What Is a Map?

A map is a representation of a place or area. Maps allow us to show locations, distances, directions, and physical features in a much smaller space than the real world.

Maps can represent:

• A classroom or school.
• A town or city.
• A country or continent.
• A hiking trail.
• A transportation network.
• The entire world.

To use a map effectively, we need to understand several mathematical ideas, including scale, coordinates, direction, distance, and routes.

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Important Parts of a Map

Maps often contain several features that help us interpret them.

These may include:

• Title – tells us what the map represents.
• Scale – connects map distances to real-world distances.
• Legend or key – explains symbols.
• Coordinates or grid – identifies locations.
• Compass rose – shows directions.
• Labels – identify places and features.

Not every map contains all of these features, but each can provide useful information.

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Map Symbols and Legends

Maps cannot draw every real object exactly as it appears.

Instead, they use symbols.

For example, a map might use symbols for:

• Roads.
• Rivers.
• Hospitals.
• Schools.
• Airports.
• Railway stations.
• Parks.

The legend, sometimes called the key, explains what these symbols mean.

Always check the legend before interpreting unfamiliar symbols.

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The Compass Rose

A compass rose shows directions on a map.

The four main directions are called the cardinal directions:

North (N)

East (E)

South (S)

West (W)

A useful memory aid for moving clockwise around a compass is:

North → East → South → West

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Intermediate Directions

Between the four cardinal directions are four intermediate directions:

Northeast (NE)

Southeast (SE)

Southwest (SW)

Northwest (NW)

Together, these provide eight commonly used directions.

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Determining Direction on a Map

On a standard map:

North → up

South → down

East → right

West → left

However, you should always check the map's compass or orientation because not every map places north at the top.

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Example: Describing Direction

Suppose a school is directly above a park on a map.

If north is at the top:

The school is north of the park.

The reverse statement is:

The park is south of the school.

Direction depends on the starting point.

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Combining Directions

Suppose the library is:

• To the right of the school.
• Above the school.

The library is:

northeast of the school

If the sports field is left and below the school, it is:

southwest of the school

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Map Scale

A real city or country cannot usually be drawn at its actual size.

Maps therefore use a scale.

A map scale shows the relationship between:

distance on the map

and:

distance in the real world

For example:

1 cm represents 5 km

means that every centimetre measured on the map represents 5 kilometres in reality.

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Using a Map Scale

Suppose a map uses:

1 cm : 4 km

Two towns are:

6 cm apart

on the map.

Calculate the real distance:

6 × 4 = 24 km

Therefore, the towns are:

24 km apart

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Finding Map Distance

We can also work backward.

Suppose two towns are:

35 km apart

and the scale is:

1 cm : 5 km

Calculate:

35 ÷ 5 = 7 cm

The towns should therefore be:

7 cm apart

on the map.

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Scale Bars

Some maps use a scale bar instead of a written scale.

A scale bar might show:

0 |——| 5 km

If the length of the bar represents 5 km, we can compare distances on the map with that bar.

Scale bars are especially useful because they provide a visual reference for distance.

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Ratio Scales

Some maps use a ratio such as:

1 : 50 000

This means:

1 unit on the map represents 50 000 of the same units in reality.

If we use centimetres:

1 cm on map = 50 000 cm in reality

Since:

100 000 cm = 1 km

then:

50 000 cm = 0.5 km

Therefore:

1 cm represents 0.5 km

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Example with a Ratio Scale

A map has a scale of:

1 : 100 000

A road measures:

8 cm

on the map.

Actual distance:

8 × 100 000 = 800 000 cm

Convert to kilometres:

800 000 cm = 8 km

Therefore:

actual distance = 8 km

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Unit Conversions in Map Problems

Map calculations often require unit conversions.

Useful relationships include:

100 cm = 1 m

1000 m = 1 km

Therefore:

100 000 cm = 1 km

For example:

300 000 cm = 3 km

Always check that the final answer uses an appropriate unit.

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Coordinates on Maps

Maps often use a coordinate grid to describe locations.

Coordinates provide a precise way of identifying positions.

A simple coordinate map might contain:

School = (2, 4)

Park = (6, 4)

Library = (6, 7)

Sports field = (2, 7)

Each ordered pair identifies a particular location.

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Reading Coordinates

Coordinates are normally written:

(x, y)

Remember:

x first

then:

y

The x-coordinate describes horizontal position.

The y-coordinate describes vertical position.

For:

(5, 3)

move:

5 horizontally

then:

3 vertically

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Grid References

Some maps use grid references rather than ordinary coordinate pairs.

A map might contain:

• Letters across the horizontal direction.
• Numbers down the vertical direction.

For example:

B4

could identify a particular square on the map.

The exact system depends on the map, so always check how its grid is labelled.

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Coordinates Describe Exact Positions

Coordinates are useful because descriptions such as:

"near the park"

are not very precise.

A coordinate such as:

(7, 4)

gives a much more exact location within a defined grid.

This is important in:

• Maps.
• Navigation.
• Computer systems.
• Surveying.
• Engineering.
• Scientific research.

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Finding Direction from Coordinates

Coordinates can also help determine direction.

Suppose:

A = (2, 3)

B = (7, 3)

The y-coordinate stays the same while x increases.

Therefore, B is:

east of A

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Another Direction Example

Suppose:

A = (4, 2)

B = (4, 8)

The x-coordinate stays the same while y increases.

Therefore, B is:

north of A

assuming the positive y-direction represents north.

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Diagonal Direction from Coordinates

Suppose:

A = (2, 2)

B = (6, 5)

From A to B:

• x increases → move east.
• y increases → move north.

Therefore, B is:

northeast of A

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Coordinates and Direction

If north corresponds to positive y and east corresponds to positive x:

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Routes

A route is a path from one location to another.

There may be several possible routes between the same two places.

For example, one route might be:

School → Library → Park → Sports Centre

Another might be:

School → Park → Sports Centre

The shortest-looking route is not always the best route in real life because roads, rivers, buildings, terrain, and traffic can affect travel.

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Describing a Route

A route can be described using:

• Direction.
• Distance.
• Turns.
• Landmarks.
• Coordinates.

For example:

Travel 3 km east, turn north, and continue for 2 km.

This combines measurement and direction.

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Calculating Route Distance

Suppose a route consists of:

3 km east

then:

4 km north

The total distance travelled is:

3 + 4 = 7 km

Therefore:

route distance = 7 km

This is the actual path travelled.

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Distance Is Not the Same as Displacement

Suppose you travel:

3 km east

then:

4 km north

Your route distance is:

7 km

However, the straight-line distance between the starting and ending positions is shorter.

The movement forms a right triangle.

Using the Pythagorean theorem:

3² + 4² = d²

9 + 16 = d²

25 = d²

d = 5 km

Therefore:

distance travelled = 7 km

but:

straight-line distance = 5 km

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Straight-Line Distance

On some maps, we want the direct distance between two locations.

If the points form a horizontal or vertical line, this is easy to calculate.

Suppose:

A(2, 5)

and:

B(8, 5)

Horizontal difference:

8 − 2 = 6 units

If each grid unit represents:

2 km

then:

6 × 2 = 12 km

Therefore, the locations are:

12 km apart

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Diagonal Distances

The Pythagorean theorem can be used when two locations form a right triangle on a coordinate grid.

Suppose one location is:

A(1, 2)

and another is:

B(4, 6)

Horizontal change:

4 − 1 = 3

Vertical change:

6 − 2 = 4

Straight-line distance:

d² = 3² + 4²

d² = 9 + 16

d² = 25

d = 5 units

If each unit represents 2 km:

5 × 2 = 10 km

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Estimating Distances

Real roads and paths are not always perfectly straight.

For a curved road, you can estimate distance by:

• Dividing the route into shorter sections.
• Measuring each section.
• Adding the measurements.
• Applying the map scale.

The more carefully the curved route is divided, the better the estimate.

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Worked Example: Map Distance

A map uses the scale:

1 cm : 12 km

Two cities are:

7.5 cm apart

Calculate:

7.5 × 12 = 90 km

Therefore:

The cities are 90 km apart.

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Worked Example: Finding Map Distance

Two locations are actually:

48 km apart

The map scale is:

1 cm : 8 km

Calculate:

48 ÷ 8 = 6 cm

Therefore, the locations should appear:

6 cm apart

on the map.

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Worked Example: Coordinates and Direction

A campsite is at:

A(3, 2)

A lake is at:

B(7, 6)

From A to B:

Horizontal change:

7 − 3 = 4

so move east.

Vertical change:

6 − 2 = 4

so move north.

Therefore, the lake is:

northeast of the campsite

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Worked Example: Route Distance

A hiking route travels:

5 km north

then:

3 km east

then:

2 km south

Total route distance:

5 + 3 + 2

= 10 km

The hiker has travelled:

10 km

Notice that this is not necessarily the straight-line distance from the starting point.

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Worked Example: Final Position

A traveler begins at:

(2, 3)

and moves:

4 units east

then:

5 units north

East increases x:

2 + 4 = 6

North increases y:

3 + 5 = 8

Therefore, the final position is:

(6, 8)

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Finding a Route from Coordinates

Suppose:

Start = (2, 1)

Destination = (7, 5)

One possible route is:

5 units east

then:

4 units north

because:

7 − 2 = 5

and:

5 − 1 = 4

Another route could travel north first and then east.

Both routes reach the same destination.

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Comparing Routes

Suppose Route A is:

4 km east + 3 km north

Total:

7 km

Route B is:

2 km east + 2 km north + 2 km east + 1 km north

Total:

7 km

Although the routes look different, their total distances are equal.

This illustrates why routes should be compared using measurements rather than appearance alone.

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Scale and Route Distance Together

Suppose a route on a map consists of:

2 cm north

then:

3 cm east

then:

4 cm north

Total map distance:

2 + 3 + 4 = 9 cm

If:

1 cm : 5 km

then:

9 × 5 = 45 km

Therefore:

route distance = 45 km

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Bearings

For more precise navigation, directions can be described using bearings.

A bearing is measured:

• From north.
• Clockwise.
• In degrees.

Bearings are normally written using three digits.

For example:

000° = North

090° = East

180° = South

270° = West

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Intermediate Bearings

Some useful intermediate bearings include:

045° = Northeast

135° = Southeast

225° = Southwest

315° = Northwest

Bearings provide more precision than simply saying north, south, east, or west.

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Example Using Bearings

Suppose a boat travels on a bearing of:

090°

It is travelling:

east

If it changes to:

180°

it is now travelling:

south

Bearings are especially useful in:

• Aviation.
• Marine navigation.
• Surveying.
• Orienteering.

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Latitude and Longitude

Real-world maps often use latitude and longitude rather than a simple x-y coordinate grid.

Latitude measures position north or south of the Equator.

Longitude measures position east or west of the Prime Meridian.

Together, latitude and longitude can identify locations anywhere on Earth.

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GPS and Coordinates

The Global Positioning System (GPS) uses satellite signals to determine locations on Earth.

Modern navigation systems can combine:

• Geographic coordinates.
• Digital maps.
• Road information.
• Distance calculations.
• Direction.
• Speed.
• Route information.

The mathematics behind navigation therefore connects geometry, measurement, coordinates, and technology.

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Maps Are Models

It is important to remember that a map is a model of reality.

A map may simplify:

• Road shapes.
• Coastlines.
• Building locations.
• Terrain.
• Distances.

Different maps are designed for different purposes.

For example, a road map may emphasize roads, while a topographic map emphasizes elevation and terrain.

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Applying Geometry to Navigation

Many geometry skills can be applied directly to navigation.

These include:

• Measuring lengths.
• Using scale factors.
• Reading coordinates.
• Calculating distances.
• Identifying angles.
• Using bearings.
• Recognizing directions.
• Applying the Pythagorean theorem.

Navigation provides a practical application for many mathematical ideas.

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Planning a Journey

Suppose a map shows:

Home = (1, 2)

Store = (5, 2)

Park = (5, 6)

School = (2, 6)

If one grid unit represents:

0.5 km

a route from Home to Store to Park has:

Home → Store:

4 units = 2 km

Store → Park:

4 units = 2 km

Total route:

4 km

Coordinates and scale can therefore be combined to calculate an entire journey.

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Choosing Between Routes

When comparing routes, several factors may matter.

Mathematically, we can compare:

• Distance.
• Direction.
• Number of turns.
• Straight-line distance.
• Scale measurements.

In real navigation, other factors may also matter, such as:

• Roads.
• Terrain.
• Traffic.
• Safety.
• Accessibility.

Therefore, the mathematically shortest route is not always the most practical route.

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A Reliable Map-Scale Strategy

When calculating real-world distance:

Step 1: Measure the map distance.

Step 2: Read the scale.

Step 3: Multiply by the scale factor.

Step 4: Convert units if necessary.

Step 5: Include the correct unit.

Step 6: Check whether the answer is reasonable.

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A Reliable Coordinate-Navigation Strategy

When navigating using coordinates:

Step 1: Identify the starting coordinates.

Step 2: Identify the destination coordinates.

Step 3: Compare the x-values.

Step 4: Compare the y-values.

Step 5: Determine horizontal and vertical movement.

Step 6: Convert the movement into directions.

Step 7: Apply the map scale if a real-world distance is required.

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Common Mistakes

Ignoring the Map Scale

A distance of:

5 cm

on a map does not usually mean the locations are 5 cm apart in reality.

The scale must be applied.

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Multiplying When You Should Divide

Suppose:

1 cm represents 5 km

If you know the map distance and want the real distance:

multiply by 5

If you know the real distance and want the map distance:

divide by 5

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Forgetting Unit Conversions

A ratio scale may use centimetres while the answer is required in kilometres.

Always check the units.

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Reading Coordinates Backward

Remember:

(x, y)

means:

horizontal first, vertical second

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Confusing East and West

On a standard north-up map:

east = right

west = left

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Confusing Route Distance with Straight-Line Distance

If a traveler follows several roads, the total route distance is the sum of those road sections.

The direct distance between the starting point and destination may be shorter.

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Assuming North Is Always at the Top

North is commonly at the top, but not always.

Check the map's compass or orientation.

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Measuring from the Wrong Points

When calculating map distances, measure between the correct locations.

A small measurement error on the map can represent a much larger error in the real world.

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Forgetting That Maps Are Simplified Models

A straight line on a map may cross:

• Buildings.
• Rivers.
• Mountains.
• Private property.

The shortest geometric path may therefore not be a usable route.

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Check Your Understanding

1. What is a map?

2. What is the purpose of a map legend?

3. What does a compass rose show?

4. Name the four cardinal directions.

5. Name the four intermediate directions.

6. On a standard map, which direction is usually at the top?

7. A library is directly right of a school. What direction is the library from the school?

8. A park is above and left of a school. What direction is the park from the school?

9. What does a map scale represent?

10. A map uses 1 cm : 5 km. Two towns are 7 cm apart. Find the actual distance.

11. A map uses 1 cm : 8 km. Two cities are 48 km apart. How far apart should they appear on the map?

12. A map has a scale of 1:100 000. What real distance does 1 cm represent?

13. Two locations are 6 cm apart on a 1:100 000 map. Find the actual distance.

14. Explain why unit conversions may be necessary when using map scales.

15. A school is at (3, 4). A park is at (8, 4). In what direction is the park from the school?

16. A campsite is at (5, 2). A lake is at (5, 8). In what direction is the lake?

17. A town is at (2, 3). Another is at (6, 7). What general direction is the second town from the first?

18. A traveler starts at (3, 2) and moves 5 units east. What are the new coordinates?

19. A traveler starts at (4, 6) and moves 3 units south. What are the new coordinates?

20. A route travels 4 km north, 3 km east, and 2 km south. Find the total distance travelled.

21. A person travels 6 km east and then 8 km north. What is the total route distance?

22. For Question 21, calculate the straight-line distance between the starting and finishing points.

23. Explain the difference between route distance and straight-line distance.

24. What direction is represented by a bearing of 090°?

25. What direction is represented by 180°?

26. What direction is represented by 270°?

27. What is the approximate bearing for northeast?

28. Explain how coordinates and map scales can be used together to calculate real-world distances.

29. Give three situations in which map and navigation skills are useful.

30. Explain why the shortest geometric route between two locations may not be the best real-world route.

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Key Terms

• Map – representation of a geographic area or location.
• Scale – relationship between map distance and real-world distance.
• Scale bar – visual representation of distance on a map.
• Legend – guide explaining symbols used on a map.
• Coordinate – value used to describe a location.
• Grid reference – code identifying a location within a map grid.
• Compass rose – diagram showing directions.
• Cardinal directions – north, east, south, and west.
• Intermediate directions – northeast, southeast, southwest, and northwest.
• Route – path followed between locations.
• Bearing – direction measured clockwise from north in degrees.
• Latitude – position north or south of the Equator.
• Longitude – position east or west of the Prime Meridian.
• GPS – satellite-based system used to determine geographic position.
• Straight-line distance – shortest direct distance between two points.

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Key Takeaways

• Maps are scaled representations of real places.
• A map's scale connects measurements on the map with real-world distances.
• Map scales may be written as statements, ratios, or scale bars.
• Coordinates provide a precise way to describe locations.
• A compass rose shows direction.
• The four cardinal directions are north, east, south, and west.
• Intermediate directions include northeast, southeast, southwest, and northwest.
• Coordinates can be used to determine direction and movement.
• Route distance is the total distance travelled along a path.
• Straight-line distance may be shorter than route distance.
• The Pythagorean theorem can sometimes be used to calculate direct distances.
• Bearings provide more precise directions using angles measured clockwise from north.
• Latitude and longitude are used to describe real locations on Earth.
• GPS and digital navigation systems combine coordinates, measurement, geometry, and map information.
• Effective navigation requires careful attention to scale, units, direction, coordinates, and the available route.

A useful summary is:

Map scale tells us how far.

Coordinates tell us where.

Directions and bearings tell us which way.

Routes tell us how to get there.