Transcript GPS Activity

Using GPS in Algebra I
One activity that
interests and engages
students in
mathematics.
When, Where, and Why
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Conducted the activity at the end of the spring
semester in 2005 at one small, rural Mississippi
secondary school.
I wanted students to see how ordered pairs and
the distance formula can be used in the real
world.
I also wanted students to use a GPS handheld
and discuss some of its limitations.
http://www.colorado.edu/geography/gcraft/notes/gps/gps_f.html
David and Norma Royster’s “GPS and
Mathematics NCTM 2003”
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Educational with respect
to the history and
technical information
surrounding GPS.
Provides activities and
information gathering
worksheets to use in the
classroom.
Describes a very similar
activity discussed in this
presentation.
Trilateration is the Key
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Trilateration is a basic geometric principle that
allows you to find one location if you know its
distance from other, already known locations
(Royster & Royster, 2003)
My Activity in Mississippi
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First, I found a
satellite image of
our school campus
from the U.S.
Geological Survey
available online in
2005.
Then, students
determined
distances between
various points on
campus using the
provided scale and
image.
Satellite
Images
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There are several websites that provide satellite
imagery.
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National Geographic:
http://plasma.nationalgeographic.com/mapmachine/index.html
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Google Earth:
Google Maps:
http://earth.google.com/
http://maps.google.com/maps?tab=wl
Students and Coordinates
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Next, the
students
estimated
latitudinal and
longitudinal
coordinates
using the
provided grid.
Then, the
students
collected these
coordinates
using a handheld
GPS devise.
33.5580¼
33.5570¼
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33.5560¼
89.0911¼
89.0891¼
89.0871¼
GPS Coordinates
Student Comments
Students and Distance
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Latitude and
Longitude
coordinates were
then collected from
various places
around campus.
The students
discussed the
conversion of
degree measures to
American Standard
units of measure.
Converting
GPS Degrees to Feet
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The spherical geometry
of the Earth can be
ignored due to the
relatively small area this
activity investigates.
Therefore, 2D
calculations and
assumptions are made
throughout this activity.
Latitude Measurements
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There is some
variance between the
number of miles per
degree latitude from
the equator to the
poles.
However, a standard
figure of 69.172 miles
per degree is
accepted.
Latitude Measurements
(Continued)
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Latitude Measurements can be in decimal
degrees or converted to a sexagesimal system
of Degrees, Minutes, and Seconds (DMS),
where one minute is a 60th of a degree and a
second is a 60th of a minute.
Therefore, since one degree latitude is 69.172
miles:
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One Degree would equal 365,228.16 ft.
One Minute would equal 6087.136 ft.
One Second would equal 101.45227 ft.
Longitude Measurements
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Calculating longitude
measurements are
more detailed than
latitude.
The reason comes
from the fact that
longitude lines
converge from the
equator to the poles.
Geometer’s Sketchpad
Geometer’s Sketchpad is one way a
teacher can demonstrate or have
students create a dynamic example of the
change in longitudinal measurements.
 Link to Sketch
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Longitude Measurements
(Continued)
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At the equator, one degree of longitude is the
same as latitude (69.172 miles). However, this
measurement shrinks to zero at the poles.
Many websites can help one convert
longitudinal degrees to miles and feet based on
the latitudinal measurements.
However, using some data values collected
from another website gives students an
opportunity to use Excel© and quadratic
regression to establish a usable formula.
Longitudinal Change
Latitude
0
30
35
40
45
50
55
60
Miles per
Feet per one Feet per one
one degree minute
second
longitude
longitude
longitude
69.17
6087
101.50
59.96
5274
88.00
56.73
4992
83.20
53.06
4669
77.80
49.00
4312
71.87
44.55
3920
65.34
39.77
3500
58.33
34.67
3051
50.85
Quadratic Regression
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First, input data into
spreadsheet.
Then, highlight and
select an XY
scatterplot chart
formatting as desired.
Finally, create a
polynomial trendline of
degree 2 displaying an
equation with r2-value.
Miles per Degree
80
y = -0.0088x2 - 0.05x +
69.209
2
R = 0.9999
70
60
Longi tude
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50
40
30
20
10
0
0
10
20
30
40
Latitude
50
60
70
The Mississippi Activity
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During the activity in Mississippi, I utilized
an online converter to establish
conversion factors for my students. I
would have students perform the previous
Excel© operations in the future, so the
students could work with functions to
determine their standard units of
measure.
The Distance Formula
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The students completed
the activity by calculating
distances between
various locations on
campus.
The students used the
distance formula and the
GPS coordinates to
calculate these
distances.
Calculating Distance
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Once students established a difference in latitude and
longitude coordinates, they were to convert these
degree differences into standard units of measure.
Calculating Distance
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Some students failed to convert their
measurements resulting in unreasonable
answers.
Student Comments
• Students originally used the scale at the
bottom of the map to determine distance. After
the distance formula was used, students
commented on the observed differences.
Now let us conduct this
activity here with a twist.
Determine the Latitude and Longitude coordinates
of above points (Use decimal degrees):
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Point A: Latitude _______________ Longitude _______________
Accuracy _____________
Point B: Latitude _______________ Longitude _______________
Accuracy _____________
Point C: Latitude _______________ Longitude _______________
Accuracy _____________
Point D: Latitude _______________ Longitude _______________
Accuracy _____________
Use the following functions to determine the number
of feet one degree, minute, and second of longitude
equals at the latitude coordinate of Point A.
Degree: f(Lat) = -46.502x2 - 263.97x + 365421
Feet per Degree: _______________
 Minute: f(Lat) = -0.7738x2 - 4.4597x + 6090.1
Feet per Minute: _______________
 Second: f(Lat) = -0.0129x2 - 0.0747x + 101.56
Feet per Second: _______________
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Calculate the distances for the above lines using the
coordinates for each point, the feet per degree
conversion factor, and the distance formula.
Line AB: _______________
 Line AC: _______________
 Line CD: _______________
 Line BD: _______________
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Using the calculated distances, what is the area of
the trapezoid surrounding the Region 10
Educational Service Center?
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Area: _______________