Scale drawings and maps · Higher
| English | 中文 | Pinyin |
|---|---|---|
| scale/skeɪl/ | 比例尺 | bǐ lì chǐ |
A map at 1:25,000 shows a 4 cm route. The written scale compares centimetres with centimetres, not centimetres with kilometres.
- A map at 1:25,000 shows a 4 cm route. The written scale compares centimetres with centimetres, not centimetres with kilometres.
- This lesson studies scale 比例尺: The ratio of a drawing length to its corresponding real length.
Choose the mathematical structure
- In a scale 1:n, one drawing unit represents n of the same real unit. Multiply a drawing length by n to obtain the real length; divide a real length by n to draw it. Then convert the unit. Measure only when the diagram explicitly supplies an accurate scale.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines scale?
The ratio of a drawing length to its corresponding real length.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
At 1:25,000, 4 cm represents 100,000 cm=1000 m=1 km. A real 1.5 km path is 150,000 cm, so it measures 150,000/25,000=6 cm on the map. A room 6 m by 4 m drawn at 1:100 becomes 6 cm by 4 cm because each metre is 100 cm. Its drawing diagonal is √(6²+4²)≈7.21 cm and the real diagonal is about 7.21 m. Enlarging the printed map changes its numerical scale: doubling drawing lengths halves the scale denominator. A scale bar printed with the map enlarges with it.
Scale drawings and maps
In a scale 1:n, one drawing unit represents n of the same real unit
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
At 1:25000, find the real length in km for 4 cm.
4×25000/100000=1 km.
Test a tempting shortcut
- A ratio compares matching units. Never measure a diagram marked not to scale. A photocopied numerical scale can become invalid even though its scale bar still works.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A scale of 1:25000 means 1 cm represents 25000 km. This claim is false. Explain which definition or assumption it violates.
Find the drawing length in cm of a 1.5 km route at that scale.
1.5 km=150000 cm; divide by 25000.
A scale of 1:25000 means 1 cm represents 25000 km.
A ratio compares matching units. Never measure a diagram marked not to scale. A photocopied numerical scale can become invalid even though its scale bar still works.
Interpret a new situation
- AQA R2 includes maps, scale factors and geometric problems. Label drawing and real dimensions separately; reverse the calculation to verify the scale. Use an exact ratio until the context asks for rounding.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the drawing width in cm of a 4 m room at 1:100.
4 m=400 cm; 400/100=4 cm.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.3. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The ratio of a drawing length to its corresponding real length. Choose the relationship, show the method, check its assumptions and interpret the result.