Lesson 2: Map It - Scale, Coordinates, and Bearings

Achievement Objectives

GM4-7: Communicate and interpret locations and directions, using compass directions, distances, and grid references.

Learning Objectives

By the end of this lesson, students will be able to:

  • Understand and apply map scales to convert between map distances and real distances
  • Use Cartesian coordinates to locate specific points on a map
  • Apply polar coordinates using bearings and distances to navigate
  • Interpret latitude and longitude coordinates

Description of Mathematics

This lesson investigates three mathematical concepts in the context of maps:

1. Scale

A map is a reduced version of a real landscape. Scale compares the size of lengths on a map to those in the real landscape.

Key Concept: If a map has a scale of 1:50,000, this means that 1 cm on the map represents 50,000 cm (or 500 m) in real life.

Example 1: Working with Scale

A map has a scale of 1:25,000.

  1. If two towns are 3 cm apart on the map, how far apart are they in real life?

Solution: - Map distance = 3 cm - Real distance = 3 × 25,000 = 75,000 cm = 750 m = 0.75 km

  1. If two landmarks are 2 km apart in real life, how far apart would they be on the map?

Solution: - Real distance = 2 km = 200,000 cm - Map distance = 200,000 ÷ 25,000 = 8 cm

2. Cartesian (Rectangular) Coordinates

Cartesian coordinates refer to the location of a specific point using a combination of horizontal distance and vertical distance.

Example: The star below is located at (7, 4).

[Imagine a coordinate grid with x-axis from 0 to 9 and y-axis from 0 to 6. A star is plotted at the point where x=7 and y=4.]

With Real Maps: Latitude and Longitude

  • Longitude is the number of degrees ‘around the world’ from the Prime Meridian
  • Latitude is measured in degrees north or south from the equator

Example 2: Using Grid References

On a map with a grid system: - Location A is at grid reference (345, 678) - Location B is at grid reference (348, 682)

If each grid unit represents 100 meters, how far apart are A and B?

Solution: Using the distance formula: \[d = \sqrt{(348-345)^2 + (682-678)^2}\] \[d = \sqrt{3^2 + 4^2}\] \[d = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ grid units}\]

Real distance = 5 × 100 = 500 meters

3. Polar Coordinates (Bearings)

Polar coordinates specify a location using the angle or bearing from a given point and the distance from that point.

Bearings are measured clockwise from North and are written as three-digit numbers: - North = 000° (or 360°) - East = 090° - South = 180° - West = 270°

Example 3: Using Bearings

From point A, a landmark is located on a bearing of 045° at a distance of 3 km.

  1. In which general direction is the landmark?

Solution: 045° is halfway between North (000°) and East (090°), so it’s Northeast.

  1. If you walk on a bearing of 045° for 3 km, how far East and North have you traveled?

Solution: Using trigonometry: - East component = 3 × sin(45°) = 3 × 0.707 ≈ 2.12 km - North component = 3 × cos(45°) = 3 × 0.707 ≈ 2.12 km

Practical Activities

Activity 1: Local Area Mapping

Using a map of your local area:

  1. Identify your school on the map
  2. Measure the distance from your school to a local landmark
  3. Use the map scale to calculate the real distance
  4. Record the grid reference or coordinates of both locations

Activity 2: Treasure Hunt Coordinates

Create a treasure map with: - A coordinate grid system - At least 5 landmarks marked with coordinates - Write instructions using coordinates to move from one landmark to another

Activity 3: Navigation Challenge

Using a compass or compass app:

  1. Stand at a starting point
  2. Follow these instructions:
    • Walk 20 paces on bearing 090°
    • Walk 15 paces on bearing 180°
    • Walk 20 paces on bearing 270°
    • Walk 15 paces on bearing 000°
  3. Where should you end up? Why?

Te Reo Māori Vocabulary

English Te Reo Māori
Latitude Ahopae
Longitude Ahopou
Map Mahere
Scale drawing Tuhinga āwhata
Scale Āwhata
Cartesian coordinates Taunga tukutuku
North Raki
South Runga
East Rāwhiti
West Rātō

Cultural Connections

Traditional Polynesian Navigation

Tipuna (Ancestors) had clever ways to navigate in the absence of sophisticated GPS systems. Polynesian navigators used:

  • Ocean currents
  • The sun and stars
  • The migration of whales and birds
  • Wave patterns and swells

These traditional methods allowed Pacific peoples to travel vast distances across the ocean with remarkable accuracy.

Practice Problems

Problem Set 1: Scale

  1. A map has a scale of 1:50,000. What real distance does 4.5 cm on the map represent?

  2. Two cities are 150 km apart. How far apart would they be on a map with a scale of 1:1,000,000?

  3. On a treasure map, the scale is 1 cm = 5 m. If the treasure is 7.5 cm from the starting point on the map, how far is it in real life?

Problem Set 2: Coordinates

  1. Plot the following locations on a coordinate grid:
    • School: (2, 5)
    • Library: (7, 5)
    • Park: (2, 1)
    • Shop: (7, 1)
  2. Using your plotted points from question 4, calculate the distance from:
    1. School to Library
    2. School to Park
    3. Library to Shop

Problem Set 3: Bearings

  1. From your starting position:

    • Point A is on a bearing of 045° at 5 km
    • Point B is on a bearing of 135° at 5 km
    • Point C is on a bearing of 225° at 5 km
    • Point D is on a bearing of 315° at 5 km

    Sketch these points. What shape do they form?

  2. You walk 3 km on a bearing of 060°. How far North and how far East have you traveled?

  3. To get from home to school, you walk 400 m East then 300 m North.

    1. What is the direct distance from home to school?
    2. What bearing would take you directly from home to school?

Extension Activities

For Advanced Learners

  1. Magic Carpet Journey: Plan a journey from your location to a Pacific Island nation. Provide:

    • Starting and ending coordinates (latitude/longitude)
    • Bearing from start to finish
    • Distance of the journey

  2. Multiple Navigation: Starting from point A, follow these instructions:

    • 5 km on bearing 030°
    • 3 km on bearing 120°
    • 4 km on bearing 210°

    Calculate your final position relative to point A using coordinates.

  3. Map Making Project: Create a detailed scale map of your school grounds including:

    • A clear scale
    • Grid reference system
    • All major buildings and landmarks
    • Compass rose showing North

Digital Tools

Consider using: - Google Maps for exploring scale and real-world coordinates - Compass apps on mobile phones for practicing bearings - GPS tracking apps to see how movement in real life corresponds to movement on a map - Online mapping tools to create custom maps with coordinate grids

Assessment Ideas

Students could demonstrate their understanding by:

  1. Creating a detailed treasure map with scale, coordinates, and bearing-based instructions
  2. Planning and executing a navigation course using bearings and distances
  3. Converting between map distances and real distances using various scales
  4. Using latitude and longitude to identify locations around New Zealand and the Pacific

Home Learning Connection

Letter to Parents and Whānau:

This week in maths, we are learning about maps. We’re exploring how maps use mathematics including: - Scale (to show big places on small maps) - Coordinates (to locate specific places) - Bearings (to give directions)

You can help by: - Looking at maps together at home (road maps, street directories, online maps) - Asking your child to explain how the scale works - Having them show you how to describe where things are using coordinates - Discussing how GPS and digital maps work

If you have any paper maps at home, ask your child to show you the important parts and demonstrate how to describe locations and give directions using the map.

Required Materials

  • Compasses (for finding north, not for drawing circles) or phones with compass app
  • Protractors
  • Rulers
  • Pencils and paper
  • Maps of local area or access to online maps
  • Calculator (for trigonometry in bearing calculations)

Learning Outcomes

By completing this lesson, students will have developed skills in: - Mathematical thinking: Understanding scale relationships and coordinate systems - Problem solving: Converting between map and real distances - Communication: Using mathematical language to describe locations and directions - Cultural awareness: Appreciating traditional navigation methods of Pacific peoples - Practical application: Using mathematical concepts in real-world contexts

Reflection Questions

  1. Why is it important to understand map scales?
  2. How do coordinates help us communicate locations precisely?
  3. What are the advantages of using bearings instead of just saying “go northeast”?
  4. How do digital maps like Google Maps use these mathematical concepts?
  5. What can we learn from traditional Polynesian navigation methods?

Previous Lesson: Lesson 1: Introduction to Cartesian Coordinates
Next Lesson: Lesson 3: The Distance Formula