College Prep Essential Math Chapter 6: Measurements

CHAPTER 6: MEASUREMENTS Chapter Objectives By the end of this chapter, students should be able to: Identify terms used in measurement in both metric and U.S. customary systems Distinguish between 1, 2, and 3 dimensional measures. Convert U.S. customary unit measures. Convert between U.S. and metric systems. Solve applications of unit measurements.

Contents CHAPTER 6: MEASUREMENTS ...... 1 SECTION 6.1 MEASUREMENT ...... 2 A. U.S. CUSTOMARY SYSTEM ...... 2 I. Weight ...... 2 II. Length...... 3 III. Area ...... 5 IV. Volume ...... 6 B. METRIC SYSTEM ...... 9 I. Weight ...... 9 II. Length...... 9 C. REVIEWING U.S. CUSTOMARY SYSTEM AND METRIC SYSTEM ...... 12 EXERCISES...... 13 SECTION 6.2 CONVERSIONS ...... 14 A. CONVERT U.S. CUSTOMARY UNITS ...... 14 B. CONVERT METRIC UNITS ...... 17 C. CONVERT BETWEEN U.S. AND METRIC SYSTEMS ...... 21 EXERCISES...... 24 CHAPTER REVIEW ...... Error! Bookmark not defined.

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SECTION 6.1 MEASUREMENT

We can measure different attributes of an object. Think about a desk. You can measure how high it is off the ground, how wide the seat is, the area of the desktop, and the space it takes up in the room. You can measure many things like length, area, volume, and weight. In the United States, two main systems of measurement are used: the metric system and the U.S. customary measurement system. The U.S. Customary System is derived from the British system of measure and will be familiar to you. The Metric system is more commonly used around the world. A. U.S. CUSTOMARY SYSTEM

I. Weight When you mention how heavy or light an object is, you are referring to its weight. The tables below list common units of measurement for weight for the U.S. Customary System.

U.S. Customary Measurement

Ounces (표푧)

Pounds (푙푏)

Tons (푡)

The table below describes a unit of weight and provides an example to illustrate the size of the unit of measurement.

Ton Some species of whales can reach weights of up to 200 tons.

Meat is a product that is usually sold by the pound. One pound of beef Pound makes about four hamburger patties.

Ounces are used to measure lighter objects. A stack of 11 pennies is Ounces equal to about one ounce.

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Media Lesson U.S. Customary Units: Weight (Duration 3:39)

View the video lesson, take notes below.

Unit Items that could be measured in this unit.

Pounds

Ounces

Tons

It is important to note there are different types of tons. You might have heard about the small ton and the long ton. The small ton refers to the U.S. Customary ton. The long ton is the British ton. The U.S. Customary ton is 2,000 pounds while the British ton is 2,240 pounds. The metric ton is called the tonne. It is 1,000 kilograms. In this chapter we will use the small ton.

YOU TRY: a) Determine what units would be appropriate to use to measure the following weights.

Item U.S. Customary Unit

The weight of an elephant

The weight of bananas The weight of a lipstick

II. Length

Length can be thought of as the distance between two points. The table below lists common units of measurement of length for the U.S. Customary System.

U.S. Customary Measurement Inches (𝑖푛) Foot (푓푡) Yard (푦푑) Mile (푚)

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The table below describes a unit of length and provides an example to illustrate the size of the unit of measurement.

Some people donate their hair to be made into wigs for cancer patients who Inches have lost hair as a result of treatment. One company requires hair donation to be at least 8 inches long. Feet The length of your bed, couch, and table can be measured in feet.

Soccer fields vary in their size. An official field can be any length between Yards 100 and 130 yards.

Miles A marathon is 26.2 miles long.

Media Lesson U.S. Customary Units: Distance (Duration 5:10)

View the video lesson, take notes below.

Unit Items that could be measured in this unit.

Feet

Inches

Yard

Mile

YOU TRY: b) Determine what units would be appropriate to measure the following lengths.

Item U.S. Customary Unit

The distance from home to campus

The height of a water bottle

The length of an ant

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III. Area Area is the amount of space within the boundaries of a 2-dimensional shape. The table below lists common units of measurement for area.

U.S. Customary Measurement

square inches (풊풏ퟐ) (a square that has 1-inch long sides)

square feet (풇풕ퟐ) (a square that has 1-foot long sides)

square yard (풚풅ퟐ) (a square that has 1-yard long sides)

The table below describes a unit of area and provides an example to illustrate the size of the unit of measurement.

A quarter is about 1 inch wide. Imagine drawing a square around a Square inch quarter. That square would be about the size of 1 square inch.

Square feet The front of your school binder has an area close to 1 square foot.

Poster papers are 2 feet by 3 feet. They are a little less than 1 square Square yard yard.

The unit you use to measure area depends on the unit you would use to measure the length of your object. A tile has sides you would measure in inches, so its area would be measured in square inches. YOU TRY:

c) What units would be appropriate to use to measure the following?

Item U.S. Customary Unit

The floor of your living room

The area of a sheet of paper

The area of a post-it note

The lot size of your home

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IV. Volume Volume is the space taken up by a 3-dimensional object. The table below lists common units of measurement.

U.S. Customary Measurement Teaspoon Table spoon

Fluid Ounce (푓푙 표푧)

Cup (푐)

Pint (푝푡)

Quart (푞푡)

Gallon (푔) Cubic inch (𝑖푛3) (a cube that has 1-inch long sides)

The table below describes each unit of volume and provides an example to illustrate the size of the unit of measurement.

Fluid Ounce The amount of liquid medicine is often measured in fluid ounces.

Cup The volume of a measuring cup is one cup.

Pint Cartons of ice cream are often measured in pints.

Quart You often see quarts of milk sold in supermarkets.

Gallon Gas is sold by gallons.

Cubic Inch This is about the size of a stack of 15 quarters.

Media Lesson U.S. Customary Units: Fluid Volume (Duration 7:20)

View the video lesson, take notes below. Unit Items that could be measured in this unit. Teaspoon Table spoon Fluid Ounce

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Cups

Pints

Quarts

Gallon

Example: The table below lists appropriate units of measurement for each item. Item U.S. Customary Unit The amount of water in a bathtub gallons The amount of coffee in a cup cups and cubic inches The amount of fluid in a drop of water ounces

YOU TRY:

a) What units would be appropriate to measure the following?

Item U.S. Customary Unit The amount of water in a pool The amount of water in a bottle The amount of fluid in an allergy shot

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Capacity 1 gallon = 4 quarts = 16 cups 1 quart = 2 pints = 4 cups 1 pint = 2 cups 1 cup = 8 fluid ounces

3 teaspoon = 1 tablespoon

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B. METRIC SYSTEM The metric system is more commonly used around the world. You may have heard of kilograms, centimeters, millimeters, and liters. These are units of the metric system. The metric system uses three basic units: meters, grams, and liters. The meter is the basic unit of length. The gram is the basic unit of weight. The liter is the basic unit of volume. I. Weight Common units of weight of the Metric System are kilograms, grams, and milligrams.

Media Lesson Metric System: Units of Weight (Duration 4:25)

View the video lesson, take notes below.

Unit Items that could be measured in this unit. Gram

Kilogram

Milligram

YOU TRY:

a) Determine what units would be appropriate to use to measure the following weights. Item Metric Unit The weight of an elephant The weight of bananas The weight of a lipstick

II. Length Common units of length of the Metric System are kilometers, meters, centimeters, and millimeters.

Media Lesson Metric System: Units of Distance(Duration 6:55)

View the video lesson, take notes below. Unit Items that could be measured in this unit. Meter

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Kilometer

Millimeter

YOU TRY:

b) Determine what units would be appropriate to use to measure the following lengths.

Item Metric Unit The distance from home to campus The height of a water bottle The length of an ant

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Area Like the U.S. Customary System units of area in the metric system are square units. Common units of area of the Metric System are square meters and square centimeters. If you would measure the length of an object in centimeters, then it’s area would be measured in square centimeters.

Example: Item Metric Unit Surface of a television square centimeters Area of a fence square meters The cover of a book square centimeters

YOU TRY:

c) What units would be appropriate to use to measure the following?

Item U.S. Customary Unit The floor of your living room The area of a sheet of paper The area of a post-it note The lot size of your home

Volume Common units of volume of the metric system are kiloliters, liters, and milliliters. Like the U.S. Customary System volume can also be measured in cubic units. You could measure an object in cubic centimeters and cubic meters.

Media Lesson Metric System: Units of Volume (Duration 4:13)

View the video lesson, take notes below. Unit Items that could be measured in this unit. Liter

Milliliter

Kiloliter

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YOU TRY:

d) What units would be appropriate to measure the following volumes?

Item Metric Unit The amount of water in a pool The amount of water in a bottle The amount of fluid in an allergy shot

C. REVIEWING U.S. CUSTOMARY SYSTEM AND METRIC SYSTEM The following table compares units of the Metric System to the U.S. Customary System. Centimeter A centimeter is a little less than half an inch. Meter A meter is slightly larger than a yard. Kilometer A kilometer is about one and half of a mile. Gram 28 grams is about the same as 1 ounce. Kilogram A kilogram is a little more than 2 pounds. Liter A liter is slightly larger than a quart. 4 liters is a little more than 1 gallon.

Media Lesson U.S. Customary and Metric Units |Ratios, Proportions, Unit, and Rates (Duration 5:09) View the video lesson, take notes below. Sort the following units of measurement into two categories: U.S. Customary Units and Metric Units. Then indicate whether each unit measures length, weight, or volume. Liter Centimeter Gallon Cup Decigram Meter Millimeter Pound Foot Inch Kilogram Ounce Centiliter Yard

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EXERCISES 1) Determine what units would be appropriate to use to measure the following lengths. Item U.S. Customary Unit Metric Unit The length of a football field The height of a textbook The length of an eyelash

2) For each of the following items, state which U.S. Customary units and which metric units would be the most appropriate. a. The weight of a dog b. The distance from Phoenix to Lost Angeles c. The area of foundation of a house

d. The length of a lady bug

3) Determine whether the following describe length, area, or volume. a. The diagonal of a television b. The screen of a television c. The bottom surface of a mug d. The amount of coffee that a mug will hold

4) Which of the following sets of three units are all metric measurements of length? a. Inch, foot, yard b. Kilometer, centimeter, millimeter c. Kilogram, gram, milligram d. Kilometer, foot, meter

5) Which of the following sets of three units are all measurements of volume? a. Inch, foot, yard b. square inches, square feet, square yards c. cubic inch, cubic feet, liters d. gram, pound, ounce

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SECTION 6.2 CONVERSIONS

A. CONVERT U.S. CUSTOMARY UNITS Sometimes we know a measurement in a particular unit, but we are interested in the value of the measurement in a different unit. Suppose we know that the length of a table is 7ft, but we want to know what the value of the measurement is in inches. The process of converting a measurement from one unit to another is called unit conversion. The table below shows some equivalencies between units.

U.S. System Units Conversions Length Volume 1 ft = 12 in 1 c = 8 oz 1 yd = 3 ft 1 pt = 2 c 1 mi = 5,280 ft 1 qt = 2 pt 1 qt = 32 oz 1 gal = 4 qt

Weight Time 1 lb = 16 oz 1 min = 60 sec 1 ton = 2,000 lb 1 hr = 60 min 1 day = 24 hr 1 wk = 7 days 1 yr = 52 wk 1 yr = 12 mo 1 yr = 365 days

To convert between units of measure we will be multiplying by a conversion factor written in fraction form. Conversion factors come from the equivalencies in the table. We write our conversion factors with the units we want to convert in the denominator. When we multiply by this conversion factor the units divide out and leave us with a new unit. Example: Convert 7ft to inches Looking at our table we see 1 푓푡 = 12 𝑖푛 We write this equivalency in fraction form. Since we are converting 7 ft, feet will be in the denominator. 12 𝑖푛푐ℎ푒푠

1 푓표표푡 Once you have a conversion factor you will multiply 7 ft by it. Note, you can write 7ft as a fraction. 7 푓푒푒푡 12 𝑖푛푐ℎ푒푠 84 𝑖푛푐ℎ푒푠 × = = 84 𝑖푛푐ℎ푒푠 1 1 푓표표푡 1

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7 feet is multiplied by the numerator of 12 inches and is then divided by the denominator of 1 foot. The unit of feet is divided out and we are left with inches.

Media Lesson Single Step Unit Conversion (US Standard – Unit Fractions) (Duration 4:18) View the video lesson, take notes and complete the problems below. a. 4 lb = ______oz

b. 10 yd = ______ft

c. 2.4 pt = ______c

Media Lesson Converting Between Different Units of Weight in Standard or American System (Duration 2:32) View the video lesson, take notes and complete the problems below. Convert the measurement. Round to the nearest tenth if needed. 240 oz = ______lbs

3 T = ______lbs = ______oz

YOU TRY:

Find the conversions below. e) 4 pounds to ounces f) 10 yards to feet

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Sometimes converting from one unit to another may have several steps. We will need to use several conversion factors. Example: Convert 66 inches to yards. We do not have an equivalency between inches and yards. We will first need to convert inches to feet and then convert our feet to yards. 1 푓푡 = 12 𝑖푛푐ℎ푒푠 Use these equivalencies for the conversion factors. 1 푦푑 = 3 푓푒푒푡

66 𝑖푛 1 푓푡 3 푦푑 Cancel out units. × × 1 12 𝑖푛 1 푓푡

66 × 1 × 3 푦푑 Multiply.

1 × 12 × 1 198 푦푑 Simplify.

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16.5 푦푎푟푑푠

66 inches is 16.5 yards.

Media Lesson Multi-Step Conversions (US Standard – Unit Fractions) (Duration 4:48)

View the video lesson, take notes and complete the problems below. Some conversions require more than one step. See how the single-step conversion process is expanded in each of the following examples. a) How many minutes are in a week?

b) Bryan needs 10 cups of fruit juice to make Sangria. How many quarts of juice should he buy at the grocery store?

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YOU TRY:

Find the conversions below. g) 9 weeks to minutes

h) 250,00 miles to yards

B. CONVERT METRIC UNITS

Convert Using Prefixes In the metric system, units are related by powers of 10. The root names reflect this relation. For example the basic unit for measuring length is the meter. One kilometer is 1,000 meters. The prefix kilo- means thousand. One centimeter is 1 of a meter. The 100 prefix centi - means one-hundredth. This makes conversions easy in the metric system. The table below shows the relationship between different units.

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Metric Chart KILO HECTO DEKA DECI CENTI MILLI 1,000 100 × 푏푎푠푒 10 × 푏푎푠푒 Base . 1 × 푏푎푠푒 . 01 × 푏푎푠푒 . 001 × 푏푎푠푒 Unit × 푏푎푠푒 Kilometer Hectometer Dekameter Meter Decimeter Centimeter Millimeter (km) (hm) (dam) (m) (dm) (cm) (mm) Kiloliter Hectoliter Dekaliter Liter (l) Deciliter Centiliter Milliliter (kl) (hl) (dal) (dl) (cl) (ml) Kilogram Hectogram Dekagram Gram (g) Decigram Centigram Milligram (kg) (hg) (dag) (dg) (cg) (mg)

It is better to remember the relationships between the prefixes and the base unit. These relationships will help us convert to different units just by moving our decimal point. Metric Chart KILO HECTO DEKA DECI CENTI MILLI 1,000 100 10 × 푏푎푠푒 Base Unit . 1 × 푏푎푠푒 . 01 . 001 × 푏푎푠푒 × 푏푎푠푒 × 푏푎푠푒 × 푏푎푠푒

Example: Nick ran a 10-kilometer race. How many meters did he run? Looking at the table we see km is 3 spaces away from our base unit of meters.

We will move the decimal point by 3 decimal spaces in the same direction to convert to meters.

Nick ran 10,000 meters.

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Media Lesson Examples: Converting Between Metric Units (Duration 4:30)

View the video lesson, take notes and complete the problems below. Metric Chart KILO HECTO DEKA DECI CENTI MILLI 1,000 100 10 Base Unit . 1 . 01 . 001

Convert the measurement. 4 m = ______cm

2.5 mm = ______m

250 ml = ______kl

1.25 kg = ______g

YOU TRY:

Convert the metric units.

i) 4,200 g = ______mg

j) 320 ml ______l

k) 1.2 kg = ______mg

Convert Using Conversion Factors The table below lists some equivalencies. We can use these equivalencies to convert between metric units like we did for the U.S. system of measurement. Metric Conversions 1 centimeter = 10 millimeters 1 meter = 100 centimeters 1 kilometer = 1,000 meters

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Let us revisit the example from before. Example: Nick ran a 10-kilometer race. How many meters did he run? Based on the table 1 kilometer is 1,000 meters. 1 푘푚 = 1,000 푚 Use this equivalency for the conversion factor.

10 푘푚 1,000 푚 Cancel out units. × 1 1 푘푚

10 × 1,000 푚 Multiply.

1

10,000 푚 Simplify.

1

10,000 푚

Nick rand 10,000 meters.

Media Lesson Introduction to Metric Conversions (Duration 8:30)

View the video lesson, take notes and complete the problems below. a. 4,200 g = ______mg

b. 45 cm = ______m

c. 7,236,137 ml = ______kl

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YOU TRY:

Use conversion factors to convert the metric units. l) 3,200 g to kilograms

m) Herman bought a rug 2.5 meters in length. How long is that in centimeters?

C. CONVERT BETWEEN U.S. AND METRIC SYSTEMS Although the U.S. relies heavily on our standard measurement system, we do use some metric units. Therefore, we need to know how to move back and forth between the systems. We will use conversion factors to achieve this process. A table of some common U.S./Metric conversions are below. Note that many of these conversions are approximations. Length Weight Volume 1 mi = 1.61 km 1 kg = 2.2 lb 1 L = 1.1 qt 1 yd = 0.9 m 1 g = 0.04 oz 1 gal = 3.8 L 1 in = 2.54 cm 1 L = 2.1 pt 0.621 mi = 1 km 0.424 kg = 1 lb 1 cubic yd = 0.76 cubic m 1.094 yd = 1 m 1 oz = 28.3 g 1 cubic in = 16.4 cubic cm 0.394 in = 1 cm

Like before we will convert between units using conversion factors.

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Media Lesson Examples: Convert Lengths Between the Standard and Metric System (Duration 2:36) View the video lesson, take notes and complete the problems below. Convert the measurement. Round to the nearest tenth if needed. 16 in ≈ ______cm

25 m ≈ ______ft

Media Lesson Examples: Convert Weight Between the Standard and Metric System (Duration 3:21) View the video lesson, take notes and complete the problems below.

Convert the measurement. Round to the nearest tenth if needed. 180 푙푏푠 ≈ ______푘푔

12 푘푔 ≈ ______표푧

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YOU TRY:

Find the conversions below. Round to the hundredths place.

n) 10 kilometers to miles o) Express 2 liters in terms of quarts.

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EXERCISES In the following exercises, convert the U.S. Customary Units. 1) A park bench is 6 feet long. 2) A floor tile is 2 feet wide. Convert Convert the length to inches. the width to inches.

3) Carson is 45 inches tall. Convert 4) Jon is 6 feet 4 inches tall. Convert his height to feet. his height to inches.

1 5) An empty bus weighs 35,000 6) Rocco waited 1 hours for his pounds. Convert the weight to 2 appointment. Convert the time to tons. seconds.

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7) Crista will serve 20 cups of juice at 8) April’s dog, Beans, weighs 8 her son’s party. Convert the pounds. Convert his weight to volume to gallons. ounces.

In the following exercises, convert the Metric Units. Use the method you prefer. 9) Ghalib ran 5 kilometers. Convert 10) A bottle of wine contained 750 the length to meters. milliliters. Convert this to liters.

11) The width of the wading pool is 12) Mount Whitney is 3,072 meters 2.45 meters. Convert the width to tall. Convert the height to centimeters. kilometers.

13) June’s multivitamin contains 1,500 14) A typical ruby-throated milligrams of calcium. Convert to hummingbird weighs 3 grams. grams. Convert this to milligrams.

15) One serving of gourmet ice ream 16) A bottle of medicine contained has 25 grams of fat. Convert this to 300 milliliters. Convert to liters. milligrams.

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In the following exercises, convert the U.S. and Metric System. Round answers to the hundredths place. 17) Bill is 75 inches tall. Convert his 18) A 5K run is 5 kilometers long. height to centimeters. Convert this length to miles.

19) Jackson’s backpack weighs 15 20) Each American throws out an kilograms. Convert the weight to average of 1,650 pounds of pounds. garbage per year. Convert this weight to kilograms.

21) Ozzie put 14 gallons of gas in his 22) A planter holds 1.5 liters of soil. truck. Convert the volume to liters. Convert this to gallons.

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Online Quiz

Log on to Canvas to take the section quiz

Directions: It is very useful to save your math exercise work and use it as a chapter test review when you study for your chapter test and final. 1) Write each question on the screen down to for your record

2) Solve the problem step by step below each question

3) Double check your work to see whether your answer make sense

4) Enter your answer in the answer box in Canvas. Make sure you click on the “Preview” button to make sure you enter the right format before you submit your answer. If you are not sure how to enter your answer with the correct format, ask your instructor.

5) If you did not answer the question correctly, solve the question again from the beginning below your 1st attempt. Sometimes, it is better to start a problem again from the beginning and compare your steps with your 1st attempt to figure out your mistake.

6) Insert your work at the end of each section in your workbook so that you can use it to study for your chapter test later.

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