How to solve a proportion Recall that a proportion is simply
the equality of two ratios or two fractions. Such as:
Slide 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.
Slide 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.
Slide 5
Identify the scale factor. Example 1: Finding a Scale Factor
Scale Drawings and Scale Models RoomBlueprint Length (in.)20025
Width (in.)10012.5 blueprint length room length = 1818 1818 The
scale factor is. Write a ratio using one of the dimensions.
Simplify. 25 200 =
Slide 6
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. Example 2 Scale Drawings and Scale Models 4141
Think: poster photo = 5151 20.5 w = 4141 4w = 20.5 w = 5.125 Write
a proportion to find the width w. Find the cross products. Divide.
The photo is 5.125 in. wide.
Slide 7
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? Scaling on a Map Let d be
the actual distance between the cities. 1.5 60 = 7.5 d 1.5 d = 60
7.5 1.5d = 450 1.5d 1.5 = 450 1.5 d = 300 The distance between the
cities is 300 miles. Write a proportion. Find the cross products.
Multiply. Divide.
Slide 8
Insert Lesson Title Here 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 = 7d7d 1 d = 50 7 1d = 350
d = 350 The distance between the cities is 350 kilometers. Write a
proportion. Find the cross products. Multiply. Scaling on a
Map
Slide 9
Try this one: Insert Lesson Title Here 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 Scaling on a Map
Slide 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.
Slide 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 1 in = 10 mi 1 : 25 000
0 500m
Slide 12
Map Scale 1 : 24,000 - primary scaled used by USGS for mapping
the United States in topographic form. 1 inch on the map equals
24000 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 : 63,360 - 1 inch equals 1 mile 1 : 50,000 - 1 cm
equals.5 km 1 : 250,000 - 1 cm equals 2.5 km, 1 in equals
approximately 4 miles 1 : 1,000,000 - 1 cm equals 10 km Any scale
can be used for a map, but a few common scales have been settled on
for use by most organizations:
Slide 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 1/250000 - 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 = 316800 inches.
316800 inches / 250000 = 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.
Slide 14
Finding Distances on a Map Using a Scale Factor Suppose the
scale on a map is given as 1 : 250000. You measure a distance of
3.5 inches. What would be the actual distance? Since the scale is 1
: 250000, we first multiply the measured distance by 250000. 3.5 x
250000 = 875,000 inches in actual distance. (Note that I actually
used a proportion here) While this gives the correct distance it
really doesnt 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 =
72916.6667 72916.6667/5280 = 13.8 miles
Slide 15
Finding Distances on a Map Using a Scale Factor Recall in our
earlier discussion that a scale of 1 : 250000 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).
Slide 16
Finding Distances on a Map Using a Scale Factor Suppose the
scale on a map is given as 1 : 250000. 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-) 250000/100000 = 2.5. 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
Slide 17
How to measure distances on a map 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. 1. Straight
line distances
Slide 18
Lets measure the distance between the caravan park and the
windmill on this map. 0 500 1Km 2Km As you can see the distance is
2Km.
Slide 19
How to measure distances on a map 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. 2. Curved Distances
Slide 20
Now Lets measure the distance along the railway on this map. 0
500 1Km 2Km 1.Note the points where the direction changes.
2.Measure the distance between each one. 3.Add them up then convert
using the scale bar. This would be 7 Km
Slide 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
Slide 22
Cardinal Directions
Slide 23
North East South West Never Eat Shredded Wheat N E S W NE SE SW
NW For any map, if you hold the major lettering upright, north is
at the top of the map.
Slide 24
Bearing
Slide 25
A Reminder from Geometry A circle has 360 o A right angle has a
measure of 90 o A straight angle or line has a measure of 180
o
Slide 26
N S EW N S EW N S EW 060 o 145 o 230 o 315 o 60 o 145 o 230 o
315 o N S EW 090 o 360/000 o 270 o 180 o Bearing Measure from North
Measure in a clockwise direction
Slide 27
N S E W 090 o 360/000 o 270 o 180 o SE 135 o SW 225 o NW 315 o
NE 045 o We use a protractor to measure bearing. Note that when
measuring in quadrants III and IV we will have to add 180.
Slide 28
360/000 o 090 o 180 o 270 o W E N S 1 2 10 9 8 4 7 6 5 3 040 o
250 o 280 o 120 o 195 o 010 o 325 o 155 o 235 o
Slide 29
Bearings Measuring the bearing of one point from another. 1.
Draw a straight line between both points. 2. Draw a North line at
A. 3. Measure the angle between. N 060 o To Find the bearing of B
from A. B A
Slide 30
N 240 o B A Bearings Measuring the bearing of one point from
another. To Find the bearing of A from B. 1. Draw a straight line
between both points. 2. Draw a North line at B. 3. Measure the
angle between.
Slide 31
N 060 o B A N 240 o How are the bearings of A and B from each
other related? Bearings Measuring the bearing of one point from
another.
Slide 32
Activity 1.Please get out one of the small maps. 2.You may draw
on the maps. 3.Measure the scale. 4.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?
Slide 33
Here is my solution
Slide 34
Activity 1.Please get out one of the small maps. 2.You may draw
on the maps. 3.Measure the scale. 4.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?
Slide 35
Here is my solution
Slide 36
Compass
Slide 37
Parts of the Compass
Slide 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.
Slide 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 lets 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.
Slide 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)
Slide 41
Needle Parallax View from Above View from Behind 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.
Slide 42
Needle Parallax Good Bad Keep the needle parallel to the
meridian lines.
Slide 43
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, 104.0 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.
Slide 44
Compass Use Example: To find the direction-of-travel based on a
bearing of 320 from a specific location:
Slide 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. 320 Direction- of-travel arrow
Slide 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
46
Slide 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.
Slide 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.