📚 Decision Maths 1 Mark Scheme High-Scoring Tips | D1 评分标准高分技巧
In Edexcel A-Level Further Mathematics, Decision Maths 1 (D1) often feels different from pure or mechanics modules. Success depends not only on finding a correct answer but also on demonstrating a clear, logical method that matches the mark scheme. Examiners award marks for individual steps, correct terminology, and accurate tracing of algorithms. Understanding how these marks are allocated can transform a borderline result into a top score.
在爱德思 A-Level 进阶数学中，决策数学 1（D1）往往与纯数或力学模块感觉不同。要拿到高分，不仅要得出正确答案，还需要展示清晰、符合评分标准的逻辑步骤。考官会为每一步骤、正确的术语使用以及准确的算法追踪分配分数。理解这些分数的分配方式，可以把边缘成绩提升为顶尖分数。
1. Understanding the Mark Scheme | 理解评分标准
The D1 mark scheme is built around method marks (M), accuracy marks (A), and sometimes independent marks (B). A method mark requires you to show a correct process, even if a numerical slip occurs later. An accuracy mark depends on obtaining the right value after a valid method. Never skip working lines – a missing stage can lose an M mark that would otherwise be earned.
D1 的评分标准围绕方法分（M）、准确分（A）以及有时出现的独立分（B）构建。方法分要求你展示正确的过程，即使后续出现数值错误也可得分。准确分取决于在有效方法后得到正确的数值。绝对不要跳步——缺少一个步骤可能导致丢失本可拿到的方法分。
- English: Always write down the initialisation step of an algorithm (e.g., ‘Start at vertex A’ or ‘Unvisited set = {A,B,…}’). Marks are often awarded for this setup.
- 中文： 始终写下算法的初始化步骤（例如“从顶点 A 开始”或“未访问集合 = {A, B, …}”）。这一步通常有对应的分值。
- English: If you make an error, you can still earn follow-through (ft) marks provided your subsequent working is consistent with the mistake. Label your values clearly so the examiner can follow your logic.
- 中文： 如果你犯了错误，只要后续计算与错误保持一致，仍可获得跟进分（ft）。请清晰标注你的数值，让考官能跟随你的思路。
2. Algorithm Tracing: Show Every Step | 算法追踪：展示每一步
D1 algorithms such as Dijkstra, Prim, or the binary search must be traced meticulously. Examiners want to see the state of lists, tables, or priority queues after each pass. A single missing row in a Dijkstra table can cost multiple marks, because the entire table is marked holistically.
D1 中的算法如 Dijkstra、Prim 或二分搜索必须仔细追踪。考官希望看到每次迭代后列表、表格或优先队列的状态。Dijkstra 表格中缺少一行就可能丢失多分，因为整个表格是整体评分的。
A typical Dijkstra table heading is: Vertex | Status | Shortest distance from S | Previous vertex. Fill it in line by line. Use ∞ for infinity and write working values at the vertices. For Prim’s algorithm on a matrix, show the chosen column and row deletion step-by-step.
典型的 Dijkstra 表头为：顶点 | 状态 | 从 S 出发的最短距离 | 前驱顶点。逐行填写。用 ∞ 表示无穷大，并在顶点处写下工作值。对矩阵上的 Prim 算法，要逐步展示所选列和行的删除。
| Vertex | Status | Dist from A | Previous |
|---|---|---|---|
| A | perm | 0 | – |
| B | temp | 4 | A |
| C | temp | ∞ | – |
The above snippet would be one step. Always update the distance when a shorter path is found and relabel the vertex as ‘perm’ (permanent) once finalised. Missing a status change loses the corresponding mark.
上表为一步快照。一旦找到更短路径就更新距离，并将顶点状态改为“perm”（永久）。漏掉状态变化会丢失相应分数。
3. Sorting Algorithms: Efficiency and Accuracy | 排序算法：效率与准确
For questions on bubble sort, shuttle sort, or quick sort, marks are given for passes and comparisons. When performing a bubble sort, write ‘Comparison 1’, ‘Swap’ or ‘No swap’, and list the modified list after each pass. A quick sort requires clear selection of pivots, sublists, and recombination.
对于冒泡排序、穿梭排序或快速排序的问题，分数会给在趟数和比较上。执行冒泡排序时，写下“比较 1”、“交换”或“不交换”，并在每一趟后列出修改后的列表。快速排序要求清晰选择枢轴、划分子列表并重组。
Example Bubble Sort pass: 7 4 9 2 → compare 7,4 (swap) → 4 7 9 2 → compare 7,9 (no swap) → 4 7 9 2 → compare 9,2 (swap) → 4 7 2 9.
冒泡排序示例：7 4 9 2 → 比较7和4（交换）→ 4 7 9 2 → 比较7和9（不交换）→ 4 7 9 2 → 比较9和2（交换）→ 4 7 2 9。
In the mark scheme, the final sorted list alone is not enough; you must show the sequence of comparisons and swaps. When counting comparisons and swaps for efficiency questions, label the total clearly at the end.
在评分标准中，仅给出最终的排序列表是不够的；你必须展示比较和交换的过程。在效率问题中计算比较次数和交换次数时，在结尾清晰标注总数。
4. Graph Algorithms: Prim, Kruskal and Dijkstra | 图算法：Prim、Kruskal 和 Dijkstra
Prim’s algorithm using a distance matrix requires crossing out the initial row, then repeatedly choosing the smallest entry in the remaining columns, crossing out the chosen row, and listing the edge. Kruskal’s algorithm requires sorting edges by weight, then selecting edges in order while avoiding cycles. The mark scheme demands a list of edges in the order they are selected, with a note of any rejected edge and why.
使用距离矩阵的 Prim 算法要求先划掉起始行，然后反复选择剩余列中的最小元素，划掉所选行，并列出边。Kruskal 算法要求先按权重排序边，然后依次选择边同时避免回路。评分标准要求按选择顺序列出边，并注明任何被拒绝的边及原因。
For Dijkstra’s algorithm, always give the final value at each vertex and write the route (e.g., 36 = A → D → F → H). If a question asks for the shortest path, you must state the length and the sequence of vertices. A common error is to write only the length, losing the A mark for the route.
对于 Dijkstra 算法，务必给出每个顶点的终值和路径（如 36 = A → D → F → H）。如果题目要求最短路径，你必须同时写出长度和顶点序列。常见错误是只写长度，丢失路径的准确分。
5. Critical Path Analysis: Float and Cascade | 关键路径分析：浮时与级联图
Precedence tables must be translated correctly into an activity-on-arc network. Dummy activities are frequently needed to maintain logical dependencies. The mark scheme looks for correct direction, a unique start and finish node, and no danglers. Once early and late event times are calculated, the critical path(s) must be identified, and the float of each activity should be stated clearly.
前导表必须正确转化为箭线图网络。常需引入虚活动以保持逻辑依赖。评分标准关注方向正确、唯一的开始与结束节点，以及没有悬挂节点。一旦计算出事件的最早和最晚时间，就必须标明关键路径，并清晰给出每个活动的浮时。
Calculate total float using: Total float = LFThead – ESTtail – duration. In the exam, present your results in a table or a box next to the activity. Missing one critical activity can cost several marks, so double-check by tracing the longest path.
计算总浮时用：总浮时 = 头部最晚时间 − 尾部最早时间 − 持续时间。考试中在表格或活动旁的方框内呈现结果。漏掉一个关键活动会损失多分，因此通过追踪最长路径反复检查。
6. Linear Programming: Feasible Region and Ruler Method | 线性规划：可行域与直尺法
Formulate constraints correctly and label all lines. The mark scheme awards marks for drawing each line accurately, shading the unwanted region, and clearly indicating the feasible region. When using the objective line method, draw a line of equal profit (e.g., P = 3x + 2y = 0), then slide it parallel to the optimal vertex.
正确列出约束并为所有直线标注。评分标准为每条直线绘制准确、阴影画出不可行区域并清晰标出可行域而给分。使用目标函数线方法时，画出等值线（如 P = 3x + 2y = 0），然后平行移动至最优顶点。
If you use a vertex test, you must show substitution of each vertex coordinate into the objective function. Write down the calculated value for each vertex and conclude with the maximum/minimum point. Using a ruler to find the optimal integer point near the boundary also requires clear documentation.
若使用顶点测试，必须展示每个顶点坐标代入目标函数的过程。写下每个顶点的计算值，并得出结论。使用直尺寻找边界附近最优整数点也需清晰记录。
7. Matching and Allocation: Bipartite Graphs | 匹配与分配：二分图
In matching problems, whether applying the Hungarian algorithm for allocation or using the augmenting path algorithm for maximum matching, the initial matching must be stated. When using alternating paths, write ‘unmatched →’ and show the alternating sequence. The mark scheme rewards a clear record of status changes (unmatched/matched).
在匹配问题中，无论是运用匈牙利算法进行分配还是用增广路径算法求最大匹配，都必须给出初始匹配。使用交替路径时，写下“未匹配 →”并展示交替序列。评分标准会奖励对状态变化（未匹配/匹配）的清晰记录。
For the Hungarian algorithm, the key steps: subtract row minima, then column minima, then cover zeros with the minimum number of lines. If an optimal assignment is not reached, augment using the smallest uncovered element. Present each reduced cost matrix in a table to gain all M and A marks.
对于匈牙利算法，关键步骤为：减去行最小值，再减去列最小值，然后用最少条直线覆盖所有零。若未能达到最优分配，则利用最小未覆盖元素进行调整。将每个约简矩阵用表格呈现以获取全部方法和准确分。
8. Travelling Salesman Problem: Bounds and Heuristics | 旅行商问题：界限与启发式
Finding an initial upper bound using the nearest neighbour method requires showing the route from the start vertex, adding the closest unvisited vertex each step, and returning to the start. List the edge weights added sequentially. The mark scheme also expects you to state the total length clearly.
用最近邻法求初始上界需要展示从起始顶点出发的路线，每一步添加最近的未访问顶点，最后返回起点。依次列出添加的边权。评分标准还期望你明确写出总长度。
For a lower bound, delete a vertex, find the minimum spanning tree of the remaining network, then add the two smallest edges from the deleted vertex. Show the MST edges, their weights, and the final sum. Many candidates lose marks by not writing the two addition edges separately.
求下界时，删除一个顶点，求剩余网络的最小生成树，然后加上从被删顶点出发的两条最小边。展示最小生成树的边、权重以及最终总和。许多考生因未单独列出两条附加边而丢分。
9. Scheduling and Resource Histograms | 调度与资源直方图
When constructing a Gantt chart from an activity network, schedule activities using their earliest start times subject to resource constraints. A clear key and consistent time scale are essential. The mark scheme checks whether the correct number of workers is used and whether all activities are placed within their total float.
根据活动网络构建甘特图时，在资源限制下按照最早开始时间排程活动。清晰的图例和一致的时间尺度至关重要。评分标准会检查是否使用了正确数量的工人，以及所有活动是否在各自总浮时内安排。
For the resource histogram, tally the number of workers needed in each time interval. Present the histogram as a bar chart, and if you need to reschedule to reduce the peak, show the new Gantt chart and the smoothed histogram. Always indicate the minimum number of workers achievable.
对于资源直方图，统计每个时间区间所需的工人数。以条形图形式呈现；若需重新排程以降低峰值，展示新的甘特图和平滑后的直方图。始终标明可实现的最小工人数。
10. Exam Technique and Common Pitfalls | 考试技巧与常见错误
Read the question carefully for the specific algorithm variant: e.g., Prim on a matrix or on a network; quick sort with the middle-left pivot or the first element. Using the wrong pivot can invalidate an otherwise correct process. Always underline or circle the chosen element in each step.
仔细读题，留意所需的具体算法变体：如 Prim 使用矩阵还是网络；快速排序选择左中位枢轴还是第一个元素。选错枢轴会使本来正确的过程无效。每一步都应在所选元素下划线或画圈。
Common pitfalls: omitting the final working value on a Dijkstra vertex; forgetting to write ‘STOP’ in a bubble sort when no swaps occur; mixing up EST and LFT in float formulas; not shading the feasible region correctly; or failing to state the reason an edge is rejected in Kruskal. Checking these small details can secure 5–10 extra marks.
常见错误：漏写 Dijkstra 顶点的最终工作值；冒泡排序未发生交换时忘记写“停止”；在浮时公式中混淆最早开始时间和最晚完成时间；未正确用阴影表示可行域；Kruskal 算法中忘记声明拒绝边的理由。检查这些细节可多拿 5–10 分。
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