Purpose

This unit uses explores the mathematics of maps, including scale, coordinates and bearings.

Achievement Objectives
GM4-7: Communicate and interpret locations and directions, using compass directions, distances, and grid references.
Specific Learning Outcomes
  • Draw a scale map of the classroom
  • Find the location of an object using Cartesian coordinates or bearings
Description of Mathematics

This unit 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. 
  2. Cartesian (or rectangular) co-ordinates refer to the location of a specific point. The combination of horizontal distance and vertical distance represent a location. For example the star below is located at (7,4).
    coordinates
    With real maps the co-ordinates often refer to latitude and longtitude. Longtitude is the number of degrees 'around the world' from the Prime Meridian, and latitude is measured in degrees north or south from the equator.
  3. Polar co-ordinates specify a location using angle or bearing from a given point, and the distance from that point.
 
Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

  • Use maps of your local area as much as possible. Experience with local landmarks will help students connect maps to the environment. Physically moving and connecting that movement to position, and path, on a map is particularly helpful.
  • Pose open challenges that allow students to attempt tasks in their own way. For example, “Give a friend instructions to get from your home to school”, or “Create a pirate treasure map that has a coordinate system for locating treasure” are examples of open challenges. Student can be prompted to complete the tasks with different levels of sophistication.
  • Use digital technology for mapping. Mobile phones are particularly useful in helping students to recognise how scale works. As they pace out a distance in real life, they can see the movement of their icon on a map. Phones also have apps for identifying compass directions. By locating their phone in front of them students can turn to a given bearing and proceed a given number of steps.

The context of mapping is very realistic for all students. Adaptation of the context should include locations that students are less familiar with, after competence is developed in local surrounds. Students will enjoy using digital tools to plan a magic carpet ride to locations within New Zealand and overseas. “The guidance system on your magic carpet requests instructions to travel to Samoa. Give coordinates for the destination, and bearing and distances for the journey.”

Required Resource Materials
  • Compasses or phones with a compass app (for finding north, not for drawing circles)
  • Protractors
  • Rulers
  • Pencils and paper
  • Maps of local area or access to online maps
Activity

Session 1: Introduction to the mathematics of maps

This lesson provides an introduction to the context of the mathematics of maps, and the idea of reading maps.

  1. Present the class with a map of the local area.

  2. Ask them to identify what it is.

  3. Ask them if they can explain why you would be looking at a map in mathematics time.

  4. Encourage students to identify the maths involved in reading and drawing maps (directions, scale, distances etc.)

  5. Challenge students to name things about maps that are part of mathematics.

  6. Hand out some examples of maps and ask students to find features that they think involve mathematical ideas.

  7. Some of the ideas that you should get are:
    • Scale
    • Coordinates
    • Compass bearings
  8. Discuss students understanding of each of these, and explain that you will be looking at all three over the next few sessions and then having a guest speaker come in to tell the class about how they use maps in their life.

Session 2: Scale

In this session we look at the scale of maps, what they mean, and how to interpret and apply them. We draw a scale map of the classroom.

  1. Show the students a map with a scale clearly labeled. Ideally use a map of the local area or somewhere the students are familiar with. Possibly copies of the map shared in small groups or a digital version of the map would make it easier for all students to see.

  2. Ask students to tell you what the scale of the map is.

  3. There are two distinct types of scale you may find on maps, a ratio, such as 1:100 000; and a graphical bar that shows how far 1km, for example, is on the map. Ensure that students have the opportunity to see both.

  4. For this session we will be working with a ratio type scale.  Ask students to describe what the scale means.

  5. Ensure that all students understand that the scale tells how many times larger distances are in reality than they are in the map.  For instance, if the scale is 1:100 000, then every cm on the map represents 100 000 cm, or 100km in reality.

  6. Try some questions around the map you have given the class to check that they understand:  How far is it from …?

  7. Explain to the students that they are going to draw a scale map of the classroom (Check that the class will fit onto whatever paper you intend to use, and adjust the scale accordingly).  As an alternative or extension you could draw a scale map of the whole school if you think your class could manage this.

  8. Ask students what this scale will mean. (1m in the class will be 2cm on the map)

  9. Start by measuring the dimensions of the class and drawing the outline of the room to scale to make sure that it fits on the paper.  (Possibly you could do this as a class to ensure that students understand the process.)

  10. Allow students the rest of the session to draw the remainder of the room in as much detail as possible. Ensure that they include landmarks, such as desks and doorways in the correct places.

Session 3: Coordinates

In this session we will draw a grid on the map of the classroom and use it to identify places within the class. Note that there are two ways in which location is represented on maps, grid references and coordinates. In both cases the representation is an ordered pair, e.g. F4 as a grid reference, and (6,4) as a pair of coordinates. The convention is to state the horizontal reference first, and the vertical reference second.

In the grid system (left image) the location of the person is an area. In the coordinate system (right image) the location of the person is a specific point at the intersection of the lines. 

  1. Refer back to the map the whole class looked at in the session 1.
    What are the vertical and horizontal lines on the map are there for?
    What do the numbers at the ends of the lines represent?

  2. On larger scale maps, they are likely to be lines of longitude and latitude, but on smaller scale maps they will be grid lines, which are used for finding and indicating specific positions on the map.
    How do they work?

  3. See if students can explain how they would describe a given point on the map using the coordinates.

  4. Once students have given suggestions teach them how it works: First give the number on the x-axis (along the top or bottom of the map), and then the number on the y-axis (up the sides) that lines up with the point chosen.

  5. Allow students to try a few examples.

  6. Now explain that we want to be able to do this with our maps of the class, so we will need to draw a grid over the maps that we drew last session.

  7. Ensure that students use a consistent grid such as 1 line per cm, starting in a specific corner of the room. As the maps are all to the same scale this should make the coordinates match up between maps.

  8. There is now the opportunity to play a game of eye spy, with the format “I spy an object at coordinates (3,4)”, either as a class or in pairs. This will reinforce students understanding of coordinates.

Session 4: Polar Co-ordinates

In this lesson we draw a north arrow on our maps, and introduce polar co-ordinates.

  1. On the map of the local area used in the previous sessions point out the North arrow and ask what it is for.

  2. There may be two arrows, one for Magnetic North and one for True North.
    • True North is the direction towards the North Pole, the point where the axis that the earth spins around passes through.
    • Magnetic North is the direction that a compass points to, and is very close to but not exactly the same as True North. Magnetic North moves slowly and inconsistently.
  3. Explain that we are going to draw a north arrow on our maps, using Magnetic North.

  4. Use a compass, or phone, to find Magnetic North, and ensure that all students accurately draw a North arrow on their maps. Include the other three cardinal directions, east, south and west, as well.

  5. As a few questions such as:
    What is north of the teacher’s desk?
    What is east of the door?

  6. Now explain that this idea can be used more accurately, if we break down the spaces between the compass points:
    What is north-east of the mat?
    What is south-west of the bookshelf?

  7. This can be made more accurate yet by breaking the gaps one more time:
    What is north-north-west of the middle of the whiteboard?
    What is west-south-west of the doorway?

  8. If you think your students will be able to cope you can introduce direction given as an angle, and include a distance. Ensure that students know that the angle is called ‘a bearing’, and that the number of degrees is measured clockwise from north.
    What object is 5m on a bearing of 1300 from where I am standing?

  9. This could again be made into a game with students in pairs challenging each other. Ideally you should use compasses to find the bearings, but a protractor used on the maps could also be used.

Session 5: Guest speaker

In this session we invite a member of a local orienteering club, or possibly a local surveyor to come and talk to the class about how they use maps.

  1. Before this session organize a guest speaker to come in and talk about maps and how they use them. A member of a local orienteering club would be ideal, or possibly a local surveyor. You can get a contact for a local orienteering club from http://www.nzorienteering.com/

  2. Have students think about questions they might want to ask, and possibly write them down so the guest can be given some idea in advance of what they might be asked.

  3. Have a few selected students show the guest the maps of the classroom and explain what the class has been working on.

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Level Four