How to solve a proportion • Recall that a proportion is simply the equality of two ratios or two fractions. Such as:
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.
Scale • A scale is the ratio between two sets of measurements. Scales can use the same units or different units. • A scale drawingis a proportional drawing of an object. Both scale drawings and scale models can be smaller or larger than the objects they represent.
25 200 1 8 The scale factor is . Scale Drawings and Scale Models Example 1: Finding a Scale Factor Identify the scale factor. blueprint length room length Write a ratio using one of the dimensions. = 1 8 Simplify. =
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 5.125 in. wide.
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. Find the cross products. 1.5 · d = 60 · 7.5 1.5d = 450 Multiply. 1.5d 1.5 450 1.5 = Divide. The distance between the cities is 300 miles. d = 300
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. Find the cross products. 1 · d = 50 · 7 1d = 350 Multiply. d = 350 The distance between the cities is 350 kilometers.
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
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.
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
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 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
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.
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 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 = 72916.6667 72916.6667/5280 = 13.8 miles
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).
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
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.
Let’s 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.
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.
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. 0 500 1Km 2Km This would be 7½ Km
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
The 4 point compass North Never East West Eat Wheat 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 SE SW S
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
360/000o N N N N 60o 270o 090o W W W W E E E E S S S S 180o 145o 315o 230o Bearing Measure from North Measure in a clockwise direction 060o 315o 230o 145o
360/000o N 315o 045o NW NE 270o 090o W E SW SE 225o 135o S 180o We use a protractor to measure bearing. Note that when measuring in quadrants III and IV we will have to add 180.
N 360/000o 010o 7 325o 040o 8 1 4 280o 2 090o E 270o W 3 5 250o 10 9 120o 235o 6 195o 155o 180o S
Bearings Measuring the bearing of one point from another. N To Find the bearing of B from A. B A 060o 1.Draw a straight line between both points. 2.Draw a North line at A. 3.Measure the angle between.
N 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. 240o 2.Draw a North line at B. 3.Measure the angle between.
Bearings Measuring the bearing of one point from another. N N 060o B How are the bearings of A and B from each other related? 240o A
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?
Here is my solution
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?
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.
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.”
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)
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 Behind View from Above
Needle Parallax Good Bad Keep the needle parallel to the meridian lines.
Magnetic north and true north are not in the same place. For example in 1994the average position of the North Magnetic Pole was located on the Noice Peninsula, southwest EllefRingnes 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.
Compass Use Example: To find the direction-of-travel based on a bearing of 320° from a specific location:
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°
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
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.
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.