Navigation Activity for Law of cosines

Navigation Activity for Law of cosines

Need:

• A large field or open area
• A magnetic compass
• A placement marker
• Pencil and Paper
• A protractor

Pre-Activity:

• Review the law of cosines for triangle ABC

• Review directional vectors (direction and magnitude)
• Review Navigational vectors (heading and speed).  Remember: Navigational vectors begin at due north, rather than the usual positive x-axis for directional vectors.
• Review the proper use of a magnetic compass.

Activity:

(Teacher-Led)

1. On a large field, find a central location and mark it with something that will be readily visible from afar (I use my blue insulated coffee mug).  You will need to return to this point to complete a triangle.  Call it point A.
2. Using you compass, select a heading (azimuth) between 0 and 360.  Proceed on that heading a specified number of paces.  Record the heading and distance.  (Note:  It is important that each pace is reasonably identical.  To avoid interpretations, one person should be the designated pacesetter and maintain the same pace throughout the activity).  Call this location point B.
3. Repeat step 2.  Be sure not to pick a heading that will take you back to point A.  Call this location point C.
4. From point C, estimate the heading and distance back to point A.  Using the compass, determine the actual heading to point A and proceed there, counting the paces.  Record the information.  How close was your estimation?

(Student-Led)

5. Break the students up into smaller groups, depending on the number of available compasses.  Have them repeat steps 1-4 above, choosing different headings and distances.

Follow-up:

1. Using the law of cosines, accurately sketch and determine the triangle ABC formed from the teacher led activity.
2. Have each group accurately sketch and determine their own triangle ABC using similar calculations.
3. Questions:

o        How close were your actual measurements to the calculated results?

o        What are some reasons why your results were off?

o        What are some things we could have done to increase our accuracy?

o        Was the Law of cosines necessary?  How could we have done the same activity without it?

o        Why/where/to whom might this type of calculation be important?

Sample:

1. Starting at point A, travel 27 paces to point B on a heading of 30 degrees.
2. From point B, travel 42 paces to point C on a heading of 135 degrees.
3. From point C, estimate the distance and the heading back to point A.
4. Estimate the heading using the compass, and count the paces to A.
5. Calculate the actual distance and heading from C to A.
6.  

Notes:

• A,B,C refer to the interior angles of the triangle; a,b,c are their respective opposite sides.

• Remember alternate interior angles are congruent.

Calculations:

1. Find B

2. Find b

3. Find C

4. Find A

5. Determine the heading at C

6. Write the directions from point C to A as

44 units at 315 degrees