Where Are We?
Author: Brink Harrison
Time:
3 class periods
Preparation
Time:
5 min copying Mercator Projections
5 min copying Robinson Projections
5-10 min selecting coordinates
5-10 min copying Epicenters
Materials:
Mercator Projection maps
(day 1)
Robinson Projection maps
(day 2)
Navigation and Distance sheet (day 2)
Earthquake Epicenters
(day 3)
Abstract
Mapping locations on the globe allows students to use Cartesian coordinates while becoming
familiar with the basic terms of navigation
Purpose – Students use Cartesian coordinates or degrees of latitude and longitude to locate
important locations within their specific countries.
Objectives
Day 1
Students will be able to:
1. Use a grid to apply the concept of latitude and longitude, or positive and negative Cartesian
coordinates, to determine the position of an object in the room.
2. Determine the “quadrant” of the earth in which a specific location lies given the coordinates
in degrees of latitude and longitude, or in positive or negative Cartesian coordinates, on a
Mercator Projection of the World.
3. Find a specific location on a map given the coordinates in degrees of latitude and longitude,
or in positive or negative Cartesian coordinates, on a Mercator Projection of the World.
Day 2
Students will be able to:
1. Determine the “quadrant” of the earth in which a specific location lies given the coordinates
in degrees of latitude and longitude, or in positive or negative Cartesian coordinates, on a
Robinson Projection of the World
2. Find a specific location on a map given the coordinates in degrees of latitude and longitude,
or in positive or negative Cartesian coordinates, on a Robinson Projection of the World
3. Calculate the distance traveled between two locations either on the same meridian of
longitude or the same line of latitude.
Day 3
Students will be able to:
1. Find the coordinates for a location of the epicenter of an earthquake
National Science Education Standard:
Geometry
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Specify locations and describe spatial relationships using coordinate geometry and other
representational systems.
Use visualization, spatial reasoning, and geometric modeling to solve problems.
Number and Operations
Compute fluently and make reasonable estimates
Teacher Background
Teachers should be familiar with the use of Cartesian coordinates to plot points on graph paper.
See attached sheet for background on Navigation
Sources:
http://encarta.msn.com/encyclopedia_761563211/Latitude_and_Longitude.html
http://www-ocean.tamu.edu/~dkobilka/navigation.html
See attached sheet for background on World Maps
http://geography.about.com/library/weekly/aa030201c.htm
http://www.cnr.colostate.edu/class_info/nr502/lg2/projection_descriptions/robinson.html
http://www.cnr.colostate.edu/class_info/nr502/lg2/projection_descriptions/mercator.html
Related and Resource Websites
http://alabamamaps.ua.edu/world/world
http://geography.about.com/library/weekly/aa030201c.htm
http://www.cnr.colostate.edu/class_info/nr502/lg2/projection_descriptions/robinson.html
http://www.cnr.colostate.edu/class_info/nr502/lg2/projection_descriptions/mercator.html
http://www.geophys.washington.edu/tsunami/general/historic/historic.html
http://www.prh.noaa.gov/ptwc/olderhmsg
http://www-istp.gsfc.nasa.gov/stargaze/Slatlong.htm
Activity
Day 1
Begin by asking the students:
“How would you tell somebody in England exactly where Bangladesh is in the world? How do
ships know their location out in the middle of the Pacific Ocean?”
Hopefully somebody will mention navigation and bring up the ideas of latitude and longitude. If
nobody has a clue, then use the following “activity” to get the idea across of laying out a grid that
everybody can agree upon and use that grid for finding the location of something.
Quickly have the students arrange the desks in straight rows and columns and move to a specific
desk. Ask the students, “How would you describe my location in the room?” Have the students
discuss how to do this, and, hopefully, someone will point out that you need a “starting point” or
“origin” to give distances and directions to your location. The class will also need to agree which
direction is which (north, south, east, and west) in order for the coordinates to make sense. Pick a
specific student as the “origin” of the coordinate plane and have that student give the coordinates
of your location. Do this several times with different students as the origin until the class
understands the concept of the coordinates. To assess how well the students understand, pick an
origin and wander around asking students at random, “What are your coordinates?” Insist that the
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students give their coordinates in terms of two directions, such as, “3 desks north and 4 desks
east.”
If somebody does mention latitude and longitude, ask the class, “How can we model these
concepts using what we have here in the classroom?” Allow time for students to discuss ideas.
They could use the desks as described in the paragraph above or ceiling tiles or floor tiles if
applicable. Be open and accept any reasonable answer that has a rectangular grid connected to
it. Again, insist that the students give their coordinates in terms of two directions, such as, “3 tiles
south and 6 tiles west.”
Tell the students that there are maps of the world that are laid out on a similar rectangular grid,
just like the grid they used in the classroom. Pass out the Mercator Projection maps. (This map
can be found at http://alabamamaps.ua.edu/world/world/world2.pdf) Explain that Mercator
Projection is different because the grid is rectangular and this causes distortion in the size of the
countries as you move closer to the poles. (See Teacher Background, Mercator Projections.)
Have the students find and darken the lines representing the equator and Prime Meridian on the
map. Tell the students that the point of intersection of these two lines is the origin from which
coordinates are calculated. The darkened lines also divide the projection into four quadrants,
which are labeled quadrants I, II, III, and IV in their standard positions as if on a coordinate plane.
Ask the students to tell you the quadrant the following coordinates are in:
1.
2.
3.
20 N ,16W
80S , 30 E
15S , 49W
Tell the students that the quadrants also allow us to use positive and negative Cartesian
coordinates to find locations. Ask them, “Looking at the map, which coordinates would you expect
to be positive?” Which would be negative? (North latitudes and East longitudes would be positive
while South latitudes and West longitudes would be negative). Discuss the importance of which
direction comes first in the coordinate pairs. (Degrees of latitude come first and then degrees of
longitude, which are backwards from the normal ways of plotting points as the first number gives
the location vertically and the second number gives you the location horizontally) Ask the
students the range of numbers valid for each coordinate. (Degrees of latitude can vary from
 90 to 90 and degrees of longitude can vary from  180 to 180 ) Ask the students to tell you
the quadrant the following coordinates are in:
 23, 54
1.
2.
74,  120
 67,  102
3.
Give the students the following coordinates to locate on their map: (these are just examples; you
may choose your own.)
29 N , 95W (Houston, Texas)
1.
2.
3.
1S , 36 E
(Nairobi, Kenya)
41 N , 50W (Site of Titanic Sinking)
Homework
(Since the students will be approximating the locations of tectonic plates later in science by
plotting the locations of earthquakes, you may use the following website to select the coordinates
in degrees latitude and longitude of current earthquakes: http://neic.usgs.gov/neis/bulletin/ . If
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you want the coordinates in positive or negative Cartesian coordinates, use the following website
instead: http://earthquake.usgs.gov/recenteqsww/Quakes/quakes_all.html. Round the degrees to
the nearest whole number.
In social studies the students will be looking at tsunami and their locations. Below is a list of
coordinates for earthquakes that have caused tsunami in the past. Have the students plot their
locations on their maps and identify the country as well:
1.
2.
3.
4.
5.
55 N , 134W
38S , 73W
4S , 135 E
53 N , 164W
53 N , 160 E
35 N , 121W
(Alaska)
(Chile)
(Indonesia)
(Aleutian Islands)
(Kamchatka Peninsula, USSR)
6.
(USA-California coast)
Source: http://www.prh.noaa.gov/ptwc/olderhmsg
Day 2
Go over the students’ maps and locations from the day before. Discuss any questions the
students might have. Tell the students that there are many different types of maps, each having
both good and bad aspects. Pass out the Robinson Projection maps, which may be found at
http://alabamamaps.ua.edu/world/world/world3.pdf , and ask, “How is this map different from the
Mercator Projections we used yesterday?” Allow time for students to discover and discuss
differences. Pass out Latitude and Longitude maps as well to help the students find lines of
latitude and longitude on the Robinson Projection as they are curved this time rather than being
straight lines. (http://alabamamaps.ua.edu/world/world/page006.pdf)
For practice, have the students find the location of the following cities given their coordinates:
1. 36 N , 115W
(Las Vegas, Nevada)
2. 31 N , 121 E
(Shanghai, China)
3. 37S , 145 E
(Melbourne, Australia)
4. 24 N , 66 E
(Karachi, Pakistan)
5. 6S , 106 E
(Jakarta, Indonesia)
(I just chose these cities at random. You can get the coordinates of many cities at
http://www.indo.com/distance/)
Ask the students, “What else are maps used for besides finding locations?” (finding distance)
Now we need to talk about how distances on maps can be related to latitude and longitude.” For
convenient measurement of distance, mariners developed the nautical mile (nm), to fit measures
of latitude and longitude.
By definition, 1 minute of longitude at the equator = 1 nautical mile (  1.15 miles ).
Since there are 60 minutes in one degree, this means at the equator, one degree of longitude =
60 nm = 69.05 miles. To convert nautical miles to miles, use the website
http://209.182.50.103/length/nautical-miles-to-miles.htm or the conversion 1 nm  1.15 miles .
Have the students convert the following distances (you could introduce unit analysis here)
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1.
2.
3.
4.
9 nm  _____ miles
1200 nm  ________ miles
4000 miles  ___________ nm
852 milles  ________ nm
Ask the students, “What’s the problem of using longitude to define 1 nautical mile?” Point out that
the distance covered by one degree of longitude shrinks as we move away from the equator and
towards the poles, where all the lines of longitude converge to one point. On the other hand,
minutes of latitude do not shrink which is why even modern mariners use the following
relationship:
one degree of latitude = 60 nm = 69.05 mile everywhere on earth, or
one minute of latitude = 1nm  1.15 miles everywhere on earth
Source: http://www-ocean.tamu.edu/~dkobilka/navigation.html
Have the students calculate the number of nautical miles (and miles) you travel between the
following two locations. (Keep the meridian the same so only the latitude is changing)
1.
34 N , 28 E to 82 N , 28 E
2.
12 N , 35W to 78S , 35W
 60 nm 

48lat   2880 nm  3312 miles
 1 lat 
 60 nm 

90 lat   5400 nm  6210 miles
 1 lat 
Tell the class, “We are now going to calculate the distance between two locations where only the
longitude is different. Looking at the map you can see that the distance covered by one degree of
longitude at 35 N latitude is different from the distance covered by one degree of longitude at
50N latitude because the meridians are father apart at 35N latitude than they are at 50N
latitude.” To calculate the distance one degree of longitude at any latitude use the formula
60 nmcoslatitude  ____ nm .
For example, the distance covered by one degree of longitude at
35N latitude is
60 nm cos 35  60 nm .82 
49.2 nm
1 lat at 35 N
and the distance covered by one degree of longitude at 50 N latitude is
60 nm cos 50  60 nm .64 
38.4 nm
1 lat at 50 N
Now have the students calculate the distance from 35 N , 5 E to 35 N , 75 E
 49.2 nm 

70  3444 nm
 1 lat at 35 N 
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Remind the students that they must calculate the distance covered by one degree of longitude at
the given latitude before they can calculate the number of nautical miles traveled by changing the
longitude.
Homework
Do the Distances, Latitude, and Longitude sheet for homework
Day 3
Begin by going over the homework from Day 2 and allow time for discussion of the “Thought
Question.” Assuming the world is flat, calculate the number of nautical miles you travel “vertically”
by using the change in the degrees of latitude and the nautical miles you travel “horizontally” by
using the change in the degrees of longitude. This gives the students two sides of a right triangle
and they would then use the Pythagorean Theorem, from geometry, to find the length of the
hypotenuse, which is the distance they are looking for. (It’s much more complicated if you use a
curved surface.)
Tell the students, “We needed to use geometry to get an approximate answer for the thought
question”. Today we are also going to use some more geometry, specifically circles, to find the
coordinates of the location of the epicenter of an earthquake.” Substantial parts of this lesson
are derived from “Earthquakes” http://mimp.mems.cmu.edu/~ordofmag/earthqua/earqua.htm
I would recommend putting Figure 6-2, the seismogram from Dallas, on the same page but above
Figure 6-1. This will put the data from Dallas directly above the graphs and should make it easier
to explain/demonstrate the steps the students need to follow to complete the example.
Guide the students through each step, allowing them the time to complete the step before moving
on to the next. The students may have some trouble making the dots on the tracing paper. Tell
them that it does not matter where the first dot is placed, but that the second dot must be placed
so there is a 3.3 minute gap between the dots. Demonstrate sliding the tracing paper across the
graph until the two dots line up with the curves; one dot should be on the S-wave curve and the
other dot should be on the P-wave curve. Drop a vertical line down to the horizontal axis, and the
value on the horizontal axis will tell you the distance from Dallas to the epicenter (roughly 1200
miles).
Homework
Have the students follow the same steps for the data from the other three seismograms, Figure
6-3. using the location of one of the cities as the center, draw a circle with the appropriate length
radius (the distance from the city to the epicenter). Do this for the other two cities as well. The
point of intersection of the three circles is the epicenter of the earthquake. Have the students
estimate the location of the epicenter in degrees latitude and longitude or positive and negative
Cartesian coordinates.
Embedded Assessment
Day 1 can be assessed by informal discussion on how to locate objects on a grid in the
classroom. Wandering around the room asking students their location also gives the students the
opportunity for self-assessment on issues of direction from the origin (“Am I north or south of the
origin”, etc.). Informal observation as the students plot locations on maps can be used to assess
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how well they are able to carry over the idea of using a grid to locate objects on the Mercator
Projection.
Day 2 can be assessed by using informal observations to see how well the students plot locations
on the Robinson Projection maps. Guided instruction, through examples, will give you an idea of
how well the students can calculate the distance moving vertically along the meridian of longitude
or horizontally along the line of latitude.
Day 3 can be assessed by using informal observations and guided instruction to see how well the
students can use the calculated distances and coordinates of the seismographs to find the
epicenter of the earthquake.
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