COORDINATE SYSTEMS

AND THE SACRED MOUNTAINS OF THE NAVAJO

Trignometry

By Vernon Willie

Introduction

Coordinate Systems are a part of mathematics, science, engineering and especially real-world concepts. We will look at several types of coordinate systems and apply certain real-world applications, some of which are usually taken for granted. At times we don't actually know we are using coordinate systems.

The Coordinate System we will look at is The Cartesian coordinate system. Applications will involve utilizing a topographical map.

PUT THE SECTION ON COORDINATE SYSTEMS (RED) IN "MATH TOPICS" Link titled as is.

The Cartesian Coordinate System, (C.C.S.)

The Cartesian coordinate system is also known as the Rectangular Coordinate System, the "X-Y" coordinate system, and the Coordinate Plane. When we superimpose a horizontal vertical number line at both origins; the zero value on the x-number line and the zero value on the y-number line, (meet at the same point), the result is a coordinate plane such as we see below.

Y

Quadrant II

Quadrant I

Quadrant III

X

Quadrant IV

There are several terms that go along with the C.C.S. The x-axis is the x-number line, which increases in unit value from left to right. The y-axis is the y-number line, which increases in unit value from bottom to top. The origin is the intersection of the two perpendicular number lines. There exist four Quadrants on the coordinate system. They are Quadrants I, II, III, and IV, which start at the top right corner and continue counterclockwise respectfully. There is one and only one point on a number line associated with each real number. Also, there is a unique pair of numbers associated with each point in the coordinate plane,

x: represents units on the x-axis.
y: represents units on the y-axis.
(x,y): called the Ordered Pair.

Plotting the coordinate point, (a,b)

Y

X

is recognized on the x-axis and is recognized on the y-axis. Thus, the intersection of and is the "address" or "location" of the coordinate point.

Example

Given the following Cartesian coordinate system, plot the following ordered pairs.

i. (2,3)

ii.(-3,1)

iii.(-2,-2)

iv.(4,2)

Y

-3 -2 -1 1 2 3

X

-1

-2

-3

Nitsáhákees

Observations regarding locating places and utilization of coordinate systems

Questions

A.) What is your location?

B.) How would you describe your location?

C.) Do you know of other ways to locate places?

D.) Do you think that coordinate systems and maps go together?

E.) Have you ever seen a topographical map?

F.) How do you think that mathematics might be used to help locate and identify places?

G.) Have you ever considered the geometry involved in relative positions of places in or near Diné Bikéyah?

Nahat'á

Learn or review mathematical concepts that are needed to study locations in or near Diné Bikéyah (see Menu on the left).

Terms that may be new to you in this study are listed below and defined.

Latitude: The measure from the equator, from 0 ° to 90 ° north or from 0 ° to 90°south , depending on which direction you decide to go.

Longitude: The measure from Greenwich (England from) 0° to 180° east or from 0 ° to 180° west, depending on which direction you decide to go.

Topographical maps are highly detailed maps of small areas, showing elevation above sea level (both graphically and numerically), vegetation density, bodies of water and the locations of roads and populated areas. They are widely used by hikers and climbers, hunters and anglers, surveyors, construction companies, lawyers and real estate firms. (http://www.globecorner.com)

Iiná

Applying mathematical models within the Cartesian Coordinate System to objects in our every day lives (such as maps and physical locations), for example, maps of the Navajo Nation, a community, or a topographical map used for location of places

1. Take out a sheet of paper and construct a Cartesian coordinate system that will be appropriate to plot the points given below.

i.) (-105.4849, 37.5778)

ii.) (-107.6078, 35.2386)

iii.) (-111.6772, 35.3463)

iv.) (-108.0883, 37.4450)

2. Plot the four points on the your coordinate system. Once you have plotted your ordered pairs, compare with each other and see if you have similar plots.

3. Internet Exercise:

In the next step, we will plot the given points on a certain website. The website we are interested in is http://www.topozone.com

Once on the website, we would like to Get A Map. Click on Decimal degrees. As you read the instructions, notice that the two terms"longitude" and "latitude" are used.

Also, notice that in the ordered pairs given, the first number is of longitude value and the second number is latitude value,

(longitude,latitude )

i.) (-105.4849 °, 37.5578 °)

ii.) (-107.6078 °, 35.2386 °)

iii.) (-111.6772 °, 35.3463 °)

iv.) (-108.0883 °, 37.4450 °)

Type in the blank boxes the first point given but in reverse order. Once you have placed both latitude and longitude values in the blank boxes, press "Map It". A topographical map will appear on your screen.

What have you located on your topographical map?

Repeat this process for each of the points given above.

You should have seen four maps each with a centered location.

Identify each of these locations.

When completed, you should see that you have located the four sacred mountains of the Navajo on your maps:

Sis Naajini, mountain to the East, (Blanca Peak, CO.)

Tsoodzil, mountain to the South, (Mount Taylor, NM.)

Dook'o'oosliid, mountain to the West, (San Francisco Peaks, AZ.)

Dibé Nitsaa, mountain to the North, (Hesperus Peak, CO.)

4. Using the coordinates given, find the equation of the line joining the sacred mountains of the east and west.

5. Similarly, find the equation of the line joining the sacred mountains of the north and south.

6. Sketch the graphs of these two lines on your coordinate system.

7. Now find the point of intersection of the two lines.

8. Internet Exercise: Go back to the Internet Site, www.topozone.com. Enter the coordinates of the point of intersection, then click "Map It".

9. Identify the location on your map.

Is this near any of the Inner Sacred Mountains?

Internet Exercise:

Click "Back" on the website and enter these coordinates, (-107.8425, 36.4230). What does this locate?

You should see that you have located one of the interior sacred mountains, Huerfano Mountain, or Dzil Ná Oodithle.

In the remaining exercises, we will find the approximate distance from the point of intersection of your lines and Huerfano Mountain. Use this information: the average radius of the earth is approximately 3,959 miles.

11. Find the difference in the latitudes of the intersection and the mountain, then convert this to radian measure.

12. Use the formula for arc length to find the north-south distance between the intersection and the mountain.

13, 14. Repeat #11 for the longitudes, then repeat #12 for east-west distance.

15. Assume that the earth is flat around the region in this exercise and use the Pythagorean Theorem to find the approximate distance from the point of intersection of your lines and Huerfano Mountain.