Download Getting your Fix: How to determine one`s location using lines of

Getting your Fix:
How to determine one’s location using lines of position.
ED 5661 Mathematics & Navigation Teacher Institute
August 2011
By Serena Gay
Target: Geometry (grades 9 & 10)
Lesson Overview:
In this lesson, students will learn about navigation as it applies to the compass and how bearings to
three different objects can be used to identify the location of an unknown object or person. They
will learn the difference between magnetic and true headings by constructing a compass rose
(reinforcing previous knowledge of central angles in a circle) and use it to draw three lines of
position and identify an unknown location on a map of the local area. They will also apply the
geometric constructions for the perpendicular bisector of a segment and erecting a perpendicular
through a given line and point on that line as part of an activity that uses a Snellius construction
(which is an alternate method of using three known bearings to identify an unknown location). With
this lesson, I hope to generate enthusiasm for navigation and show an application of the
constructions they’ve learned. The lesson should take no more than two 80-minute class periods,
allowing time for warm-ups, discussion and working on their projects.
Sources:
 Lessons and practical exercises from the Mathematics & Navigation Teacher Institute at
Michigan Tech University, summer 2011
The activities and discussions on using bearings to establish a fix struck me as really cool and
served as the basis for this idea – thanks Captain!
 Bill Brogdon, Boat Navigation for the Rest of Us: Finding Your Way by Eye and Electronics
(International Marine 2001)
The discussions about true vs. magnetic heading, along with lines of position for getting a fix,
helped me to formalize the teaching and formal activity parts of my lesson.
 http://www.sailingissues.com/navcourse4.html
This site reinforced my thoughts about using the three bearings to establish a fix, but it also
showed me the wicked-cool Snellius construction that allowed me to really tie this lesson to the
compass-straightedge constructions in the State of Michigan content expectations for
Geometry. The extension activity that stems from the Snellius construction also serves as a
review of congruent triangles and angles in circles.
Materials Needed (all based on one item per group of 3-5 students):
1) For the Creating the Compass Rose activity:
 construction or card stock paper (8.5’’ by 11’’)
 180 or 360 protractor
 copy of a compass dial (smaller than the protractor)
 tape or glue stick
 
laminated maps of local area that show latitude and longitude, as well as local landmarks


2) For




the Getting Your Fix activity:
parallel rule or rolling rule
straight edge
vis-à-vis markers (4-color set)
laminated maps from compass rose activity
3) For




the Snellius Construction activity (these materials are one per student, not group):
straightedge
compass
180 protractor
Snellius construction handout
New Vocabulary:
 true north – refers to the geographic north pole
 magnetic north – tends to shift and refers to the pole of the Earth’s magnetic field
 declination – the difference between true north and magnetic north
 compass rose – two concentric circles, each broken into 360, where the outer circle
measures true directions and the inner circle measures magnetic directions
 bearing – the direction from one object to another (e.g. from you to a fixed object)
 heading – the direction one is pointed at the moment

 line of position (LOP) – a line drawn between two objects that is used to indicate the bearing
from one object to another.
 parallel rule – a tool used to draw parallel lines. It consists of two unmarked rulers
connected by hinges that keep them parallel at all times.
Focus Question(s):
I will ask students what they know about GPS and then introduce them to a situation where they
are somewhere in the Battle Creek area with only a compass and a map of the area – how could they
estimate their location on the map without the aid of a GPS? After they’ve made their compass
rose and applied the process of using bearings and lines of position to determine a specific location,
I will introduce them to an alternate method of determining a fix using three bearings, called the
Snellius construction, that will allow them to apply the compass and straightedge constructions
they’ve learned previously in the course.
Learning Objectives:
By the end of this lesson, students will have learned about heading and bearing, reinforcing their
prior knowledge of measured degree “turns” along a circle. They will also learn how to use a
compass rose, a parallel rule (reinforcing prior knowledge of parallel lines), and the implication of
intersections of three lines of position forming a triangle as it pertains to accuracy. Finally, they
will see an application for the “by-hand” compass and straightedge constructions they’ve had to
learn (which they’ll appreciate because it helps answer that annoying question that haunts so many
educators “When are we EVER gonna use this?!”) The extension to the Snellius Construction will
also draw out and reinforce congruence of triangles and angles associated with circles.
State of Michigan benchmarks addressed:
 G.TR.05.01 Associate an angle with a certain amount of turning; know that angles are
measured in degrees.
 G1.1.4: Given a line and a point, construct a line through the point that is parallel to the original
line using straightedge and compass. Given a line and a point, construct a line through the
point that is perpendicular to the original line. Justify the steps of the constructions.
 G1.2.5: Solve multistep problems and construct proofs about the properties of medians,
altitudes and perpendicular bisectors to the sides of a triangle, and the angle bisectors of a
triangle. Using a straightedge and compass, construct these lines.
 G1.6.3: Solve problems and justify arguments about central angles, inscribed angles, and
triangles in circles.
Classroom Activities:
1.
After a warm-up reviewing inscribed and central angles of a circle and the above-mentioned
focus questions, we’ll review how to read a compass and discuss the difference between true
and magnetic headings, along with the concept of declination. I’ll show them an example of a
compass rose on a nautical chart and how to read it. We’ll also discuss the difference between
a heading and a bearing, along with lines of position and how they can be used to pinpoint one’s
location on a map.
2. Next, we’ll create our own compass rose for the Battle Creek area. Divide students into groups
of three to five students. Using either the 180 or 360 protractor, make a full circle on the
construction or card stock paper and make tick marks every 5 degrees (starting with 0 at due
north and working clockwise around). Draw lines marking north-south and east-west through
the circle. Then, using the magnetic declination found by calling a local surveyor, draw a line


from the center of the circle out to the edge of the circle that marks off where 0 would be
for the magnetic declination. For example, the magnetic declination for our area is 539'W . For
simplicity, the students will round to 5 degrees and draw a line that hits 355 on the circle.
Then, using a reduced copy of the compass dial (which I got from Bluewater Books online – the

ones Capt. Roblee showed us in the Navigation institute), line up the centersand rotate the
smaller dial so that its zero-degree mark lines up with the adjusted
 line on the larger compass
and glue/tape it on. Now they have a compass rose to use on the local map of the area for their
next activity. Line up true north with one of the longitude lines on the map and tape it on the
map.
Note: if time is an issue, you can use two copies of the compass dial, one larger and one
smaller, instead of having them trace a circle and make tick marks along the edge.
3. Once the compass rose is done and taped to the group’s copy of the area map, they will do their
Getting a Fix activity. First, I’ll demonstrate how to use the parallel rules under the document
camera or on the whiteboard. Then, each group will be assigned two sets of three bearings they
will use to create lines of position (using the vis-à-vis markers) and get an estimate of the two
unknown locations on their maps. As an extension, if time allows, each group will be given a
location on their map and be asked to determine three good reference points that could be used
to pinpoint the given location from their individual bearings (i.e. do the process backwards).
Then, they can give their bearings to a different group to see if they can identify the unknown
location. This could be done as a warm-up for the next day.
4. Once they’ve created lines of position using magnetic bearings, we’ll discuss another method for
locating one’s position using three bearings. In this method, the Snellius construction, relative
angles are used (i.e. the angle between two locations found by calculating the difference
between their compass bearings). The process is mapped out in the attached activity (The
Snellius Construction), which the students will do individually as a guided activity. Here they
will need their compass (not the dial!) and straightedge for the formal constructions of
perpendicular bisector of a segment and line perpendicular to another line, through a given point
on the line.
5. A possible extension to the last activity is to have them draw additional segments on the
diagram and investigate the various triangles formed by the segments, as well as the central
and inscribed angles formed within the two circles.
Assessments:
 Creating the Compass Rose activity – students will receive a group grade based on the
accuracy of their compass rose, most specifically the placement of their inner compass dial
 Getting Your Fix activity – students will receive a group grade for correctly identifying their
mystery location from the assigned bearings. I will look for the lines of position drawn in and
the resulting intersection, hopefully as small of a triangle as possible. (Each assigned “mystery
spot” is a specific landmark on the Battle Creek area map, so they should be very close to their
spot.) If time permits and we are able to do the extension (doing the process of lines of
position backwards), I will look for accuracy in their chosen bearings and how well the other
group was able to locate the indicated spot using them.
 Snellius Construction activity: this will be an individual grade, as each student will be
performing the constructions on their own paper. I will grade them as I did their formal
constructions earlier in the term, which was to look for good construction marks and labels, as
well as how close they get to the unknown position. One thing that works really well for me is to
make a transparency of my constructions “answer key” and place it over theirs to see how well
they match up. It makes for a quick and easy check for those key construction characteristics.
For instance, the arcs used in the perpendicular bisector construction do not have to have the
same radius as mine, but the bisector line should be in the same place. If time permits and we
are able to do the extension (forming the triangles and central/inscribed angles within the
circles), I will grade them on their responses to the leading questions.
The Snellius Construction Activity
The following process can be used to locate an unknown position using the known compass
bearings to three fixed objects. Using a straightedge, compass and protractor, follow the
steps to complete the construction on the handout (Figure 1).
1. The compass bearings from the unknown position to each of the three fixed objects
are as follows: 320 to A, 360 to B, and 30 to C.
2. Calculate the angles between each pair of points based on their bearings.


A and B: _______

B and C: _______
3. Draw segments AB and BC on the diagram.
4. Construct (using your compass and straightedge!) the perpendicular bisectors of
segments
 AB and
 BC . Sketch them lightly and don’t forget to add your markings that
show you have a perpendicular bisector!
5. From
 point
 A, use your protractor to construct a line inland of AB that measures the
same as the angle between A and B. Label this angle measure on the diagram.

6. From point C, use your protractor to construct a line inland
of BC that measures the
same as the angle between B and C. Label this angle measure on the diagram.
7. Use your compass and straightedge to construct a line
perpendicular to the
construction line at point A that passes through point A. Extend it to intersect the
perpendicular bisector of AB . Label this intersection point D.
8. Use your compass and straightedge to construct a line perpendicular to the
 point C that passes through point C. Extend it to intersect the
construction line at
perpendicular bisector of BC . Label this intersection point E.
9. Use your compass to construct circle D with radius AD . Also construct circle E with

radius CE . The offshore
intersection of these two circles is your unknown position!
Label it point P.


Figure 1
B
C
A
The Snellius Construction – Extending to triangles and angles in
circles.
Now that you’ve located your unknown position, do the following to discover some connections to
some of the other geometry you’ve learned so far!
1.
Label the intersection of the two segments, AB and BC , with their perpendicular bisectors as
F and G, respectively. Then lightly draw in the following segments: AP , BP , CP , AD , FD , BD ,

BE , EG , and EC .

 The
Snellius
 Construction
  Activity,
2. Given the measure of the angle you constructed in step 5 of


 be the measure of FAD ? Explain – without measuring! What about the measure of
what must
GCE ? Label these measures on your diagram.
 are ADF , BDF , BEG , and CEG ? Explain.
3. What type of triangles

4. What are the measures of ADF and CEG ? (Again, without measuring!) Explain how you
know.




special point lies on both circle D and circle E?
5. Other than point P,what other
6. What is the relationship between ADF and BDF ? Between BEG and CEG ? JUSTIFY
your conclusions!

? Explain.
7. What is the measure of FDB
? Themeasure of BEG
8. What is the measure of ADB ? The measure of BEC ? What type of angles are these in

terms of a circle?


 in terms of a circle?
9. What type of angles
are APB and BPC
10. Find the measures of APB and BPC WITHOUT measuring them. Explain how you know
these measures.


 told at the beginning of the Snellius Construction activity, do these
11. Given the bearings
you were
angle measures make sense? Now use your protractor to measure them and see how close you
were to the correct measurements.
Answer key to The Snellius Construction – Extending to triangles
and angles in circles.
1.
see attached Snellius Construction diagram
2.
mFAD  50 and mGCE  60 Since AD and CE are perpendicular to the construction lines
drawn at points A and C, the angles formed by the construction lines and the segments AB and
 FAD
 and GCE .

to angles
BC are complementary

 angle.
3. The are all right triangles because FDAB and GEBC , giving them each a right

4.
  30 because
 they are complementary to the other acute angle in
mADF  40 and mCEG

their respective right triangles.

5. Point B


6. They are congruent. Since AD and BD are radii of the same circle, they are congruent. Since
FD is the perp. bisector of AB , this makes AF  BF . Therefore, since the triangles are also

the Hypotenuse-Leg Theorem (or you could use SSS by stating
right, they are congruent
by
FD  FD by the reflexive

property). This argument holds for both pairs of triangles.




7.
mFDB  40 and mBEG  30 by CPCF (in Chicago series) or CPCTC (in other books)
8.
mADB  80 and mBEC  60 (by angle addition) These are central angles because the vertex
of the angle
 is the center of the circle.
9. These areinscribed angles because the vertex of the angle lies on the circle and its legs or
sides intersect the circle in two distinct places.
10. mAPB  40 and mBPC  30 The measures of arcs AB and BC are equal to the measures of
the central angles that intercept them ( ADB and BEC ) and the measures of the inscribed

angles that
 intercept these arcs ( APB and BPC ) are equal to half the measures of the arcs.
the differences

11. They should make sense because
between the bearings given are the measures
 they willbe very close to the correct measurements!
of these angles. Hopefully,
Snellius Construction diagram