📚 IGCSE Mathematics: Graphs and Kinematics (Teacher's Guide Unit 300) | IGCSE 数学：图像与运动学（教师用书 G-K 单元 300）
This teacher's guide unit covers the essential links between algebraic graphs and kinematics in the IGCSE Mathematics syllabus. It focuses on interpreting distance-time and speed-time graphs, connecting gradients and areas to motion, and using graph shapes to solve equations. The aim is to build both procedural fluency and conceptual understanding for Core and Extended candidates.
本教师用书单元涵盖 IGCSE 数学大纲中代数图像与运动学之间的核心联系。重点包括解读距离-时间图和速度-时间图，将斜率与面积同运动联系起来，以及利用图像形状求解方程。目标是帮助核心和拓展考生同时建立解题熟练度与概念理解。
1. Coordinates and the Cartesian Plane | 坐标与笛卡尔平面
The Cartesian plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Every point has coordinates (x, y), where x gives the horizontal position and y gives the vertical position from the origin (0, 0).
笛卡尔平面由两条互相垂直的数轴组成：x 轴（水平）和 y 轴（垂直）。每个点都有坐标 (x, y)，其中 x 表示相对于原点 (0, 0) 的水平位置，y 表示垂直位置。
Plotting points accurately is a prerequisite for drawing graphs. The signs of x and y determine the quadrant: (+, +) is the first quadrant, (−, +) the second, (−, −) the third, and (+, −) the fourth.
准确描点是绘制图像的前提。x 和 y 的符号决定象限：(+, +) 为第一象限，(−, +) 为第二象限，(−, −) 为第三象限，(+, −) 为第四象限。
When reading coordinates from a graph, always check the scale on each axis. A point such as (2, 3) means moving 2 units along the x-axis and 3 units up the y-axis.
从图像读取坐标时，始终检查每个坐标轴上的刻度。例如点 (2, 3) 表示沿 x 轴移动 2 个单位，并沿 y 轴向上移动 3 个单位。
2. Straight-Line Graphs: y = mx + c | 直线图像：y = mx + c
s the gradient and c is the y-intercept. The graph is a straight line crossing the y-axis at (0, c).
任何非竖直直线都可以写成 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。图像是一条在 (0, c) 处穿过 y 轴的直线。
To draw a straight line, find three points by substituting x-values into the equation. Three points include a check for arithmetic errors; a straight line should pass through all three.
要绘制直线，代入 x 值求出三个点。三个点包含算术错误的检验；一条直线应穿过全部三个点。
and 2 to obtain (0, 1), (1, 3), and (2, 5). Plot these points and join them with a ruler.
例如，要绘制 y = 2x + 1，代入 x = 0、1 和 2，得到 (0, 1)、(1, 3) 和 (2, 5)。描出这些点并用直尺连接它们。
3. Gradient and Intercept | 斜率与截距
e gradient rises from left to right, a negative gradient falls, a zero gradient is horizontal, and an undefined gradient is vertical.
斜率 m 衡量陡峭程度：m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)。正斜率从左到右上升，负斜率下降，零斜率为水平，未定义斜率为竖直。
f y when x = 0. If a line passes through the origin, its equation is y = mx because c = 0.
y 轴截距 c 是 x = 0 时 y 的值。如果直线经过原点，其方程为 y = mx，因为 c = 0。
To find the gradient from two points, subtract the y-coordinates and divide by the difference in x-coordinates. Keep the order the same in numerator and denominator.
要从两个点求斜率，先求 y 坐标之差，再除以 x 坐标之差。分子和分母中保持相同的顺序。
4. Distance-Time Graphs | 距离-时间图像
ing point over time. The gradient of the graph gives the speed: speed = Δdistance / Δtime.
距离-时间图像显示物体从起点出发所行驶的距离随时间的变化。图像的斜率表示速度：速度 = 距离变化量 ÷ 时间变化量。
A horizontal section means the object is stationary because distance is not changing. A steeper straight section indicates a higher constant speed, while a curved section indicates acceleration or deceleration.
水平段表示物体静止，因为距离没有变化。较陡的直线段表示较高的匀速，而曲线段
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