math in the middle of oceans navigation · these islands. many shipwrecks were caused by poor...

Math in the Middle... of Oceans NAVIGATION LESSON 7

Math Concepts• Integers• Decimals• Computation/Estimation• Ratios/Rates• Proportion• Variables/Solving equations• Distance/Rate/Time formula• Measurement• Navigational methods• Longitude, latitude• Angles, degrees, minutes• Bearing• Triangles• Range (meaning in navigation)• Parallel lines

Objectives••••• Use basic navigational techniques

to determine course direction• Measure distance using dividers• Determine angles using parallel

rules and compass

Math in the Middle ... of Oceans

Navigation

Electronic FieldtripsExperience charting a course to findsunken treasure with a sea captain

Visit a captain aboard a ferry in the NorthCarolina Ferry System

Key Terms• compass rose• D = R x T••••• range


NCTM Standards ConnectionsMATHEMATICS AS PROBLEM SOLVING

• Using basic navigational techniques to determine course direction• Determining locations at sea

MATHEMATICS AS COMMUNICATION• Finding locations, longitude and latitude, using navigational tech-

niques

MATHEMATICS AS REASONING• Using landmarks and navigational techniques to determine location

of a boat at sea• Using parallel rules to construct parallel lines and determine course

direction

MATHEMATICS AS CONNECTIONS• Using landmarks and navigational techniques to determine location

of a boat at sea

COMPUTATION AND ESTIMATION• Using parallel rules and dividers to find locations and distances at

sea

MEASUREMENT• Measuring distance using dividers••••• Determining angles using parallel rules and compass


Getting Started - Discussion Topics

The Navigation lesson is designed to encourage students to try hands-on navigation exer-cises based on a similar chart that is used in the video lesson. We realize that many of theteachers that will use this material have never used navigational methods before. However,we believe that you will find the basics of navigation to be very straightforward as well asbeing a very interesting way to teach the uses of geometry and pre-algebra.

This lesson will follow a different course on the chart than the one charted in the videolesson. This lesson will, however, use and reinforce the same methods of navigation thatwere introduced in the video lesson. We strongly recommend referring back to the videolesson for helpful ideas. This lesson elaborates upon themes presented in the video lesson.This lesson also introduces more advanced navigation, such as the forming of isoscelestriangles to determine your boat’s position.

Introduction to NavigationNavigating a boat at sea is both an art and a science. Sailors have used the principles ofgeometry for centuries in order to cross hundreds of miles of water, to avoid potentiallydeadly rocks and sand bars, and to find secret fishing locations. The work you are about todo involves some of the basic techniques captains use to determine their location and findtheir way through dangerous waters. Many of these techniques can also be used to findyour way on trails through the forest, especially in the mountains where you can see forgreat distances. These techniques involve the use of measurement, parallel lines, angles,and the definitions of various types of triangles. These types of math are all importantaspects of navigation.

Navigation has played a fascinating role in shaping history throughout the ages. Manybattles have been won because the winners’ boats arrived at a location in time, whereas thelosers were off course and late. Some countries have prospered because the navigationtechniques their sailors learned enabled them to trade with other countries thousands ofmiles away. Many of the islands in the Pacific, including Hawaii, would never have beeninhabited if it was not for the exceptional navigation skills of the people who first discoveredthese islands. Many shipwrecks were caused by poor navigation skills - captains thoughtthey were someplace else than where they actually were.

Before we start doing some actual hands-on navigation problems, we will first go over someconcepts about navigation. Navigation is different from just traveling from one place toanother. When you navigate you are trying to figure out exactly where you are at all times.

Much of the navigation exercises we will do today are based on using landmarks to tell uswhere we are. A landmark is an object that can be seen from a long distance away anddoes not move. Examples of landmarks include tall antennas, water towers, and chimneys.You might have used smaller landmarks when you were trying to remember your way on abike path. You know how to get there because you remember certain landmarks, like to turnright when you get to a certain house or a store. You often remember where to turn becauseyou remember the landmark.


Knowing where you are when you are in a boat at sea is more important than knowing yourlocation when you are drive in a car or on your bike. At least when you are in car you areon a road that is connected to other roads. When you are at sea you can go in any direction.If you get lost when driving in a car it will take you longer to get to where you want to go.However, if you get lost when you are on a boat, you may be risking hitting a rock andsinking, or causing damage to the enviroment - remember the Exxon Valdez? Also, if youare not sure where you are you may not be able to find your way back to shore if a stormcomes up suddenly.

It is also more difficult to figure out where you are when you are in a boat at sea. At seathere are no signs to tell you where you are. You do not know how far you have gonebecause there are no signs that say “Columbia 12 Miles”. Boats do not even have speed-ometers to tell you how fast you are going. And you cannot pull over at the next gas stationand ask directions!

One thing that does help at sea is the fact that you can see landmarks from a great distanceaway. There are no trees or buildings or hills to block your view. Because of this, ships cansee lighthouses and other tall structures from many miles away.

Most of the work you will be doing will be based on using a nautical chart. Nautical chartsare like maps made for boating. They are different from maps in several ways. First of all,they show mainly water features and information. Whatever information they have on landis based on what you see on land from a boat in the water. Secondly, charts are made tobe written on - we write on charts to determine where we are and where we want to go.

Getting Started - Discussion Topics continued


Activity - Plotting CoursesLook at the copy of the chart you will be working on. The numbers scattered in the wateron the chart are the water depths in feet. The marks on both sides of the chart is theLatitude Scale. The marks at the top and bottom of the chart is the Longitude Scale. Wewill use these scales to find various locations on the chart. Mark each of these scales inpencil so you will remember which is which. Both the longitude and latitude scales aredivided into degrees and into minutes (60 minutes per degree). Minutes are further dividedinto seconds (for our chart, seconds are divided into tenths). Sixty nautical miles = 1degree. Therefore each minute = 1 nautical mile. This makes measuring distance veryconvenient when you use a chart. Each minute on the latitude scale = 1 nautical mile.

One important tool we use when working with charts are called dividers. Notice how theyare basically the same thing as the compass you use in math class except they have twosharp points rather than one. As we spread the dividers out we can measure distance aswell as degrees and minutes on our longitude and latitude scale.

Let’s try using the dividers to find a location on the chart. Find “Little Gull Island” on thechart. What is its longitude and latitude? To find this we spread the dividers so that onepoint lies on Little Gull Island and the other point on the closest line to the island thatintersects the latitude scale (this will be a horizontal line). We then move the dividers overto the latitude scale - letting one point follow the line that is perpendicular to the latitude scale- and see where the other point falls. It should read 41 degrees 12.4 minutes.

Go though the same procedure to determine longitude (remember, the longitude scale is onthe top and bottom of the chart so the reference line that one point of your dividers willintersect will be a vertical line). Longitude: 72 degrees 6.4 minutes.

Now try this backwards - we have been given the location of a sunken wreck that is a greatspot to fish. Sunken wrecks make for good fishing because small plant and animal lifeattaches to the wreck and attracts fish that eat this life. The wreck is located at -

Longitude: 71 degrees 53.5 minutes; Latitude: 40 degrees 59 minutes

Plotting a position from a known longitude and latitude is just a little more work than the otherway around. First, find the longitude. Measure the distance with your dividers from 71degrees 53.5 ‘ on the longitude scale to the closest line that is perpendicular to the longitudescale. Then come up that line to the general area where your latitude (horizontal line) isgoing to fall. Mark the measurement you made with your dividers on the lines that run acrossthe chart. Make a mark on the line above and below the latitude you are trying to plot.Connect these two lines with your ruler. Then measure the latitude with your dividers, andplot your position by transferring this measurement over to the line formed by your ruler.


Locate the wreck so we know where to steer the boat.

Notice the big circle on the chart indicating which directions are North, South, East, andWest. This is call a “compass rose”. The numbers on the compass rose on the chart arethe same as you will find on a ship's magnetic compass. All boats that go into the oceanhave magnetic compasses attached to them near the steering wheel. These work the sameway as the magnetic compasses that Girl and Boy Scouts learn to use. These work basedon the Earth’s magnetism which keeps the compass always pointing towards the North. Youwill notice that both compasses display 360 degree marks, just like a circle is 360 degreesin circumference. To do most of our problems, we will have to draw parallel lines. One ofthese parallel lines will pass directly through the very middle of the compass rose. Draw thisline lightly because you will need to erase it every time you draw a new line through thecompass rose.

The other parallel line will be either your imaginary course you want the boat to take or theparallel line will be the direction you are looking as you spot a landmark from your boat. Thiswill all fit together as we start working on the problems, so do not worry if you do notunderstand all of this right now.

When you go to sea, a chart is one of the most important pieces of equipment to haveaboard. For this activity, we will be using the same chart as was used in the video lesson.This is a training chart used by students practicing navigation. The area it covers includesthe eastern end of Long Island, Block Island, Block Island Sound, and part of the AtlanticOcean.

Our exercises will take us over the wreck of the Mary-Lee, a fishing boat that sunk in heavyseas in 1956.

We are departing from Pt. Judith Harbor. Find this on the chart.

Note: Part of good chart work includes knowing the locations of major geographical pointson a chart. For this reason, we suggest that you do not tell your students where thesegeographical points are located. It will also be fun for your class to see who is the first tolocate points referred to in lesson.

Why is it important to know the geography of a chart you are using?

ANSWER: If you are out in a boat and an accident happens, you may need to travelto a location you are not familiar with to get immediate care. You also might hear onthe radio that someone is in trouble at a location you have never been to. Also, if youare off course and are on the radio with the Coast Guard, they may refer you to alocation unknown to you. While longitude and latitude can be given over the radio,knowing the name of a location can be easier to hear on a bad radio connection andcan be easier to remember than longitude and latitude numbers you may be given.

Activity - Plotting Courses continued


Therefore, being able to quickly locate geographical features on your chart is part ofbeing a good navigator. St. Judith Harbour is located in the far northeast corner(upper right) of the chart.

Notice how shallow water is shaded blue, while deeper water is white. Why do you think the people thatmade the charts made them this way? At what depth does the color change from blue to white?

ANSWER: It is important to know where shallow water is because it means that youare close to hazards that can sink your boat. These include rocks, wrecks, and sandbars. The blue coloring makes shallow water stand out more. Any water depth thatis 30 feet or less is marked in blue.

We are starting out by heading towards the Northern tip of Block Island. Locate the buoymarked G “1Bl” located just north of Block Island. Use a ruler or yardstick to draw (on yourchart) a line from the entrance of Pt. Judith Harbor to this buoy. This is your course line.

What direction is this - North, South, East, or West?

ANSWER: South (a little bit West of due South)

What is the length (in nautical miles) of this leg of our trip?

ANSWER: Spread dividers out from starting point to destination point; hold thisspread. Go over to the latitude scale on side of chart. Remember, each degree oflatitude is divided into 60 minutes. Each minute = 1 nautical mile. The minutes(nautical miles) are symbolized by the changing dark-white-dark-white marks on thelatitude scale and are represented with minute marks ( ‘ ).

Determine how far the spread of the dividers is in nautical miles. The first leg of our journeyis 6.8 nautical miles.




Boats do not have speedometers. Therefore, we must use the time-speed-distance formulae todetermine how fast we are going. If it takes us 18.5 minutes to travel the first leg of our journey, howfast is the boat going?

ANSWER:Time - Speed - Distance Formulae(Using time in minutes)

D=DistanceS=SpeedT=Time

D = S x T S = 60min/hr x D T= 60min/hr x D 60 min/hr T S

Speed = 60 X D / Time = 60 min/hr X 6.8 nautical miles/ 18.5 minutes = 22 knotsWe are traveling at 22 knots. A knot = one nautical mile per hour. Nautical miles perhour are part of the definition of a knot - it is incorrect to say knots per hour.

But how do we know which direction to steer the boat? It looks like a lot of water out thereto get lost in! The way we decide in what direction to steer the boat is by using both ourchart and our ship’s magnetic compass.

We need to find what direction, in degrees, our course line is heading. If we could move ourcompass rose from where it is on the chart over to our desired course line, then we wouldknow what our correct “course” would be. A “course” is the direction (in degrees) in whichwe steer the boat. We cannot move our compass rose, so we need to move our course lineover to the compass rose. We do this by way of a navigation tool called a “parallel rules”.Parallel rules transfer parallel lines from one part of the chart to the other. No matter howfar you go, these line will always be parallel.

We place one of the outer edges of the parallel rule on the course line, then “walk” the twosides over to the closest compass rose to get our degrees. Intersect the crossbar in themiddle of the compass rose with the first edge of the parallel rules that enter the compassrose. Draw a line along this edge that goes through the crossbar, extending the line throughthe degree measurement of the compass rose. Extend this line in the direction that the boatwill be going (rather than the opposite side of the compass rose which will be 180 degreesopposite your desired number). For our exercises we are going to use the outer circle of thecompass rose.



What course should we follow?

ANSWER:205 degrees

To review how parallel rules work we are going to do a short exercise. For this exercise, usethe compass rose that is in the bottom left corner of your chart. Using the parallel rules,draw a line starting at the crossbar in the center of the compass rose and extending up-wards to intersect due North (0 degrees). Then “walk” the parallel rules across the chartand draw 4 parallel lines in different sections of the chart. A boat that is moving from thebottom to the top of these lines is moving North no matter which line they are traveling upon.

Parallel rules are highly recommended for these plotting exercises. They are a very elegantmeans to illustrate parallel lines in a hands-on setting. If they are not available there areother ways to transfer parallel lines from the course line to the compass rose. These waysare useful to mention, in any case, to reinforce the rules of parallel lines. One method totransfer the lines is by way of drawing a rectangle. Start with your course line and draw twoperpendicular lines from this course line so that each perpendicular line will pass on eachside of the closest compass rose. Then draw a line through the crossbar of the compassrose making sure that the line intersects these two lines at right angles. This will producethe same parallel lines as would parallel rules.

Another method would involve folding paper. Start with one edge of a piece of paper on thecourse line then fold the paper over (evenly) until you reach a compass rose on the chart.You will need two sheets of paper to reach compass roses that are out of reach. Longpaper such as poster board works the best. If the paper is big enough you do not have tofold it (with two sheets). Have the widths of the paper line up with the piece that is lined upwith the course line as you slide the paper width over width of the other paper. You cantransfer the paper down or up a chart by drawing a line (with a ruler or yardstick) parallel tothe edge of the paper and extending down the chart towards the compass rose. To avoidmaking too many marks which you later need to erase, you may want to just make a coupleof marks on each end of the ruler and use these as your marks for moving paper edges overto the compass rose. Obviously, these methods are very tedious compared to parallelrules, but they do illustrate other properties of geometry.

Now we know the position of the buoy - but how do we get there? We simply line up ourboat with 205 degrees as shown on our magnetic compass and we keep the compass on205 degrees for 18.5 minutes.

When you are traveling in a boat at sea, knowing the time - speed - distance formulae is ofcritical importance. You have probably worked word problems where you have determinedhow far you would travel in a certain amount of time at a certain speed if you were drivinga car. While being able to solve these problems is helpful when you are on the road, the useof the time - speed - distance formulae can make the difference between life and deathwhen you are operating a boat at sea.



While at sea you also use these formulae much more often than when driving a car forseveral reasons. For one, boats do not have speedometers to tell your speed, and neitherdo they have odometers to tell you the distance you have traveled. Also, there are no roadsigns to tell you the distance to certain points. You have to determine these variables fromthe information you have.

We will now use an example to show you how important the time - speed - distance formulaeare when you are operating a boat at sea:

You are planning a trip from the “Great Salt Pond” on Block Island to “West Harbor” on thenorthwest side of Fishers’ Island. To get to West Harbor you will have to pass by the FisherIsland Lighthouse, which is located about a half mile off the western tip of Fishers’ Island.Locate this lighthouse and draw a line with a yardstick from the Great Salt Pond to a point1/2 mile to the south of the lighthouse where you see the abbreviations “hrd”. Finding thelighthouse will let us know that we have reached Fishers’ Island. But we want to keep adistance from the lighthouse because it is sitting on top of a large rock pile. That is why weare heading to a point 1/2 mile to the South. The lighthouse is 67 feet high so will easily seeit from 1/2 mile away.

We decide that since it is a clear day we do not have to bother determining how long it shouldtake us to arrive near the Fisher Island Lighthouse. We will not use the time-speed-distanceformulae. We should be able to see it from several miles away if we are off course a littlebit.

But as we head towards Fisher Island fog starts to roll across the Sound. We keep goinguntil we hear a fog horn off to our right. The Fisher Island Lighthouse has a horn and wewanted to stay to the left of the lighthouse. Everything seems OK so we keep going.Suddenly we hear breaking waves! Waves only break when they get to shallow water.From the way we planned our trip we shouldn’t be anywhere near shallow water! We getcaught in a breaking wave and almost get tossed into a rock! That could have broken theboat in two. What has happened?

Wind, waves, and water current had pushed the boat way to the North into the area ofWatch Hill Reef. The horn they heard was actually from the Watch Hill Point Lighthouse notthe lighthouse on the western tip of Fishers’ Island. If the crew had used the time-distance-speed formulae they would have known something was wrong when they first heard thehorn. If they had been using the formulae they would have known that they should not behearing the Fisher Island Lighthouse horn that early in their trip. Instead they proceeded intodangerous waters not knowing they were way off course.This is what the captain of the boat should have done before heading towards Fisher IslandLighthouse:



1. Determine the distance from the Great Salt Pond to Fisher Island Lighthouse;then,

2. Enter this distance in the time-speed-distance formulae to determine when theyshould have been expecting to approach the Fisher Island Lighthouse.

To find distance on a chart you use dividers. Take the dividers over to the latitude scale andspread them to a set distance. Five miles is a good distance for this measurement. Be surethe dividers needles do not become spread apart more or less from this measured distanceof 5 miles (or whatever distance you chose). Then take the dividers over to the start of yourcourse line (in this case it is the entrance to the Great Salt Pond) and swing them one pointover the next, counting the different sections. Be sure one point is in contact with yourcourse line at all times. When you get to where the next swing-over will take you past yourdestination (in this case, the lighthouse off of Fishers’ Island), pull the dividers in until they hityour destination. Then measure this last distance on the latitude scale. Find the overalldistance by multiplying the number of sections you measured by five then add the finalshorter distance. This will give you the overall distance.

The distance from the Great Salt Pond on Block Island to the lighthouse located off thewestern tip of Fishers’ Island is 21.1 nautical miles. (Remember, the latitude scale is innautical miles.) If we know that the boat is traveling 20 knots, then we can determine howlong it should take us to get to Fisher Island Lighthouse.

T = 60 min/hr x D/ST = 60 min/hr x 21.1 nautical miles/20 knotsT = 63.3 minutes

We should expect to arrive off of the Fisher Island Lighthouse in about 63 minutes. If wethink we have gotten there much earlier than this amount of time, we should suspect that weare off course. If we do not see the lighthouse very soon after 63 minutes have passed,then we should also suspect we are off course and that we have passed the lighthouse.

We will now go back to our fishing trip to the wreck of the Mary Lee. We have reached thebuoy on the North side of Block Island and will now proceed to the buoy at SouthwestLedge. Locate this on your chart - it is just to the Southwest of Block Island and marks thelocation of this shoal. Draw this course line.

From here we are going to make a course for “Montauk Shoals” and try some fishing there.Locate Montauk Shoals on the chart. Draw your course line from Southwest Ledge toMontauk Shoals. A shoal is a shallow area surrounded by deep water. There is excellentfishing at Montauk Shoals because the shoal abruptly rises out of deep water. Currents risewith this underwater hill and bring nutrients which supplies a rich food chain.


Notice the water depth between Southwest Ledge and Montauk Shoals. The current runsvery strong between Block Island and Long Island. We are going to use a special navigationtechnique through this area so we can make sure we do not drift with the current and end upoff course. Draw a line between the buoy at Southwest Ledge and the middle of MontaukShoals. Then, take a piece of typing paper that is thin enough for you to see the featuresof your chart through the paper. Place this sheet of paper over your course line and writedown the water depth every .3 miles (use your dividers to measure .3 miles). Notice thedepth lines on your chart are in 30 foot intervals. Use the closest depth mark that is withinthe line of your .3 marks.

As we head through this area we measure our depth using the ship’s depth sounder (ourdepth sounder gives us the depth of the water that we are in). We measure this every .3mile and come up with the following information.

Depth: 69, 56, 82, 82, 96, 104, 175, 175, 137, 98, 83, 79.

Are we on the right course?

ANSWER: No! The tide is falling and our boat has drifted to the South.

How can we tell where we are?

ANSWER: Draw another course line on the typing paper, entering the depths re-corded by the ship’s depth sounder every .3 miles. Then place this sheet of paperover the chart and move it around the general area we have gone through until itroughly matches up with the depth on the chart.

The intended course depth reading (on a line from Southwest Ledge to Montauk Shoals)should have read: 69, 82, 82, 83, 175, 175, 103, 67, 46, 49, 43.

We have drifted with the tide and are off course. We roughly re-establish our course basedon the depth line that was just formed and we estimate our position to be Long.: 71 degrees44.6 minutes; Lat.: 41 degrees 3.5 minutes

Plot our location on your chart. What course do we need to take to get to Montauk Shoals?

ANSWER: 248 degrees

As we arrive in the area where we thought we would find Montauk Shoals, our depthinformation tells us we are in 72 feet of water. We believe the tide has caused the boat todrift towards the south again.



Notice on your chart that there is no buoy marking Montauk Shoals. That makes it moredifficult to find its location. We are going to have to find it by taking bearings of landmarkson Long Island.

A bearing is the direction, in compass degrees, in which an object is seen from your boat.One very obvious structure in the distance is False Point Lighthouse which is 168 feet tall.How many degrees does False Point Lighthouse appear from Montauk Shoals? To find thisinformation, on our chart we draw a line connecting the lighthouse to the middle of MontaukShoals. Then we use the parallel rules and the compass rose to determine where thelighthouse lies on the line from the perspective of Montauk Shoals. This line is called a “lineof position” and should be marked on this line as “LOP” followed by the degrees we find itto be from the shoals. This turns out to be 335 degrees. By extending this line a couple ofmiles further south we can use this LOP to go toward the shoal. We determine that if westeer the boat west for another 1/5 mile we will be on the same line with False PointLighthouse and Montauk Shoals. Once we sight the lighthouse at 335 degrees from ourposition we just steer the boat lining up the lighthouse with the reading from the boats’compass as it falls on 335 degrees. If we stay on that line we should go right over MontaukShoals.

Refer to Figure 7A. Illustration # 1

But we need to know more information to know when we are over Montauk Shoals - wereally can’t tell that we are there from depth information because the shoal area is just a littleshallower that the surrounding waters. We are on a line of position of 335 with the FalsePoint Lighthouse but we do not know where we are on that line. A line of position tells usthat we are located somewhere on this line but that it doesn’t tell us where we are on thisline; we could be one mile from shore or we could be 4 miles from shore. To determine ourexact location we need to form another line of position between our location and anotherreference point. Where these two lines intersect will be our exact position. Determiningposition by way of intersecting lines is a guiding principle of navigation.

Refer to Figures 7B & 7C. Illustration #2 and #3

We take a bearing off of a tall building that is located in the town of Montauk and find it tobe 273 degrees from Montauk Shoals. (On the chart this building is marked as “tall bldg.”.)Plot this bearing on the chart, using your parallel rules to measure the bearings from MontaukShoal to these two landmarks. We head our boat to where we determined these bearingsintersected and find ourselves right over Montauk Shoals - its time to go fishing!



Activity - Plotting Courses continuedTo review, a line of position is an imaginary line formed between a landmark on shore andyour location at sea. This line can tell you what angle your boat is located from certainreference points. A reference point is any object that is fixed - it will not move from year toyear. These reference points can be found on your chart and include objects in the water,such as buoys, and tall landmarks on land, such as lighthouses, radio antennas, watertowers, and steeples. Since their position is known, we can use them to determine ourposition. What are some good landmarks on Long Island that can be used as referencepoints? Answer: a TV antenna, a lighthouse, water towers, a steeple, and a tall building.

Would a fishing pier make a good reference point? Why or why not?

ANSWER: It only would if you were relatively close to shore (within about 2 miles).Tall features that have well defined centers make good reference points. Fishingpiers are not very tall and do not have a defined center. Would high hills make goodreference points (locate hills on Long Island - they are marked with circular contourlines which refer to their height)? They would if they have a defined peak that standsup above the rest of the hill.

After some good fishing at Montauk Shoals we decide to travel on to the wreck of the MaryLee. The wreck of the Mary Lee is located at

longitude: 71 degrees 57.4'latitude: 40 degrees 55'

Find this on your chart and draw in our course line. What is our course, in degrees, from MontaukShoals to the Mary Lee?

ANSWER: 219 degrees

What would happen if our ship’s compass was off 5 degrees when we left Montauk Shoals?Draw a line showing the course you would have taken from Montauk Shoals if our compasswas off 5 degrees to the West.

HINT: Add 5 to the number of degrees of your first course line to the Mary Lee and find thatnumber on the compass rose. Then draw a parallel line from the line you just made; beginat Montauk Shoals and end just past the Mary Lee.


Activity - Plotting Courses continuedHow far would we miss the Mary Lee if we went on this course? Place dots on the original line tothe Mary Lee every two miles. Do the same for the other course. Record how far we are off courseevery two miles. How much does this distance increase every two miles?

ANSWER: We would be off course by 7/10 of a mile. We become .2 miles more offcourse every 2 miles.

If a 5 degree compass error would put us .9 miles off course after traveling on a 10 miletrip, determine the answer to the following question.

You are heading for a small island 100 miles out in the ocean. The lighthouse on this island can beseen from 15 miles away from the island. If your compass is off 10 degrees, will you still be closeenough to the island to see the lighthouse and find the island? You are in bad shape if you cannotfind the island!

ANSWER: Using the rate of increase above of .9 miles for 5 degree compass errorover 10 miles, we multiply .9 by 2 (since 10 degrees is double 5 degrees) and multiplythis by 10 (10 miles x 10 = 100 miles; 100 miles is the distance to the island). Wewould be off course 18 miles and would be unable to see the lighthouse - we wouldbe lost at sea!

After some great fishing, we decide to head towards Long Island so everybody can see thebeach from the boat. We want to approach the beach near Amagansett. Find this on yourchart. By looking over the chart closely, we see that our current position is very close to a“range”. A range is a line of position formed by lining up two fixed structures, such as anantennae and a buoy or a water tower and a church steeple. Find the TV antennae locatedin East Hampton that is 450' high. Draw a line from this antennae through the fuel tankmarked by a “F” and then out past where our boat is located. If we steer our boat towardsthe west, we will meet up with this line of position and can follow it all the way to Amangansett.We do not even have to use our compass. The TV Antennae should appear directly behindthe fuel tank.

What would be some other ranges you can locate based on reference points on Long Island?

ANSWER: There are numerous ranges students can identify. After this exercise isover, have them draw in some of these ranges on the chart. Ranges can be formedby lining up any two visible reference points.


A range makes for a better line of position than lines formed from taking compass bearingsoff of landmarks. Why would this be? The answer is because magnetic compasses cansometimes be off. Metal located on a boat can pull them off. Compass error can also resultfrom the compass mechanism itself. Since a compass is not used to follow a range line,then a range line of position is better to follow than a line of position formed from a compassbearing.

Refer to Figure 7D. Illustration #4

We follow the range line toward Amangansett and stop a couple of miles from shore in orderto determine our position. We sight a smokestack (marked “G”) at a bearing of 337degrees from our position. Plot this LOP on the chart and determine our exact position onthe chart. Where the two LOP’s intersect is our position. What is this position?

Long.: 72 degrees 4.1'Lat.: 40 degrees 56.9'

We decide we would like to try a little trolling. This is where you drag your fishing line on topof the water as the boat travels slowly through the water. It is a good way to catch bluefish.We want to troll back toward Montauk Shoals before we head back to the dock. Whatcourse will take us to Montauk Shoals? Draw a course line between our present positionand Montauk Shoals then use the parallel rules to determine which way we head to steer theboat. The answer is 065 degrees.

We catch a couple of nice bluefish but then the weather starts to get rough.

The wind has picked up and the waves are starting to get bigger. The sky is getting darkerand it is becoming more difficult to see landmarks. As we get closer to Montauk Shoals wewant to be sure that we are a safe distance away from the shore - if our engines stalled rightnow and our anchor couldn’t hold the bottom we would be pushed to shore real fast. A lotof times when you are operating a boat at sea you want to be able to pay attention tosteering the boat and nothing else. This is especially true when it gets rough like it is now.That is why we are going to use a navigation technique, based on properties of an isoscelestriangle, to give us our distance from shore without having to look at our chart too often. Wedo this by forming an isosceles triangle on the chart. The three legs of the triangle are thecourse line (C) and the two lines of position (B1 and B2) that we form off of an object onshore. As we approach False Point Lighthouse on our course line of 065 degrees, we wantto take a bearing off of the lighthouse when it appears 045 degrees off of our course line andthen later when it appears at 90 degrees off of our course line. Draw the first line of positionon the chart from the lighthouse to our course line as the lighthouse falls at 45 degrees fromour course line. This will occur when the lighthouse bears 020 degrees from our course (65- 45 = 20). We now want to use the time-speed-distance formulae to determine thedistance covered between this point and the point from which we take our next bearing.



If we are going 20 knots how long will it take us on this course until we arrive at a point where thelighthouse bears 090 degrees from our course line?

ANSWER: We first have to determine at what point on our course line we will viewthe lighthouse at an angle of 90 degrees from our course line. To do this we have tosubtract 90 from 65 (we are on a course of 065). This gives us a -25. We can addthis to 360 (due North) to get 335 degrees. We then form a line of position of 335degrees between the lighthouse and our course line. By doing this we have formeda line that is 90 degrees from our course line.

What is the distance between the two bearings we have made on the lighthouse if we covered thisdistance in 8.1 minutes at 20 knots?

ANSWER: 2.7 nautical miles

From the information that we have gathered, what is our distance from False Point Lighthouse?

ANSWER: 2.7 miles

Our distance from shore is the same as the distance we ran between bearings. We plannedour bearings so that we would form a triangle on the chart with angles of 45-90-45. Sincethe triangle formed by our two LOP’s and our course line is an isosceles triangle, the sideof the triangle that is the distance run between bearings is the same length as the distancefrom shore from the second bearing.

If we were in a boat, the advantage to using this method is that once we plot our LOP’s wewould not have to look back at the chart. Referring back to the chart can be cumbersomeat sea when dangerous weather is consuming all of your attention. After taking his firstbearing, all the captain would have to do was determine his distance between the twobearings based on the boat’s speed and then wait until the lighthouse appeared at 090degrees from the boat. He would then know the distance from shore. Remember, knowingthe distance from shore is very important information to a captain.

As we pass over Montauk Shoals we change course to head toward Shagwong Rock. Wehave decided that it is too rough to try to pass through the waters between Long Island andBlock Island to get back to Pt. Judith Harbor. The waves in this area are very high. We wantto get on the back side of Long Island where the waves are not nearly as big. Instead ofheading back to Pt. Judith Harbor, we will go visit some friends that live in Threemile Harboron the sound side of Long Island. To get there we are going to go around the north side ofGardiners’ Island.



Activity - Plotting Courses continuedWe are next to the buoy that marks Shagwong Reef. From here we want to head to thebuoy off the northern end of Gardiner’s Island.This buoy is marked: G “1GI”

Fl G 4sGong

[G means it is a green buoy; 1 GI is painted on the buoy, it stands for #1 buoy at Gardiner’sIsland; Fl G 4s denotes that the buoy has a flashing green light that flashes every 4seconds; Gong denotes that there also is a gong on the buoy so its location can be deter-mined in fog. The first two features distinguish this buoy from others in the area so thatmariners will know which buoy they are next to and, thus, know their position.]

What course do we head in to get to the buoy just north of Gardiner�s Island? How many miles isit?

ANSWER: We should set a course for 281 degrees; the distance is 10.9 miles

If we are still traveling at 22 knots, how long will it take us to get to the buoy off of Gardiner�sIsland?

ANSWER: About 29.72 minutes, rounded off to 30 minutes

From the buoy off of Gardiner’s Island we will be heading to the buoy marked [R W “TM”;Mo (A); Bell] which marks the entrance to Threemile Harbour.

[The R W on the buoy denotes that it is a red and white striped buoy. TM identifies it as thebuoy marking the entrance to Threemile Harbour. Mo(A) denotes that there is a light on thebuoy that flashes in Morse code as in the letter “a”. The magenta colored disk at the bottomof each buoy denotes that they are lighted buoys.]

What course should we head in to get to this buoy? How far is it?

ANSWER: We should set a course for 196 degrees; the distance is 6.6 miles.



After traveling towards Threemile Harbour for 5 minutes we come across a boat that has broken down.The wind and waves have picked up so that we decide we should not try to tow the boat into ThreemileHarbour; it would be too difficult for our small boat to do that. We need to call the Coast Guard andlet them know our location. What is the longitude and latitude of our location?

ANSWER: We have traveled for 5 minutes at a rate of 22 knots. Therefore we candetermine that we have traveled 1.83 miles from the buoy at Gardiner’s Island.Rounded off to 1.8 miles we measure down our course line from the buoy to plot ourposition. Place a dot at this location and put a small circle around the dot. Now wedetermine the longitude and latitude of this position.

We determine that we are at: longitude: 72 degrees 9.6'latitude: 41 degrees 7.2'

We call these coordinates in to the Coast Guard, wait for them to arrive to tow the boat, andthen proceed on to Threemile Harbour.

Note the small circle at the bottom of the buoy symbol. This is the actual location of the buoyand all course lines should intersect these circles. Buoy locations on old charts wererepresented as dots. A good question for the class would be “why did the makers ofnautical charts, the National Ocean Service, change over to using small circles instead ofdots to mark the location of buoys”. The answer is because dots misrepresented theaccuracy with which buoys can be charted. Buoys are anchored to the bottom and theyswing on their anchors as the current and wind changes. A circle shows the range of areathat the buoy could swing. The difference between a small dot and the circles presentlyused is obviously very slight. This is a good example, however, of the need for precisenessin making and using charts. A very tiny charting error can result in grave consequences.This is why precise measurement and calculations is essential to proper navigation.

After spending some time with our friends in Threemile Harbour, we get back in our boat andstart heading back to Pt. Judith Harbour. We start out by heading back to the buoy at theend of Gardiner’s Island.

What direction, in degrees should we be heading? Can this be determined without using theparallel rulers and the compass rose?

ANSWER: We should head on a course of 016 degrees. Since we are going in theopposite direction from the last leg of our course, all we have to do is subtract 180from our previous course of 196 degrees. Therefore, we should head in direction of016 degrees.


From the tip of Gardiner�s Island we chart a direct course for Pt. Judith Harbour. What courseshould we take and how many mile is it?

ANSWER: Course: 066 degrees for 31.9 miles

Discussion - Occupational Uses of Navigation TechniquesWho else besides someone who operates a boat uses these techniques that we have justlearned? Airplane pilots use very similar skills. Other people that use some of the tech-niques you have just learned include surveyors (to determine property lines) and engineers(for numerous reasons including to determine if dams are shifting and to determine the rateof erosion in an area). Explorers and hunting guides also use these techniques to find outwhere they are located.

Submarine captains use many of these same techniques and engineers that guide space-craft do also. The principle of determining where you are by intersecting lines is used intracking satellites, space probes and with all modern electronic navigation.

While all modern electronic navigation systems can give you a more accurate position thanthe methods we just learned, it is still essential that captains learn the traditional ways offinding their position. The most important rule of navigation is to not rely solely on any singlesource of information to determine your position. Sometime boats lose electrical power sothat their electrical navigation systems are not working. This is why the techniques we havejust practiced are still taught by the US Navy and Coast Guard. If you were interested inbecoming a ship’s captain, you would have to pass a test in which you would have to usethese same navigation skills.

Video ChallengeUse some of the techniques you have learned about navigation to map your schoolyard orthe “course” you take from homeroom to your other classes — or have a treasure hunt inyour schoolyard. Or make and map out your own course — then, give a friend a magneticcompass and have them follow your map to navigate the course.



LES

SO

N 7

-N

AVIG

ATIO

N P

RIN

TF

igur

e 7A

:

“Lin

e of

Pos

ition

”

light

hous

e

0 9018

0

270

lighthouse

tallbuilding

MONTAUK SHOALS

Boat's position is marked with a dotwith a circle around it ( ).

LESSON 7 - NAVIGATION PRINTFigure 7B

LESSON 7 - NAVIGATION PRINTFigure 7C: Determining position from 2 lines of position

Aerial Perspective Perspective on Chart

R BN 286

168 ft. 24 Mlighthouse

radioantennae

FL 5HORN

!

LESSON 7 - NAVIGATION PRINTFigure 7D: Example of a range

View from Boat View on Chart

water tower withantennae

Radio Tower

Tank

math in the middle of oceans navigation · these islands. many shipwrecks were caused by poor...

Documents