1. Scaling

2. How to solve a proportion
Recall that a proportion is simply the equality of two ratios or two fractions. Such as:

3. How to solve a proportion
Suppose we have the following proportion with a missing element:
To solve the proportion, we “cross multiply” then divide to get the results.

4. Scale
A scale is the ratio between two sets of measurements. Scales can use the same units or different units.
A scale drawing is a proportional drawing of an object. Both scale drawings and scale models can be smaller or larger than the objects they represent.

5. Example 1: Finding a Scale Factor
Scale Drawings and Scale Models
Example 1: Finding a Scale Factor
Identify the scale factor.
Room
Blueprint
Length (in.)
200 25
Width (in.)
100
12.5
blueprint length
room length 25
200
Write a ratio using one of the
dimensions. = 1 8 =
Simplify. 1 8
The scale factor is

6. Scale Drawings and Scale Models
Example 2
A photograph was enlarged and made into a poster. The poster is 20.5 inches by 36 inches. The scale
factor is Find the width of the photograph. 4 1
poster
photo 5 1
Think: =
20.5 w 4 1 =
Write a proportion to find the width w.
4w = 20.5
Find the cross products.
w = 5.125
Divide.
The photo is in. wide.

7. Scaling on a Map
On a road map, the distance between Pittsburgh and Philadelphia is 7.5 inches. What is the actual distance between the cities if the map scale is 1.5 inches = 60 miles?
Let d be the actual distance between the cities.
1.5 60
7.5 d =
Write a proportion.
1.5 · d = 60 · 7.5
Find the cross products.
1.5d = 450
Multiply.
1.5d
1.5 =
450
1.5
Divide.
The distance between the cities is
300 miles.
d = 300

8. Insert Lesson Title Here
Scaling on a Map
On a road map, the distance between Dallas and Houston is 7 inches. What is the actual distance between the cities if the map scale is 1 inch = 50 kilometers?
Let d be the actual distance between the cities. 1 50 7 d =
Write a proportion.
1 · d = 50 · 7
Find the cross products.
1d = 350
Multiply.
d = 350
The distance between the cities is 350 kilometers.

9. Insert Lesson Title Here Scaling on a Map
Try this one:
On a road map, the distance from Green Bay to
Chicago is 11 cm. What is the actual distance
between the cities if the map scale is 3 cm = 90 km?
330 km

10. Map Scales
Large scale maps show the most detail but only cover a small area e.g. road maps, town plans.
Small scale maps show less detail but cover a larger area e.g. maps of the whole of the US.

11. How is scale shown on a map?
As a written statement
As a ratio or fraction
Using a scale line.
4 cm = 1 Km in = 10 mi
1 : m

12. Map Scale
Any scale can be used for a map, but a few common scales have been settled on for use by most organizations:
1 : 24,000 - primary scaled used by USGS for mapping the United States in topographic form. 1 inch on the map equals inches in the real world, which is the same as 2,000 feet. This scale is used on the over 54,000 quadrangle maps covering the entire country.
1 : 63, inch equals 1 mile
1 : 50, cm equals .5 km
1 : 250, cm equals 2.5 km, 1 in equals approximately 4 miles
1 : 1,000, cm equals 10 km

13. The smaller the number on the bottom of the map scale, the more detailed the map will be. A 1:10,000 map will show objects ten times as large as a 1:100,000 map but will show less land area on the same sized map.
Here is an example of a Bar Scale found on a map. The scale shows that about 1.25 inches equals 5 miles. The smaller increments to the left of zero are each 1 mile and are used to estimate smaller distances. Notice the scale is 1/ that means 1 inch on the map is equal to 250,000 inches on the real land. (5 miles = 5*5280 feet = 5*5280*12 inches = inches inches / = 1.27 inches)
By including a map scale like the image below, if the map is photocopied and reduced in size, the scale can still be used. Otherwise, 1 inch would no longer equal what it should.

14. Finding Distances on a Map Using a Scale Factor
Suppose the scale on a map is given as 1 : You measure a distance of 3.5 inches. What would be the actual distance?
Since the scale is 1 : , we first multiply the measured distance by x = 875,000 inches in actual distance. (Note that I actually used a proportion here)
While this gives the correct distance it really doesn’t answer the question. A better answer would be given by converting to miles. To do this we divide by 12, to convert to feet, then divide by 5280 to convert to miles.
875,000/12 =
/5280 = 13.8 miles

15. Finding Distances on a Map Using a Scale Factor
Recall in our earlier discussion that a scale of 1 : is about 1 inch equals 4 miles. We could also use this to determine the distance.
Since 1 inch is equal to four miles, we simply multiply by 4.
3.5 x 4 = 14 miles (you can see some error here).

16. Finding Distances on a Map Using a Scale Factor
Suppose the scale on a map is given as 1 : You measure a distance of 6.8 centimeters. What would be the actual distance?
Metric measurements are much easier since we can divide the scale factor by the conversion factor needed to change centimeters to kilometers.
Since there are 100 centimeters in 1 meter and 1000 meters in a kilometer. We divide the scale by 100,000.(Think: hundred thousand centi- kilo-)
250000/ = So the conversion is 1 cm = 2.3 km.
Now all we have to do is multiply our centimeters by 2.5 to obtain the actual distance.
6.8 x 2.5 = 17 km

17. How to measure distances on a map
1. Straight line distances
The shortest distance between two points is sometimes known as the distance ‘as the crow flies’.
This can be measured with a ruler then converted to the correct scale with reference to the scale bar given on the map.

18. Let’s measure the distance between the caravan park and the windmill on this map.
Km Km
As you can see the distance is 2Km.

19. How to measure distances on a map
2. Curved Distances
You may need to measure the distance along a road or river that does not travel in a straight line.
To do this you ideally need a piece of string (or you can use a strip of paper).
You lay the string down to follow the shape then measure the total length before converting back using the scale.
If using paper you need to pivot the paper each time the path changes direction.

20. Now Let’s measure the distance along the railway on this map.
Note the points where the direction changes.
Measure the distance between each one.
Add them up then convert using the scale bar.
Km Km
This would be 7½ Km

21. Activity
On the large Oklahoma map what is the straight line distance from Shawnee to Woodward?
160 miles
See if you can find the road distance by traveling along the following route.
1. Travel to I-40 turn west toward Oklahoma City
2. Take the US-281-SPUR north, EXIT 108, toward GEARY/WATONGA
3. Turn west at Watonga onto OK 33 (also marked as US-270 and US-281)
4. Continue to follow this road into Woodward

22. Cardinal Directions

23. The 4 point compass The 8 point compass
North
Never
West
East
Wheat
Eat
South
Shredded
The 8 point compass N NW NE
For any map, if you hold the major lettering upright, north is at the top of the map. E W SW SE S

24. Bearing

25. A Reminder from Geometry
A circle has 360o
A right angle has a measure of 90o
A “straight angle” or line has a measure of 180o

26. Bearing Measure from North Measure in a clockwise direction 360/000o

27. We use a protractor to measure bearing
We use a protractor to measure bearing. Note that when measuring in quadrants III and IV we will have to add 180. N S E W
090o
360/000o
270o
180o NW
315o NE
045o SW
225o SE
135o

28. 360/000o
090o
180o
270o W E N S
010o
325o 7
040o 8 1 4
280o 2
250o
235o
195o
120o
155o 3 10 9 5 6

29. Bearings B A Measuring the bearing of one point from another.
To Find the bearing of B from A. B A
060o N
1. Draw a straight line between both points.
2. Draw a North line at A.
3. Measure the angle between.

30. Bearings B A Measuring the bearing of one point from another.
To Find the bearing of A from B. N B A
1. Draw a straight line between both points.
240o
2. Draw a North line at B.
3. Measure the angle between.

31. Bearings B How are the bearings of A and B from each other related? A
Measuring the bearing of one point from another. N
060o B A
240o
How are the bearings of A and B from each other related?

32. Activity
Please get out one of the small maps.
You may draw on the maps.
Measure the scale.
Follow the following directions.
Start at Shawnee and take a bearing of 335 degrees for 150 km
From that point take a bearing of 215 degrees for 225 km
What county are you in?

33. Here is my
solution

34. Activity
Please get out one of the small maps.
You may draw on the maps.
Measure the scale.
Follow the following directions.
Start at Woodward and take a bearing of 145 degrees for 225 km
From that point take a bearing of 45 degrees for 225 km
What county are you in?

35. Here is my solution

36. Compass

37. Parts of the Compass

38. How a Compass Works
There is a huge magnetic field around the earth. It is huge, but it is not very strong. The magnetized needle in a compass is aligned with this magnetic field. As the image to the right shows, the composition of the earth acts as a huge bar magnet sitting upside down in the middle of the planet. Since its South end is at the north pole and its North end is at the south pole, the North end of a compass needle is pulled north. Your compass has to have a very light needle sitting on a pivot that has almost no friction. This is because the earth's magnetic field is weak and would not be able to turn the needle.

39. Basic Compass Reading
No matter the compass, one end of the needle always points North. On a mountaineering compass, it is almost always the RED end, but its a good idea to test your compass before starting to use it. Make certain that your dial reads 0 degrees and let’s all turn and face north.
To read your compass.
Hold your compass steadily in your hand so the base plate is level and the direction-of-travel arrow is pointing straight away from you.
Hold it about halfway between your face and waist in a comfortable arm position with your elbow bent and compass held close to your stomach.
Look down at the compass and see where the needle points.
The red end of the needle should be inside the little house. We call this “Red in the Shed.”

40. Please turn 90 degrees to the left.
The compass needle should be on the E. Does this mean you are facing EAST?
Sorry, but no!
To find your direction, you must turn the compass dial (the one with the degree markings on it) until the North mark and the "Orienting Arrow" are lined up with the North end of the needle (see bottom compass).
Then I can read the heading that is at the Index Pointer spot (the butt of the direction-of-travel arrow).
Remember: RED IN THE SHED
Now we know we are really heading West (270 degrees)

41. Needle Parallax
When you align your needle align it with the orientation arrows. When you look at the compass from behind or from the side, it may appear that the needle is not pointing true north.
View from Above
View from Behind

42. Keep the needle parallel to the meridian lines.
Needle Parallax
Good
Bad
Keep the needle parallel to the meridian lines.

43. Magnetic north and true north are not in the same place
Magnetic north and true north are not in the same place. For example in 1994 the average position of the North Magnetic Pole was located on the Noice Peninsula, southwest Ellef Ringnes Island, at 78.3 degrees North, minutes West. The yearly motion of the pole has increased, and is now 15 kilometers per year.”
This difference is referred to as magnetic declination. You can correct for it, but that is a topic for another time. At our position, we will be off about three degrees.

44. Compass Use
Example: To find the direction-of-travel based on a bearing of 320° from a specific location:

45. Travel Direction for a Bearing of 320°
Rotate the housing on the compass until the 320° mark lines up with the direction-of-travel arrow.
Direction-of-travel arrow
320°

46. Bearing of 320°
Rotate the entire compass until the compass needle lines up with the orientation lines.
Remember red in the shed.
The direction of travel arrow shows the direction.
If you move your feet so that you are behind the compass, as long as you keep the alignment correct, you travel in the correct direction.
Direction-of-travel arrow
Needle in the North Alignment Position

47. Measuring Distances
One way we can measure distances is to see how many steps it takes for us to cover 100 feet.
We are going to go outside and set out a 100 foot course. You should walk the course at least three times to see how many steps it takes to cover the 100 feet.
We will use an average to determine your scale factor
100 feet = ? steps.
Walk at a natural gait.

48. Practice Run
Once you finish getting your scale. I will give you a bearing and distance for practice.
Lets suppose your scale was 100 feet = 45 steps and I asked you to go 160 feet at a bearing of 270 degrees
First you would need to convert 160 feet to steps by using a proportion.
Solve the proportion and you get 72 steps
So you would travel west for 72 steps.