📚 Bearings and Scale Drawings | 方位与比例绘图
Bearings and scale drawings are practical tools in KS3 Cambridge mathematics. They connect direction, measurement, and ratio to real-world maps and navigation.
方位与比例绘图是剑桥 KS3 数学中的实用工具。它们将方向、测量和比例与现实世界的地图和导航联系起来。
1. Understanding Bearings | 理解方位角
A bearing is a three-figure angle measured clockwise from the north line.
方位角是从正北方向按顺时针测量的三位数角度。
Bearings always use three digits. For example, 60° is written as 060°, and 8° is written as 008°.
方位角总是使用三位数字。例如，60° 写作 060°，8° 写作 008°。
The bearing of north is 000°, east is 090°, south is 180°, and west is 270°.
正北的方位角是 000°，正东是 090°，正南是 180°，正西是 270°。
000° ≤ bearing < 360°
2. Measuring Bearings with a Protractor | 用量角器测量方位角
- Draw a north line from the starting point. 从起点画一条正北线。
- Place the protractor centre on the starting point and align 0° with north. 将量角器中心放在起点，并使 0° 与正北对齐。
- Measure clockwise to the target direction. 按顺时针方向测量到目标方向。
- Write the answer as three figures. 将答案写成三位数。
Always check that you are measuring the clockwise angle, not the smaller anticlockwise angle.
务必检查你测量的是顺时针角度，而不是较小的逆时针角度。
3. Working with Reverse Bearings | 反方位角计算
A reverse bearing is the bearing from the destination back to the starting point.
反方位角是从终点返回起点的方位角。
If the original bearing is less than 180°, add 180°. If it is 180° or more, subtract 180°.
如果原方位角小于 180°，则加 180°；如果原方位角大于等于 180°，则减 180°。
Back bearing = (Bearing + 180°) mod 360°
For example, a bearing of 075° has a reverse bearing of 075° + 180° = 255°.
例如，方位角 075° 的反方位角是 075° + 180° = 255°。
4. Scale Drawings and Maps | 比例绘图与地图
A scale drawing is a drawing that shows a real object with its sizes reduced or enlarged by a fixed ratio.
比例绘图是以固定比例缩小或放大实际物体的图纸。
The scale is usually written as a ratio, such as 1 : 50000. This means 1 cm on the map represents 50000 cm in real life.
比例尺通常写成比的形式，如 1 : 50000。这意味着地图上的 1 cm 代表实际中的 50000 cm。
Scale = map distance : actual distance = 1 : n
5. Converting Map Distances to Real Distances | 将地图距离换算为实际距离
To find the actual distance, multiply the map distance by the scale factor n.
要计算实际距离，将图上距离乘以比例因子 n。
Actual distance = map distance × n
000 map, a path is 4 cm long. The actual path is 4 cm × 50000 = 200000 cm = 2000 m = 2 km.
在一张 1 : 50000 的地图上，一条小路长 4 cm。实际小路长度为 4 cm × 50000 = 200000 cm = 2000 m = 2 km。
Remember to convert cm to m or km when appropriate. There are 100 cm in 1 m and 100000 cm in 1 km.
记住在需要时把 cm 转换为 m 或 km。1 m = 100 cm，1 km = 100000 cm。
6. Converting Real Distances to Map Distances | 将实际距离换算为地图距离
To find the drawing distance, divide the actual distance by the scale factor n.
要计算图上距离，将实际距离除以比例因子 n。
Map distance = actual distance ÷ n
: 200. First convert 10 m to cm: 10 m = 1000 cm. Then map distance = 1000 cm ÷ 200 = 5 cm.
一面实际长度为 10 m 的墙，比例尺为 1 : 200。先把 10 m 转换成 cm：10 m = 1000 cm。图上距离 = 1000 cm ÷ 200 = 5 cm。
7. Scale Drawing Construction | 比例绘图作图
- Choose a suitable scale that fits the page. 选择一个适合页面的比例尺。
- Convert every real length into drawing length. 将每个实际长度转换为图上长度。
- Draw the base line, then use a protractor for angles or bearings. 画出基线，然后用量角器画角度或方位角。
- Label the scale and key points clearly. 清楚标注比例尺和关键点。
A clear scale drawing shows all measurements in proportion and keeps angles identical to those in the real object.
清晰的比例绘图展示所有成比例的测量结果，并保持与实际物体相同的角度。
8. Area and Scale Factor | 面积与比例因子
When a shape is enlarged or reduced by a length scale factor, its area changes by the square of that factor.
当一个形状按长度比例因子放大或缩小时，其面积按该因子的平方变化。
Area scale factor = n²
length scale is 1 : 100, then 1 cm² on the drawing represents 100² = 10000 cm² in reality.
如果长度比例为 1 : 100，则图上 1 cm² 代表实际 100² = 10000 cm²。
a map area of 3 cm² at scale 1 : 100 gives an actual area of 3 × 10000 = 30000 cm² = 3 m².
例如，在 1 : 100 的比例下，图上面积 3 cm² 对应的实际面积是 3 × 10000 = 30000 cm² = 3 m²。
9. Common Mistakes and Exam Tips | 常见错误与应试技巧
- Forgetting to use three figures for bearings. 忘记方位角要写成三位数。
- Measuring anticlockwise instead of clockwise. 测量了逆时针角度而不是顺时针角度。
- Mixing units when converting distances. 转换距离时混淆了单位。
- Using the length scale factor for area instead of squaring it. 计算面积时直接使用长度比例因子，而没有平方。
- Not aligning the protractor with the north line. 没有将量角器与正北线对齐。
In exams, underline the north line and write the scale as a ratio before you start calculating.
考试时，先标出正北线并把比例尺写成比的形式，然后再开始计算。
10. Practice Ideas | 练习思路
Try this question: A ship sails from port P on a bearing of 140° for 5 km. Using a scale of 1 cm : 1 km, draw the journey and find the bearing of P from the ship.
试试这道题：一艘船从港口 P 沿方位角 140° 航行 5 km。使用比例 1 cm : 1 km 画出航程，并求从船看 P 的方位角。
The ship' ne at 140° from the north line. The back bearing from the ship to P is 140° + 180° = 320°.
船的航程可以画成一条与正北线成 140° 的 5 cm 线段。从船到 P 的反方位角是 140° + 180° = 320°。
Another quick practice: A map has scale 1 : 25000. Two towns are 7.5 cm apart. Find the actual distance in km.
另一个快速练习：一张地图的比例尺是 1 : 25000。两个城镇相距 7.5 cm。求实际距离，以 km 为单位。
Actual distance = 7.5 × 25000 = 187500 cm = 1.875 km.
实际距离 = 7.5 × 25000 = 187500 cm = 1.875 km。
11. Summary and Key Formulae | 总结与关键公式
| Relationship | Formula |
| Bearing | Three-figure clockwise angle from North |
| Reverse bearing | (Bearing + 180°) mod 360° |
| Actual distance | Map distance × n |
| Map distance | Actual distance ÷ n |
| Area scale factor | n² |
Review these formulae before your assessment and practise drawing accurate scale diagrams with a ruler and protractor.
在评估前复习这些公式，并用直尺和量角器练习绘制准确的比例图。
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