Scale, distance and exact location

Grade 9 Social Studies Worksheet - Challenge Me

Circle the best answer.

1. Read this first

   Use this

   Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km.

2. 1.

   Why does the great-circle route from Phoenix to Charleston bend on a Mercator map, Ari asks?

   • A Mercator map keeps the shortest paths straight and bends compass headings
   • Mercator keeps constant headings straight, so the shortest path curves
   • The route really bends, since Earth’s surface bends the shortest path
   • A Mercator map is flat, so any route drawn on it must bend
3. 2.

   A captain holds one compass heading between two ports. How does that route compare with the great-circle route, Bo asks?

   • It is longer in every case, so the two never match
   • It is never shorter, since the great circle is the shortest path on a sphere
   • It is shorter, since a constant heading avoids the bend in the great circle
   • It is the same length, since a straight line on the map is the shortest
4. 3.

   Two ports lie on 40°N. A ship sails due east along the 40°N parallel. Is that the shortest route, Esme asks?

   • Yes: staying on one latitude holds one heading, which is the shortest
   • No: the shortest route bulges south of the line toward the equator
   • Yes: each parallel is a great circle, so it is the shortest route
   • No: the shortest route bulges north of the 40°N line toward the pole

5. Use this

   Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km. Card B. A flat-plane check. Using the city coordinates, a student gets Phoenix to Charleston as 2,794 km with a right triangle on a flat map. Coordinates are rounded to the minute, and one minute of latitude is about 1.85 km.

6. 4.

   A flat-plane method gave 2,794 km, and the sphere gave 2,780 km. What does the 14 km gap show, Kira asks?

   • A flat triangle is close here: 14 km is about half a percent
   • The gap shows Earth is flat over distances of 2,800 km
   • The sphere figure is wrong by 14 km, so the flat figure is right
   • The two must agree exactly, so one of them has an error

7. Use this

   Card C. East–west distance. One degree of longitude spans about 111 km times the cosine of the latitude. The cosine of 60° is 0.50. Two towns lie on the 60°N parallel, 10° of longitude apart.

8. 5.

   Two towns on the 60°N parallel are 10° of longitude apart. How far apart are they along the parallel, Milo asks?

   • About 55 km: 10° at 5.5 km a degree
   • About 555 km: 10° at 111 × 0.50 km a degree
   • About 1,110 km: 10° at 111 km a degree
   • About 1,110 km along the parallel, and 555 km in a straight line

9. Use this

   Card C. East–west distance. One degree of longitude spans about 111 km times the cosine of the latitude. The cosine of 60° is 0.50. Two towns lie on the 60°N parallel, 10° of longitude apart. Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km.

10. 6.

    For those two towns, how does the great-circle distance compare with the 555 km along the parallel, Noah asks?

    • It is the same, since both stay on the 60°N line
    • A little shorter: the great circle bulges toward the pole
    • It is longer, since the great circle bulges and adds distance
    • It is half as long, since the pole shortens the path

11. Use this

    Card D. Drawing roads. On a 1:1,000,000 map, 1 mm stands for 1 km. A country road about 10 m wide is drawn as a line at least 0.3 mm wide.

12. 7.

    Why is a 10 m road drawn 0.3 mm wide on a 1:1,000,000 map, Pia asks?

    • Widening a line makes the map’s scale wrong everywhere on it
    • The road is really 300 m wide, which is what 0.3 mm stands for
    • At true scale it would be 1 mm, so drawing it 0.3 mm shrinks it
    • At true scale it would be 0.01 mm, too thin to see, so it is widened
13. 8.

    Which measurement does drawing roads at 0.3 mm distort most on that map, Ren asks?

    • A road’s width; the distance between two town centers stays reliable
    • The length of the road, since a thick line is longer than a thin one
    • The distance between two town centers, since a wide line moves the centers
    • The map’s ratio scale, since line widths are part of a ratio

14. Use this

    Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km.

15. 9.

    Tala’s note says: “On a Mercator map the shortest route is always a straight line.” Which fix is right?

    • Leave it: Mercator is accurate, so a straight line is the shortest
    • Change it: the shortest route is straight on a globe, so it is straight on a map
    • Change it: a constant compass heading is straight; the shortest route usually curves
    • Change it: the shortest route is straight, and a constant heading curves

16. Use this

    Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km. Card B. A flat-plane check. Using the city coordinates, a student gets Phoenix to Charleston as 2,794 km with a right triangle on a flat map. Coordinates are rounded to the minute, and one minute of latitude is about 1.85 km.

17. 10.

    Zee’s note says: “Phoenix to Quito is 5,130 km, so Quito is 5,130 km south of Phoenix.” Which fix is right?

    • Change it: the south part is 5,130 + 544 km, adding Charleston’s gap
    • Leave it: Quito is directly south, so the two distances match
    • Change it: 5,130 km is direct; the south part is about 3,740 km
    • Change it: Quito is 5,130 km east of Phoenix, not south