1. Multiply both sides of the equation by "X in".
   Left side: 2.54 cm x X in / 1 in: Now cancel out "in" leaving 2.54 cm x X
   Right side: 10 cm x X in / X in: Now cancel out the "X in"s leaving 10 cm
   Equation left: 2.54 cm x X = 10 cm
2. To leave X by itself, divide both sides by "2.54 cm".
   Left side: 2.54 cm x X / 2.54 cm = X Right side: 10 cm / 2.54 cm = 3.94
   So 10 cm = 3.94 in
   

CHALLENGE 9: Convert 50 centimeters to inches. Use the two methods and see if they agree.

EXAMPLE 5: A certain outdoor racetrack is 880 yards long. How many feet is that?

THINK: In the previous example we converted a little unit to a larger unit. Here we must convert a larger unit to a smaller unit. Remember that 1 yard equals 3 feet.
We could think that 880 yards is 880 sets of 3 feet. So the number question is 880 X 3 feet = ?
COMPUTATION: Feet = 880 X 3 feet = 2640 feet
CHECK your thinking and computations.

CHALLENGE 10: Ned ran the 100-yard dash. How many feet is that?

Distance is a measure of the "gap" between two things such as two towns or two stars or whatever. It can be measured in inches, feet, yards, rods, or miles in the US Sytem or in meters or kilometers in the metric system.

CHALLENGE 11: The distance between two villages in France is 4 kilometers. How many miles is that?
Back to the Beginning

[Eight standard metal paper clips, a foot ruler, and two pencils will be needed.]

When we step on the scales, we can read our weight. We can think of weight in two different ways. In one sense it is a measure of the force of gravity exerted on something by the Earth. A person who weighs 100 lb on Earth would weigh only 17 lb on the Moon because the Moon is much smaller and exerts much less gravitational force. A scientist may think of weight as a measure of the mass of an object which would be the same whether measured on the Earth or the Moon. Mass is the quantity of matter of "stuff" in an object.

A chemist measure mass with scales called balances. They balance the mass of a chemical in one pan against a mass of known quantity such as little brass weights. Many scales in doctors' offices are balances. The nurse balances your mass against a little leveraged metal mass moved along a bar.

Our US bathroom scales measure weight in pounds, which is a unit in our US System of measures. (Butter, for example, usually comes in 1 lb packages.) In most of the rest of the world scales measure weight in kilograms or a thousand grams, which is a metric System unit of measure.

EXAMPLE 1: How many grams does a dime weigh? Here is a clue: A metal paper clip weighs about 1 gram. How can you use that clue to weigh a dime using a ruler and 2 pencils?

Put one pencil under the ruler at the 6-inch mark. Put another pencil under the left end and put the dime at the 1/2-inch mark. Put the clips one at a time at the 11-1/2-inch mark. See how many clips it takes to raise the left end.

CHALLENGE 1: How many grams does a penny weigh? Tip: Since you now know how much a dime weighs, you can use a dime and paper clips for the known weights.

EXAMPLE 2: A 1-pound carton of butter usually contains 4 sticks of butter. How many ounces does each stick weigh? The pound (lb.) is equal to 16 ounces (oz).

We divide 16 oz into 4 parts. The weight of 1 part is the weight of 1 stick.
16 oz ÷ 4 = 4 oz per stick

CHALLENGE 2: Jimmy has 2 pounds of peanuts. If he divides it into equal-sized bags for his 4 friends, how many ounces will each friend get?

EXAMPLE 3: If you have a kitchen scale that has a dial or a pointer, weigh a one-pound package of something such as butter.

The first thing to do is to check to see that the scale shows 0 when nothing is on it. You may find a little knob to adjust to 0. There may be one scale that shows the weight in pounds and ounces. Another scale may show the weight in grams.

On the pound scale notice how many little spaces are marked off between 0 and 1 pound. If there are 16 spaces, what weight does each space equal?

CHALLENGE 3: Carefully measure 1 pint or 2 cups of water in a measuring cup using the U. S. system and weigh the cup with the water. To determine the net weight of the water, what do you weigh next? "Net" means without the container.

As to water at least, do you agree that a "pint's a pound the world around?"

EXAMPLE 4: Mrs. Jones made a birthday cake for Jimmy. The recipe came from Europe where the metric system is used. The recipe calls for 250 grams of sugar. How many ounces is that? Use your calculator.

THINK: There are 28.4 grams per ounce. So how many sets of 28.4 grams are there in 250 grams?
COMPUTE: 250 gm ÷ 28.4 gm (or 1 oz) = 8.8028169. This shows once again why it is necessary to round off your answer when you use a calculator to make computations with measurements. Our answer can't be more precise than the numbers we use to compute it. Since the 28.4 is a rounded number (rounded to three digits), we will round off our answer to 8.80 oz.
CHECK: Set up a proportion: 1 oz ÷ 28.4 gm = X ÷ 250
Cross-multiply: 28.4 x X = 1 x 250; 28.4X = 250, X = 250 ÷ 28.4 = 8.80 oz

CHALLENGE 4: A can of imported coffee shows the weight to be 1 kilogram. How many pounds is that?
Back to the Beginning

Anything that has width as well as length has a two-dimensional surface, and the amount of surface is called area.

EXAMPLE 1: Draw a square that measures 1 inch by 1 inch. Also draw a rectangle that is 2 inches long and 1/2 inch high. Do both have the same area?

A rectangle has four sides. The opposite sides are parallel and equal in length, and all four angles are right angles. Your computer screen is a rectangle with 4 right angles at the corners. Parallel means that the lines are the same distance apart at any point on the lines like railroad tracks. A square is a rectangle with four equal sides.

The unit of measure that we will use in this problem is the square inch. The square that you drew is an inch square, and it contains 1 square inch of area. The 2 x 1/2 in. rectangle that you drew is not an inch square, but it also has 1 square inch of area. How could you prove that with a pair of scissors and your drawings?

EXAMPLE 2: Draw a square that measures 3 inches by 3 inches (a 3-inch square). Mark it off in 1-inch squares. What is the area of the square?

THINK: How can you determine the area by counting?
How could you find the area by multiplication? Hint: How many rows of 1-inch squares are there? How many 1-inch squares are there in each row? So how do you compute the area?
Here is the formula for computing the area of a rectangle: Area = Length x Width

Since the length of each side of the square is the same, we simply multiply the length of one side by itself. When we multiply a number by itself, we call the product the square of that number. The square of 3 is 9. Working backwards, the square root of 9 is 3. What is the square of 4?

Since each side of a square is the same length, here is the formula for the area of a square: Area = Side squared
We can write "side squared" by following S with a little raised 2, which is the exponent. On the Web we may write S^2. Now let's solve the problem by plugging in the amounts in the formula:
Area = S^2 = 3^2 = 3 x 3 = 9.

You will someday discover that we sometimes square a number when it has nothing to do with the area of a square.

CHALLENGE 2: John's bedroom is 10 ft. long by 10 ft. wide. Using the formula for the area of a square, compute in your head the area of the floor.

CHALLENGE 3: Draw a rectangle that is 4 inches long by 2 inches wide and compute the area using the formula for a rectangle. To check your answer, divide it into inch squares and count the squares.

EXAMPLE 3: John's bedroom is actually 10 ft 2 in. long by 10 ft wide. What is the area?

THINK: Since one dimension is in feet and inches, we need to express the inches as a fraction of a foot. What is the fraction? 2 in. is 2/12 ft. How do we change 2/12 to a decimal fraction?
COMPUTE AND CHECK: You divide 2 by 12 on a calculator, and the quotient is 0.1666666. Round that off to 0.2, so the length of the room is 10.2 feet.
Area = 10.2 ft x 10 ft = 102 ft^2 or 102 square feet

CHALLENGE 4: Measure your bedroom and compute the floor area.

EXAMPLE 4: Draw a rectangle that is 4 inches long and 2 inches wide. Starting at the lower left-hand corner, draw a line diagonally across to the upper right-hand corner. You now have two right triangles. A triangle has three angles and three sides. What is the area of each triangle?
Area of any triangle = Base (length) x Height (width) ÷ 2
Why do you divide by 2? Take your time and look at the two triangles.
Is the base and height of the triangle the same as the length and width of the rectangle? How many triangles are there in the rectangle?
So this is the area of each triangle:
Area = = Base (length) x Height (width) ÷ 2 = 4 in. x 2 in. ÷ 2 = 4 in.^2 or 4 square inches

CHALLENGE 5: Draw a line 4 inches long. At the middle of that line measure 2 inches vertically and make a dot. From each end of the line draw lines connecting to the dot. You have drawn an isosceles triangle, which has two equal sides and two equal angles. Compute the area using the formula for computing the area of a triangle and compare it to the area of the right triangle.

CHALLENGE 6: Draw a line 4 inches long. Starting one inch from the end of that line measure vertically 2 inches and make a dot. From each end of the line draw lines connecting to the dot. Compute the area of that scalene triangle and compare it with the area of the right triangle.

EXAMPLE 5: Draw an inch square. Draw a circle inside of the square that is 1 inch in diameter. What is the area of your circle?

Now you will learn about Pi, and it's not your favorite dessert. Pi is the ratio of the circumference of a circle to its diameter, and its rounded value is 3.14. If the diameter of a circle is 1 inch, the circumference of the circle is 3.14 in.

You should also know that the radius (r) of a circle is 1/2 of its diameter. In this case the radius is 0.5 in.

Here is the formula for the area of a circle: Area of circle = Pi x r^2.
Area = 3.14 x 0.5^2 = 3.14 x 0.25 = 0.79 in^2 or square inches.

To better understand the circle formula, look at the square. The ratio of the "circumference" or perimeter of the square to its "diameter" or one side is 4 to 1. Notice that the circumference of your circle is a little less than 4 because the circle "rounds off the corners." If we treat the square like a circle, this is the area of a square:
Area of square = 4 x r^2 = 4 x (0.5 x 0.5) = 4 x 0.25 = 1 sq. in.

CHALLENGE 7: Jim wants to fertilize his circular flower bed, but first he must compute the area so that he will know how much fertilizer he needs. The circle is 20 feet in diameter. What is the area of the circle in square feet?
Back to the Beginning

Volume is the amount of space in something. Since space has three dimensions, length, width, and height, volume is expressed in cubic units such as cubic feet or ft^3.

When we figure the volume of containers, we call it capacity, and we usually use measures such as quart or liter instead of a cubic measure.

EXAMPLE 1: Jane has a flower box that is 1 foot wide, 1 foot long, and 1 foot high. What volume of soil will it hold?

She has a 1-foot cube of soil, and the volume is 1 cubic foot or 1 ft cubed 1 ft with the little raised exponent 3 following the ft. On the Web we may write 1ft^3. The cube of a number is the third power of a number. In this case it is 1 x 1 x 1. What is the cube of 2? What is the cubic root of 8?

Here is the formula for volume: Volume = Length x Width x Height.

CHALLENGE 1: Jimmy's sandbox is 2 feet long, 2 feet wide, and 1 foot deep. How many cubic feet of sand are required to fill it?

EXAMPLE 2: John has a can that has a volume of 1 cubic foot. What is the capacity of the can measured in gallons?

We measure liquid volume in gallons, quarts, and pints in the U.S. system. Dry volume or capacity is measured in bushels and pecks. In the metric system both dry and liquid are measured in liters. What we need to know in this problem is that a gallon is equal to 231 cubic inches. So the first step is to convert 1 cubic foot to cubic inches.
1 ft^3 = 12 in. x 12 in. x 12 in. = 1,728 in.^3 or cubic inches
How many sets of 231 in.^3 can be separated from 1,728 in.^3?
Capacity in gallons = 1,728 in.^3 ÷ 231 in.^3 = 7.5 gal.

CHALLENGE 2: A wading pond is 10 feet long, 10 feet wide, and 1 foot deep. How many gallons of water will it hold?

EXAMPLE 3: Bryan bought what he thought was a quart of juice. He discovered that it was actually a liter, which is a metric measure. How can he figure out, without measuring, whether this is more or less than a quart?

One way to solve this problem is to compare the weight of a liter of water with that of a quart. A liter is equal to 1,000 milliliters (ml). A ml of water weighs 1 gram, so a liter of water weighs 1,000 gm. There are 454 gm in a pound. So how many sets of 454 can be separated from 1,000? 1,000 ÷ 454 = 2.2 pounds. Bryan remembers the little jingle about the weight of water: "A pint's a pound the world around." Since there are two pints in a quart, a quart of water weighs 2 pounds compared to 2.2 pounds for a liter of water.

CHALLENGE 3: How many milliliters are there in a quart? Use the weight of water to make your computation.
Back to the Beginning

Temperature is a measure of the intensity of heat or the coldness or hotness of something. Zero weight means no weight, but a temperature of zero means a certain point on a temperature scale.

There are two common temperature scales. Thermometers that we have in the home in the U. S. usually measure temperature on the Fahrenheit scale. On this scale water freezes at 32 degrees F and boils at 212 at sea level, which is 180 degrees above freezing. Our body temperature measured under the tongue is 98.6 degrees F, and it is 99.6 measured in the other place.

Nations using the metric system use the Celsius scale, which was adopted in 1948, in the home. It is a centigrade scale in that there is a 100 degree difference between the freezing and boiling points of water. Water freezes at about 0 degrees Celsius and boils at about 100 at sea level. Our normal body temperature measured under the tongue is about 37 degrees C.

There may be two types of analog thermometers around your house. The bulb type shows the temperature with a column of mercury or red liquid. A second type has a needle that turns around the dial. There also may be a digital thermometer that electronically displays the temperature with a decimal number.

EXAMPLE 1: A typical outdoor bulb-type thermometer has 5 spaces between 0 and 10 degrees. What does each space mean?

If 5 spaces equals 10 degrees, then 1 space equals 10 ÷ 5 or 2 degrees. When reading any analog thermometer, the first thing we must do is to see what each mark on the scale means.

CHALLENGE 1: Find a suitable thermometer and place the thermometer outside in the shade. After five minutes read the temperature. Then place it in the sun and after 5 minutes read it again.

EXAMPLE 2: The weather report said the temperature in Paris was 30 degrees Celsius. What would that be on the Fahrenheit scale?

Draw two vertical lines about 6 inches long and an inch apart. Label the first line F and the second C. At the top of the F scale write 212. At the top of the C scale write 100. About an inch from the bottom of the F scale write 32. At the same point on the C scale write 0.

On the F scale water boils at 212 degrees or 180 degrees above freezing, which is 32 degrees. Water boils at 100 degrees C, so a range of 180 degrees F equals 100 degrees C. One degree C equals how many degrees F?

To convert a temperature above freezing from Celsius to Fahrenheit, we first multiply the Celsius reading by 1.8 and, second, add 32 degrees. Here is the formula for converting Celsius (above zero or freezing) to Fahrenheit: F = 1.8C + 32

Temperature F in Paris = (1.8 x 30C) + 32 = 54 + 32 = 86 F in Paris

CHALLENGE 2: A nurse told Harry that his temperature was 38 degrees. Which temperature scale do we hope was being used? What would be his temperature on the F scale?

CHALLENGE 3: In January the temperature got down to 0 degrees F. What would the reading be in Celsius? Hint: Roughly show these temperatures on your drawing of the two scales side by side. The formula won't work because 0 degrees F is below freezing.
(Ans. C = minus 32/1.8)

NOTE: The measure of the intensity of heat is temperature, which is not the same as the amount of heat. One measure of the amount of heat is the calorie. The "gram calorie" is the amount of heat required to raise the temperature of 1 gram of water by 1 degree Celsius. When we say that a piece of cake contains 400 "calories," we really mean kilocalories or kcalories (1,000 calories). A kcalorie is enough energy to raise the temperature of 1 KILOgram of water 1 degree. So just 100 kcalories is enough energy to bring 1 quart of water at the freezing point (but not frozen) to a rolling boil. (Fat produces 9 Kcalories per gram and carbohydrates and protein produce 4 per gram. The average adult needs to eat enough food to produce about 2,000 kcalories energy per day. Five pieces of that cake would supply the calories but would not supply the nourishment needed.)
Back to the Beginning

[Go to and print out the clock face graphic. Review often and add other problems until telling time is mastered.]

Some units of time are natural units such as the day, the week, the month, and the year. What in nature is the basis for each of these units of time?

A day is the time it takes for the earth to turn completely around on its axis like a top. A week is roughly the length of time of each of the four phases of the moon. The month is roughly based on the lunar month, which is the time it takes for the moon to circle the earth. The year is the time it takes for the earth to circle the sun.

When people found they needed smaller units of time than the day, they divided the day into two 12-hour periods. The 12 hours before noon is AM, ante meridiem, and the 12 hours after noon is PM, post meridiem. Then they divided the hour into 60 minutes and the minute into 60 seconds.

Clocks measure time and show it in either a digital display or on a dial. Let's see how the time is shown on a dial. Look at the black "hands" on the picture of the clock.

1. Point to the shortest black hand, which is the hour hand. It takes 12 hours for it to sweep around the dial, so there are twelve numbers.
   
2. Point to the longest black hand, which is the minute hand. It takes one hour or 60 minutes for it to sweep around the dial. Point to the little black marks around the outside of the dial. There is one mark for each minute. There are no numbers for minutes, so we use the hour numbers multiplied by five to show the minutes. Point to the number where the minute hand would be for 10 minutes past the hour?

EXAMPLE 1: Look at the picture of the clock and determine the time of day.

Here are the steps for telling the time shown on a dial clock:

1. See where the hour hand is pointing. It is pointing between 2 and 3, so the time is some minutes past 2 o'clock.
   
2. See where the minute hand is pointing. It is pointing between 9 and 10. The number of minutes at 9 is 5 x 9 or 45 minutes past the hour. We count from 45 on to where the minute hand is pointing: 46, 47, 48, 49. The minute hand shows 49 minutes past the hour.
   
3. So the time is 49 minutes past 2 or 2:49. We might also say that the time is 11 minutes 'til 3 o'clock.

CHALLENGE 1: Draw a clock face with the numbers. What time will it be when both the hour hand and minute hand are on 12?

CHALLENGE 2: What time will it be when the hour hand is half way between 12 and 1, and the minute hand is on 6?

CHALLENGE 3: What time will it be when the hour hand is between 2 and 3 and the minute hand is on 11?

CHALLENGE 4: Tell me where each hand will be on a clock face for each of the following times:

12:04, 1:32, 3:14, 4:59, 6:06

If you saw these times on a digital watch, how would you read them?

EXAMPLE 2: Peter started doing his homework at 6:50 PM and finished at 9:20 PM. How long did he work?

You subtract minutes from minutes and hours from hours, but in this case you must first borrow 1 hour, change it to 60 minutes, and add the borrowed 60 minutes to the 20 minutes.
9:20 = 8:20 + 60 or 8:80; 8:80 - 6:50 = 3 hr. and 30 min.

CHALLENGE 4: Bill left New York on a plane at 4:50 PM Eastern Standard Time. He arrived in San Francisco at 9:40 by his watch, which was still set on Eastern Standard Time. How long did the flight take?

EXAMPLE 3: Bill arrived in San Francisco at 9:40 Eastern Standard Time. What time was it in San Francisco by their local time, which is Pacific Standard Time?

There are four time zones across the continental United States: Eastern, Central, Mountain, and Pacific. Why do we need time zones?
We want clock time to roughly reflect Sun time so that when it is 12:00 noon local time the Sun will be about at its highest point. Earth turns on its axis once in 24 hours. So 24 time zones around the world were drawn, but there are lots of local modifications. The Sun spends about one hour in each time zone.

Going from east to west, the time changes to one hour earlier in each time zone. Why is the time change an hour earlier rather than later?

Although there are four time zones, there is only a three-hour difference between the Eastern Standard Time zone and the Pacific Standard Time zone. Central time is one hour earlier than Eastern ST, Mountain ST is two hours earlier, and Pacific ST is 3 hours earlier. So Bill must reset his watch back to 6:40 PM.

CHALLENGE 5: Mr. Smith lives in Denver and the local time, which is Mountain Time, is 4:00 PM. He wants to call a business office in New York. What time will it be in New York?

EXAMPLE 4: Kevin reset his watch on the first Sunday in April to switch from Standard Time to Daylight Savings time. The time shown on his watch was 8:45 AM. What time should he change it to?

When we reset our watches, remember that in the spring we "spring forward" and in the fall we "fall back." Kevin should reset his watch by "springing forward" to 9:45.

CHALLENGE 6: On the last Sunday in October, Daylight Savings time ended. Kevin's watch showed that the time was 8:10 AM. What time should he reset it to?

EXAMPLE 5: Jake, who is a soldier, said he would arrive at 1830 hours. What time is that in civilian time?

In the Armed Forces, time is measured from midnight to midnight on a 24-hour basis. The first two digits show the hour and the second two digits show the minutes. Notice that 1200 hours is noon; 1300 hours is 1:00 PM. So, if the military time is 1300 hours or later, we must subtract 1200 from it to convert to civilian time.
1830 - 1200 = 600 or 6:00 PM

Why do you think the Armed Forces use the 24-hour basis?
Back to the Beginning

Jimmy is licking his ice cream cone at the rate of 20 licks of per minute. Notice that licks per minute contains a time unit of measure. Measures that contain a time unit are called rate. Another example of a measure of rate is the speed of a certain car that is traveling at the rate of 60 miles per hour (mph).

EXAMPLE 1: Fred left home at 3:00 PM in his car and arrived at his friends home, which was 100 miles away, at 5:00 PM. What was his average rate of speed measured in miles per hour?

Since the rate of speed in the U. S. is expressed in miles per hour (mph or miles/hour), we divide the miles traveled by the hours of travel to find the average speed traveled.

Here is the formula for rate of speed: Speed = Distance ÷ Time.
Speed in miles per hour = Distance in mi. ÷ Time in hrs
Speed = 100 mi. ÷ 2 hr = 50 mph

CHALLENGE 1: Bill drove his car 90 miles in one hour and 30 minutes. What was his average rate of speed?

EXAMPLE 2: What is your pulse rate?

Point to the red second hand on the picture of the clock face. It takes 60 seconds for the second hand to go around the dial. We count seconds the same way we count minutes. Find a clock or watch that measures seconds of time.

Each time your heart beats it sends a pulse of blood through your arteries. Find the place on the inside of your wrist just below your first thumb joint where you can feel the pulse. Look at a clock or watch that shows seconds. Start counting when the second hand is on 12 or 6 and count the number of beats that you feel in 30 seconds and multiply that number by two.

CHALLENGE 2: How many breaths per minute are you taking?
Back to the Beginning

[You will need a ruler, and a map of your area. Go to and print out the protractor graphic.]

Stand in one place and turn slowly all the way around until you are facing in the original direction again. You have just turned through an angle of 360 degrees or one revolution. Why do you suppose they originally decided on 360 degrees of angle for one revolution?

The earth makes its trip around the sun in close to 360 days, so they decided to say that the angle that it moves through in one day is 1 degree. When the earth turns completely around on its axis, how many degrees of angle does it turn through?

EXAMPLE 1: Draw a line parallel to the bottom of the page. Now draw a vertical line that is parallel to the side of the page. What is the angle between the horizontal line and the vertical line? Look at the protractor and determine the angle.

The angle between the horizontal line and the vertical line is 90 degrees. If you put your ruler on the horizontal line and turn it to be along the vertical line, you will turn the ruler through 90 degrees of angle.

CHALLENGE 1: Look at a clock face. How many degrees does the minute hand move through in one hour?

How many degrees does the hour hand move through in one hour?

EXAMPLE 2: Look at the picture of the protractor on top of a right triangle with sides measuring as follows: The vertical side A is 3 inches; the bottom horizontal side B is 4 inches; and the diagonal side C (hypotenuse) is 5 inches. This is a special triangle because the lengths of the sides just happen to be consecutive whole numbers. What are the angles between sides A and B, B and C, and C and A?

Angle between sides A and B: This is a right angle, so it is a 90-degree angle.
Angle between sides B and C, which is the smallest angle: The reading on the outside scale of the protractor is 37 degrees.
Angle between sides A and C: Since the angles between B and C and between A and C must add up to 90 degrees, the angle between A and C must be 90 minus 37 or 53 degrees.

CHALLENGE 2: Suppose we double the length of all sides of the triangle in Example 2. What would be the angle between sides B and C? Look again at the protractor and see if the length of the sides would make any difference in the angles assuming that the shape of the triangle is not changed.

EXAMPLE 3: Stargazers compare the size of objects in the sky by measuring their angular width. Why would you compare distances in the sky by angular width instead of miles or some other linear measure of distance?

When the stars are out, look at the two top bowl stars in the Big Dipper. The angular distance between them is about 10 degrees as viewed from our position on Earth. Hold up your fist at arm's length between the two stars. It should fill the space. If it does, then you have a convenient measure for 10 degrees of angular distance. Your pinky at arm's length is a measure of about 1 degree. You will find that the Moon is 1/2 degree in angular width.

CHALLENGE 3: Measure the angular distance between two points on the horizon with your fist.

EXAMPLE 4: Slope on the ground is usually measured in percent. A highway grade that rises 1 foot in 100 feet has a slope of 1 per 100 or 1%. Freeways seldom have slopes greater than about a 5% grade.

CHALLENGE 4: What is a 100% slope expressed in degrees of angle? Make a drawing.