"Search and Rescue, Part I" Teacher Support

Search and Rescue, Part I Archived PoW || Student Version

Search and Rescue, Part I is no longer the current ESCOT Problem of the Week. The student version allows teachers to use the problem with their students without giving the students access to the archived answers. Teachers can use the link to the archived problem to get ideas of student thinking.

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Search and Rescue, Part I asks students to explore concepts of degrees and angles based on a helicopter flight school theme. Students are asked to figure out appropriate headings and distances to reach specific destinations. Students are encouraged to choose the best routes and positions for their travel.

This ESCOT PoW could be used as an introductory exploration of angles and degrees.

If you have something to share with us as you use any of the links or suggestions on this page (something you tried and changed or a new idea), we would love to hear from you. Please email us.

Alignment to the NCTM Standards - Grades 6-8

Geometry
- use visualization, spatial reasoning, and geometric modeling to solve problems

Measurement
- apply appropriate techniques, tools, and formulas to determine measurements

Problem Solving
- solve problems that arise in mathematics and in other contexts

Communication
- communicate their mathematical thinking coherently and clearly to peers, teachers, and others
- use the language of mathematics to express mathematical ideas precisely

Connections
- recognize and apply mathematics in contexts outside of mathematics

Possible Activities:

Pre-Activity
- Get students acquainted with concepts of angles. Ask them what kinds of angles they encounter in their everyday lives (90-degree angles in doors and windows; 45-degree angles in rooftops; 20-degree angles on ramps, etc.). Then, referring to navigation, have students describe how to determine, given one angle, the angle that is in the opposite direction (180 degrees clockwise or counterclockwise). For instance, for angles less than 180, the angle in the opposite direction is 180 degrees more; for angles more than 180 degrees, the angle in the opposite direction is 180 less. Discuss what might be signified by angles greater than 360 degrees.

- Ask students to pair up. Have one person walk a certain path and call out headings, and have the other person respond with the path "in the opposite direction." This could be extended to looking for other symmetric paths, like ones that are a 90-degree rotation.

- The Traveling Monkey - Ivars Peterson's MathLand
To use the idea covered in The Traveling Monkey, give students a map with two ice cream stands on it. Ask the students,

- how many ways are there of visiting each stand and returning home?
- which way is best, according to minimum distance?

Then ask the students to try 3, 4, and 5 ice cream stands. They will see that the number of possible combinations quickly increases. Students could design their own maps with the goal of making it difficult for a classmate to find the shortest way.

Post-Activity
- Pythagorean Explorer - Shodor Organization

- Have students investigate another way of thinking about angles. For example, if I am at a 90-degree heading, how much do I have to rotate to get to a 30-degree heading? Have students explain how to navigate around different shapes, starting with a square, then a triangle (different kinds), then a polygon. This reinforces some angle concepts and leads them to the sum of exterior angles theorem ("In any triangle, if one of the sides is extended, then the exterior angle is greater than either of the opposite interior angles").

Suggested Activities from Susan Socha:

Ask the students to give the bearing and distance to get from one city to another in the U.S. They will first need to have a basic understanding of latitude and longitude. Here is what they can do (there are two parts, one easier than the other):

1. A simpler problem can be to go to Indo.Com, the Bali Indonesia Travel Portal's distance calculator How Far Is It. This site calculates the distances and gives the bearings for getting from one place to another in both directions. From the helicopter applet, they should have noticed that to get back to a starting point, they just add 180 degrees. If you take cities like Washington, DC, and Richmond, VA, the bearings going and returning are close to 180 degrees apart, but when you go from Washington, DC to Savannah, GA they are more than 180 degrees. The students should think about why this happens in the real world. They could try a series of cities, noting the bearings and distances.

2. The second project is more complicated. Students can go to the U.S. Census Bureau's U.S. Gazetteer and find the latitude and longitude of a selected city, say, Washington, DC. They would then be given a target city like New York City, and told to try to estimate the distance and heading.

Then students could go to the Federal Communications Commission (FCC) page, Find Terminal Coordinates, which allows the visitor to enter the initial latitude and longitude, (Washington, DC), a bearing, and a distance in kilometers. The applet then gives the terminal position in terms of latitude and longitude. Students could enter their own estimates. When they enter an estimate, they should be reminded that the first box is for degrees, the second box for minutes, and the third for seconds.

After that, students can check their estimate by going to the Etak Guide's Find Location page. This finds a location in the US given the latitude and longitude. Students can enter the latitude and longitude of their estimated positions, and see where in the U.S. they end up - you hope, close to New York. It shows an actual map of their terminal position. There are some things they need to know when entering latitude and longitude: south is entered as a negative in front of the degrees and minutes, colons must appear between degrees, minutes, and seconds, and west must be entered as a negative in front of the minutes and seconds.

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