📚 Graphs and Kinematics (G-K) | 图像与运动学（G-K）
Graphs and kinematics form an essential part of the IGCSE Mathematics syllabus. Students learn to translate motion into visual form and interpret the shape of a graph to describe real-world events. This teacher’s guide provides a structured approach to teaching distance-time and speed-time graphs, including common misconceptions, worked examples, and classroom strategies.
图像与运动学是 IGCSE 数学大纲中非常重要的一部分。学生需要学会将运动转化为图像，并通过图像形状来描述现实事件。本教师用书提供了讲授距离-时间图和速度-时间图的结构化方法，包括常见误区、例题和课堂策略。
1. Learning Outcomes and Prerequisites | 学习目标与前置要求
By the end of this unit, students should be able to: interpret a distance-time graph; calculate speed from the gradient; interpret a speed-time graph; find acceleration from the gradient; find distance from the area under the graph; and solve problems involving multi-stage journeys.
学完本单元后，学生应能够：解读距离-时间图；由梯度求速度；解读速度-时间图；由梯度求加速度；由图下方面积求距离；并解决包含多阶段运动的问题。
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Ratio and proportion
比与比例
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Gradient of a straight line
直线的斜率
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Area of rectangles, triangles and trapeziums
矩形、三角形和梯形的面积
2. The Distance-Time Graph | 距离-时间图
A distance-time graph shows how far an object is from a fixed point over time. Time is always plotted on the horizontal axis and distance on the vertical axis.
距离-时间图表示物体在一段时间内与固定点的距离变化。时间始终画在横轴上，距离画在纵轴上。
The gradient (slope) of a distance-time graph represents speed. A horizontal line means the object is stationary. A steeper line means a faster speed. If the gradient is negative, the object is returning towards the starting point.
距离-时间图的梯度（斜率）代表速度。水平线表示物体静止。直线越陡，速度越快。若梯度为负，则物体正在返回起点。
speed = distance ÷ time = gradient of distance-time graph
Example: A cyclist travels 30 m in 10 s, then stops for 5 s, then travels 20 m in 4 s. The first segment has gradient 3 m/s, the second segment gradient 0, and the third segment gradient 5 m/s.
例如：一名骑车人 10 秒内行驶 30 米，然后停下 5 秒，再在 4 秒内行驶 20 米。第一段梯度为 3 m/s，第二段梯度为 0，第三段梯度为 5 m/s。
3. Gradients as a Rate of Change | 梯度：变化率
The concept of gradient is central. For a straight-line segment between two points (t₁, d₁) and (t₂, d₂), the gradient is (d₂ – d₁) / (t₂ – t₁).
梯度是核心概念。对于直线上两点 (t₁, d₁) 和 (t₂, d₂)，梯度为 (d₂ – d₁) / (t₂ – t₁)。
A positive gradient shows motion away from the origin; a negative gradient shows motion back toward it; zero gradient shows no motion. The unit of the gradient is distance unit per time unit, such as m/s or km/h.
正梯度表示远离起点；负梯度表示返回起点；零梯度表示静止。梯度的单位是距离单位每时间单位，例如 m/s 或 km/h。
4. Speed-Time Graphs | 速度-时间图
In a speed-time graph, speed is on the vertical axis and time is on the horizontal axis. The gradient of a speed-time graph gives acceleration. A horizontal line therefore means constant speed, not rest.
在速度-时间图中，速度在纵轴上，时间在横轴上。速度-时间图的梯度表示加速度。因此，水平线表示匀速运动，而不是静止。
The area between the graph and the time axis gives the total distance travelled. This is the main difference from a distance-time graph, where the gradient gives speed and no area interpretation is used.
速度-时间图与时间轴之间的面积表示总路程。这是与距离-时间图的主要区别：距离-时间图中梯度给出速度，而不使用面积。
5. Acceleration and Deceleration | 加速度与减速度
a = (v – u) / t
Here, u is the initial speed, v is the final speed and t is the time taken. The standard unit is m/s².
其中，u 是初速度，v 是末速度，t 是所用时间。标准单位是 m/s²。
A positive gradient means acceleration, while a negative gradient means deceleration. In IGCSE questions, students should state the sign clearly. For example, if a car slows from 15 m/s to 5 m/s in 2 s, the acceleration is (5 – 15) / 2 = -5 m/s², so the deceleration is 5 m/s².
正梯度表示加速，负梯度表示减速。在 IGCSE 题目中，学生应清楚说明正负号。例如，如果汽车在 2 秒内从 15 m/s 减速到 5 m/s，则加速度为 (5 – 15) / 2 = -5 m/s²，因此减速度为 5 m/s²。
6. Area Under a Speed-Time Graph | 速度-时间图下方的面积
To find distance from a speed-time graph, find the area between the graph and the time axis. For constant speed, distance = speed × time, so the area is a rectangle.
要从速度-时间图求距离，需要计算图像与时间轴之间的面积。匀速时，距离 = 速度 × 时间，因此面积为矩形。
For an accelerating segment, the graph is a triangle or a trapezium. Use the standard area formulas.
对于加速段，图像为三角形或梯形。请使用标准面积公式。
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