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Objectives • Perform calculations involving scientific notation and conversion factors. • Identify the metric and SI units used in science and convert between common metric prefixes. • Compare and contrast accuracy and precision. • Relate the Celsius, Kelvin, and Fahrenheit temperature scales. 1.3 Measurement How old are you? How tall are you? The answers to these questions are measurements. Measurements are important in both science and everyday life. It would be difficult to imagine doing science without any measurements. 1.3 Measurement Using Scientific Notation Why is scientific notation useful? 1.3 Measurement Using Scientific Notation Why is scientific notation useful? Scientists often work with very large or very small numbers. Astronomers estimate there are 200,000,000,000 stars in our galaxy. 1.3 Measurement Using Scientific Notation Scientific notation is a way of expressing a value as the product of a number between 1 and 10 and a power of 10. For example, the speed of light is about 300,000,000 meters per second. In scientific notation, that speed is 3.0 × 108 m/s. The exponent, 8, tells you that the decimal point is really 8 places to the right of the 3. 1.3 Measurement Using Scientific Notation For numbers less than 1 that are written in scientific notation, the exponent is negative. For example, an average snail’s pace is 0.00086 meters per second. In scientific notation, that speed is 8.6 × 10-4 m/s. The negative exponent tells you how many decimals places there are to the left of the 8.6. 1.3 Measurement Using Scientific Notation To multiply numbers written in scientific notation, you multiply the numbers that appear before the multiplication signs and add the exponents. The following example demonstrates how to calculate the distance light travels in 500 seconds. This is about the distance between the sun and Earth. 1.3 Measurement Using Scientific Notation When dividing numbers written in scientific notation, you divide the numbers that appear before the exponential terms and subtract the exponents. The following example demonstrates how to calculate the time it takes light from the sun to reach Earth. 1.3 Measurement Using Scientific Notation Using Scientific Notation A rectangular parking lot has a length of 1.1 × 103 meters and a width of 2.4 × 103 meters. What is the area of the parking lot? 1.3 Measurement Using Scientific Notation Read and Understand What information are you given? 1.3 Measurement Using Scientific Notation Read and Understand What information are you given? 1.3 Measurement Using Scientific Notation Plan and Solve What unknown are you trying to calculate? What formula contains the given quantities and the unknown? Replace each variable with its known value 1.3 Measurement Using Scientific Notation Look Back and Check Is your answer reasonable? Yes, the number calculated is the product of the numbers given, and the units (m2) indicate area. 1.3 Measurement Using Scientific Notation Look Back and Check 1. Perform the following calculations. Express your answers in scientific notation. a. (7.6 × 10-4 m) × (1.5 × 107 m) b. 0.00053 ÷ 29 1.3 Measurement Let’s Practice • Scientific Notation – as a class • On Your Own – Practice Exercises 1.3 Measurement SI Units of Measurement What units do scientists use for their measurements? 1.3 Measurement SI Units of Measurement Scientists use a set of measuring units called SI, or the International System of Units. • SI is an abbreviation for Système International d’Unités. • SI is a revised version of the metric system, originally developed in France in 1791. • Scientists around the world use the same system of measurements so that they can readily interpret one another’s measurements. 1.3 Measurement SI Units of Measurement If you told one of your friends that you had finished an assignment “in five,” it could mean five minutes or five hours. Always express measurements in numbers and units so that their meaning is clear. These students’ temperature measurement will include a number and the unit, °C. 1.3 Measurement SI Units of Measurement Base Units and Derived Units SI is built upon seven metric units, known as base units. • In SI, the base unit for length, or the straightline distance between two points, is the meter (m). • The base unit for mass, or the quantity of matter in an object or sample, is the kilogram (kg). 1.3 Measurement SI Units of Measurement Seven metric base units make up the foundation of SI. 1.3 Measurement SI Units of Measurement Additional SI units, called derived units, are made from combinations of base units. • Volume is the amount of space taken up by an object. • Density is the ratio of an object’s mass to its volume: 1.3 Measurement SI Units of Measurement Specific combinations of SI base units yield derived units. 1.3 Measurement SI Units of Measurement To derive the SI unit for density, you can divide the base unit for mass by the derived unit for volume. Dividing kilograms by cubic meters yields the SI unit for density, kilograms per cubic meter (kg/m3). A bar of gold has more mass per unit volume than a feather, so gold has a greater density than a feather. 1.3 Measurement SI Units of Measurement Metric Prefixes The metric unit is not always a convenient one to use. A metric prefix indicates how many times a unit should be multiplied or divided by 10. Metric Conversions Ladder Method T. Trimpe 2008 http://sciencespot.net/ Ladder Method 1 2 KILO 1000 Units 3 HECTO 100 Units DEKA 10 Units DECI 0.1 Unit Meters Liters Grams How do you use the “ladder” method? CENTI 0.01 Unit MILLI 0.001 Unit 4 km = _________ m 1st – Determine your starting point. Starting Point 2nd – Count the “jumps” to your ending point. How many jumps does it take? 3rd – Move the decimal the same number of jumps in the same direction. Ending Point 4. __. __. __. = 4000 m 1 2 3 Conversion Practice Try these conversions using the ladder method. 1000 mg = _______ g 1 L = _______ mL 160 cm = _______ mm 14 km = _______ m 109 g = _______ kg 250 m = _______ km Compare using <, >, or =. 56 cm 6m 7g 698 mg Metric Conversion Challenge (worksheet) Write the correct abbreviation for each metric unit. 1) Kilogram _____ 4) Milliliter _____ 7) Kilometer _____ 2) Meter _____ 5) Millimeter _____ 8) Centimeter _____ 3) Gram _____ 6) Liter _____ 9) Milligram _____ Try these conversions, using the ladder method. 10) 2000 mg = _______ g 15) 5 L = _______ mL 20) 16 cm = _______ mm 11) 104 km = _______ m 16) 198 g = _______ kg 21) 2500 m = _______ km 12) 480 cm = _____ m 17) 75 mL = _____ L 22) 65 g = _____ mg 13) 5.6 kg = _____ g 18) 50 cm = _____ m 23) 6.3 cm = _____ mm 14) 8 mm = _____ cm 19) 5.6 m = _____ cm 24) 120 mg = _____ g Compare using <, >, or =. 25) 63 cm 26) 536 cm 6m 53.6 dm 27) 5 g 28) 43 mg 508 mg 5g 29) 1,500 mL 30) 3.6 m 1.5 L 36 cm Objectives • Perform calculations involving scientific notation and conversion factors. • Identify the metric and SI units used in science and convert between common metric prefixes. • Compare and contrast accuracy and precision. • Relate the Celsius, Kelvin, and Fahrenheit temperature scales. 1.3 Measurement SI Units of Measurement For example, the time it takes for a computer hard drive to read or write data is in the range of thousandths of a second, such as 0.009 second. Using the prefix milli- (m), you can write 0.009 second as 9 milliseconds, or 9 ms. 1.3 Measurement SI Units of Measurement A conversion factor is a ratio of equivalent measurements used to convert a quantity expressed in one unit to another unit. To convert the height of Mount Everest, 8848 meters, into kilometers, multiply by the conversion factor on the left. 1.3 Measurement SI Units of Measurement To convert 8.848 kilometers back into meters, multiply by the conversion factor on the right. Since you are converting from kilometers to meters, the number should get larger. In this case, the kilometer units cancel, leaving you with meters. Conversion Factors Fractions in which the numerator and denominator are EQUAL quantities expressed in different units Example: 1 in. = 2.54 cm Factors: 1 in. and 2.54 cm 2.54 cm 1 in. How many minutes are in 2.5 hours? Conversion factor 2.5 hr x 60 min 1 hr = 150 min cancel By using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the numbers! Sample Problem • You have $7.25 in your pocket in quarters. How many quarters do you have? 7.25 dollars X 4 quarters 1 dollar = 29 quarters Learning Check A rattlesnake is 2.44 m long. How long is the snake in cm? a) 2440 cm b) 244 cm c) 24.4 cm Solution A rattlesnake is 2.44 m long. How long is the snake in cm? b) 244 cm 2.44 m x 100 cm 1m = 244 cm Learning Check How many seconds are in 1.4 days? Unit plan: days hr min seconds 1.4 days x 24 hr x60 min x 60 s 1 hr 1 min 1 day = 1.2 x 105 s Wait a minute! What is wrong with the following setup? 1.4 day x 1 day 24 hr x 60 min x 60 sec 1 hr 1 min English and Metric Conversions • If you know ONE conversion for each type of measurement, you can convert anything! • I will provide these equalities, but you must be able to use them: – Mass: 454 grams = 1 pound – Length: 2.54 cm = 1 inch – Volume: 0.946 L = 1 quart Steps to Problem Solving Read problem Identify data Make a unit plan from the initial unit to the desired unit (good practice at beginning, not necessary as you get comfortable with this) Select conversion factors Change initial unit to desired unit Cancel units and check Do math on calculator Give an answer using significant figures Dealing with Two Units If your pace on a treadmill is 65 meters per minute, how many seconds will it take for you to walk a distance of 8450 feet? HINT: Always start with the simplest label. You’re looking for seconds, so you can’t start there. 65 m/min has two labels so that’s not very simple. Best STARTING place is 8450 feet! What about Square and Cubic units? • Use the conversion factors you already know, but when you square or cube the unit, don’t forget to cube the number also! • Best way: Square or cube the ENTIRE conversion factor • Example: Convert 4.3 cm3 to mm3 4.3 cm3 ( ) 10 mm 1 cm 3 = 4.3 cm3 103 mm3 13 cm3 = 4300 mm3 Learning Check • A Nalgene water bottle holds 1000 cm3 of dihydrogen monoxide (DHMO). How many cubic decimeters is that? Solution 1000 cm3 1 dm 3 10 cm ( ) = 1 dm3 So, a dm3 is the same as a Liter ! A cm3 is the same as a milliliter. Let’s Practice • Dimensional Analysis Notes • Dimensional Analysis Worksheet 1 • Find the percentage error. Objectives • Compare and contrast accuracy and precision. • Utilize the rules of significant figures. • Relate the Celsius, Kelvin, and Fahrenheit temperature scales. 1.3 Measurement Limits of Measurement How does the precision of measurements affect the precision of scientific calculations? 1.3 Measurement Limits of Measurement Precision Precision is a gauge of how exact a measurement is. Significant figures are all the digits that are known in a measurement, plus the last digit that is estimated. 1.3 Measurement Limits of Measurement The precision of a calculated answer is limited by the least precise measurement used in the calculation. 1.3 Measurement Limits of Measurement A more precise time can be read from the digital clock than can be read from the analog clock. The digital clock is precise to the nearest second, while the analog clock is precise to the nearest minute. Significant Figures ► When using our calculators we must determine the correct answer; our calculators are mindless drones and don’t know the correct answer. ► There are 2 different types of numbers – Exact – Measured ► Exact numbers are infinitely important ► Measured number = they are measured with a measuring device (name all 4) so these numbers have ERROR. ► When you use your calculator your answer can only be as accurate as your worst measurement…Doohoo Chapter Two 53 Exact Numbers An exact number is obtained when you count objects or use a defined relationship. Counting objects are always exact 2 soccer balls 4 pizzas Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches. 54 Learning Check A. Exact numbers are obtained by 1. using a measuring tool 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool 2. counting 3. definition 55 Solution A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool 56 Learning Check Classify each of the following as an exact or a measured number. 1 yard = 3 feet The diameter of a red blood cell is 6 x 10-4 cm. There are 6 hats on the shelf. Gold melts at 1064°C. 57 Solution Classify each of the following as an exact (1) or a measured(2) number. This is a defined relationship. A measuring tool is used to determine length. The number of hats is obtained by counting. A measuring tool is required. 58 Measured Numbers ►Do you see why Measured Numbers have error…you have to make that Guess! ►All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate. ►To indicate the precision of a measurement, the value recorded should use all the digits known with certainty. 59 Note the 4 rules When reading a measured value, all nonzero digits should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not. ► RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, 94.072 g has five significant figures. ► RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four. Chapter Two 60 ► RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m. ► RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point. Chapter Two 61 Practice Rule #1 Zeros 45.8736 6 •All digits count .000239 3 •Leading 0’s don’t .00023900 5 •Trailing 0’s do 48000. 5 •0’s count in decimal form 48000 2 •0’s don’t count w/o decimal 3.982106 4 1.00040 6 •All digits count •0’s between digits count as well as trailing in decimal form Rounding Off Numbers ► Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified. ► How do you decide how many digits to keep? ► Simple rules exist to tell you how. Chapter Two 63 ► Once you decide how many digits to retain, the rules for rounding off numbers are straightforward: ► RULE 1. If the first digit you remove is 4 or less, drop it and all following digits. 2.4271 becomes 2.4 when rounded off to two significant figures because the first dropped digit (a 2) is 4 or less. ► RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 is 4.6 when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater. ► If a calculation has several steps, it is best to round off at the end. Chapter Two 64 Practice Rule #2 Rounding Make the following into a 3 Sig Fig number 1.5587 1.56 .0037421 .00374 1367 1370 128,522 129,000 1.6683 106 1.67 106 Your Final number must be of the same value as the number you started with, 129,000 and not 129 Examples of Rounding For example you want a 4 Sig Fig number 0 is dropped, it is <5 4965.03 4965 780,582 780,600 8 is dropped, it is >5; Note you must include the 0’s 1999.5 2000. 5 is dropped it is = 5; note you need a 4 Sig Fig RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers. Chapter Two 67 ►RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers. Chapter Two 68 Multiplication and division 32.27 1.54 = 49.6958 49.7 3.68 .07925 = 46.4353312 46.4 1.750 .0342000 = 0.05985 .05985 3.2650106 4.858 = 1.586137 107 1.586 107 6.0221023 1.66110-24 = 1.000000 1.000 Addition/Subtraction 25.5 +34.270 59.770 59.8 32.72 - 0.0049 32.7151 32.72 320 + 12.5 332.5 330 Addition and Subtraction .56 __ + .153 ___ = .713 __ .71 82000 + 5.32 = 82005.32 82000 10.0 - 9.8742 = .12580 .1 10 – 9.8742 = .12580 0 Look for the last important digit Mixed Order of Operation 8.52 + 4.1586 18.73 + 153.2 = = 8.52 + 77.89 + 153.2 = 239.61 = 239.6 (8.52 + 4.1586) (18.73 + 153.2) = = 12.68 171.9 = 2179.692 = 2180. 1.3 Measurement Limits of Measurement If the least precise measurement in a calculation has three significant figures, then the calculated answer can have at most three significant figures. • Mass = 34.73 grams • Volume = 4.42 cubic centimeters. • Rounding to three significant figures, the density is 7.86 grams per cubic centimeter. 1.3 Measurement Limits of Measurement Accuracy Another important quality in a measurement is its accuracy. Accuracy is the closeness of a measurement to the actual value of what is being measured. For example, suppose a digital clock is running 15 minutes slow. Although the clock would remain precise to the nearest second, the time displayed would not be accurate. 1.3 Measurement Measuring Temperature A thermometer is an instrument that measures temperature, or how hot an object is. 1.3 Measurement Measuring Temperature Celsius (centigrade) temperature scale Fahrenheit scale Capillary tube Colored liquid The liquid moves up and down the capillary tube as the temperature changes. Bulb The bulb contains the reservoir of liquid. Scale The scale indicates the temperature according to how far up or down the capillary tube the liquid has moved. 1.3 Measurement Measuring Temperature Compressed scale Liquid rises less in a wide tube for the same temperature change. Liquid rises more in a narrow tube for the same temperature change. Expanded, easy-to-read scale 1.3 Measurement Measuring Temperature The two temperature scales that you are probably most familiar with are the Fahrenheit scale and the Celsius scale. • A degree Celsius is almost twice as large as a degree Fahrenheit. • You can convert from one scale to the other by using one of the following formulas. 1.3 Measurement Measuring Temperature The SI base unit for temperature is the kelvin (K). • A temperature of 0 K, or 0 kelvin, refers to the lowest possible temperature that can be reached. • In degrees Celsius, this temperature is –273.15°C. To convert between kelvins and degrees Celsius, use the formula: 1.3 Measurement Measuring Temperature Temperatures can be expressed in degrees Fahrenheit, degrees Celsius, or kelvins. Measurement and Significant Figures ► Every experimental measurement has a degree of uncertainty. ► The volume, V, at right is certain in the 10’s place, 10mL<V<20mL ► The 1’s digit is also certain, 17mL<V<18mL ► A best guess is needed for the tenths place. Chapter Two 81 What is the Length? 1 • • • • • 82 2 3 We can see the markings between 1.6-1.7cm We can’t see the markings between the .6-.7 We must guess between .6 & .7 We record 1.67 cm as our measurement The last digit an 7 was our guess...stop there 4 cm Learning Check What is the length of the wooden stick? 1) 4.5 cm 2) 4.54 cm 3) 4.547 cm ? 8.00 cm or 3 (2.2/8) 84 Let’s Practice • What is the dimensions of an index card in inches? • Convert each dimension to centimeters. The conversion factor is 1 in. = 2.54 cm. • Check your answer. • Measure with various instruments using accuracy and precision in your measurements. • Activity Relating to 1.3 1.3 Measurement Assessment Questions 1. A shopping mall has a length of 200 meters and a width of 75 meters. What is the area of the mall, in scientific notation? a. b. c. d. 1 × 103 m2 1.5 × 103 m2 1.5 × 104 m2 1.75 × 104 m2 1.3 Measurement Assessment Questions 1. A shopping mall has a length of 200 meters and a width of 75 meters. What is the area of the mall, in scientific notation? a. b. c. d. 1 × 103 m2 1.5 × 103 m2 1.5 × 104 m2 1.75 × 104 m2 ANS: C 1.3 Measurement Assessment Questions 2. A student measures the volume and mass of a liquid. The volume is 50.0 mL and the mass is 78.43 g. What is the correct calculated value of the liquid’s density? (A calculator reads 1.5686.) a. b. c. d. 1.6 g/cm3 1.57 g/cm3 1.569 g/cm3 1.5686 g/cm3 1.3 Measurement Assessment Questions 2. A student measures the volume and mass of a liquid. The volume is 50.0 mL and the mass is 78.43 g. What is the correct calculated value of the liquid’s density? (A calculator reads 1.5686.) a. b. c. d. 1.6 g/cm3 1.57 g/cm3 1.569 g/cm3 1.5686 g/cm3 ANS: B 1.3 Measurement Assessment Questions 3. How can you convert a temperature expressed in kelvin (K) to degree Celsius (°C)? a. b. c. d. add 32 subtract 32 add 273 subtract 273 1.3 Measurement Assessment Questions 3. How can you convert a temperature expressed in kelvin (K) to degree Celsius (°C)? a. b. c. d. add 32 subtract 32 add 273 subtract 273 ANS: C 1.3 Measurement Assessment Questions 1. The SI base unit for length is the mile. True False 1.3 Measurement Assessment Questions 1. The SI base unit for length is the mile. True False ANS: F, meter 1.3 Measurement Homework • Section 1.3 Measurement