Scale: map and reality

What is it about?

A map shows reality reduced. The scale tells you by how much. A hiking map says, for example, 1 : 25,000 (read “one to twenty-five thousand”). This means:

The number after the colon is the scale number. You can calculate in both directions:

• Map → reality: map length times the scale number. Example: $4\,\text{cm} \cdot 25\,000 = 100\,000\,\text{cm} = 1\,000\,\text{m} = 1\,\text{km}$.
• Reality → map: convert the real length to cm, then divide by the scale number. Example: $2.5\,\text{km} = 250\,000\,\text{cm}$ and $250\,000\,\text{cm} : 25\,000 = 10\,\text{cm}$.
• Finding the scale: put both lengths in the same unit and divide. Example: $6\,\text{cm}$ on the map are $3\,\text{km} = 300\,000\,\text{cm}$; $300\,000 : 6 = 50\,000$, so $1 : 50\,000$.

A map with a small scale number (e.g. $1 : 5\,000$) shows a small area with many details – you can see every house. A map with a large scale number (e.g. $1 : 100\,000$) shows a large area with few details. The scale bar at the bottom left of the map is divided into pieces of $1\,\text{cm}$ each and shows directly how far one centimetre is in reality.

Very small things are enlarged. Then the larger number comes first: 10 : 1 means $10\,\text{cm}$ in the picture are $1\,\text{cm}$ in reality. A $7\,\text{mm}$ ladybird is $7\,\text{cm}$ long in the picture; back to reality: $7\,\text{cm} : 10 = 0.7\,\text{cm} = 7\,\text{mm}$.

Note: The centimetres in the simulation are the centimetres of the drawn ruler – on screen they may be a little larger or smaller than real centimetres, depending on the device. Lengths are always measured to $1\,\text{mm}$, just like with a school ruler.

Try it

1. The line from the church to the station is $4\,\text{cm}$ long on the map. How far is that in reality? Compare with the scale bar. Load example
2. Switch between the scales. What happens to the length on the map, and what to the real distance? Which details disappear at $1 : 100\,000$?
3. Tap two places on the map one after the other (e.g. church and lookout tower). How long is the straight line at $1 : 50\,000$? Load example
4. Measure the way from the church to the bathing spot once as a straight line and once along the road. Why is the route longer? Route
5. Drag both handles exactly on top of each other. What does the calculation show now?
6. Under Enlarge, look at a grain of sand at the scale $50 : 1$. How big is it in the picture, how big in reality? Grain of sand

Tasks

Task 1: From the map to reality

On a map at the scale $1 : 50\,000$, two places are $3\,\text{cm}$ apart. How far apart are they in reality? Load task

Task 2: Town plan

On the town plan ($1 : 5\,000$), measure the straight line from the school to the town hall and convert it. Load task

Task 3: From reality to the map

The bathing spot is $2.5\,\text{km}$ from the church as the crow flies. How long is this line on a map at the scale $1 : 25\,000$? Load task

Task 4: Finding the scale

On a map, two places are $6\,\text{cm}$ apart; in reality they are $3\,\text{km}$ apart. What is the scale of the map? Load task

Task 5: Enlargement

An ant is $5\,\text{mm}$ long. How long is it in a picture at the scale $10 : 1$? And a grain of sand with a diameter of $0.4\,\text{mm}$ at the scale $50 : 1$? Work it out yourself first – the Enlarge view shows the result straight away.

Task 6: Straight line or route?

From the church to the bathing spot, the straight line on the map ($1 : 25\,000$) is $10\,\text{cm}$. Measured piece by piece along the road, the route is $10.7\,\text{cm}$ in total. How long are both in reality, and how much longer is the route? Load task

Ideas for the classroom

• Use: Grade 5, measures (length, scale). Use the map as a shared introduction on the projector and the “New measuring task” with the result hidden for practice in pairs.
• Link to a real map: Afterwards, have the children measure their way to school on a hiking map of the area and convert it.
• Model: The landscape is invented. As on real maps, houses and map symbols are drawn somewhat larger than they would be to scale. The map length is measured to the nearest $1\,\text{mm}$; the calculation uses this measured value.
• Common stumbling blocks:
  • The unit is forgotten: $4 \cdot 25\,000 = 100\,000$ are centimetres, not metres.
  • The number of zeros is mixed up when converting cm to km – correct is $1\,\text{km} = 100\,000\,\text{cm}$.
  • “Large scale” is confused with “large scale number”: $1 : 5\,000$ is the larger scale; it shows more details.
  • When finding the scale, lengths in different units are divided ($3\,\text{km} : 6\,\text{cm}$).
  • With enlargements, students multiply instead of dividing ($7\,\text{cm} : 10$ is correct, not $7\,\text{cm} \cdot 10$).