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Eastings and northings name a square by its southwest corner and a point to a tenth of it, and turn movement and distance into arithmetic.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to give four- and six-figure grid references, read them back, and use them to find distances.
You can name a grid square by two numbers, across first and then up, and you can turn a distance on a map into a distance on the ground with its scale. This lesson makes the grid precise enough to find a single spring on a hillside.
| Term | What it means |
|---|---|
| Easting | A grid line running north-south, numbered along the bottom of the map. |
| Northing | A grid line running east-west, numbered up the side of the map. |
| Four-figure reference | A square named by the easting and northing at its southwest corner. |
| Six-figure reference | A point named to a tenth of a square: easting and tenths, then northing and tenths. |
| Topographic map | A detailed map showing the shape of the land as well as roads, streams and buildings. |
| UTM grid | The Universal Transverse Mercator grid of lines a kilometer apart printed on U.S. Geological Survey maps. |
A topographic map is ruled with numbered grid lines. Eastings run north-south and are numbered along the bottom; northings run east-west and are numbered up the side.
The order never changes: along the corridor, then up the stairs.
Another way: picture
Picture a square of the grid as a small room. To say where a chair stands in it, you say how far across the room it is, in tenths, and then how far up. The four figures name the room; the two extra figures place the chair.
Another way: steps
Take the six-figure reference $457234$.
| Figures | Meaning |
|---|---|
| 45 | the easting line west of the point |
| 7 | seven tenths of a square east of that line |
| 23 | the northing line south of the point |
| 4 | four tenths of a square north of that line |
Split it in half: the first three figures say how far east, the last three how far north.
Every square has four corners, and four different pairs of lines touch it. Mapmakers agreed long ago to name a square by the lines at its bottom-left corner, which are the ones you reach first when you count along the bottom and up the side.
The rule matters because the next square to the northeast shares one corner with it. Name a square by the wrong corner and you point your reader one square away, which on a real map is a kilometer.
The U.S. Geological Survey, or USGS, has published topographic maps of the whole country. Modern USGS maps carry the Universal Transverse Mercator grid, called UTM, with lines one kilometer apart, labeled in the margins.
Search and rescue teams, foresters and wildland firefighters give locations as UTM grid references, because a reference to a hundred meters can be read by anyone with the same map, without landmarks or long descriptions.
Nobody draws the tenths on the map; you estimate them. Picture the side of the square cut into ten equal parts: halfway is five, a little past halfway is six or seven.
A small ruler, or a card with a scale called a romer, makes the estimate exact. Hikers and rescuers carry such cards so that two people reading the same point write the same six figures.
Because each tenth is one hundred meters, references can be subtracted. Two campsites on the same northing with eastings $412$ and $437$ are $25$ tenths apart, which is $2500$ meters.
Points that differ in both easting and northing are farther apart than either difference alone; the straight distance is the diagonal, found with the Pythagorean theorem you will meet in mathematics.
Walking east increases the easting; walking north increases the northing. A hiker who starts at easting $457$ and walks east at one hundred meters a minute is at easting $457 + m$ after $m$ minutes.
That is a simple function, and it shows why the grid is useful: movement on the ground becomes arithmetic on the numbers.
A six-figure reference names a square one hundred meters across, not a single spot. For finer work, eight or ten figures split each tenth again into tenths.
Grid references also say nothing about height. Two points with the same reference can be a valley floor and a cliff top in rugged country, which is why topographic maps add contour lines, the subject of the next lesson.
You already know latitude and longitude, which name any point on Earth in degrees. A grid does the same job over a smaller area, in kilometers instead of degrees, and its lines are straight and evenly spaced on a flat map.
That makes a grid easier for measuring distance: a kilometer is a kilometer anywhere on the map, while a degree of longitude shrinks toward the poles.
Checking an answer. Read your reference back onto the map: go along the bottom, then up the side. If you do not land on the feature, you have swapped the halves or used the wrong corner.
Naming a square by its southwest corner is allowed because it is the agreed rule, and a reference only works if writer and reader share the rule. Estimating tenths is allowed because the grid lines are evenly spaced, so the distance between them can be split evenly.
Subtracting references to find distances is allowed because each tenth stands for the same number of meters everywhere on the grid.
When a hiker is injured in the backcountry, rescuers need an exact location. A phone may show latitude and longitude, but a helicopter crew or a ground team holding a USGS map often works in UTM grid references.
A reference given in the wrong order sends the team to a point that can be many kilometers away, so rescue training drills the order until it is automatic.
The most common slip is putting the northing before the easting. Another is naming a square by its northeast corner instead of its southwest corner.
A third is forgetting that each tenth is one hundred meters when finding a distance, and giving a count of tenths as if it were meters. A fourth is dropping a leading figure, writing $57234$ instead of $457234$.
The rest of this unit adds height to the grid. Contour lines show how high the land is; gradient says how steep it is between two points; a cross-section draws the shape of the land along a line.
Each of those starts with finding two points on the map, and grid references are how you name them.
Suppose a hiker in Shenandoah National Park, Virginia, twists an ankle on a trail and calls for help. The ranger who answers opens the park's USGS topographic map and asks the hiker's companion to read the location from their own map, and the companion reads a six-figure reference.
Six figures narrow the search to a square one hundred meters across, small enough for a team to find the hiker quickly. The ranger subtracts the reference of the nearest trailhead to find the distance, and multiplies the tenths by one hundred meters to plan how long the carry-out will take.
If the companion had read the northing first, the reference would point to a spot that could be many kilometers away, perhaps on the other side of a ridge. That is why search and rescue volunteers practice the order of a grid reference until it is automatic, and why they read every reference back before acting on it.
The U.S. Forest Service has used fire lookout towers since the early 1900s. A lookout who spots smoke takes a compass bearing to it and reports the fire's location to a dispatcher, who plots it on a map with the bearings from other towers.
Firefighters on the ground and pilots in the air then need the same location in a form they can all read. Modern crews exchange grid references and coordinates, so that a helicopter, a fire engine and a crew on foot all head for the same hundred-meter square.
Crews also use the grid to measure. Subtracting the eastings and northings of a fire's edge and a road tells them how far a fire line must be cut, and how long it will take a crew to reach it. The grid turns a map into a tool for arithmetic, which in a fast-moving fire can matter as much as the map itself.
It is easy to read the side of the map first, since we read pages from the top down, and to name a square by whichever corner catches the eye. Either slip moves the reference to a different square, not a nearby spot in the right one.
Always go along the corridor before up the stairs: easting first, from the square's southwest corner. Then read your reference back onto the map to check.
A square has easting $31$ on its west side and northing $87$ on its south side. Find its southwest corner.
$31, 87$
Bottom-left names the square.
Write the easting first.
$31$
Along the bottom.
Add the northing.
$3187$
Then up the side.
Check by reading it back.
$\text{east of } 31, \text{ north of } 87$
The same square.
A spring lies eight tenths east of easting $31$. Write the three-figure easting.
$318$
Line, then tenths.
It lies five tenths north of northing $87$. Write the three-figure northing.
$875$
Line, then tenths.
Join them, easting first.
$318875$
Six figures.
Say how precise it is.
$100\ \text{m}$
One tenth of a kilometer square.
Check the four-figure square it lies in.
$3187$
Drop the tenths.
Two points share northing $234$; their eastings are $412$ and $437$. Subtract.
$437 - 412 = 25$
Tenths apart.
Convert to meters.
$25 \times 100 = 2500$
One hundred meters a tenth.
Convert to kilometers.
$2.5\ \text{km}$
A thousand meters each.
Say the direction between them.
$\text{due east-west}$
Same northing.
Estimate a walking time at five kilometers an hour.
$\dfrac{2.5}{5} = 0.5\ \text{h}$
Thirty minutes on flat ground.
Say what could lengthen it.
$\text{hills between them}$
The grid shows no height.
Write the three-figure easting.
$186$
Line, then tenths.
Write the three-figure northing.
$562$
Line, then tenths.
Join them, easting first.
On a topographic map, a square has the easting line $74$ along its west side and the northing line $12$ along its south side. What is its four-figure grid reference?
Complete the worked solution: a spring lies $6$ tenths of the way east from easting line $18$, and $2$ tenths of the way north from northing line $56$. Find its three-figure easting, its three-figure northing and its six-figure grid reference.
Build the three-figure easting.
$\text{easting line, then tenths} =$ a
Hundreds of meters east.
Build the three-figure northing.
$\text{northing line, then tenths} =$ b
Hundreds of meters north.
Join them, easting first.
$\text{easting, then northing} =$ r
The six-figure reference.
Say how precise it is.
$\text{to one hundred meters}$
Tenths of a one-kilometer square.
A trail junction is at grid reference $457234$. Match each part of the reference to what it tells you.
| the easting line west of the point | tenths of a square east of that line | the northing line south of the point | tenths of a square north of that line | |
|---|---|---|---|---|
| 45 | ||||
| 7 | ||||
| 23 | ||||
| 4 |
On a map extract, a spring lies $741$ hundred meters east and $127$ hundred meters north of the grid's origin, a trail junction $735$ east and $133$ north, and a summit $748$ east and $121$ north. Write each feature's six-figure grid reference.
| six-figure grid reference | |
|---|---|
| the spring | |
| the trail junction | |
| the summit |
A hiker starts at a point whose three-figure easting is $318$ and walks due east at one hundred meters a minute. Write the hiker's three-figure easting as a function of the minutes $m$ walked.
Answer:
Two campsites lie on the same northing. Their three-figure eastings are $412$ and $437$. How far apart are they, in meters?
Answer: m
Suppose a topographic map of the Ozark National Forest, Arkansas shows a ranger station and a fire lookout on the same easting, the lookout due north. Their three-figure northings are $342$ and $369$. How far must a ranger hike north to reach the lookout, in meters, measured on the map?
Answer: m
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A hiker starts at a point whose three-figure easting is $318$ and walks due east at one hundred meters a minute. Write the hiker's three-figure easting as a function of the minutes $m$ walked.
Answer:
You can give a six-figure grid reference. Explain why reading the northing first sends someone to the wrong square.
24. Your turn: a summit lies six tenths east of easting $18$ and two tenths north of northing $56$. What is its six-figure reference?, step 3
$186562$
The six-figure reference.