📚 Bearings and Scale Drawings in KS3 Cambridge Maths | KS3 剑桥数学：方位角与比例绘图
In the Cambridge KS3 mathematics curriculum, bearings and scale drawings bring together geometry, ratio and real-world measurement. A bearing describes a direction as a three-figure angle measured clockwise from north, while a scale drawing shows a real object or distance at a fixed ratio. This topic is common in checkpoint tests because it tests practical skills such as using a protractor, converting units and applying map scales.
在剑桥 KS3 数学课程中，方位角与比例绘图把几何、比和实际测量结合在一起。方位角用一个从正北顺时针测量的三位数角度来表示方向，而比例绘图则按照固定比例展示真实物体或距离。这个主题在 Checkpoint 考试中很常见，因为它考查使用量角器、换算单位和应用地图比例尺等实用技能。
1. What is a Bearing? | 什么是方位角？
A bearing is a way of describing the direction from one point to another. It is always measured from north in a clockwise direction and is written as a three-figure number. For example, due east is written as 090°, due south as 180°, due west as 270°, and due north as 000°.
方位角是描述从一个点到另一个点方向的一种方式。它总是从正北开始按顺时针方向测量，并写成三位数的形式。例如，正东写作 090°，正南写作 180°，正西写作 270°，正北写作 000°。
Bearings are useful in navigation, map reading and construction because they give an exact direction without needing words like 'north-east' or 'south-west'. A bearing of 045° means exactly halfway between north and east, while 225° means halfway between south and west.
方位角在航海、地图阅读和建筑绘图中很有用，因为它能给出准确方向，而不必使用“东北”或“西南”这样的词语。方位角 045° 表示正北与正东的中间方向，而 225° 表示正南与正西的中间方向。
2. Measuring Bearings with a Protractor | 用量角器测量方位角
To measure the bearing of B from A, first draw the north line at point A. Then place the centre of the protractor on A and line up 000° with the north line. Measure clockwise from the north line to the line segment AB. If the angle is less than 100°, add a leading zero so the bearing has three digits.
要测量从 A 到 B 的方位角，首先在点 A 处画出正北线。然后把量角器的中心放在点 A，并将 000° 与正北线对齐。从正北线顺时针测量到线段 AB。如果角度小于 100°，就在前面补一个零，使方位角成为三位数。
For example, if the angle from north to AB is 75° clockwise, the bearing is 075°. If the angle is 130°, the bearing is simply 130°. A common mistake is measuring anticlockwise or forgetting to use three figures.
例如，如果从正北到 AB 的顺时针角度是 75°，方位角就是 075°。如果角度是 130°，方位角就是 130°。常见的错误是逆时针测量，或者忘记使用三位数表示。
3. Three-Figure Bearings and Direction Rules | 三位数方位角与方向规则
All bearings must be written as three-figure numbers, so 8° becomes 008° and 95° becomes 095°. The rule makes bearings easy to read in navigation and avoids confusion with angles measured in other ways. When a bearing is given, it does not tell you the distance; it only tells you the direction.
所有方位角都必须写成三位数，因此 8° 写成 008°，95° 写成 095°。这条规则使方位角在导航中易于读取，也避免与其他方式测量的角度混淆。给出方位角时，它并不告诉你距离，只告诉你方向。
Bearings can be reversed. The bearing of A from B is not usually the same as the bearing of B from A. For example, if the bearing of B from A is 060°, then the bearing of A from B is 240° because you turn around and continue clockwise from north at B. In general, reversing a bearing means adding 180° if the original is less than 180°, or subtracting 180° if it is 180° or more.
方位角可以反向。从 B 到 A 的方位角通常与从 A 到 B 的方位角不一样。例如，如果从 A 到 B 的方位角是 060°，那么从 B 到 A 的方位角就是 240°，因为你在 B 点掉头后继续从正北顺时针旋转。一般来说，反向方位角的方法是：如果原方位角小于 180°，就加上 180°；如果大于或等于 180°，就减去 180°。
4. Drawing Bearings Accurately | 准确绘制方位角
To draw a bearing from a point, start by drawing a vertical north line through that point. Place your protractor with 000° pointing north and mark the required clockwise angle. Then draw a straight line through the mark. Use a ruler to make the line long enough to reach the target or another given point.
要从一个点绘制方位角，先通过该点画一条竖直的正北线。将量角器的 000° 对准正北，标出所需的顺时针角度，然后通过标记画一条直线。用直尺把线画得足够长，以到达目标或另一个已知点。
ng a scale of 1 cm = 2 km, you would draw 4 cm in the direction 120° to represent the ship's position.
如果还给出了比例尺，你可以沿着方位线标出实际距离。例如，一艘船以 120° 的方位角行驶了 8 km。使用 1 cm = 2 km 的比例尺，你需要在 120° 方向上画 4 cm 来表示船的位置。
5. Understanding Scale Drawings | 理解比例绘图
A scale drawing is an accurate representation of a real object or area drawn smaller or larger than the original. The scale tells you how the drawing length relates to the real length. For example, a scale of 1 : 100 means that 1 unit on the drawing represents 100 units in real life.
比例绘图是按原物体或区域缩小或放大后绘制的精确表示。比例尺告诉你绘图长度与实际长度的关系。例如，比例尺 1 : 100 表示图纸上的 1 个单位代表实际中的 100 个单位。
Scales are often written as ratios, such as 1 : 50000 on a map. They can also be written as statements, such as '1 cm represents 5 km'. The key skill is converting between the drawing length and the real length by multiplying or dividing by the scale factor.
比例尺通常写成比，例如地图上的 1 : 50000。它也可以写成文字说明，例如“1 cm 代表 5 km”。关键技能是通过乘以或除以比例因子，在图纸长度和实际长度之间进行换算。
6. Scale Factor and Map Scales | 比例因子与地图比例尺
m in real life. Since 25000 cm = 250 m, the map scale tells us that 1 cm represents 250 m.
比例因子是你将图纸长度乘以它来得到实际长度的数。如果比例尺是 1 : 25000，那么地图上的 1 cm 就是实际中的 25000 cm。因为 25000 cm = 250 m，所以这个地图比例尺告诉我们 1 cm 代表 250 m。
To convert a drawing length to a real length, use the formula:
要将图纸长度换算成实际长度，可使用以下公式：
Real distance = Drawing distance × Scale factor
To convert a real length to a drawing length, divide by the scale factor:
要将实际长度换算成图纸长度，则除以比例因子：
Drawing distance = Real distance ÷ Scale factor
: 200000 and the map distance is 3 cm, the real distance is 3 × 200000 = 600000 cm = 6 km.
换算单位时要格外小心。如果比例尺为 1 : 200000，地图距离为 3 cm，则实际距离为 3 × 200000 = 600000 cm = 6 km。
7. Converting Real Distances and Areas | 实际距离与面积换算
If a map scale is 1 : 100, then 1 cm² on the map represents 100² = 10000 cm² in real life.
使用地图比例尺进行距离换算很简单，因为它只涉及长度。然而，处理面积时，比例因子必须平方。如果地图比例尺是 1 : 100，那么地图上的 1 cm² 表示实际中的 100² = 10000 cm²。
= 2 × 25000000 = 50000000 cm². Converting to m², divide by 10000, so the area is 5000 m².
例如，一块田地在一张 1 : 5000 的地图上的面积是 2 cm²。实际面积是 2 × 5000² = 2 × 25000000 = 50000000 cm²。换算成 m² 时要除以 10000，所以面积是 5000 m²。
0 cm and area uses square units. This is a common source of error in checkpoint questions.
记住 1 m² = 10000 cm²，因为 1 m = 100 cm，而面积使用平方单位。这是 Checkpoint 题目中常见的错误来源。
8. Combining Bearings with Scale Drawings | 方位角与比例绘图的结合
Many exam questions ask you to use a bearing and a distance to plot a point on a scale drawing, or to work out the bearing and distance between two points already marked. This combines accurate protractor work with ruler measurement and scale conversion.
许多考试题目要求你利用方位角和距离在比例图上标出一个点，或者计算已标记的两点之间的方位角和距离。这结合了准确的量角器作图、直尺测量和比例换算。
at the tower, mark 210° clockwise, then measure 3 cm along that line because 9 ÷ 3 = 3 cm.
例如，一张藏宝图的比例尺是 1 cm = 3 km。一块岩石在塔的 210° 方位角上，距离 9 km。要标出岩石，先在塔处画正北线，顺时针标出 210°，然后沿该线量出 3 cm，因为 9 ÷ 3 = 3 cm。
Similarly, if two points are shown on a scale drawing, measure the angle from north at the starting point and the length between them. Then convert the length using the scale to find the real distance.
同样，如果比例图上已经标出两个点，则测量从起点正北方向到两点连线的角度以及它们之间的长度。然后使用比例尺换算出实际距离。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Several mistakes appear regularly when students work with bearings and scale drawings. The most common is measuring a bearing anticlockwise from north instead of clockwise. Always follow the clockwise direction and double-check with the three-figure rule.
学生在处理方位角和比例绘图时，经常会出现几个错误。最常见的是从正北逆时针测量方位角，而不是顺时针。一定要始终沿顺时针方向测量，并用三位数规则进行双重检查。
Another mistake is forgetting to write a bearing as three figures, for example writing 60° instead of 060°. This may not change the direction but it loses method marks. Also, when reversing a bearing, many students simply add 180° to any value, which is incorrect when the result exceeds 360°.
另一个错误是忘记将方位角写成三位数，例如写作 60° 而不是 060°。这可能不会改变方向，但会丢掉方法分。另外，在反向方位角时，许多学生对任何数值都简单地加上 180°，当结果超过 360° 时，这是错误的。
With scale drawings, mixing up multiplication and division is very common. If you know the drawing length and want the real length, multiply. If you know the real length and want the drawing length, divide. Use the formulas to check your operation.
在比例绘图中，把乘法和除法搞混是非常常见的。如果你知道图纸长度而想求实际长度，就乘以比例因子。如果你知道实际长度而想求图纸长度，就除以比例因子。用公式来检查你的运算。
10. Worked Example: Bearings and Distance | 例题：方位角与距离
point P. Using a scale of 1 cm = 2 km, draw the journey and find the bearing of H from P.
一艘船离开港口 H，沿 140° 方位角行驶 12 km 到达点 P。使用 1 cm = 2 km 的比例尺，绘制这次航行，并求从 P 到 H 的方位角。
6 cm. Draw a north line at H, measure 140° clockwise, and draw a line 6 cm long. Label the end P.
首先，绘图距离是 12 ÷ 2 = 6 cm。在 H 处画正北线，顺时针量出 140°，画一条 6 cm 长的线段，末端标记为 P。
is the reverse bearing of 140°. Since 140° + 180° = 320°, the bearing of H from P is 320°.
要求从 P 到 H 的方位角，在 P 处画一条新的正北线。从 P 到 H 的方位角是 140° 的反向方位角。因为 140° + 180° = 320°，所以从 P 到 H 的方位角是 320°。
This example shows how a scale drawing and a bearing calculation work together. In a checkpoint question, you might need to measure the drawn angle rather than calculate it, but the principle is the same.
这个例子展示了比例绘图和方位角计算如何结合使用。在 Checkpoint 题目中，你可能需要测量图中角度而不是计算，但原理是相同的。
11. Practice Skills for Checkpoint Success | 针对 Checkpoint 成功练习技能
To master this topic, practise drawing and measuring bearings with a protractor until you can do it quickly and accurately. Use squared paper or plain paper and check that your north lines are truly vertical and parallel to each other.
要掌握这个主题，请反复练习用量角器绘制和测量方位角，直到你能快速准确地完成。使用方格纸或白纸，并检查你的正北线是否真正竖直且彼此平行。
When working with scale drawings, always write the scale as a ratio or statement first. Then decide whether you need to multiply or divide. Underline the given units and convert them before calculating.
使用比例绘图时，首先要把比例尺写成比或文字说明。然后确定需要用乘法还是除法。在计算前给已知单位划线并先换算单位。
A useful revision exercise is to draw a simple journey with two or more legs, such as 5 km on a bearing of 060° followed by 8 km on a bearing of 150°. Plot both legs on a scale drawing and then measure the final bearing and distance from the starting point.
一个有用的复习练习是绘制包含两段或更多段的简单旅程，例如先沿 060° 方位角行驶 5 km，再沿 150° 方位角行驶 8 km。在比例图上画出两段旅程，然后测量从起点到终点的最终方位角和距离。
- Always write bearings as three digits.
- Always measure bearings clockwise from north.
- To reverse a bearing, add or subtract 180°.
- For lengths, multiply to find real distance and divide to find drawing distance.
- For areas, square the scale factor.
- 方位角始终写成三位数。
- 方位角始终从正北顺时针测量。
- 反向方位角加或减 180°。
- 长度换算：求实际距离用乘法，求图纸距离用除法。
- 面积换算：比例因子要平方。
12. Key Summary and Exam Tips | 核心总结与考试技巧
Bearings and scale drawings test your ability to link measurement, direction and proportional reasoning. The key formulas to remember are the real distance and drawing distance conversions, and the reverse bearing rule.
方位角与比例绘图考查你将测量、方向和比例推理联系起来的能力。要记住的关键公式是实际距离与图纸距离的换算，以及反向方位角规则。
In an exam, show your working clearly. Draw north lines fine and parallel, write all bearings in three figures, and label your scale drawing. Even if your final answer is wrong, correct method marks can be earned.
考试时，要清晰地写出解题步骤。正北线画细且平行，所有方位角写成三位数，并标注你的比例图。即使最终答案错误，正确的解题方法也能获得步骤分。
Before moving to the next question, check whether the question asks for a bearing or a distance. Bearing answers need degrees and three figures; distance answers need units such as km, m or cm. This simple check can prevent losing marks.
在进入下一题之前，检查题目问的是方位角还是距离。方位角答案需要度数和三位数；距离答案需要单位，如 km、m 或 cm。这个简单的检查可以防止丢分。
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