Using GPS in Algebra I

One activity that interests and engages students in mathematics.

When, Where, and Why 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”

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

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

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

There are several websites that provide satellite imagery. National Geographic:

http://plasma.nationalgeographic.com/mapmachine/index.html Google Earth: http://earth.google.com/ Google Maps: http://maps.google.com/maps?tab=wl

Students and Coordinates Next, the students estimated latitudinal and longitudinal coordinates using the provided grid.

Then, the students collected these coordinates using a handheld GPS devise.

89.0911º 89.0891º 89.0871º

33.5580º

33.5570º

33.5560º

GPS Coordinates

Student Comments

Students and Distance 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 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 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) 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: One Degree would equal 365,228.16 ft. One Minute would equal 6087.136 ft. One Second would equal 101.45227 ft.

Longitude Measurements

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

Longitude Measurements (Continued) 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 ChangeLatitude Miles per

one degree longitude

Feet per one minute longitude

Feet per one second longitude

0 69.17 6087 101.50

30 59.96 5274 88.00

35 56.73 4992 83.20

40 53.06 4669 77.80

45 49.00 4312 71.87

50 44.55 3920 65.34

55 39.77 3500 58.33

60 34.67 3051 50.85

Quadratic Regression 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

y = -0.0088x 2 - 0.05x + 69.209

R2 = 0.9999

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70

Latitude

Longitude

The Mississippi Activity 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 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 Once students established a difference in latitude and longitude coordinates, they were to convert these degree differences into standard units of measure.

Calculating Distance 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):

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 + 365421Feet per Degree: _______________

Minute: f(Lat) = -0.7738x2 - 4.4597x + 6090.1Feet per Minute: _______________

Second: f(Lat) = -0.0129x2 - 0.0747x + 101.56Feet per Second: _______________

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: _______________

Using the calculated distances, what is the area of the trapezoid surrounding the Region 10 Educational Service Center?

Area: _______________