• Students will combine skills in Geography, Geometry and Trigonometry to write their own original Ambiguous map problem for two cities with identical names.
• Students will make geometric constructions (especially perpendicular bisectors).
• Students will measure distances on a map and make proper conversions.
• Students will measure angles on a map.
• Students will demonstrate the ability to find locations on a map.
• Students will minimize computational errors and understand how estimates from rounding or measuring affect the amount of error that does occur.

Procedures:

Students should have already had a few lessons on the Ambiguous case in their Trigonometry class. Use this lab as a follow-up to the class discussions. Students can work in pairs on this lab/project if desired. There should be some incentive for students to create a problem with the least error and the most unique solution. Use the step-by-step instructions provided below to show how an Ambiguous map problem is created. Students should then conduct their own research to find two cities with identical names. The lab pairs must submit all of the following to meet the objectives of this project: construction/creation work on the Atlas or map, the statement of the Ambiguous problem that arises from their creation, the complete Trigonometric solution (including sketch and use of the “Law of Sines”) to the problem, a sentence answer to the problem and a small explanation of any error that occurred in calculations. Students will be graded using the rubric that follows this lab.

Enrichment Activity:

Students should be encouraged to find identical cities between countries. One example would be Charleroi, PA and Charleroi, Belgium. Have students explain some of the new issues that would be encountered in creating an Ambiguous map problem over such large distances.

Creating an Ambiguous Map Problem:

Use the following process (and illustrated example) to create your own Ambiguous Map Problem from an Atlas or a map, then write the problem, solve the problem and submit the solution including all work used to create the problem and all calculations and drawings used to solve it. Check the accuracy of the problem via measurement and discuss any errors that occurred (provide reasons for any errors that may have occurred also).

1. Find two cities with identical names within an Atlas such that they are contained on the same page of a map (the cities can cross states lines as long as the map that is being used contains multiple states on the same page).
   Palmyra, PA to Palmyra, VA will be utilized to illustrate this process.
2. Use a ruler to connect the two cities with the same name to a totally different city that lies along the same line. In this case, connect both Palmyra’s with Dillwyn, VA (there were other options).
3. Find the Perpendicular Bisector of the line containing Palmyra, PA to Palmyra VA in order to find an additional city to create the Ambiguous case (use any one of the options listed below for constructing Perpendicular lines):
   
1. Patty Paper
   
   1. Place the Patty Paper or Wax Paper on top of the map and trace the points that mark Palmyra, PA and Palmyra, VA.
   2. Pick up the Patty Paper (or Wax Paper) and fold it so that the two points marking the cities coincide (are on top of each other).
   3. When the points are lined up properly, crease the Patty Paper (or Wax Paper) along the fold.
   4. Open the Patty Paper (or Wax Paper) and draw a line along the crease. This line is the perpendicular bisector of the two points marking Palmyra, VA and Palmyra PA.
   5. Place the Patty Paper over the map and observe the crease until it hits another city (Upper Marlboro, MD lines up fairly well in this case).
      
2. Compass and Ruler
   
   1. Place the pointed end of the compass on Palmyra, PA. Open the compass a little more than half the distance to Palmyra, VA.
   2. With the pointed end on Palmyra, PA, swing an arc in the region between Palmyra, PA and Palmyra, VA on both sides of the line containing both cities.
   3. Without changing the opening in the compass, put the pointed end on Palmyra, VA and swing another arc in the regions between both cities until the arcs intersect in two places.
   4. Use a ruler to draw the line containing both points of intersection. This is the perpendicular bisector. Locate a city along this line (Upper Marlboro, MD lines up fairly well in this case).
      
3. Mira
   
   1. Place the Mira upright with the beveled edge face down over the map between Palmyra, PA and Palmyra VA.
   2. Look through the Mira from the Palmyra, PA side (again the beveled edge is face down).
   3. Slide the Mira until the point marking Palmyra, PA lines up with the point marking Palmyra, VA.
   4. When the two points coincide (lie on top of each other), draw a line along the beveled edge of the Mira. This is the perpendicular bisector. Locate a city along this line (Upper Marlboro, MD lines up fairly well in this case).
      
4. Connect Dillwyn, VA to Upper Marlboro MD with a ruler.
5. Verify that the distances from Palmyra, PA and Palmyra VA are approximately equal distant to Upper Marlboro, MD (they were both around 8.5 cm on the map used for this lab).
6. Measure the distance from Dillwyn, VA to Upper Marlboro, MD and use the map’s conversion to compute the unit of measurement. For this sample, 1 cm = 19.6 km. The distance was 10.4 cm or 203.84 km.
7. Find the distance from Upper Marlboro, MD to either Palmyra city. The distance was already calculated in step 5, so 8.5 cm is 166.6 km.
8. Assume that there was a North, South, West and East line placed directly over Dillwyn, VA. Hopefully the Atlas or map used for this lab has NORTH, SOUTH, WEST and EAST lines labeled. Use a protractor to find the angle from Dillwyn to Palmyra, VA and Palmyra PA. In this case, Palmyra was approximately 26.5º northwest of Dillwyn.
9. Now find the angle from Dillwyn to Upper Marlboro, MD the same way. In this example, Upper Marlboro is approximately 45º northwest of Dillwyn. At this point, everything necessary to write the Ambiguous case problem is available.
10. Remember that angles and bearings can be radically different for each Ambiguous problem created. Angles might be southeast, northeast, southwest, etc. In fact, the problem gets more creative every time angles and bearings become more unique. The choice of how the problem is stated can also add variations to the presented facts. Is the angle given from Dillwyn to Upper Marlboro or from Upper Marlboro to Dillwyn for example?

Statement of the Ambiguous Case Problem:

Dillwyn, VA is 203.84 km from Upper Marlboro, MD. Upper Marlboro, MD is 166.6 km from Palmyra. Palmyra is 26.5º northeast of Dillwyn, VA, and Upper Marlboro, MD is 45º northeast of Dillwyn. How far is it from Dillwyn, VA to Palmyra?

Note: Notice that we don’t specify a state for Palmyra because that is the portion of this problem that is AMBIGUOUS.

Sketch:

Solution:

Answer:
Palmyra, PA is approximately 346.8 kilometers from Dillwyn, VA and Palmyra, VA is approximately 39.7 kilometers from Dillwyn, VA.

Verification:
The distance on the map from Dillwyn to Palmyra, PA is 17.7 cm (which scales to 17.7 * 19.6 or 346.92 kilometers). The distance on the map from Dillwyn to Palmyra, VA is 2 cm (which scales to 2 * 19.6 or 39.2 kilometers). The error is very small and probably comes from measurement and rounding errors.

Grading Rubric:

Map Example Illustration for this Lab/Project:

Susquehanna Township School District

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© J. Jay Yohe Jr. -- July 2002