This unit explores the mathematics of maps, including scale, coordinates, and bearings.
- Draw a scale map of the classroom
- Find the location of an object using Cartesian coordinates or bearings
This unit investigates three mathematical concepts in the context of maps.
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.
- Cartesian (or rectangular) coordinates 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).
With real maps the coordinates often refer to latitude and longitude.
Longitude is the number of degrees 'around the world' from the Prime Meridian, and latitude is measured in degrees north or south from the equator.
- Polar coordinates specify a location using angle or bearing from a given point, and the distance from that point.
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:
- using 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.
- posing 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. Students can be prompted to complete the tasks with different levels of sophistication.
- using 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 working with local surroundings. 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.”
Students might be interested in traditional forms of navigation. Tipuna (Ancestors) had clever ways to navigate in the absence of the sophisticated GPS systems used nowadays. Polynesian navigators used ocean currents, the sun and stars, and migration of whales and birds to find their way around the Pacific Ocean.
Te reo Māori vocabulary terms such as ahopae (latitude), ahopou (longitude), mahere (map), tuhinga āwhata (scale drawing), āwhata (scale), taunga tukutuku (cartesian coordinates), raki (north), runga (south), rāwhiti (east), and rātō (west) could be introduced in this unit and used throughout other mathematical learning.
- Compasses or phones with a compass app (for finding north, not for drawing circles)
- Pencils and paper
- Maps of local area or access to online maps
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. You could start this session with investigation into the traditional navigation methods used by Māori and Pasifika peoples. Consider inviting local iwi members to your classroom to discuss where the local iwi heralded from. These locations, and the locations of other iwi relevant to your students, could be identified on a map.
- Present the class with a map of the local area (Google Maps might be used).
- Ask them to identify what it is.
- Ask them if they can explain why you would be looking at a map in mathematics time.
- Encourage students to identify the maths involved in reading and drawing maps (directions, scale, distances etc.)
- Challenge students to name things about maps that are part of mathematics.
- Hand out some examples of maps of Aotearoa or the local area and ask students to find features that they think involve mathematical ideas.
- Encourage and support students to identify the following features:
- Compass bearings
- Discuss each of these features, and explain that you will be looking at all three over the next few sessions. You will invite a guest speaker to come in to tell the class about how they use maps in their life and work.
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.
- Show the students a map with a scale clearly labelled. Ideally use a map of the local area or somewhere the students are familiar with. You might use a source that provides free maps, such as Request a Map (print or request large touring maps), Google Maps (free online maps), or the AA (you can collect maps from your local AA site or order them online). Ensure that you have enough copies of the map to be shared in small groups. A digital version of the map could make it easier for all students to see.
- Ask students to tell you what the scale of the map is.
- 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.
- For this session we will be working with a ratio type scale. Ask students to describe what the scale means. For example, a ratio of 1:50 000 means each centimetre represents 50 000 centimetres (500 metres in real life). If the scale is 1:100 000, then every cm on the map represents 100 000 cm, or 1000 m or 1 km in reality.
- 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, you might model this by showing a range of maps, and explaining what their scales means in reality.
- Pose some questions around the map you have given the class to check that they understand: How far is it from …? Model calculating this first, and then support students to independently figure out the distances. Calculators can be used to help students make accurate calculations with larger numbers.
- 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). At 1:20 a 6m x 5m classroom will be drawn the size of a 30cm x 25cm. As an alternative or extension you could draw a scale map of the whole school if you think your class could manage this. More knowledgeable students could work independently, or in small groups, to draw scale models of specified buildings, and you could work with less knowledgeable students to draw scale models of specified buildings in a scaffolded environment.
- Ask students what this scale will mean. (1m in the real classroom will be 5cm on the map)
- 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.)
- 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.
- Students might work from the Figure It Out resource, who lives where? to practise using the scale on a map.
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.
- Refer back to the map the whole class looked at in the session 1.
What are the vertical and horizontal lines on the map there for?
What do the numbers at the tick marks (ends of the lines) represent?
- 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?
- See if students can explain how they would describe a given point on the map using the coordinates.
- Once students have given suggestions teach them how coordinates work: 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. "X is a cross, Y is in the sky" may be a helpful mnemonic for students to remember the order in which to read coordinates.
- Allow students to try a few examples.
- 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.
- 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.
- 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' understandings of coordinates.
- You may give students the Figure It Out activity Map Mysteries to practise finding coordinates in a pirate context. If you have a large world map with coordinates on the x-axis and y-axis, then you could use this as an opportunity for students to identify countries they whakapapa to. In turn, this could help establish connections and cultural understanding between your students. If your students are ready to be extended in their use of coordinates, they could complete this activity independently using a printed world map, and compare their maps to those completed by their peers. They could also investigate the places their ancestors travelled from to arrive in New Zealand (e.g. from Samoa) and record this information using coordinates.
Session 4: Polar Coordinates
In this lesson we draw a north arrow on our maps, and introduce polar coordinates.
- On the map of the local area used in the previous sessions point out the North arrow and ask what it is for.
- 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.
- Explain that we are going to draw a north arrow on our maps, using Magnetic North.
- 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.
- As a few questions such as:
What is north of the teacher’s desk?
What is east of the door?
- 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?
- 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?
- 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?
- 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.
- To apply the idea of bearings and distance further show students how to use orienteering compasses or the compass app on their phone. In small groups students prepare a set of up to ten clues for another group to locate a position. For example, a clue might be “Walk 45 metres at a bearing of 76°. You might use a local legend about a lost person, usually a tale of love, such as the story of Matakauri and Manata from Lake Wakatipu or Hinemoa and Tutanekai from Rotorua. The clues are given to another group who try to find the lost person.
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.
- Before this session, organise a guest speaker to come in and talk about maps and how they use them. A member of a local orienteering club or Land Search and Rescue would be ideal, or possibly a local surveyor. You can get a contact for a local orienteering club from http://www.nzorienteering.com/
- 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.
- Have a few selected students show the guest the maps of the classroom and explain what the class has been working on.