SECTION 11.1 – LINEAR MEASUREMENT

History of U.S. Measurement

BOY…THAT’S ONE BIG FOOT!

In ancient times, the body ruled when it came to measuring. The length of a foot, the width of a finger, and the distance of a step were all accepted measurements.

Inch: At first an inch was the width of a man's thumb. In the 14th century, King Edward II of England ruled that 1 inch equal 3 grains of barley placed end to end lengthwise.

Hand: A hand was approximately 5 inches or 5 digits (fingers) across. Today, a hand is 4 inches and is used to measure horses (from the ground to the horse's withers, or shoulder).

Span: A span was the length of the hand stretched out, about 9 inches.

The English system of measurement grew out of the creative way that people measured for themselves. Familiar objects and parts of the body were used as measuring devices. For example, people measured shorter distances on the ground with their feet.

Foot: In ancient times, the foot was 111/42 inches. Today it is 12 inches, the length of the average man's foot.

Yard: A yard was originally the length of a man's belt or girdle, as it was called. In the 12th century, King Henry I of England fixed the yard as the distance from his nose to the thumb of his out-stretched arm. Today it is 36 inches, about the distance from nose to out-stretched arm of a man.

Cubit: In ancient Egypt, a cubit was the distance from the elbow to the fingertips. Today a cubit is 18 inches.

Lick: A Lick was used by the Greeks to measure the distance from the tip of the thumb to the tip of the index finger.

Pace: The ancient Roman soldiers marched in paces, which were the length of a double step, about 5 feet; 1,000 paces was a mile. Today, a pace is the length of one step, 21/2 to 3 feet.

Unfortunately, these creative measuring devices allowed for different measurements to be obtained when different people measured the same items. Eventually, a standard was set so that all measurements represented the same amount for everyone.

Standard Measures of Length

• Inch = 0.083 feet • Foot = 12 inches • Yard = 3 feet or 36 inches • Mile = 5,280 feet or 1,760 yards

Conversions - Units of Length

Sometimes it is necessary to convert from one unit of measure to another similar unit –from feet to inches, feet to yards, yards to feet, etc. The following method is the easiest way to do this:

Look up the equivalent values for the two units and then decide whether to multiply or divide.

The table below shows the standard equivalencies for length.

HMMMM… DO I MULTIPLY? DIVIDE? FIND OUT ON THE NEXT PAGE!

1 foot = 12 inches 1 yard = 3 feet or 36 inches 1 mile = 1760 yards or 5280 feet

CONVERTING FROM A LARGER UNIT TO A SMALLER UNIT

When converting from a larger unit to a smaller unit, you need to multiply.

For example, to convert 8 feet to inches, look at the table of equivalencies to find that 1 foot equals 12 inches. Multiply the number of feet (8) by the number of inches in 1 foot (12) to find that 8 feet = 96 inches.

EXAMPLES:

0.2 miles = _____ feet Answer: 1056 feet Because we are converting from a larger unit to a smaller unit, we must multiply the number of miles given (0.2) by the number of feet in one mile (5280), so that 0.2 x 5280 = 1056 feet. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 18 yards = _____ inches

Answer: 648 inches

Because we are converting from a larger unit to a smaller unit, we must multiply the number of yards given (18) by the number of inches in one yard (36), so that 18 x 36 = 648 inches. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

NOW DO YOU GET IT? GOOD, ‘CAUSE THERE’S MORE!

CONVERTING FROM A SMALLER UNIT TO A LARGER UNIT

When converting from a smaller unit to a larger unit, you need to divide.

For example, to convert 48 feet into yards, look at the table of equivalencies to find that 1yard equals 3 feet. Divide the number of feet (48) by the number of feet in 1 yard (3) to find that 48 feet = 16 yards.

EXAMPLES:

219 feet = _____ yards Answer: 73 yards Because we are converting from a smaller unit to a larger unit, we must divide the number of feet given (219) by the number of feet in one yard (3),

so that ftft

3219 = 73 yards.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 64 inches = _____ yards Answer: 1.78 yards Because we are converting from a smaller unit to a larger unit, we must divide the number of inches given (64) by the number of inches in one yard

(36), so that inin

3664 = 1.78 yards.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

SEE ATTACHED WORKSHEETS FOR MORE PRACTICE PROBLEMS

The Metric System

Used in nearly every country in the world, the Metric System was devised by French scientists in the late 18th century who recognized the need for a standard unit of linear measurement. The goal of this effort was to produce a system that did not rely on miscellaneous separate standards, and to use the decimal system rather than fractions.

Today, the metric system is used for many different types of measurements such as mass, area, volume, density, etc.; however, because the section being covered deals with linear measurement, this lesson will focus on the different units of length in the metric system.

Converting Metric Units

The metric system has prefix modifiers that are multiples of 10.

• A kilometer (km) is 1000 meters

• A hectometer (hm) is 100 meters

• A decameter (dam) is 10 meters

• A meter (m) is the basic unit of length

• A decimeter (dm) is 1/10 meter

• A centimeter (cm) is 1/100 meter

• A millimeter (mm) is 1/1000 meter

As we move down the units, the next unit is one tenth as long. As we move upward, each unit is 10 times as long.

It is possible to convert units by using the chart below. An easy way to remember the set-up of the chart is by using the first letter of each unit in a sentence such as “King Henry Danced Merrily Down Center Main”. Conversions among metric lengths are accompanied by multiplying or dividing by powers of ten as shown on the “steps”, however, there is a much simpler way of figuring conversions, but you must learn the chart below. 10 3 kilo 10 2 hecto 10 1 deka 10 meter 10 1− deci 10 2− 10 3−

centi milli

Step 1: What is being converted? Go to that “stair-step” on the chart. Step 2: What are you converting to? Move up or down the stair-steps. Step 3: Count the number of stair-steps you moved. Step 4: If you moved DOWN the stair-steps, move the decimal point to

the RIGHT that many steps. If you moved UP the stair-steps, move the decimal point to the

LEFT that many steps.

EXAMPLES: Using the chart above, convert the following: 1.4 km = _____ m Answer: 1400 m Because we moved 3 steps down from km to m, we moved the decimal point 3 spaces to the right. 285 mm = _____ m Answer: 0.285 m Because we moved 3 steps up from mm to m, we moved the decimal point 3 spaces to the left. 0.03 km = _____cm Answer: 3000 cm Because we moved 5 steps down from km to cm, we moved the decimal point 5 spaces to the right.

SEE ATTACHED WORKSHEETS FOR MORE PRACTICE PROBLEMS

Distance Around a Plane Figure The perimeter of a simple closed curve is the distance around the figure. In a regular polygon, it is the sum of the lengths of its sides.

To find the perimeter of a regular polygon like in the examples shown below, add together the lengths of all of the sides of the figure.

Examples: 4 cm 4 cm 4 cm Perimeter: 4cm + 4cm + 4cm = 12cm

8 cm 3 cm 3 cm 8 cm Perimeter: 3cm + 8cm + 8cm + 3cm = 22cm 6 cm 7 cm 7 cm 3 cm Perimeter: 7cm + 6cm + 3cm + 7cm = 23cm 2cm 2cm 2cm 2cm 2cm 2cm Perimeter: 2cm + 2cm + 2cm + 2cm + 2cm + 2cm = 12cm

See attached worksheets for

practice perimeter problems

Circumference of a Circle The circumference of a circle is the distance around a circle, also known as the perimeter of a circle. Before determining the circumference of a circle, either the radius or the diameter of the circle must be known. The radius of a circle is the distance of the center of a circle to any point on the circumference.

The diameter of a circle is a straight line passing through the center of a circle that touches both sides of the circumference.

About π (pi): The ancient Greeks discovered that if they divided the circumference of any circle by the length of its diameter, they always came up with approximately the same number (3.14). Today, the ratio of circumference C to diameter d is symbolized as π (pi).

The approximate values of π (pi) are 722 , 3

71 , or 3.14

Formulas for determining the circumference of a circle: If radius is given, use C = 2πr (circumference = 2 times pi times radius) If diameter is given, use C = πd (circumference = pi times distance) EXAMPLES: Problem: Find the circumference of a circle if the radius is 2m. Solution: C = 2πr = (2)(3.14)(2m) = 12.56 m

Problem: Find the circumference of a circle if the diameter is 6cm Solution: C = πd = (3.14)(6cm) = 18.84 cm Problem: Find the circumference of a circle if the radius is 3.5cm Solution: C = 2πr = (2)(3.14)(3.5cm) = 21.98 cm

SEE ATTACHED WORKSHEETS FOR MORE PRACTICE PROBLEMS

THE END…FINALLY!

Circumference of Circles

Find the Circumference for each. Assume = 3.14

1.

g = 20 yd ________________

2.

m = 46 cm ________________

3.

s = 11 ft ________________

4.

e = 4 cm ________________

5.

s = 10 cm ________________

6.

m = 83 m ________________

7.

g = 13.9 in ________________

8.

e = 26.8 mi ________________

Perimeter Find the perimeter.

1.

e = 7 m f = 13 m _________________

2.

v = 3 ft t = 7 ft r = 12 ft s = t ________________

3.

The side d of this square is 39 yd ________________

4.

a = 3 ft c = 5 ft b = c ________________

5.

All sides equal 7 yd ________________

6.

m = 14 yd All sides are equal ________________

7.

v = 4 ft t = 6 ft r = 12 ft s = t ________________

8.

The side d of this square is 21 m ________________

9.

All sides equal 11 cm ________________

10.

e = 9 m f = 17 m ________________

Metric Conversions Convert the following.

1. 400009 cm = ___________ km

2. 0.00069 km = ____________ mm

3. 828 m = _____________mm

4. 0.125 m = _____________cm

5. 276 km = _____________m

6. 0.26678 m = ____________km

7. 9.651 cm = ____________mm

8. 84322 dm = ___________ m

9. 6.00042 dam = ___________cm

10. 534.255 cm = ____________ mm

11. 00254.32 mm = ____________ hm

12. 82,800 cm = ____________ m

13. 426,000,000 mm = ____________ km

14. 0.864 m = ____________ mm

15. 8 cm = ____________m

Standard Conversions Convert the following.

1. 3.5 miles = ____________ feet

2. 9 yards = ____________ inches

3. 84 inches = ____________ feet

4. 120 feet = ____________ inches

5. 4.2 miles = ____________ yards

6. 24 yards = ____________ feet

7. 6.2 feet = ____________ inches

8. 1.2 miles = ____________ feet

9. 15840 feet = ____________ miles

10. 255 inches = ___________ feet

11. 42 feet = ____________ yards

12. 6 miles = ____________ yards

13. 32 yards = ____________ inches

14. 44 yards = ____________ miles

15. 72 feet = ____________ inches

Answers to Worksheets

Circumference Metric Conversions

1. 62.8 yd 1. 4.00009 km 2. 288.88 cm 2. 690 mm 3. 69.08 ft 3. 828000 mm 4. 25.12 cm 4. 12.5 cm 5. 62.8 cm 5. 276000 m 6. 521.24 mi 6. 0.00026678 km 7. 43.646 in 7. 96.51 mm 8. 168.304 mi 8. 843220 m 9. 6000.42 cm

10. 53425.5 mm 11. .0025432 hm 12. 828 m 13. 426 km 14. 0864 mm 15. .008 m Perimeter Standard Conversions

1. 40 m 1. 18480 ft 2. 29 ft 2. 324 in 3. 156 yd 3. 7 ft 4. 13 ft 4. 1440 in 5. 21 yd 5. 73.92 yd 6. 84 yd 6. 72 ft 7. 34 ft 7. 74.4 in 8. 84 m 8. 6336 ft 9. 33 cm 9. 3 mi 10. 52m 10. 21.25 ft 11. 14 yd

12. 10560 yd 13. 1152 in

14. 0.025 mi 15. 864 in

NOTES AND ACTIVITES FOR THIS LESSON WERE PREPARED AND PRESENTED BY:

CHRISTI L. ALARCON

JODIE CORDOVA LYZA DE HOYOS

MARIA ELENA GUTIERREZ REBECCA IDROGO

Fundamentals of Math II

March 25, 2004