic easier to understand, communicate and debug. Mastering flow charts means being able to read a given chart, trace its execution with sample data, and construct clear, unambiguous diagrams from a written specification.
流程图是算法设计与流程映射中的基本工具，在 Edexcel A‑Level 决策数学中占有重要地位。它们以可视化的方式展现操作步骤、判断分支和流向，使复杂的逻辑更容易理解、交流和排错。掌握流程图意味着要能读懂已有的图表、用样本数据追踪其执行过程，并能根据文字描述画出清晰、无歧义的图。
1. What Is a Flow Chart? | 什么是流程图？
A flow chart is a diagram that uses standardised symbols connected by arrows to represent a process or algorithm. In Decision Mathematics, flow charts model the step‑by‑step logic of algorithms such as sorting, searching and optimisation routines. Unlike pseudocode, a flow chart emphasises the visual structure, which helps in spotting loops and branches at a glance.
流程图是一种使用标准符号、并用箭头连接起来表示某个过程或算法的图示。在决策数学中，流程图用于描述排序、查找和优化等算法的逐步逻辑。与伪代码不同，流程图强调可视化的结构，让人可以一眼看出循环和分支。
2. Standard Symbols and Their Meanings | 标准符号及其含义
The Edexcel specification uses a small set of unambiguous symbols. Knowing these by heart is essential for both interpreting and drawing flow charts in the exam.
Edexcel 考纲使用一套数量不多但意义明确的符号。熟记这些符号对考试中读图和画图都至关重要。
| Symbol | Name | Meaning | 中文 |
|---|---|---|---|
| 🏁 Oval / Rounded Rectangle | Start / End | Indicates the beginning or termination of the flow chart | 开始 / 结束 |
| ▭ Parallelogram | Input / Output | Shows data entering or leaving the process | 输入 / 输出 |
| ⬜ Rectangle | Process | Represents a calculation or an assignment, e.g. "x ← x + 1" | 处理步骤（如赋值或计算） |
| ⬥ Diamond | Decision | Used for a condition that leads to a Yes/No (or True/False) branch | 判断（条件分支） |
| → Arrow | Flow line | Shows the direction of logic flow | 流程线（表示执行方向） |
In Edexcel papers, these five symbols cover virtually all required diagrams. Occasionally a circle (connector) is used when splitting across pages, but on‑screen exams rarely ask for that.
在 Edexcel 试卷中，这五种符号几乎可以覆盖所有需要画的图。偶尔会用到圆圈（连接符）来跨页连接，但在机考中极少要求。
3. Start and End Terminators | 起止符号
Every flow chart must begin with a single 'Start' terminator and finish with at least one 'End' terminator. The start symbol usually contains the word 'Start' or 'Begin'. The end symbol holds 'End' or 'Stop'. These provide clear boundaries for tracing algorithms.
每一个流程图都必须以一个“开始”符号作为起点，并以至少一个“结束”符号作为终点。开始符号中通常写有“Start”或“Begin”，结束符号则写有“End”或“Stop”。这些符号为追踪算法提供了明确的边界。
In Decision Mathematics algorithms, the start box often initialises variables implicitly, while the end box simply signals completion. For instance, a bubble sort flow chart will start with a list of numbers and terminate when no swaps occur in a complete pass.
在决策数学的算法中，开始框通常隐含着变量的初始化，而结束框仅表示完成。例如，冒泡排序的流程图会从一个数列开始，并在某一轮完整扫描中没有发生交换时终止。
4. Input and Output Operations | 输入与输出操作
The parallelogram symbol is reserved for input and output steps. Typical input instructions include "INPUT N" or "READ list", while output steps may say "OUTPUT sum" or "PRINT 'Sorted'." In trace tables, these steps explain where data originates and what results are displayed.
平行四边形符号专门用于输入和输出步骤。典型的输入指令包括“INPUT N”或“READ list”，而输出步骤可以是“OUTPUT sum”或“PRINT 'Sorted'”。在追踪表中，这些步骤解释了数据的来源以及显示的结果。
When drawing a flow chart from a real‑world problem, it is important to list all required inputs at the beginning and outputs at the end. This habit prevents missing data flows and ensures the algorithm is complete.
在根据实际问题绘制流程图时，应当在开头列出所有需要的输入，并在结尾列出所有输出。这一习惯可以防止遗漏数据流，并确保算法的完整性。
5. Process Boxes and Assignments | 处理框与赋值语句
A rectangle box contains any arithmetic, logical or assignment operation. In Edexcel notation, an assignment is often written as "x ← x + 1" or "total ← total + value". The left‑pointing arrow shows the new value being stored in the variable on the left.
矩形框内包含任何算术、逻辑或赋值操作。在 Edexcel 表示法中，赋值通常写作“x ← x + 1”或“total ← total + value”。左向箭头表示新值被存入左侧变量。
Multiple process boxes can be placed sequentially. However, a single box should contain only one logical step. For example, do not try to put both a multiplication and an output in the same rectangle; split them into two boxes connected by an arrow.
多个处理框可以依次排列。但是，一个框内只应包含一个逻辑步骤。例如，不要试图把乘法和输出放在同一个矩形中，而是将它们分成两个框并用箭头连接。
6. Decision Diamonds and Branching | 判断菱形与分支
is the only symbol that creates alternative paths. A condition is written inside, such as "Is x ≤ 10?" or "swapped = True?" Two flow lines leave the diamond: one labelled 'Yes' (or 'True') and the other 'No' (or 'False'). The condition must be a question that has exactly two possible answers.
菱形是唯一能产生分支路径的符号。其内部写有条件，例如“x ≤ 10?”或“swapped = True?”。两条流程线从菱形引出：一条标记为“是”（或“True”），另一条标记为“否”（或“False”）。条件必须是一个恰好有两种可能答案的问题。
In algorithms like binary search, the decision diamond repeatedly checks whether the midpoint equals the target. Depending on the answer the search narrows to the left or right half. The clarity of the branching directly affects the traceability of the diagram.
在二分查找等算法中，判断菱形会反复检查中点是否等于目标值。根据答案，查找范围会缩小到左半部分或右半部分。分支的清晰程度直接影响到图表的可追踪性。
7. Flow Lines and Direction | 流程线的方向
Arrows connect all symbols and must indicate the direction of logic flow. Standard practice uses downward and rightward arrows; upward or leftward arrows are allowed but can make the chart harder to read. When returning to an earlier point (a loop), a line goes back upwards, but it should still be clearly arrowed.
箭头连接所有符号，并且必须标明逻辑流向。通常惯例是向下和向右画箭头；向上或向左的箭头虽然允许，但可能会降低可读性。当需要返回前面某一点（形成循环）时，线条向上画回，但仍应明确标出箭头。
Crossing lines should be avoided whenever possible. If a flow line must cross another, normal Edexcel practice is to use a small bridge or simply ensure the lines do not intersect ambiguously. Neatness is part of the assessment.
应尽可能避免线条交叉。如果流程线必须交叉，Edexcel 常见做法是使用一个小桥拱，或者确保线条不会产生歧义的交错。卷面整洁也是评分的一部分。
8. Constructing a Flow Chart from an Algorithm | 根据算法构建流程图
When given a written algorithm or a problem statement, start by identifying the inputs, main process and outputs. Draft the flow chart in rough using the standard symbols. Every loop must have a decision diamond that controls exit; indefinite iteration is represented by a decision that checks a condition after each iteration.
当给出一个书面算法或问题描述时，首先要识别输入、主要处理过程和输出。使用标准符号粗略画出流程图。每个循环都必须有一个控制退出的判断菱形；无限循环由每次迭代后检查条件的判断来表示。
Consider this simple algorithm: " integer N. Set factorial = 1. For i from 1 to N multiply factorial by i. Output factorial." The flow chart would contain: Start → Input N → Process factorial ← 1 → Process i ← 1 → Decision 'Is i ≤ N?' → Yes path leads to 'factorial ← factorial × i' and 'i ← i + 1', then loop back to the decision. The No path goes to 'Output factorial' → End. Notice the use of a count‑controlled loop expressed as a decision and an increment.
考虑一个简单算法：“输入正整数 N。设 factorial = 1。对于 i 从 1 到 N，让 factorial 乘以 i。输出 factorial。”其流程图将包含：开始 → 输入 N → 处理 factorial ← 1 → 处理 i ← 1 → 判断“i ≤ N?” → “是”分支执行“factorial ← factorial × i”和“i ← i + 1”，然后返回判断框；“否”分支走向“输出 factorial” → 结束。注意这里用判断和递增表达了一个计数控制的循环。
9. Tracing a Flow Chart with a Trace Table | 用追踪表执行流程图
Trace tables are a systematic way to test a flow chart by recording the values of variables step‑by‑step. Each column represents a variable or an output. Each row records the state after executing a process box or a decision. This dry‑run technique is frequently required in Edexcel exam questions.
追踪表是一种通过逐步记录变量值来测试流程图的系统方法。每一列代表一个变量或输出，每一行记录了执行一个处理框或判断后的状态。这种干运行技术在 Edexcel 考试题目中经常被要求使用。
When tracing, follow the flow lines strictly. For a decision diamond, write the condition's truth value (Y/N) in a separate column if needed. Stop when you reach an End terminator. A well‑laid trace table helps you spot logical errors without running actual code.
追踪时，要严格遵循流程线。对于判断菱形，必要时可将条件的真假值（Y/N）写在单独一列中。到达结束符号时停止。一个布局良好的追踪表可以帮助你在没有实际运行代码的情况下发现逻辑错误。
For the factorial example with N = 3, a partial trace table would be:
| Step | N | i | factorial | i ≤ N? | Output |
|---|---|---|---|---|---|
| 1 | 3 | – | – | – | – |
| 2 | 3 | 1 | 1 | – | – |
| 3 | 3 | 1 | 1 | Y (1≤3) | – |
| 4 | 3 | 2 | 1×1=1? Wait, factorial update: after process factorial ← factorial×i, then i←i+1. Let's step carefully: after first loop: factorial=1×1=1, i becomes 2. | – | – |
(Note: A complete trace would continue until i=4, showing factorial 6 and then output.)
(注：完整追踪会继续执行，直到 i=4，显示 factorial=6，然后输出。)
Practising trace tables on past‑paper flow charts is one of the most effective revision strategies. Always label columns clearly and use a ruler to keep rows distinct if drawn by hand.
在历年真题的流程图上练习追踪表是最有效的复习策略之一。请务必标清列标题，若手绘则借助尺子保持行间距清晰。
10. Common Pitfalls in Drawing and Interpreting Flow Charts | 画图与读图的常见误区
Examiners' reports highlight several recurring mistakes. The most frequent is a missing decision box for a loop, causing an infinite cycle that never reaches End. Another is forgetting to label the 'Yes' and 'No' branches, making the logic ambiguous. Also, students sometimes use a process box when a decision is required, or place multiple, unrelated operations in a single symbol.
考官报告指出了几类反复出现的错误。最常见的是循环缺少判断框，导致无限循环而永远无法到达结束。另一常见错误是忘记为“是”和“否”分支添加标签，使逻辑含糊不清。此外，学生有时在本该使用判断框的地方用了处理框，或者将多个不相关的操作写在同一符号内。
Input validation is another area where flow charts go wrong. If an algorithm needs to reject invalid data (e.g. a negative age), a decision loop should check the input and ask again. A simple sequence of boxes cannot enforce re‑entry; it merely processes whatever is given.
输入验证是流程图容易出错的另一个地方。如果算法需要拒绝无效数据（如负数的年龄），应当用一个判断循环来检查输入并重新要求输入。简单的顺序框无法强制重新输入，它只会直接处理给定的值。
11. Flow Charts for Standard Decision Mathematics Algorithms | 决策数学标准算法的流程图
Edexcel D1 expects familiarity with flow charts for bubble sort, quick sort, binary search and bin‑packing algorithms. For bubble sort, the chart includes nested loops: an outer loop tracking passes, an inner loop comparing adjacent items, and a decision checking for swaps. The algorithm terminates when a complete pass makes zero swaps.
Edexcel D1 要求熟悉冒泡排序、快速排序、二分查找和装箱算法的流程图。以冒泡排序为例，其流程图包含嵌套循环：外层循环跟踪趟数，内层循环比较相邻元素，并用一个判断检查是否发生过交换。当完整一趟交换次数为零时，算法终止。
Binary search flow charts are characterised by a decision diamond that compares the midpoint with the target, then adjusts low or high pointers. The loop exits either when the target is found or when low > high. The clarity of such flow charts is often tested with a trace table on an ordered list.
二分查找的流程图特点在于使用判断菱形比较中点和目标值，然后调整 low 或 high 指针。循环在找到目标或 low > high 时退出。这类流程图的清晰度常通过有序列表上的追踪表来考查。
12. Exam Tips for Flow Chart Questions | 流程图题目的考试技巧
In the Edexcel exam, you may be asked to complete a partially drawn flow chart, spot an error in a given chart, or draw a chart from scratch. Start by carefully reading the algorithm description and underlining key words such as 'repeat', 'until', 'while', 'if'. These indicate loops and decisions. Use pencil for the rough version, then finalise with a black pen if the exam is paper‑based; on‑screen tests expect clear digital drawing.
在 Edexcel 考试中，你可能会被要求补全一个部分画好的流程图，找出给定流程图中的错误，或者从零开始画图。首先要仔细阅读算法描述，并在“repeat”“until”“while”“if”等关键词下划线，这些词暗示循环和判断。铅笔起稿，纸质考试最后用黑笔定稿；机考则需要清晰的数字绘图。
When drawing, keep the diagram spaced out and symmetrical. Use standard symbols only. After finishing, perform a mental dry‑run with a small data set to verify the logic. Time management is essential: if a question asks for a flow chart and a trace table, allocate roughly 40% of the marks' time to the chart and 60% to the table, as the table often carries more detail marks.
画图时，保持图面宽松、对称。仅使用标准符号。完成后，用一小笔数据在脑中干运行以验证逻辑。时间管理很重要：如果一道题同时要求流程图和追踪表，按分值比例将约 40% 的时间分配给流程图，60% 分配给追踪表，因为表格通常占分更细。
Finally, remember that Edexcel values clarity over artistic beauty. Every symbol must be recognisable, and every arrow must have a clear direction. A correct but untidy flow chart may lose marks for ambiguity, so practice neat sketching under timed conditions.
最后，记住 Edexcel 看重的是清晰而非美术效果。每个符号都必须可辨认，每条箭头都必须方向明确。一个正确但潦草的流程图可能因歧义而失分，因此要在限时条件下练习整洁地绘制。
Published by TutorHao | Decision Mathematics Revision Series | aleveler.com
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