SCRIPT FOR MEASUREMENT MATH II
NAVIGATION
[You will need a globe or a map of the world. Go to and print out the magnetic compass graphic.]
Suppose you are lost in the woods. How can you tell directions?
There are three traditional ways to tell directions: 1) If the sun is shining, it will appear to rise in the east and set in the west, but the exact directions depend on the time of year. In the summer it rises and sets a little to the north and in the winter a little to the south. 2) If it is dark and the stars are out, the North Star (Polaris) is within one-degree of true north. 3) A magnetic compass. Why is a compass better for finding direction than using the Sun or the North Star?
The magnetic compass is one of the great inventions of all time. It measures the angle between north and the direction you are headed, which is called your compass heading. You may want to review the meaning of an angle. For what line of work do you think the compass was an extremely important invention?
The European compass first came into widespread use in the century before Columbus. Without a compass do you think he would have made his voyage to America? He thought that the world is round, that he would find land by sailing in a westerly direction, and that he could get home by sailing east.
You have seen magnets on refrigerator doors. Well, the earth is like a huge magnet. Its magnetic field causes the needle on a compass to point to magnetic north, which in most locations is not true north. The difference between magnetic north and true north is called magnetic variation.
Look at the compass in the graphic. It is used in a place where magnetic north is about 22 degrees east of true north. The compass has been corrected for deviation rotating the dial within 22 degrees. Therefore, when the needle points to "N" on the dial, the white lubber line on the cover points to true north. When you turn the compass so that the needle points to "S" on the dial, the lubber line points to true south.
EXAMPLE 1: Harold is sailing his boat, and wants to be sailing due west. What should his compass heading be?
Notice that the compass in the picture has two different scales on the dial. The inside scale is used mostly by people working with land descriptions. The compass reading for due west on that scale would be North 90 degrees West. The outside scale is the azimuth scale, which is used by sailors and others. The reading for west on the azimuth scale would be 270 degrees. That means 270 degrees in a clockwise direction from true north.
CHALLENGE 1: Harold now wants to head due south. What should be the azimuth reading on his compass?
CHALLENGE 2: Arnie found an old map that shows the location of buried treasure. It says to start at a certain big rock, and it gives the courses (directions) to walk to find the treasure.
Make a map showing the courses, but first write N for north at the top center of the page. Write E for east on the right center of the page, S for south at the bottom, and W for west on the left side. Start your first course at the center of the page, and draw your courses on your map using the scale of 100 strides equals 1 centimeter. A stride is two steps. An average man's stride walking at a normal pace is about 5-1/2 feet. You might want to check out the length of your stride. Measure the length of the sidewalk sections between "cracks" in the sidewalk and count your strides as you walk at a normal pace over 10 sections. Then "do the math."
Suppose that Arnie has a chart to find buried treasure. The compass bearing (true direction) for the first course is 90 degrees azimuth or due east, and the distance is 1,000 strides to an old oak tree.
The second course is 180 degrees or due south, and the distance is 1,000 strides to a big rock.
The third course is 270 degrees or due west, and the distance is 1,000 strides to a big stump.
The fourth course is 360 degrees or due north, and the distance is 1,000 strides. Uh oh! Where is Arnie?
EXAMPLE 2: Harold sailed out into the ocean, and a storm came up. His mast, which holds the sails up, broke. That can ruin a sailor's whole day! He called for help on his ship-to-shore radio, and he needed to give his exact location. How can he give his location at sea?
On your globe or world map find Greenwich, England. Notice it has a line drawn north and south through it. It is the zero meridian of longitude, and it is called the Prime Meridian. Look for other lines for meridians. They measure degrees of angular distance from Greenwich England as viewed from the center of the earth. Notice that longitude is measured in degrees both east and west of Greenwich, and it goes up to 180 degrees in each direction. There are also lines running east and west that measure angular distance north and south from the equator, and these lines are called parallels of latitude. They go from 0 degrees at the equator to 90 degrees north at the North Pole and south to the South Pole. So Harold can give his position in meridians of longitude and parallels of latitude as follows: 41 degrees 50 minutes N, 60 degrees 30 minutes W. There are 60 minutes in one degree. Look at a world map and find Harold's location.
Well and good, you are probably thinking, but how did Harold determine his position? A generation ago he might have used tables and an instrument called a sextant, which measures the angle between the sun or stars and the horizon (altitude). For determining longitude he would also have needed a chronometer, which is a very accurate clock. Why would he need an accurate clock in addition to a sextant to find the longitude but not latitude?
The reason is that the earth turns from west to east. If you want to find your longitude with a sextant, the angle between the sun and the horizon depends on the time of day as well your location.
Here is something you can tell your friends: The angular altitude of the North Star above the horizon at sea equals the latitude of the position of the observer in the northern hemisphere. If the altitude of the North Star is 30 degrees above the horizon, your latitude is 30 degrees north. If you are sitting on the North Pole, what is the altitude of the North Star?
Modern sailors can fix their position with an electronic global positioner or GPS. You just turn it on and give it time to pick up the signals from several satellites orbiting above the earth. It will give your latitude, longitude, and, for good measure, the elevation above sea level.
CHALLENGE 3: Find your location on the globe or world map and determine your longitude and latitude.
EXAMPLE 3: In a strong wind Harold sails 10 nautical miles in one hour. What is his speed?
Speed at sea or in the air is measured in nautical miles per hour, which is called knots. Since Harold sailed 10 nautical miles in an hour, his speed was 10 knots. The knot is a rate that does not include a time unit such as "per hour," but it is understood.
Here is an interesting thing about nautical miles. A degree of latitude is divided into 60 minutes. When you go due north or south through 1 minute of latitude, you travel approximately 1 nautical mile. This doesn't apply to longitude except at the equator. Look at the meridians of longitude on a globe and figure out what the equator has to do with it.
CHALLENGE 4: A nautical mile is 6,076.1 feet. The statute mile, which measures distance on the ground, is 5,280 feet. An airplane is flying at the speed of 400 knots. How many statute miles per hour is it flying?
Back to the Beginning
So far we have been concerned with making single measurements such as the height of a third grade boy. Now we will look at ways to summarize measurements in a group. For example, we might summarize the heights of all of the boys in a third grade class. The branch of mathematical analysis that deals with this is called statistics.
EXAMPLE 1: Suppose we want to study the weights of 40 football players on a NFL team. How can we summarize our findings?
The first thing we might do is to list them from the lightest to the heaviest. From this list we can see the range, which is a measure of the spread. We may find that the weights range from 180 to 350 pounds.
The second thing we might do is produce a single number or average that will tell us the most typical weight. When we talk about an "average," we usually are referring to the mean average. To find the mean, we sum the weights of all of the players and divide by the number of players. It is affected by extreme values. In this case one 350-pound player can raise the average significantly and give a misleading idea of the other player's weights.
CHALLENGE 1: The Ages of five boys are 8, 9, 9, 9, and 14. What is their mean average age?
CHALLENGE 2: The heights of the starting five players on a basketball team are as follows: 6 feet 6 inches, 6-7, 6-7, 6-8, and 6-11. What is their mean average height? Tip: Since all of them are 6 feet and some-odd-inches tall, just compute the average of the odd inches and add it to 6 feet.
EXAMPLE 2: The median is another measure of the average. To find the median of a group, we put the values in order by amount, and find the middle value in the order. The median of the ages of the five boys, whose ages are 8, 9, 9, 9, and 14. The median age is 9. Is the median affected by extreme values such as the 14?
CHALLENGE 3: What is the median height of the five players in Challenge 2?
EXAMPLE 3: Another measure of the average is the mode. The mode is the most common value in a series of values. It is not appropriate for measurements such as weights unless they are rounded. The most common age of the five boys in Challenge 1 is 9, so the mode is 9 years of age. Is the mode affected by extreme values?
CHALLENGE 4: What is the mode of the heights of the five players in Challenge 2?
EXAMPLE 4: If we want a summary that shows more detail, we might make a frequency distribution table. In the case of the football players, we might separate the weights into 20 lb. intervals and total the number of players within each interval.
Weight 180- : 200- : 220- : 240- : 260- : 280- : 300- : 320- intervals: 199 : 219 : 239 : 259 : 279 : 299 : 319 : 339 Number of Players: 2 4 6 9 8 7 3 1We can make our summary even more easily grasped by using a graph.
[Find a number of graphs in popular magazines and have the student interpret them.]
Back to the Beginning
[Two coins and two dice will be needed in this section.]
EXAMPLE 1: Have you heard someone say something like this: "The Yankees will definitely win the game." Any major league ball team has a chance of wining or losing on any given day. We can analyze the Yankees' strengths and weaknesses in relation to their opponent and put a number on their chances of winning. We call that number their probability of winning.
We can think of probability as a scale. At the top is absolute certainty, which is the probability of 1 or 100%. At the bottom of the scale is absolute impossibility, which is a probability of 0. In the middle is a 50-50 chance that something will happen, which is a probability of 0.5 or 50%. The probability for any future outcome will lie somewhere on this scale. After the analysis of the Yankees' chances, we might estimate that they have a probability of about 0.7 of winning. In other words, if they played 10 straight games with their opponent, they would probably win 7 of them.
CHALLENGE 1: What is the probability of the sun rising in the east tomorrow morning? What is the probability that it will rise in the west tomorrow morning?
EXAMPLE 2: The weatherman forecasted a 90% probability of rain tomorrow? How might he measure probability?
There are three ways to determine probability. The first way is the educated estimate. This is what we would do in the baseball example. It is also the weatherman's way. He measures and analyzes the important atmospheric conditions and trends. He decides on the possible outcomes, and based on the data and experience he makes a probability estimate for the most likely outcome. In this case it is a 90% chance of rain. We now have not only his prediction that it will rain, but we also know how confident he is in his prediction. This is the most common method of determining probability of real world events.
CHALLENGE 2: The weatherman forecasted a 0% probability of rain tomorrow. How confident is he in his forecast?
EXAMPLE 3: If you toss a die in the air, what is the theoretical probability that 1 dot will come up on top?
The second way to determine probability is to compute it. The first step in computing theoretical probability is to count the number of different possible outcome. There are 6 sides, so there are 6 possible outcomes. There is 1 chance per 6 possible outcomes or a 1/6 or 0.17 or 17% probability for 1 dot. This assumes that we have a fair die, one that is not weighted more on one side than on the other.
CHALLENGE 3: If I toss a coin in the air, what is the probability it will come up heads? Compute the theoretical probability.
EXAMPLE 4: Determine experimentally the probability of 1 dot coming up on top of a die in 30 tosses.
The third way to determine probability is by experiment. Here is the formula to determine probability experimentally:
Probability = occurrences of one kind of outcome ÷ total number of trials.
If you only toss the die 6 times, one dot may come up once, twice, or not at all. So, we will toss it 30 times, and compute the experimental probability. If one dot comes up 6 times, the probability is 6 ÷ 30 or 0.2 or 20%.
CHALLENGE 4: Determine experimentally the probability of a coin coming up heads. Toss the coin 10 times and record the outcome for each toss. Notice runs of three or more heads or tails.
EXAMPLE 5: If I toss 2 dice in the air, what is the computed probability that 2 single dots will come up together?
First, we compute the probability for 1 dot on one die, which we have found to be 0.17, and it would be the same for the 2nd dot on the second die. To compute the probability of both single dots coming up together, we multiply together the probabilities of each coming up alone.
Probability of 2 single dots = 0.17 x 0.17 = 0.029 or 2.9%
CHALLENGE 5: If you toss 2 coins in the air, what is the probability that 2 heads will come up together?
EXAMPLE 6: Suppose that I toss a coin and it comes up heads 4 times in a row. What is the probability that it will come up heads on the 5th toss?
For each individual toss the probability is always 0.5 because the coin has no memory and no conscience.
Back to the Beginning
Because you have traveled this far on the road to math smarts, you are going get some bonus subjects. Your first bonus is the subject of measurement error.
To begin, always remember this: No measurement is exact. You can count things exactly, but you can't measure them exactly. There is always an error of the measurement. This may sound discouraging, but you will learn how to deal with this error.
EXAMPLE 1: Charles wants to label a box of "stuff" with a complete statement of the weight. What should his statement include?
To be complete, every measurement should show 1) the measured value, 2) the unit of measurement, and 3) an estimate of the error of measurement. Since we can't avoid error in a measurement, we often need to have an estimate of the error. Here is how Charles could label the box: 500 grams +/- 1 gram. The "+/- 1 gram" is an estimate of the absolute error. He determined it by selecting a sample of 10 boxes, weighing them, and computing the average weight. He then selected the weight furthest from the average weight and determined the difference in weight. That is the absolute error.
This error can also be shown as an estimate of the relative error expressed as a per cent. Divide the absolute error of 1 gram by the total weight of 500 grams and multiply by 100, and we find that the error compared to or relative to the total weight is +/- 0.2 %.
CHALLENGE 1: The label on a package of salt says, "NET WT. 26 OZ., 737 gr. What needs to be added to make a complete statement of the weight?
EXAMPLE 2: Bryan's boss told him to measure out packets of 50 grams each of moon dust with a precision of plus or minus 1 gram. How could Bryan choose a weighing instrument that is just accurate enough for the job?
The graduations on the instrument's scale give some idea of the maximum precision of the instrument. A common kitchen scale is marked off in 5-gram graduations, so it clearly doesn't have the required precision. The precision of a chemist's analytical balance might be +/- 0.0001 gram, which is more precision than Bryan needs. A small platform balance that has a precision of +/- 0.5 grams is suitable for the job.
CHALLENGE 2: If you have a kitchen scale, see what each graduation stands for. A typical metal paper clip weighs about 1 gram. See how many paper clips you can drop on the scales before the dial moves.
CHALLENGE 3: Estimate the precision of your ruler either in centimeters or in inches. What are the smallest graduations on the centimeter and/or inch scales?
The next bonus subject is significant figures, which is related to measurement error. When we make computations, especially with a calculator, using measurements, it is important to know about significant figures.
EXAMPLE 1: Tim weighed out and divided 5.0 lb. of salt into three equal parts. Using a calculator, compute how much each part weighs.
The zero to the right of the decimal point in 5.0 means that the weight is accurate to the nearest .1 lb. We make the division on our calculator: 5 ÷ 3 = 1.6666666 lb.
How many of those 6s in the quotient mean anything, or, in other words, how many of the digits are significant figures?
If Tim's scales are accurate only to the nearest one-tenth pound, the true weight could be anywhere between 4.9 and 5.1 lb. If it is 4.9, the calculated quotient would be 1.6333333 lb. If the true weight is 5.1, the quotient would be 1.7000000 lb. Now we can see that in our first calculated quotient, 1.6666666 lb, only two figures are significant. Tim could say that each part weighs 1.7 lb and add an estimate of the error measurement.
Another way of looking at significant numbers in doing calculations is that you may only have one doubtful number in your answer. In the above case the "7" in 1.7 is a doubtful number because the true value could be 7 or 6. The 1.6666666 numeral has one reliable number (the 1), one doubtful number (the first .6), and six meaningless numbers (666666). Why are they meaningless?
Here is a rule to help you: The number of significant figures in the answer must not exceed the number of significant figures in the least reliable number used.
CHALLENGE 1: A room is 30.5 ft wide and 40.9 ft long. What is its area? Use a calculator for the computation and round off your answer.
CHALLENGE 2: Mrs. Brown bought 13 lbs of candy. She divided the candy into 3 equal parts. What was the weight of each part? Use a calculator and round off your answer.
EXAMPLE 2: Charlie bought 105 lbs of bulk fertilizer. He also bought a bag of fertilizer marked 100.5 lbs. What was the total amount purchased? Is it 205.5 lbs?
No. The rule in addition and subtraction is that the answer may only contain one doubtful number. We can see that 205.0 contains two doubtful numbers (5 and .5). So the sum is 205 lbs.
CHALLENGE 3: A merchant opened a bag containing 25.0 lbs of salt and sold 5 lbs. How much was left in the bag? If he had measured the salt more accurately and sold 5.0 lbs, how much salt would be left?
Back to the Beginning
Your next bonus subject is scientific notation. First, let's see how this helps us deal with very large numbers.
EXAMPLE 1: The average distance to the sun is about 93,000,000 miles. How can we write this without having so many zeroes?
Scientific notation gives us an easy way to handle big numbers. First, we write the base number or coeficient, which must be 1 or more but less than 10. So, it is 9.3. Second, we use a multiplier that is 10 with an exponent that shows how many 10s are multipied together. In this case we move the decimal point 7 places to the left in order to get 9.3, so the multiplier must be 10 to the 7th power (10 with an exponent of 7). We write 9.3 x 10 with a little raised 7. On the Web we write 9.3 x 10^7 or 9.3E+7.
To better understand scientific notation, study the following:
CHALLENGE 1: The average distance from earth to the moon is 239,000 miles. Write this using scientific notation.
EXAMPLE 2: Jack bought 50 lbs of salt. Is the zero a significant number or a place holder. Someone may have intended for it to be significant, but as it is written, we must assume that it is only a place holder. How can we show that the zero is a significant figure?
To show that the 0 in 50 is significant, we can simply add a decimal point (50.). We could also use scientific notation. If the zero is significant, we write 50 like this: 5.0 x 10^1. The base, 5.0, now makes it clear that the zero is significant because it is to the right of the decimal point with nothing following it. If the zero in 50 is not significant, then we would write 50 this way: 5 x 10^1
CHALLENGE 2: Write 600 three times in scientific notation assuming one, two, and three significant figures.
EXAMPLE 3: A scientist measures a particle and finds that it is 0.0007 millimeters in width. How can he write this in scientific notation?
Now you will see how scientific notation helps us with very small numbers.
First, let's see how decimal fractions are written in scientific notation:
To change 0.0007 to scientific notation, the base must be 7. How many places must you move the decimal? What exponent of 10 must we use?
We must move the decimal 4 places to the right in 0.0007 to get the base of 7, so we write the number this way: 7 x 10^-4 or 7E-4.
CHALLENGE 3: Write 0.07 in scientific notation.
EXAMPLE 4: The fourth advantage of scientific notation is that it makes comparison of orders of magnitude easy. The difference between 1 and 10 is 1 order of magnitude: 1 x 10^0 compared to 1 x 10^1. The difference in the exponents of 10 is 1 or 1 order of magnitude.
If we are comparing two rough estimates, the first thing we want to know is how close are the estimates. Are they within the same order of magnitude? What is the difference in orders of magnitude between 12,500,000 and 800,000?
12,500,000 is 1.25 x 10^7
800,000 is 8 x 10^5
We subtract 5 from 7 and find that the numbers differ by two orders of magnitude, which is a lot of difference.
CHALLENGE 4: John's wife asked him how much money he lost at the races. He said he lost $1, but he actually lost $100. How big was his lie by orders of magnitude?
Back to the Beginning
EXAMPLE 5: How much is 5.48 x 10^4 meters plus 3.7 x 10^2 meters?
What we have here is the addition of "apples and oranges" because they are of different orders of magnitude. The first thing we need to do is to change them to the same order of magnitude.
5.48 x 10^4 = 548 x 10^2
548 x 10^2 plus 3.7 x 10^2 = 551.7 x 10^2 = ? Can you finish?
[5.5 x 10^4]
Your next bonus is an opportunity to use your knowledge of scientific notation in dealing with jumbo problems. You will happily discover that very large numbers are hardly any more difficult to work with than small ones.
EXAMPLE 1: How many paper clips would it take to balance the weight of the earth? You might wonder why we would figure the weight in paper clips. Well it so happens that the average metal paper clip weighs about 1 gram.
THINK: Since a paper clip weighs about 1 gram (gm), you must find what the earth weighs in grams. So first, we need to know the earth's density, which is the weight in grams of a typical cubic centimeter (cc) of earth. The volume of a small cube of sugar is about 1 cc. (Water weighs 1 gm per cc, so its density is 1.)
Second, we need to compute the volume of the earth for which we need to know its radius (half of its diameter).
Third, the earth's weight or mass equals its density times its volume in cc.
To repeat, what do we need to know?
1) We need to know the earth's average density. Is it very dense like lead or lighter like water? You can make a fair estimate of density by finding a mineral from the earth that might have about the same density as Earth's average density. Since we know that Earth has an iron core, let's choose the iron rich mineral, magnetite, which has a density of 5.20. This means that 1 cc of magnetite weighs 5.20 gm. Since this is a rounded number consisting of 3 significant figures, we will not carry any other number out beyond 3 significant figures.
2) We need to know the earth's radius (r) at the equator. It is 6,378 or 6.38E+3 km, which must be changed to centimeters. To change it to meters, we multiply by a 1,000 or add +3 to the exponent to make it 6.38E+6 m. To change to meters to centimeters, what do we multiply by? Since there are 100 cm in a 1 m, we multiply by 100 or add +2 to the exponent to make it 6.38E+8 cm.
COMPUTE:
Here is the formula for computing the volume of a sphere: V = 4/3 x Pi x r^3
Earth's volume: V = 4/3 x 3.14 x (6.38E+8)^3
V = 1.3 x 3.14 x (6.38 x 6.38 x 6.38 E+8 x 3)
V = 4.19 x 260E+24
V = 1,090E+24
V = 1.09E+27 cm
Earth's mass = Volume x Density = (1.09E+27) x 5.20 = 5.67E+27 gm or roughly 6 followed by 27 zeroes
So, the next time you want to impress someone tell him the mass of the earth is the same as that of a pile of paper clips numbering 6 followed by 27 zeroes.
A more accurate published number for the earth's mass is 5.98E+27. How can we explain the difference? What are three possibilities?
One possibility is that we goofed in our thinking or in our computations, but we checked for errors. The second possibility is that the density of magnetite is less than the earth's average density. It turns out that the best estimate of the earth's density is 5.52, which would make our answer is a bit low. The third possibility is that the earth is not exactly a sphere, and it isn't. The earth is flattened a little at both poles, which reduces the volume somewhat. This error partially offsets our density error. Our answer is a little low, but at least we can say our answer is in the correct order of magnitude.
CHALLENGE 1: What is the weight of the moon?
Here is what you need to know about the moon: 1) density = 3.3 and 2) radius = 1738 km.
EXAMPLE 2: If you sit on the ground at the equator, how fast will you be moving around the axis of the earth?
When you fly in an airplane, you may not feel that you are moving. Likewise, when you sit on the ground, you do not feel you are moving unless there is an earthquake. But you are moving at high speeds within our Solar system in at least two "orbits!"
The first orbit is your "orbit" around the center of the earth. The earth turns on its axis one full revolution every 24 hours. So, if you are sitting on the equator, you make the trip around center of the earth in the same way as if you were sitting on the outside of a merry-go-around.
If you are sitting at the equator, how far do you travel in one 24-hour day? The earth's diameter at the equator is 7,928 miles, so how do you find its circumference?
Circumference = Pi x Diameter = 3.14 x 7930 = 24,900 miles.
Speed = Distance ÷ Time
Speed in mph = 24,900 mi. ÷ 24 hr = 1,040 mph
CHALLENGE 2: If we send a man to Mars and land him at the equator, how fast will he be moving around the axis of Mars?
Mars has a diameter at the equator of 4,217 miles. One rotation takes 24.62 hours.
EXAMPLE 3: The earth is moving in an orbit around the sun. How fast are you moving around the sun in miles per hour?
THINK: The earth's orbit is slightly elliptical, but we can assume it is circular. We need to compute the circumference of the earth's orbit in miles and the hours required for the earth to orbit the sun.
What do we know?
The earth orbits the sun once a year or once in 365.25 days.
The average distance to the sun is about 93 million miles, which is the radius of the earth's orbit around the sun.
Circumference of a circle = Pi x Diameter (2 x radius)
Speed = Distance ÷ Time
COMPUTE:
Time for one orbit = 365.25 (days in a year) x 24 hours = 8766 hours = 8.8E+3 hr
Diameter of the orbit = 2 x 9.3E+7 = 18.6E+7 mi.
Circumference of orbit = 3.14 x 18.6E+7 = 58E+7 mi.
Orbital speed of earth = 58E+7 mi. ÷ 8.8E+3 hr = 58 ÷ 8.8E+(7-3) = 6.6E+4 = 66,000 mph
So you are speeding around the Earth's axis at the speed of a fast airplane and around the sun at about three times the speed of most of our rockets. But that is not all. The Sun is speeding around the center of our Milky Way Galaxy, and the Galaxy is speeding through space.
Back to the Beginning
U. S. SYSTEM
Weight: 1 ounce = 28.35 grams = 1 pound (lb.) = 16 ounces (oz) = 454 grams; 1 (short) ton (t) = 2,000 lb. = 908 kilograms
Length: 1 inch (in.) = 2.54 cm; 1 foot (ft) = 12 inches; 1 yard (yd) = 3 ft; 1 rod = 16.5 ft; 1 mile (mi.) = 5,280 ft = 0.62 kilometer
Area: 1 acre = 43,560 sq. ft = 0.41 hectares; 1 square mile = 640 acres
Volume: 1 cubic inch = 16.387 cubic centimeters; 1 cubic foot = 1,728 cubic inches; 1 cubic yard = 27 cubic foot = 0.765 cubic meters
Capacity, liquids: 1 tablespoon (tbsp.) = 3 teaspoons (tsp.) = 1/2 fl oz; 1 cup = 8fl oz; 1pint (pt) = 2 cups; 1 quart (qt) = 2 pt = 0.94 liter; 1 gallon (gal) = 4 qt
Capacity, dry: 1 peck (pk) = 8 quarts; 1 bushel (bu) = 4 pk
Temperature: Freezing point of water is 32 degrees Fahrenheit (deg F);
boiling point of water (at sea level) is 212 deg F.
Time: 1 minute (min) = 60 seconds (sec); 1 hour (hr) = 60 min
Rate: Miles per hour (mph); miles per gallon (mpg); feet per second (ft sec);
gallons per minute (g.p.m.); revolutions per minute (RPM)
Angle: 1 degree = 60 minutes (') of angle = 1/360 of a full circle; 1 minute of angle = 60 seconds (")
Metric prefixes: kilo or k = thousand, mega or M = million, giga or G = billion, tera or T = trillion, centi or c = hundredth, milli or m = thousandth, micro = one-millionth, nano = one-billionth, pico = one-trillionth. Now you know the order of magnitude for megabytes and nanoseconds.
Weight: 1 gram (gm) = 1,000 milligrams (mg) (One gram is about the weight of
a metal paper clip.); 1 kilogram or "kilo" (kg) = 1,000 gm = 2.2 lb.; 1 metric ton = 1,000,000 gm = 1.103 short tons or 2,205 lb.
Length: 1 meter = 1,000 millimeters (mm) = 100 centimeters (cm) = 39.4 in;
1 kilometer (km) = 1,000 meters = 0.62 mi
Area: 1 square centimeter = 0.155 square inch; 1 square meter = 10.76 square feet; 1 hectare = 10,000 sq. m = 1/100 square kilometer = 2.44 acres
Volume: 1 cubic centimeter = 0.061 cu. in; 1 cubic meter = 1.35 cu. yd
Capacity, liquid and dry: 1 liter (L) = 1,000 milliliters (ml) or cubic centimeters =
1.06 qt
Temperature: Water freezes at 0 degrees Celsius (deg C) and boils at 100 deg C. A temperature change of 1 deg C equals a change of 1.8 deg F.
