Vectors and Maps

Can your students read a map? Vectors are multidimensional numbers used for making maps. Vectors are used by surveyors, archeologists, cartographers, and of course, physicists. Below are simple activities that will work for introducing vectors in elementary school as part of map making and reading exercises.

Objective: Students will learn how to write and read vectors by doing activities which use vectors.

Reformatted: January 2010

 

Part 1: Distance-Distance Vectors

Distance-Distance vectors represent how far over and how far forward.

They are properly written like: vector However for ease they are frequently written as [2, 3].
Activity 1: Classroom Vectors

Make 2 sets of signs with numbers starting with 1 going as high as you need. Place the signs on two walls every foot starting from a corner numbering every foot across the room. Have students determine what the classroom vector is for their seat. For example if they see the number 13 straight in front of them and the number 8 directly to the side the vector for their seat is [13, 8].

Once they have learned to identify the vector for their location have them make maps of the room by using graph paper and numbering the grid on the graph paper to match the room.

 

map

This activity may be extended to three dimensional vectors by labeling the feet up from the floor. For example if a student's head vector written as [ 6, 8, 4], the he is standing 6 feet over, 8 feet forward, and his head is 4 feet above the floor.

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Activity 2: Chess board Vectors

Label two edges of a chess board with the numbers 1 through 8. Have students place pieces on the board and identify the vector to the location of the piece. Be sure they count over-up not up-over. Have them discuss the difference between [ 5, 2] and [2, 5].

Activity 3: Applied Vectors and maps.

Most street maps substitute one of the dimensions as letters rather than numbers. Students may replace the letters on the map with number and give the map vector for various locations.

On a map of the world the equator and Greenwich mean time are the standard lines for marking vectors. Notice that locations to the west or south require the use of negative numbers in the vectors.

Center city Philadelphia is laid out such that vectors may be used to assign addresses. The East-West starting point is the River with street going 1 to 23 going west. City Hall is the starting point moving north and south with address going up by 100 for each block moved north.

Archeologists use vectors to map their sites. Many pictures of digs show a set of strings stretched across the dig to the the vector to objects.

Activity 4: Vector Addition

Adding distance-distance vectors is very easy. Students may discover the method using the following reasoning: The first vector represents a starting point, the second vector represents how far someone moved and the sum represents where they ended up. This may be demonstrated by moving pieces on the chessboard or having students walk out the steps in the classroom.

vector addition

Here are student started at a point 3 steps over and 5 steps forward. She then moved over 4 more steps and up 6 more steps to stop at 7 steps over, 11 forward.

 
 

Part 2: distance angle vectors

Distance-angle vectors represent how far and at what angle. They may be written as 3/45 -that's 3 followed by the angle symbol followed by 45.

Activity 5: Classroom Vectors

To do the distance-angle vectors in the classroom find the center of the room. Draw a protractor on the floor there. Zero degree should point to a significant location like the door or the teacher's desk. Angle numbers should rotate counter-clock-wise around the protractor. Have students take turns standing in the center of the room and determining what angle they must look to see any person or object in the room. They may count out steps, or measure feet from the center to determine distance. They may use polar graphing paper to make a map of the room this way.

This may be expanded to three dimension by placing pictures at different heights on the wall. Student then must also determine what angle from level they are looking.

distance & angle

Here each circle represents 4 steps from center.

Activity 6: Applied Vectors for discussion

Surveyors use distance angle vectors when they plot out land. The surveyor holding the striped pole is marking an exact point. The device that the other surveyor looks through calculates the angle and distance to the striped pole.

Look at the north pole on a globe. The line radiating from it represent angle and the circles may be used to calculate distance (Notice how this is not true at the equator where the line represent distance-distance vectors.)

Pilots use maps that show the distance and angle from various airports.

 
 

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