Measurement.pdf - Study Unit Measurement Reviewed by...

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Unformatted text preview: Study Unit Measurement Reviewed by Richard Dempsey ABOUT THE REVIEWER Richard is a high school mathematics teacher in the Scranton School District located in Scranton, Pennsylvania. He holds a Bachelor of Science degree in secondary education from Bloomsburg University and a Master of Science degree in educational administration from the University of Scranton. He is certified to teach mathematics in grades seven through twelve and to hold a principal’s position in either an elementary or secondary school setting. He resides in Scranton with his wife and four children. © PENN FOSTER, INC. 2015 INTRODUCTION CONTENTS INTRODUCTION 1 ASSIGNMENT 1: MEASUREMENT 7 A Brief History of Measurement 7 The English System vs. the Metric System 7 ASSIGNMENT 2: THE ENGLISH SYSTEM 10 Length and Area 10 Converting Measurements 11 Adding and Subtracting Measurements 17 Borrowing 20 Perimeter 23 Multiplying Measurements 26 Dividing Measurements 29 Weight 34 Volume and Capacity 40 Liquid Measure 45 Dry Measure 49 ASSIGNMENT 3: TEMPERATURE 53 The English System 53 The Metric System 56 Converting Temperatures 57 ASSIGNMENT 4: TIME, NUMBER, AND MONEY 64 Time 64 Elapsed Time 68 Military Time 70 Number 72 Money 74 © PENN FOSTER, INC. 2015 MEASUREMENT Introduction PAGE 1 INTRODUCTION CONTENTS ASSIGNMENT 5: THE METRIC SYSTEM 80 Distance and Length 82 Converting Metric Measurements 83 Operations with Metric Measurements 85 Weight 88 Volume and Capacity 96 ASSIGNMENT 6: CONVERTING BETWEEN THE ENGLISH AND METRIC S ­ YSTEMS 103 KEY POINTS 110 MIND TEASER ANSWERS 113 GLOSSARY 115 © PENN FOSTER, INC. 2015 MEASUREMENT Introduction PAGE 2 INTRODUCTION NAVIGATING THIS COURSE Your study materials are designed so that you can take ownership of your educational goals and schedule and complete your coursework whenever and wherever you can. At Penn Foster, you won’t study at the same pace as other students in the same program, but you can reach out to your peers using the Community. Your study materials are broken down into small chunks that are easy to handle, and each section is tied directly to the learning outcomes and objectives. Materials include summaries, reviews, self-checks, and activities to help you master them! You’ll find it easiest to study if you follow the plan outlined below. 1 Look over the contents page to get a general idea of what you’re going to learn in this study unit. 2 Quickly read the pages in Assignment 1. This process is called skimming. 3 Return to the beginning of Assignment 1. This time read more closely and pay careful attention to what you’re reading. Focus on main concepts and definitions. 4 Complete all exercises as you come to them and check your answers with those provided. 5 When you finish reading Assignment 1, complete any exercises at the end of the assignment. The exercises aren’t graded; they’re designed to help you test yourself to make sure you understand what you’ve read. 6 Complete each assignment in this manner; then review the material in preparation for the examination. 7 When you feel confident that you understand the material, complete the examination for this study unit. © PENN FOSTER, INC. 2015 MEASUREMENT Introduction PAGE 3 INTRODUCTION COURSE MATERIALS Your General Math 1 course includes the materials listed below: OO OO This study unit, Measurement, which includes  All of your assigned readings   elf-checks and other exercises that allow you to measure how well you S understand your course material  he study units Whole Numbers; Fractions; Decimals; Ratio, Proportion, and T Percent; and General Review © PENN FOSTER, INC. 2015 MEASUREMENT Introduction PAGE 4 INTRODUCTION UNIT OBJECTIVES What will you get from this study unit? This study unit will introduce you to both the English and the metric systems of measurement. You’ll learn how to solve problems involving length, volume, weight, time, money, and temperature. Most of us have been measuring things since we were young. Indeed, measuring things is a very important part of life. We use many tools every day to measure things: OO Teaspoons, tablespoons, and measuring cups for cooking OO Bathroom scales for measuring weight OO Mileage gauges on cars that measure distance traveled OO Gasoline gauges on cars that measure the amount of gas used OO Maps that record the distances between towns OO Rulers, yardsticks, and tape measures that measure lengths OO Timers, stopwatches, and clocks that measure time OO Calendars that measure days, months, and years OO Thermometers that measure temperature © PENN FOSTER, INC. 2015 MEASUREMENT Introduction PAGE 5 INTRODUCTION UNIT OBJECTIVES So far in your General Math 1 course, you’ve learned about whole numbers, fractions, and decimals. You’ve learned how to add, subtract, multiply, and divide all these kinds of numbers. Now, in this study unit, you’re going to use your skills in arithmetic to solve practical problems you’ll see in everyday life or on your job. When you complete this study unit, you’ll be able to 1 Convert between various types of measurements 2 Identify units of measure in the English and metric systems 3 Solve real world problems involving measurements using all ­operations © PENN FOSTER, INC. 2015 MEASUREMENT Introduction PAGE 6 ASSIGNMENT 1 MEASUREMENT ASSIGNMENT OBJECTIVES When you complete Assignment 1, you’ll be able to 1.1 Define conversion factor 1.2 Organize given measurements into either the English or metric system 1.3 Define the terms length and distance A BRIEF HISTORY OF MEASUREMENT Just about everyone has measured things during their lifetime. Every time you’ve weighed yourself, used a ruler or tape measure, checked the mileage on your car, or made change in a store, you’ve used measurements. This study unit will help you review your knowledge of measurement and help you improve your skills in converting measurements and working with the metric system. The word measure means to compare an object against an established standard. At one time in history, measurements weren’t standard. An inch, for example, was considered to be the distance between the first knuckle and the tip of a person’s thumb—but everyone’s thumb is different! An ounce was known as the weight of a certain number of pennies, but the recognized number varied from town to town. Naturally, in the marketplace, there was much room for deception and cheating in measurement! People could never be sure that they were getting what they’d traded for, whether the item was a yard of fabric or a pound of meat. For this reason, many years ago, measurements in Europe and America were standardized. Laws were passed setting the value of different units of measure such as miles, feet, acres, pounds, and gallons. These measures now stood for exactly the same amounts in all areas of the country. In this way, customers were assured of getting exactly what they’d paid for. THE ENGLISH SYSTEM VS. THE METRIC SYSTEM The English system of measurement has been used for many years in the United States, Great Britain, and Canada. The main units of measurement in the English system are the pound, the yard, and the gallon. Although the English system is still widely used in the United States, England and Canada have begun using the metric system. This is © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 1 PAGE 7 because the English system is complicated and has so many different units of measure. There’s no easy way to work with the many different units of measurement without referring to conversion charts or relying on memory. The metric system was developed more than 400 years ago by Simon Stevin, a mathematician. His system of measurement is based entirely on the number 10. In the English system, there are 12 inches in a foot, 3 feet in a yard, and 5,280 feet in a mile. In the metric system, it’s unnecessary to memorize all these strange numbers. Every measurement is a multiple or product of 10. For example, there are 10 millimeters in a centimeter, 100 centimeters in a meter, and 1,000 meters in a kilometer. This makes it much easier and faster to work with metric measures. For this reason, the metric system has been used by most countries of the world for many years. The United States, too, is now in the process of introducing the metric system to its citizens. The metric system is already familiar to many Americans. For example, most products are now labeled with both English and metric measurements. Soft drinks are packaged in two-liter bottles. Dosages of drugs and medicines are often measured in milligrams. Sporting events, such as races, are measured in meters and kilometers. In the auto and electronics industries, many products and parts are measured metrically. It’s possible that someday, the English system of measurement will no longer be the main system of measurement used in the United States. That’s why it’s very important to learn how to use both systems, English and metric. In this study unit, you’ll be learning about both the English system and the metric system in depth. Throughout the book, as each new topic is presented, you’ll be provided with tables and charts to help you solve conversion problems. Try to become familiar with the conversion of the most commonly used units, such as inches, pounds, meters, and so on. However, when you need more complicated conversion factors, it’s safer to simply refer to the tables than to rely on memory. In writing units of measure, it’s often convenient to use abbreviations instead of writing out the names of the units in full. For example, the names inch and inches can be abbreviated to in. Meter and meters are abbreviated to m. Once an abbreviation is introduced, it will be used frequently so that you’ll become familiar with it. © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 1 PAGE 8 ASSIGNMENT 1 DISCOVER MORE DISTANCE AND LENGTH The length of an object is simply its longest measurement. For example, if a room measures 14 feet by 10 feet, the longest measurement, 14 feet, is the length. Distance is a measurement of the space between two objects, such as the number of miles between two cities (Figure 1). LENGTH 1 2 3 DISTANCE 4 5 City A 6 inches City B 495 miles FIGURE 1—It’s important to understand the difference between length and distance. ASSIGNMENT 1 REFLECT AND RESPOND MEASUREMENT 1. The ____________ system of measurement is used commonly in the United States. 2. The ____________ system of measurement is used by most countries of the world. Click here to check your answers. © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 1 PAGE 9 ASSIGNMENT 2 THE ENGLISH SYSTEM ASSIGNMENT OBJECTIVES When you complete Assignment 2, you’ll be able to 2.1. Name and use the conversion factor required to convert from one measurement to another in the English system 2.2 Carry out the rules for addition and subtraction of measurements involving length, distance, and weight 2.3. Develop a strategy to solve real world problems requiring the addition and subtraction of measurements 2.4. Solve for perimeter by adding given dimensions 2.5. Use the rules for multiplying and dividing measurements 2.6. Use the rules for multiplying measurements to find the area and volume of a figure LENGTH AND AREA In the English system, distance and length are most commonly measured in inches, feet, yards, and miles. From your previous math studies, you may remember that the area of a shape equals its length times its width. For example, suppose you were laying carpeting in a room measuring 8 feet by 10 feet. The area of the room equals its length times its width (10 feet times 8 feet, or 80 square feet). So, you would need 80 square feet of carpeting to cover the floor. In the English system, area is measured in acres, square yards, square feet, and square inches. Acres are used to measure large areas such as plots of land, fields, and so on. Square yards and square feet are often used to measure surfaces like floors, walls, and ceilings, and products like carpeting, linoleum, and tile. Square inches are used to measure smaller surfaces, like sheets of paper or advertising space in a newspaper or magazine. © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 2 PAGE 10 The conversion chart below shows how the units of length in the English system are related. LENGTH Conversion Abbreviation 1 mile = 5,280 feet, or 1,760 yards 1 yard = 3 feet, or 36 inches 1 foot = 12 inches mi. yd. ft. or 9 in. or 0 1 1 inch = ⁄12 foot 1 square mile = 640 acres 1 acre = 4,840 square yards, or 43,560 square feet 1 square yard = 1,296 square inches, or 9 square feet 1 square foot = 144 square inches 1 1 square inch = ⁄144 square foot sq. mi. a. or A. sq. yd. sq. ft. sq. in. CONVERTING MEASUREMENTS Whenever you measure an object, you’re comparing it to an established, standard quantity. The standard quantity is called a unit of measure. For example, imagine you’re trying to determine the number of cups of water it will take to fill a one-gallon bottle. To do this, fill cups with water, and count as you pour them into the bottle. You’ll find that it takes sixteen cups to fill the bottle. In this case, the cup is the standard quantity, or the unit of measure. You can say that 16 cups = 1 gallon. Now, imagine that you’re trying to find out how many quarts it will take to fill the gallon bottle. If you fill quart jars with water and count as you pour them into the bottle, you’ll count four. In this case, the standard unit of measure is the quart, and you can say that 4 quarts = 1 gallon. In the preceding example, you performed two measurements—one with cups, and the other with quarts. To find out how many cups and quarts of water would fill the bottle, you had to actually perform the different measurements. In real life, however, you may need to solve measurement problems that would be difficult or impossible to actually perform by hand. For example, you may want to figure out how many gallons of water your hot-water tank holds, how many pounds of gravel you’ll need to cover a driveway, or how many ounces of medicine you can safely give to a child. Luckily, in real-life situations like these, you can make measurements using arithmetic. When solving measurement problems, you’ll often need to ­convert, or change, one unit of measure to another. To do this, you’ll need to find the conversion factor first. The conversion factor is the number of units of a smaller measurement that make up one larger unit of measurement. For example, there are 12 inches (smaller unit) in 1 foot (larger unit). So, in this case, 12 is the conversion factor. Look at the following example. © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 2 PAGE 11 ASSIGNMENT 2 DISCOVER MORE CONVERSION FACTORS Find the conversion factor for each pair of measurement units given. Smaller Unit of Measure Larger Unit of Measure Conversion Factor Inch Yard 36; There are 36 inches in 1 yard. Foot Yard 3; There are 3 feet in 1 yard. Yard Mile 1,760; There are 1,760 yards in 1 mile. Foot Mile 5,280; There are 5,280 feet in 1 mile. To change from one unit of measure to another, follow these steps. Step 1: Read the problem carefully to determine what units of measure you need to convert. Step 2: Find the conversion factor. Consult a conversion table if necessary. Step 3: To change from a smaller unit to a larger unit, divide by the conversion factor. Step 4: To change from a larger unit to a smaller unit, ­multiply by the conversion factor. © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 2 PAGE 12 ASSIGNMENT 2 DISCOVER MORE CONVERTING MEASUREMENTS Convert 3 miles to feet. Solution: Look at the chart on page 11 to find the conversion factor. There are 5,280 feet in 1 mile, so the conversion factor is 5,280. 3 mi. 3 5,280 5 15,840 ft. You’re going from a larger unit to a smaller unit, so multiply by the conversion factor, 5,280. Answer: 3 miles equals 15,840 feet Convert 696 feet to yards. Solution: There are 3 feet in 1 yard, so the conversion factor is 3. 696 ft. 4 3 5 232 yd. Your’re going from smaller units to larger ones, so divide by 3. Answer: 696 feet equals 232 yards. Convert 264 feet to miles. Solution: There are 5,280 feet in a mile, so the conversion factor is 5,280. 5,280 .05 264.00 2264.00 0 ) You’re going from smaller units to larger ones, so divide 264 feet by 5,280. Answer: 264 feet equals .05 of a mile. (Continued) © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 2 PAGE 13 ASSIGNMENT 2 DISCOVER MORE A tailor needs 2 yards 2 feet 10 inches of fabric to make a suit. How many inches is this? Solution: Convert 2 yards to inches; convert 2 feet to inches, then add all the inches together to find the answer. There are 36 inches in a yard, so the first conversion factor is 36. 2 yd. 3 36 5 72 in. You’re going from larger units to smaller ones, so multiply yards by 36. 2 ft. 3 12 5 24 in. There are 12 inches in a foot. Multiply feet by 12 72 24 + 10 106 Add all the converted inches together to find the total. Answer: 2 yards 2 feet 10 inches equals 106 inches. in. in. in. in. Kathy wants to rent office space in a new building. She needs a space measuring at least 20 feet long by 15 feet wide. What’s the area of the space she needs? Solution: Remember, to find the area of a space, multiply the length times the width. Your answer will be given in square feet. 20 ft. 3 15 ft. 100 + 200 300 sq. ft. Multiply length (20 feet) times width (15 feet). Answer: The area of the space Kathy needs is 300 square feet. An oriental carpet measures 12 feet long and 9 feet wide. What’s the area of the carpet in square feet? What’s the area in square yards? Solution: Multiply the length of the carpet times the width to find the area in square feet. Then, convert the area in square feet to square yards. 3 12 ft. 9 ft. 108 sq. ft. 108 sq. ft. 4 9 5 12 sq. yd. First, multiply length (12 feet) by width (9 feet) to find the area of the carpet. The conversion chart on page 4 tells you that there are 9 square feet in 1 square yard. Divide by the conversion factor (9). Answer: The area of the carpet is 108 square feet, or 12 square yards. © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 2 PAGE 14 ASSIGNMENT 2 WORK IT OUT CONVERTING MEASUREMENTS 1. The longest measurement of an object is called its ____________. 2. The measurement of the space between two objects is called its ___________. 3. Convert the following measurements. a. 13 miles to yards__________________________________________ b. 67 square feet to square inches_______________________________ c. 4,800 inches to feet________________________________________ d. 87,120 square yards to acres_________________________________ e. 3 miles to yards___________________________________________ f. 54 inches to yards_________________________________________ g. 12 yards to feet____________________________________________ h. 7 acres to square feet_______________________________________ Click here to check your answers. © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 2 PAGE 15 ASSIGNMENT 2 WORK IT OUT MIND TEASER 1 Mind Teasers are little puzzles you’ll see throughout the study unit. Remember, these puzzles are just for fun and won’t be graded. True measurement isn’t always what your eyes see. Look at the two lines AB and CD. Which line is longer, based on what you see?________________________________ A B C D Look at the hat in the following illustration. Which is longer, the distance from the top of the hat to the base or the distance from one end of the brim to the other?___________ Sometimes the shapes we see can distort what is real. Look at the three triangles below. Which is greater, the distance from A to B or the distance from B to C?______________ A B C Now take out a ruler and measure the lines. Were you correct? Solution on page 113. © PENN FOSTER, INC. 2015 MEASUREMENT Assignment 2 PAGE 16 ADDING AND SUBTRACTING MEASUREMENTS Measurements can be added and subtracted, just like whole numbers. However, you can add and subtract only like units. In other words, you ...
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