In this unit we use scale drawings to locate ourselves, and other people and objects, in the classroom and in the local community. We also introduce the compass as an instrument that can tell us in which direction we are facing.
- draw and interpret simple scale maps
- use maps or plans to propose actions
- understand the use of a compass to specify and find directions
Scale drawings are used extensively in real life. Plans of buildings are used by builders both to make, and to remodel, buildings. Plans are also used by workmen who need to find appropriate parts of buildings if something has gone wrong. Maps are used by car drivers to find their way around town and between towns. Maps are used by pilots and yachties to enable them to navigate too. Plans and maps are used on a daily basis. Knowing how to use them is one of the necessary skills of life.
The underlying mathematics is quite simple. It is just a matter of using ratios. The scale of the plan or map is the conversion factor that changes the size in real life to the size on paper. Although the lengths change from life to its representation, the shape and angles of real life objects don’t change. This makes it easy for us to follow what the plan is representing.
Things that don’t change are called invariants. Invariants are important objects of study in their own right. Much of mathematics is involved in looking at invariant properties of objects. For instance, the sum of the angles of a triangle is invariant under scaling, as is the shape of a given figure. A square always has four equal sides and four angles that are right angles, no matter how large or how small the square is. In higher mathematics, things called matrices affect direction. An important invariant here for individual matrices is the directions that remain fixed. These have useful ramifications. So dealing with scale drawings in the shape of plans and maps is a first step at looking into a fundamental idea that is to be found in most of mathematics.
street map of local area
compasses (for bearings)
large sheets of paper
Connected 3, 1999, Mechanical Harry make a map.
We begin the week by working in pairs to draw a scale plan of the classroom and the objects in it. To do this accurately we first need to discuss the use of scale.
- Show the students an outline shape of their classroom drawn to a scale of 10cm to 1m. (A sheet of manila paper will be about the right size for classrooms that are approximately 8 metres by 6.5 metres).
What can you tell me about this shape?
What could it represent?
- If I told you that this was a floor plan of a room that is 10 times larger than this piece of paper, what do you think it could be a plan of? (If no one guesses correctly add windows to the plan).
- Use the scale plan to work out the dimensions of the classroom.
How wide is our classroom?
Check this by measuring the actual classroom.
- As a class add the classroom door(s) to the plan.
Where will we put the door(s)?
How large will this be on the plan?
What part of the door do we need to measure? (width only)
- Have a student measure the door.
If the door is 80cm wide (which is approximately the width of most doors) how large will it be on the plan?
- Add the doors to the plan by drawing the semi-circular opening space.
- Have pairs of students prepare a floor plan of the classroom with windows and opening doors. This plan will be used in the following days as the students draw a detailed floor plan of the class as it is now and how they would like it to be in future.
Over the next 2 to 3 days the students prepare detailed scale plans of the classroom to include desks and other major features.
- Ask and discuss the students' ideas about how they will complete the task. These ideas may include:
Making a scale model of a desk and then tracing around it for the desks in the room.
One student measuring the object and the other making the scale model.
Working with other pairs to share information.
- As the students work, ask questions that focus on their understanding and use of scale.
How did you work out the size of the desk on your plan?
What remains the same when you make a scale model? (shape, angle)
What changes? (length of sides, area)
How much smaller is it?
(Where appropriate, ask the students to quantify the change in length and area, i.e, length of scale model = one-tenth of real object, area of scale = one-hundredth of real object.)
- Share and compare completed scale floor plans.
Are our plans the same?
Why are there differences?
What did you find difficult about this task?
What did you learn while completing it?
Why do you think that scale plans are useful? (for building, for designing and redesigning, for planning, etc.)
Next we give the students an opportunity to redesign the classroom floor plan with the aim of selecting one plan as the basis for a real reorganisation.
What do you think is good about the way our classroom is arranged?
What would you like to change?
- I would like to rearrange the classroom and I would like your ideas. We are going to work in pairs to create a new floor plan. We will then select one and rearrange our classroom on Friday.
What do we need to remember when we do this?
- Brainstorm and list considerations. The classroom exits must remain clear; there must be room to move between the desks, etc.
- Discuss use of cardboard scale models of the furniture so that they can be rearranged in the process of deciding on a final arrangement of the furniture.
- Display completed floor plans.
- Discuss the plans, getting the students to ask questions of the "architects" of the plans. For example:
Why have you put the teacher’s desk there?
Is there room to walk to the book corner if you are seated at X?
How would we get out quickly if the fire alarm goes?
Can everyone see the whiteboard?
- Number the plans and get students to vote for the plan that they would like.
- Rearrange the room according to the best plan.
In the final part of our Location, Location unit, we consider a map of the local region and use a compass to put a bearing on our classroom plan.
- Display and discuss a map of the local region.
Can you locate our school?
The local shop?
The swimming pool?
What do the symbols on the map mean?
- Hopefully one of the symbols is the compass bearing for North. If not display a map that does have a compass bearing. Work with the class to add a compass bearing to the classroom floor plan.