3 Scientific Measurement Scientific M Planning Guide

60A Chapter 3

Scientific Measurement Planning GuideScientific MPlanning G3

In many aspects of chemistry, it is vital to know the amount of material with which you are dealing.

Lessons and Objectives Print Resources

For the Student For the Teacher

A-2 3.1 Using and Expressing Measurements p 62–72

3.1.1 Write numbers in scientific notation.3.1.2 Evaluate accuracy and precision.3.1.3 Explain why measurements must be

reported to the correct number of significant figures.

Reading and Study Workbook Lesson 3.1

Lesson Assessment 3.1 p 72Quick Lab Accuracy and

Precision p 72

Class Activity, p 64: Precision and Accuracy

Class Activity, p 67: Olympic Times

Class Activity, p 68: Significant Zeros

A-2, E-2 3.2 Units of Measurement p 74–823.2.1 Explain why metric units are easy to use.3.2.2 Identify the temperature units scientists

commonly use.3.2.3 Calculate the density of a substance.


Lesson Assessment 3.2 p 82

Teacher Demo, p 76: Volume Measurements

Class Activity, p 77: Mass of a Penny

Teacher Demo, p 81: Density Calculations

A-2 3.3 Solving Conversion Problems p 84–913.3.1 Explain what happens when a

measurement is multiplied by a conversion factor.

3.3.2 Describe the kinds of problems that can be easily solved using dimensional analysis.


Lesson Assessment 3.3 p 91Small-Scale Lab Now What

Do I Do? p 92

Class Activity, p 85: Expanding a Recipe

Class Activity, p 89: Sports Stats

Essential Questions1. How do scientists express the degree of

uncertainty in their measurements?2. How is dimensional analysis used to solve

problems?

Study Guide p 93Math Tune-Up p 94STP p 99Reading and Study

Workbook Self-Check and Vocabulary Review Chapter 3

In many aspects of chemistry, it is vital to know the amount of material with which I t f h i t it i it l

Introducing the BIGIDEA: QUANTIFYING MATTER

Essential Questions1 How do scientists express the degree of

Study Guide p 93Math Tune-Up p 94

Assessing the BIGIDEA: QUANTIFYING MATTER

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Scientific Measurement 60B

For the StudentQuick Lab p 72 • 3 x 5 in. index cards• metric ruler

Small-Scale Lab p 92• meter stick• balance• pair of dice • aluminum can • calculator • small-scale pipet • a pre- and post-1982 penny• 8-well strip• plastic cup

For the TeacherClass Activity p 64• a small object, such as a

lead fishing weight• triple-beam balance

Class Activity p 67• almanacs or Internet access

Class Activity p 68• scientific literature• index cards

Class Activity p 76• sets of Erlenmeyer flasks• buret• graduated cylinder beaker• volumetric flask• water• food coloring

Class Activity p 77• balance with a precision

of at least 0.01 g • sets of 10 pennies each

separated according to minting dates between 1970 and the present (Be sure that sets include pre- and post-1982 minting dates.)

Class Activity p 81• 3 or 4 different-sized

cubes of a material such as wood, metal, or marble

• metric ruler• balance

Class Activity p 85• recipe• lists of equivalents and

conversions among the following measurements: teaspoon, tablespoon, 1/4 cup, 1/2 cup, and 1 cup

Class Activity p 89• media guides containing

vital statistics, such as heights and weights, of sports players

Digital Resources

Editable Worksheets PearsonChem.com

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Significant Figures in Multiplication and Division

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Small-Scale Lab Manual Lab 3: Design and Construction of a Small-Scale Balance

Small-Scale Lab Manual Lab 4:Design and Construction of a Set

of Standardized Weights Lab 4: Mass, Volume, and DensityLab Practical 3-1: Basic

MeasurementLab Practical 3-2: Density

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Using Dimensional Analysis

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Using Density as a Conversion Factor

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Converting Ratios of Units

Exam View Assessment SuiteClassroom Resources Disc(includes editable worksheets)

• Lesson Reviews• Practice Problems• Interpret Graphs• Vocabulary Review• Chapter Quizzes and Tests• Lab Record Sheets

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Additional Digital Resources

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3Scientific Measurement

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A surveyor in Antarctica uses a device called a theodolite to measure the landscape for a future airstrip.

60 Chapter 3

Focus on ELL

1 CONTENT AND LANGUAGE Read the title of the chapter aloud to the class. List reasons why the word scientific is in front of the word measurement. Discuss what it means if a measurement is not scientific. Have students document the discussion with a KWL chart.

BEGINNING: LOW/HIGH Have students identify tools used to measure distance, mass, volume, and time, and order the tools based on their accuracy for small measurements.

INTERMEDIATE: LOW/HIGH Have students relate the measures provided with various tools to a familiar object, such as: the mass of a paperclip is about one gram.

ADVANCED: LOW/HIGH Have students practice identifying measurements as a length, a mass, a time, a temperature, or a volume.

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MATH Identify the students who struggle with math by assigning an online math skills diagnostic test. These students can then improve and practice math skills using the MathXL tutorial system.

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in which density is studied in a simulated laboratory environment.

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QUANTIFYING MATTER

Essential Questions:1. How do scientists express the degree

of uncertainty in their measurements?2. How is dimensional analysis used to

solve problems?

BIGIDEA

CHEMYSTERYJust Give Me a SignWhile traveling in a foreign country, you happen to get lost, as many tourists do. But then you spot these signs along the road. If you know the distance to your destination, you can find your way. However, in the signs shown here, the distances are listed as numbers with no units attached. For example, is Preston 8 kilome-ters away or 8 miles away? Is there any way to know for sure?

Connect to the BIGIDEA As you read the chapter, try to familiarize yourself with common metric units used in science.

NATIONAL SCIENCE EDUCATION STANDARDS

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Scientific Measurement 61

Understanding by DesignStudents are building toward understanding how to use scientific measurement as a method of quantifying matter.

PERFORMANCE GOALS At the end of Chapter 3, students will be able to answer the essential questions by applying their knowledge of scientific measurement. Students will also be able to convert between common units of measurement and solve problems involving density.

ESSENTIAL QUESTIONS Read the essential questions aloud. Ask When you make a measurement, what are some possible sources of uncertainty? (Sample answers: You might make a mistake when reading the measurement. The instrument you use might have increments so large that you are unable to obtain an accurate measurement.) Ask What do the terms “dimension” and “analysis” mean? Based on this, what do you think “dimensional analysis” is? (“Dimension” is a unit of measurement, such as meter or liter. “Analysis” is trying to explain something by examining its parts. “Dimensional analysis” is trying to explain something by using its units.)

BIGIDEA Use the photo of the surveyor to help students connect to the

concepts they will learn in this chapter. Activate prior knowledge by asking students how they decide which units to use when making a measurement. Explain that the surveyor in the photo is using the theodolite to make very accurate measurements of angles. Ask Why is it important for his measurements to be so accurate? (Sample answer: The airstrip needs to be located on a flat surface for safety.) Ask What are some sources of uncertainty in his measurements? (Sample answers: He could read the instrument markings incorrectly. The instrument may not have very small increments.)

CHEMYSTERY Have students read over the CHEMystery. Connect the

CHEMystery to the Big Idea of Quantifying Matter by engaging students in a discussion of different units they might use for different types of measurements. Point out that some descriptions of distances are not quantified. Examples include far, close, and nearby. When descriptions are quantified, a unit is needed to avoid ambiguity. Have students suggest ways that they might know for sure whether the distances on the sign are in kilometers or miles. As a hint, have them try to draw a map showing the locations of the cities relative to each other.

Introduce the ChapterIDENTIFYING PRECONCEPTIONS Students may think that measurements are either correct or incorrect. Use the activity to help them realize that measurements can be correct but have degrees of uncertainty.

Activity Divide the class into groups of two or three students, and provide each group with a meter stick and a metric ruler. Have different groups measure and record the length of a wall to the nearest meter, decimeter, centimeter, or millimeter. Afterwards, have groups compare their measurements. Ask Were the measurements for each type of unit the same? (Answers will vary.) Ask Which units do you think gave measurements closest to the actual length of the wall? (millimeters)

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Sample Problem 3.1

Using Scientific NotationSolve each problem and express the answer in scientific notation.

➊ Analyze Identify the relevant concepts. To multiply numbers in scientific notation, multiply the coefficients and add the exponents. To add numbers in scientific notation, the exponents must match. If they do not, then adjust the notation of one of the numbers.

➋ Solve Apply the concepts to this problem.

(8.0 a. × 10−2) × (7.0 × 10−5) (7.1 b. × 10−2) + (5 × 10−3)

a. (8.0 ∙ 10–2) ∙ (7.0 ∙ 10–5 ) ∙ (8.0 ∙ 7.0) ∙ 10–2 + (–5)

b. (7.1 ∙ 10–2) ∙ (5 ∙ 10–3) ∙ (7.1 ∙ 10–2) ∙ (0.5 ∙ 10–2)

∙ 56 ∙ 10–7

∙ 5.6 ∙ 10–6

∙ (7.1 ∙ 0.5) ∙ 10−2

∙ 7.6 ∙ 10–2


Multiplication and Division To multiply numbers written in scientific notation, multiply the coefficients and add the exponents.

(3 × 104) × (2 × 102) = (3 × 2) × 104+2 = 6 × 106

(2.1 × 103) × (4.0 × 10−7) = (2.1 × 4.0) × 103+(−7) = 8.4 × 10−4

To divide numbers written in scientific notation, divide the coefficients and subtract the exponent in the denominator from the exponent in the numerator.

Addition and Subtraction If you want to add or subtract numbers expressed in scientific notation and you are not using a calculator, then the exponents must be the same. In other words, the decimal points must be aligned before you add or subtract the numbers. For example, when adding 5.4 × 103 and 8.0 × 102, first rewrite the second number so that the exponent is a 3. Then add the numbers.

(5.4 × 103) + (8.0 × 102) = (5.4 × 103) + (0.80 × 103) = (5.4 + 0.80) × 103 = 6.2 × 103

3.0 × 105

6.0 × 102 = ( 3.06.0 ) × 105−2 = 0.5 × 103 = 5.0 × 102

Rewrite one of the numbers so that the exponents match. Then add the coefficients.

Multiply the coefficients and add the exponents.

Solve each problem and express the answer in 1. scientific notation.

(6.6 a. × 10−8) + (5.0 × 10−9) (9.4 b. × 10−2) − (2.1 × 10−2)

Calculate the following and write your 2. answer in scientific notation:

6.6 × 106

(8.8 × 10−2) × (2.5 × 103)

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CHEMISTRY YOU&

0.00007 m 7 10 5 mExponent is 5Decimal point moves

5 places to the right.

62 Chapter 3 • Lesson 1

Using and Expressing Measurements3.1

Key Questions How do you write numbers

in scientific notation?

How do you evaluate accuracy and precision?

Why must measurements be reported to the correct number of significant figures?

Vocabularymeasurement• scientific notation• accuracy • precision• accepted value• experimental value• error • percent error• significant figures•

Q: How do you measure a photo finish? You probably know that a 100-meter dash is timed in seconds. But if it’s a close finish, measuring each runner’s time to the nearest second will not tell you who won. That’s why sprint times are often measured to the nearest hundredth of a second (0.01 s). Chemistry also requires making accurate and often very small measurements.

Scientific Notation How do you write numbers in scientific notation?

Everyone makes and uses measurements. A measurement is a quantity that has both a number and a unit. Your height (66 inches), your age (15 years), and your body temperature (37°C) are examples of measurements.

Measurements are fundamental to the experimental sciences. For that reason, it is important to be able to make measurements and to decide whether a measurement is correct. In chemistry, you will often encounter very large or very small numbers. A single gram of hydrogen, for example, contains approximately 602,000,000,000,000,000,000,000 hydrogen atoms. The mass of an atom of gold is 0.000 000 000 000 000 000 000 327 gram. Writing and using such large and small numbers is cumbersome. You can work more easily with these numbers by writing them in scientific notation.

In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power. For example, the number 602,000,000,000,000,000,000,000 can be written in scientific notation as 6.02 × 1023. The coefficient in this number is 6.02. The power of 10, or expo-nent, is 23. In scientific notation, the coefficient is always a number greater than or equal to one and less than ten. The exponent is an integer. A positive exponent indicates how many times the coefficient must be multiplied by 10. A negative exponent indicates how many times the coeffi-cient must be divided by 10. Figure 3.1 shows a magnified view of a human hair, which has a diameter of about 0.00007 m, or 7 × 10−5 m.

When writing numbers greater than ten in scientific notation, the expo-nent is positive and equals the number of places that the original decimal point has been moved to the left.

Numbers less than one have a negative exponent when written in scien-tific notation. The value of the exponent equals the number of places the deci-mal has been moved to the right.

6,300,000. 6.3 106 94,700. 9.47 104

0.000 008 8 10 6 0.00736 7.36 10 3

Figure 3.1 Just a HairA hair’s width expressed in meters is a very small measurement.

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Key Objectives3.1.1 WritE numbers in scientific notation.3.1.2 EvaLuatE accuracy and precision.3.1.3 ExpLaiN why measurements must be

reported to the correct number of significant figures.

Additional Resources• Reading and Study Workbook, Lesson 3.1

Core Teaching Resources, Lesson 3.1 Review

EngageCHEMISTRY YOU& Have students study

the photograph and read the text that opens the section. Ask How do you think scientists ensure measurements are accurate and precise? (Acceptable answers include that scientists make multiple measurements by using the most precise equipment available. They use samples with known values to check the reliability of the equipment.)

Activate Prior KnowledgeDiscuss the various everyday activities that involve measuring, as well as the tools used to make those measurements. Ask a volunteer to be measured for height. Hand out a tape measure, a yardstick, and a metric ruler to three students. Have these students measure the height of the volunteer with their tools and state aloud their measures. Ask Which measurement appears to be the most accurate? Why? (Answers will vary based on the techniques used.)

Focus on ELL

1 CONtENt aNd LaNguagE Review the abbreviations used in the lesson. Supply the class with a listing for each abbreviation and its meaning. Read aloud each abbreviation and have students repeat.

2 FrONtLOad thE LESSON Put together a display of common and uncommon measuring tools that students can examine prior to the lesson. Ask students to group the tools on the basis of the type of measurement for which they believe the tools are used: distance, time, volume, and so forth. Have students predict which tools in each group are the most and least accurate. Revisit students’ predictions after completing the lesson.

3 COmprEhENSibLE iNput When discussing Figure 3.2, assist students in completing the associated analogy by allowing them to test their answers with an unzeroed balance.

Sample Problem 3.1

Using Scientific NotationSolve each problem and express the answer in scientific notation.

Analyze Identify the relevant concepts. To multiply numbers in scientific notation, multiply the coefficients and add the exponents. To add numbers in scientific notation, the exponents must match. If they do not, then adjust the notation of one of the numbers.

Solve Apply the concepts to this problem.

(8.0 a. 10 2) (7.0 10 5) (7.1 b. 10 2) (5 10 3)

problem.pp y p p

a. (8.0 10 2) (7.0 10 5 ) (8.0 7.0) 10 2 ( 5)

b. (7.1 10 2) (5 10 3) (7.1 10 2) (0.5 10 2)

56 10 7

5.6 10 6

(7.1 0.5) 10 2

7.6 10 2


Multiplication and Division To multiply numbers written in scientific notation, multiply the coefficients and add the exponents.

(3 104) (2 102) (3 2) 104 2 6 106

(2.1 103) 4.0 10 7) (2.1 4.0) 103 ( 7) 8.4 10 4

To divide numbers written in scientific notation, divide the coefficients and subtract the exponent in the denominator from the exponent in the numerator.

Addition and Subtraction If you want to add or subtract numbers expressed in scientific notation and you are not using a calculator, then the exponents must be the same. In other words, the decimal points must be aligned before you add or subtract the numbers. For example, when adding 5.4 103 and 8.0 102, first rewrite the second number so that the exponent is a 3. Then add the numbers.

(5.4 103) (8.0 102) (5.4 103) (0.80 103) (5.4 0.80) 103 6.2 103

3.0 105

6.0 102 ( 3.06.0 ) 105 2 0.5 103 5.0 102

Rewrite one of the numbers so that the exponents match. Then add the coefficients.

Multiply the coefficients and add the exponents.

Solve each problem and express the answer in 1. scientific notation.

(6.6 a. 10 8) (5.0 10 9) (9.4 b. 10 2) (2.1 10 2)

Calculate the following and write your 2. answer in scientific notation:

6.6 106

(8.8 10 2) (2.5 103)

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Answers1. a. 7.1 × 10–8 b. 7.3 × 10–2

2. 3.0 × 104

Foundations for ReadingBUILD VOCABULARY Have students write definitions of the words accurate and precise in their own words. As they read the text, have students compare the definitions with those of accuracy and precision given in the text.READING STRATEGY After reading each section, create a bulleted list summarizing the meaning of the key words or summarizing the rules to follow. Have students compare their lists with others. Break the class into small groups and have each group consolidate their individual efforts into a group summary.

Explain

Scientific NotationUSE VISUALS Have students study Figure 3.1. Ask How is the exponent of a number expressed in scientific notation related to the number of places the decimal point is moved to the right in a number smaller than 1? (They are equal.)

Sample Practice ProblemCalculate the following and express your answers in scientific notation.

1. (3.0 × 10–3) × (2.5 × 10–4) =? (7.5 × 10–7)

2. (4.2 × 10–3) + (7.0 × 10–4) =? (4.9 ×10–3)

Foundations for MathCALCULATIONS INVOLVING SCIENTIFIC NOTATION Tell students that solving problems in chemistry often involves numbers written in scientific notation. Explain that when multiplying or dividing numbers written in scientific notation, the exponents do not have to be the same. However, when adding or subtracting without a calculator, the exponents must be the same. Point out that another way to combine numbers written in scientific notation is to write both numbers in standard form first, perform the operation, and then convert the resulting answer back to scientific notation.

In Sample Problem 3.1b, each number can be written in standard form first. Some students may find this easier than changing the exponent for one of the numbers to make them match.

7.1 × 10–2 = 0.071

5 × 10–3 = 0.005

Now add: 0.071 + 0.005 = 0.076 = 7.6 × 10–2

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Good Accuracy, Good PrecisionCloseness to the bull’s-eye indicates a high degree of accuracy. The closeness of the darts to one another indicates high precision.

Poor Accuracy, Good PrecisionPrecision is high because of the closeness of grouping—thus, the high level of reproducibility. But the results are inaccurate.

Poor Accuracy, Poor PrecisionThe darts land far from one another and from the bull’s-eye. The results are both inaccurate and imprecise.

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Accuracy, Precision, and Error How do you evaluate accuracy and precision?Your success in the chemistry lab and in many of your daily activities depends on your ability to make reliable measurements. Ideally, measure-ments should be both correct and reproducible.

Accuracy and Precision Correctness and reproducibility relate to the con-cepts of accuracy and precision, two words that mean the same thing to many people. In chemistry, however, their meanings are quite different. Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured. Precision is a measure of how close a series of measurements are to one another, irrespective of the actual value. To evaluate the accuracy of a measurement, the measured value must be com-pared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.

Darts on a dartboard illustrate accuracy and precision in measurement. Let the bull’s-eye of the dartboard in Figure 3.2 represent the true, or correct, value of what you are measuring. The closeness of a dart to the bull’s-eye cor-responds to the degree of accuracy. The closer it comes to the bull’s-eye, the more accurately the dart was thrown. The closeness of several darts to one another corresponds to the degree of precision. The closer together the darts are, the greater the precision and the reproducibility.

Figure 3.2 Accuracy vs. PrecisionThe distribution of darts illustrates the difference between accuracy and precision. Use Analogies Which outcome describes a scenario in which you properly measure an object’s mass three times using a balance that has not been zeroed?

Determining Error Suppose you use a thermometer to measure the boil-ing point of pure water at standard pressure. The thermometer reads 99.1°C. You probably know that the true or accepted value of the boiling point of pure water at these conditions is actually 100.0°C.


Explain

Accuracy, Precision, and ErrorUSE VISUALS Have students inspect Figure 3.2. Ask If one dart in Figure 3.2c were closer to the bull’s-eye, what would happen to the accuracy? (The accuracy would increase.) Ask What would happen to the precision? (The precision would increase.) Ask What is the operational definition of error implied by this figure? (The error is the distance between the dart and the bull’s-eye.)

Explore

Class ActivityPURPOSE Students will explore the concepts of precision and accuracy.MATERIALS a small object (such as a lead fishing weight), triple-beam balancePROCEDURE Place the object and a triple-beam balance in a designated area. Set a deadline by which each student will have measured the mass of the object. After everyone has had an opportunity, have students compile a summary of all the measurements. Illustrate precision by having the students find the average and compare their measurement to it.EXPECTED OUTCOME Measured values should be similar, but not necessarily identical, for all students.USE ANALOGIES Guide students to think of other analogies that may be useful in explaining precision vs. accuracy, such as casting a fishing line, pitching horseshoes, or marching in a precision marching band.

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Striving for Scientific Accuracy

The French chemist Antoine Lavoisier worked hard to establish the importance of accurate measurement in scientific inquiry. Lavoisier devised an experiment to test the Greek scientists’ idea that when water was heated, it could turn into earth. For 100 days, Lavoisier boiled water in a glass flask constructed to allow steam to condense without escaping. He weighed the water and the flask separately before and after boiling. He found that the mass of the water had not changed. The flask, however, lost a small mass equal to the sediment he found in the bottom of it. Lavoisier proved that the sediment was not earth, but part of the flask etched away by the boiling water.

Sample Problem 3.2

KNOWNS

Experimental value 99.1°C

Accepted value 100.0°C

UNKNOWN

Percent error

100%99.1°C 100.0°C

100.0°C

Percent error 100%experimental value accepted value

accepted value

Percent error 100%error

accepted value

0.9°C

100.0°C100% 0.9%

Start with the equation for percent error.

Substitute the equation for error, and then plug in the known values.

0.0.10


There is a difference between the accepted value, which is the correct value for the measurement based on reliable references, and the experimental value, the value measured in the lab. The difference between the experimen-tal value and the accepted value is called the error.

Error experimental value accepted value

Error can be positive or negative, depending on whether the experimental value is greater than or less than the accepted value. For the boiling-point measurement, the error is 99.1°C 100.0°C, or 0.9°C.

The magnitude of the error shows the amount by which the experimental value differs from the accepted value. Often, it is useful to calculate the rela-tive error, or percent error. The percent error of a measurement is the abso-lute value of the error divided by the accepted value, multiplied by 100%.

Percent error 100%erroraccepted valuet dd ll

Evaluate Does the result make sense? The experimental value was off by about 1°C, or 1

100 of the accepted value (100°C). The answer makes sense.

Calculating Percent ErrorThe boiling point of pure water is measured to be 99.1°C. Calculate the percent error.

Analyze List the knowns and unknown.The accepted value for the boiling point of pure water is 100°C. Use the equations for error and percent error to solve the problem.

Calculate Solve for the unknown.

A student measures the depth of a 3. swimming pool to be 2.04 meters at its deepest end. The accepted value is 2.00 m. What is the student’s percent error?

READING SUPPORTBuild Reading Skills: Inference As you read, try to identify some of the factors that cause experimental error. What factors might result in inaccurate measurements? What factors might result in imprecise measurements?

Think about it: Using the absolute

value of the error means that percent

error will always be a positive value.

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AnswersREADING SUPPORT Inaccurate measurements

may be due to uncalibrated instruments or incorrect experimental procedure; imprecise measurements may be due to poor instrument operation, variable experimental conditions, or errors in reading output values.

FIGURE 3.2 poor accuracy, good precision

3. 2%

ExplainSTART A CONVERSATION Review the concept of absolute value. Ask What is the meaning of a positive error? (The measured value is greater than the accepted value.) Ask What is the meaning of a negative error? (The measured value is less than the accepted value.) Explain that the absolute value of the error is a positive value that describes the difference between the measured value and the accepted value, but not which is greater.

Sample Practice ProblemDiamonds and other gemstones are measured in carats. The accepted value for the weight of a carat is 0.2 gram. A jeweler measured the weight of a carat of opals to be 0.192 gram. What is the percent error? (4%)

Foundations for MathABSOLUTE VALUE Remind students that the absolute value of a number is its distance from zero on a number line and is always written as a positive number. So, |4.2| = 4.2 and |–4.2| = 4.2.

In Sample Problem 3.2, the numerator in the fraction of the formula is the absolute value of the error. Regardless of whether the experimental value or the accepted value is larger, the percent error will always be a positive value.

1m10 20 30 40 50 60 70 80 90

1m10 20 30 40 50 60 70 80 90

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0.77 m

0.772 m

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Figure 3.3 Degrees CelsiusThe temperature shown on this Celsius thermometer can be reported to three significant figures.

Figure 3.4 Increasing Precision Three differently calibrated meter sticks are used to measure a door’s width. A meter stick calibrated in 0.1-m (1 dm) intervals is more precise than one calibrated in a 1-m interval but less precise than one calibrated in 0.01-m (1 cm) intervals. Measure How many significant figures are reported in each measurement?

Significant Figures Why must measurements be reported to the correct number of significant figures?Look at the reading of the thermometer shown in Figure 3.3. If you use a liquid-filled thermometer that is calibrated in 1°C intervals, you can easily read the temperature to the nearest degree. With the same ther-mometer, however, you can also estimate the temper-ature to about the nearest tenth of a degree by noting the closeness of the liquid inside to the calibrations. Looking at Figure 3.3, suppose you estimate that the temperature lies between 22°C and 23°C, at 22.9°C. This estimated number has three digits. The first two digits (2 and 2) are known with certainty. But the right-most digit (9) has been estimated and involves some uncertainty. These reported digits all convey useful information, however, and are called significant figures. The significant figures in a measurement include all of the digits that are known, plus a last digit that is esti-mated. Measurements must always be reported to the correct number of significant figures because cal-culated answers often depend on the number of sig-nificant figures in the values used in the calculation.

Instruments differ in the number of significant fig-ures that can be obtained from their use and thus in the precision of measurements. The three meter sticks in Figure 3.4 can be used to make successively more pre-cise measurements.

More on precision in measurements online.

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Significant FiguresSTART A CONVERSATION Point out that the concept of significant figures applies only to measured quantities. If students ask why an estimated digit is considered significant, tell them a significant figure is one that is known to be reasonably reliable. A careful estimate fits this definition.USE VISUALS Direct students’ attention to Figure 3.3. Point out that when calibration marks on an instrument are spaced very close together (e.g., on certain thermometers and graduated cylinders), it is sometimes more practical to estimate a measurement to the nearest half of the smallest calibrated increment, rather than to the nearest tenth.CRITICAL THINKING As students inspect Figure 3.4, model the use of the top meter stick by pointing out that one can be certain that the width of the door is between 0 and 1 m, and one can say that the actual width is closer to 1 m. Thus, one can estimate the width as 0.8 m. Similarly, using the middle meter stick, one can say with certainty that the width is between 70 and 80 cm. Because the width is very close to 80 cm, one should estimate the width as 77 cm or 0.77 m. Have students study the bottom meter stick and use similar reasoning to describe the measurement and estimation process. Ask If the bottom meter stick were divided into 0.001 m intervals, as are most meter sticks, what would be the estimated width of the door in meters? (Acceptable answers range from 0.7715 to 0.7724 m.) In millimeters? (771.5 to 772.4)

Check for UnderstandingThe Essential Question How do scientists express their uncertainty in measurement?

Assess students’ understanding of the Essential Question by asking students to complete the following statement:

A measured value’s uncertainty is expressed by its _____ digit. (final; last; rightmost)

ADJUST INSTRUCTION If students are having difficulty answering, refer them to Figure 3.3. Point out that the rightmost digit in each measurement is an estimated value. The fact that the rightmost digit is estimated gives the digit its uncertainty.


Determining Significant Figures in Measurements To determine whether a digit in a measured value is significant, you need to apply the following rules.

Every nonzero digit in a reported measurement 1. is assumed to be significant.

24.7 meters 0.743 meter 714 meters

Zeros appearing between nonzero digits are 2. significant.

Leftmost zeros appearing in front of nonzero 3. digits are not significant. They act as placeholders. By writing the measurements in scientific notation, you can eliminate such placeholding zeros.

0.0071 meter 7.1 10 3 meter 0.42 meter 4.2 10 1 meter 0.000 099 meter 9.9 10 5 meter

Each of these measurements has three significant figures.

Each of these measurements has four significant figures.

Each of these measurements has four significant figures.

The zeros in these measurements are not significant.

The zeros in this measurement are significant.

Each of these measurements has only two significant figures.

This measurement is a counted value, so it has an unlimited number of significant figures.

Each of these numbers has an unlimited number of significant figures.

Zeros at the end of a number and to the right 4. of a decimal point are always significant.

7003 meters 40.79 meters 1.503 meters

43.00 meters 1.010 meters 9.000 meters

Zeros at the rightmost end of a measurement that 5. lie to the left of an understood decimal point are not significant if they serve as placeholders to show the magnitude of the number.

300 meters (one significant figure) 7000 meters (one significant figure) 27,210 meters (four significant figures)

If such zeros were known measured values, however, then they would be significant. Writing the value in scientific notation makes it clear that these zeros are significant.

300 meters 3.00 102 meters (three significant figures)

There are two situations in which numbers have 6. an unlimited number of significant figures. The first involves counting. A number that is counted is exact.

23 people in your classroom

60 min 1 hr100 cm 1 m

The second situation involves exactly defined quantities such as those found within a system of measurement.

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AnswersFIGURE 3.4 0.8 m, one significant figure;

0.77 m, two significant figures; 0.772 m, three significant figures

Explore

Class ActivityPURPOSE Students will explore how similar measurements from different eras may vary in precision.MATERIALS Almanacs or Internet accessPROCEDURE Have students look up the winning times for the men’s and women’s 100-meter dashes at the 1948 and 2008 Olympic Games. Ask Why do the more recently recorded race times contain more digits to the right of the decimal? (The technology used for timekeeping improved to allow for more precise measurements.)EXPECTED OUTCOME Students should find that the race times from 1948 were recorded to the nearest tenth of a second. The race times from 2008 were recorded to the nearest hundredth of a second.

Differentiated Instruction LPR LESS PROFICIENT READERS Have students write in their own words the rules for determining the number of significant digits. Help them if necessary. Direct them to make several measurements, and then use their rules to correctly determine the correct number of significant digits in the measurements.

L1 STRUGGLING STUDENTS Create one set of flash cards for each rule. Write three numerical values on each card—two values that follow the rule and one that does not follow the rule. Have students identify the two values that follow the rule and explain why the third values does not.

ELL ADVANCED LEARNERS Have students determine how a measurement, such as area or volume, can be precise without being accurate.

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Significant Figures in Calculations Suppose you use a calculator to find the area of a floor that measures 7.7 meters by 5.4 meters. The calculator would give an answer of 41.58 square meters. However, each of the measurements used in the calculation is expressed to only two significant figures. As a result, the answer must also be reported to two significant figures (42 m2). In gen-eral, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated.

Rounding To round a number, you must first decide how many significant figures the answer should have. This decision depends on the given measure-ments and on the mathematical process used to arrive at the answer. Once you know the number of significant figures your answer should have, round to that many digits, counting from the left. If the digit immediately to the right of the last significant digit is less than 5, it is simply dropped and the value of the last significant digit stays the same. If the digit in question is 5 or greater, the value of the digit in the last significant place is increased by 1.

Sample 3.3

Counting Significant Figures in MeasurementsHow many significant figures are in each measurement?

Count the significant figures in each 4. measured length.

0.057 30 metera. 8765 metersb. 0.000 73 meterc. 40.007 metersd.

How many significant figures are in each 5. measurement?

143 gramsa. 0.074 meterb. 8.750 c. 10 2 gram1.072 metersd.

a. three (rule 1)

b. five (rule 2)

c. five (rule 4)

d. unlimited (rule 6)

e. four (rules 2, 3, 4)

f. two (rule 5)

Apply the rules for determining significant figures. All nonzero digits are significant (rule 1). Use rules 2 through 6 to determine if the zeros are significant.

123 ma. 40,506 mmb. 9.8000 c. 104 m

22 meter sticksd. 0.070 80 me. 98,000 mf.

Analyze Identify the relevant concepts. The location of each zero in the measurement and the location of the decimal point determine which of the rules apply for determining significant figures. These locations are known by inspecting each measurement value.


Make sure you understand the

rules for counting significant

figures (on the previous page)

before you begin, okay?

Q: Suppose that the winner of a 100-meter dash finishes the race in 9.98 seconds. The runner in second place has a time of 10.05 seconds. How many significant figures are in each measurement? Is one measurement more accurate than the other? Explain your answer.

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Class ActivityPURPOSE Students will practice applying the rules governing the significance of zeros in measurements.MATERIALS textbook and scientific literature, index cardsPROCEDURE Have students search their textbooks and other sources for length, mass, volume, or temperature measurements that contain zeros. Have them include some examples written in scientific notation. Ask them to write each measurement on the front of an index card; on the back of each card, have them write (1) all the rules governing the significance of zeros that apply to the measurement, and (2) the number of significant figures in the measurement. Have pairs of students exchange index cards and agree on the appropriateness of the rules and the answers.EXPECTED OUTCOME Students should be able to apply correctly rules 2–5 listed on page 69.

ExplainDRAW A CONCLUSION Explain that due to rounding, there will often be discrepancies between actual values and calculated values derived from measurements. Ask students to draw a conclusion about whether following the rules for significant figures introduces error in calculated values. (Although rounding errors are generally small, they should be acknowledged when performing calculations.)

CHEMISTRY YOU YOYY U&& The time of 9.98 seconds has three significant figures, and the time of 10.05 seconds has four significant figures. Both times have the same accuracy because they are measured to the nearest 0.01 second.

Sample Practice ProblemHow many significant figures does the number 103,400 have? (4)

Foundations for MathZEROS IN SIGNIFICANT FIGURES Remind students that a number may contain both significant and nonsignificant zeros. In this situation, more than one rule must be applied to determine which zeros are significant and which zeros are not significant.In Sample Problem 3.3e, the number 0.07080 contains both significant and nonsignificant zeros. Rewriting this number in scientific notation makes it easier to determine which zeros are significant figures, by removing the nonsignificant zeros from immediate view:

0.07080 = 7.080 × 10–2

Now it is obvious that the zeros to the left of the 7 act as placeholders and are not significant, while the remaining two zeros are significant because of their location.

Sample Problem 3.4

a. 314.721 meters

2 is less than 5, so you do not round up.

314.7 meters 3.147 102 meters

b. 0.001 775 meters

7 is greater than 5, so round up.

0.0018 meter 1.8 10 3 meter

c. 8792 meters

9 is greater than 5, so round up.

8800 meters 8.8 103 meters


Round each measurement to three 6. significant figures. Write your answers in scientific notation.

87.073 metersa. 4.3621 b. 108 meters0.01552 meterc. 9009 metersd. 1.7777 e. 10 3 meter629.55 metersf.

Round each measurement in 7. Problem 6 to one significant figure. Write each of your answers in scientific notation.

Rounding MeasurementsRound off each measurement to the number of significant figures shown in parentheses. Write the answers in scientific notation.

314.721 meters (four)a. 0.001 775 meter (two)b. 8792 meters (two)c.

Analyze Identify the relevant concepts. Using the rules for determining significant figures, round the number in each measurement. Then apply the rules for expressing numbers in scientific notation.


Starting from the left, count the first four digits that are significant. The arrow points to the digit immediately following the last significant digit.

Starting from the left, count the first two digits that are significant. The arrow points to the digit immediately following the second significant digit.

Starting from the left, count the first two digits that are significant. The arrow points to the digit immediately following the second significant digit.

If you’re already familiar

with rounding numbers,

you can skip to Sample

Problems 3.5 and 3.6.

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Answers4. a. 4 c. 2 b. 4 d. 55. a. 3 c. 4 b. 2 d. 46. a. 8.71 × 101 m b. 4.36 × 108 m c. 1.55 × 10–2 m d. 9.01 × 103 m e. 1.78 × 10–3 m f. 6.30 × 102 m7. a. 9 × 101 m b. 4 × 108 m c. 2 × 10–2 m d. 9 × 103 m e. 2 × 10–3 m f. 6 × 102 m

Explain

Significant FiguresSUMMARIZE As a class, summarize the rules for significant figures in a bulleted list or a fishbone map. Students should state the rules in their own words and provide an example for each rule.

Misconception AlertSome students may think that because they use a calculator, that there results are shown with the proper number of significant figures. Explain that this is not the case, and even scientific calculators or graphing calculators do not round answers to the correct number of significant figures.

Sample Practice ProblemRound each measurement to two significant figures. Write your answer in scientific notation.a. 94.592 grams (9.5 × 101 g)b. 2.4332 × 103 grams (2.4 × 103 g)c. 0.007438 grams (7.4 × 10–3 g)d. 54,752 grams (5.5 × 104 g)e. 6.0289 × 10–3 grams 6.0 × 10–3 g)f. 405.11 grams (4.1 × 102 g)

Foundations for MathCOMMON ERROR Stress to students that rounding to a given number of significant figures is not necessarily the same thing as rounding to that many decimal places. For example, if 7.376 is rounded to two significant figures, the result is 7.4. Students often confuse this process with rounding to two decimal places, in which case the result would be 7.38. This number has three significant figures, not two.

In Sample Problem 3.4b, none of the zeros are significant. Stress that the answer is not rounded to two decimal places. Rather, the number will be rounded to four decimal places because the second significant figure (7) is in the ten ten-thousandths place.

Sample Problem 3.5

74.626 meters

28.34 meters

46.286 meters



12.52 meters

349.0 meters

8.24 meters

369.76 meters

a.

b.

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Addition and Subtraction The answer to an addition or subtraction cal-culation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. Sample Problem 3.5 gives examples of rounding in addition and subtraction.

Multiplication and Division In calculations involving multiplication and division (such as those in Sample Problem 3.6), you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measure-ments. The position of the decimal point is important only in rounding the answers of addition or subtraction problems.

Significant Figures in Addition and SubtractionPerform the following addition and subtraction operations. Give each answer to the correct number of significant figures.

12.52 meters a. 349.0 meters 8.24 meters74.626 meters b. 28.34 meters

Analyze Identify the relevant concepts. Perform the specified math operation, and then round the answer to match the measurement with the least number of decimal places.


Perform each operation. Express your answers 8. to the correct number of significant figures.

61.2 meters a. 9.35 meters 8.6 meters9.44 meters b. 2.11 meters1.36 meters c. 10.17 meters34.61 meters d. 17.3 meters

Find the total mass of three diamonds that 9. have masses of 14.2 grams, 8.73 grams, and 0.912 gram.

Align the decimal points and add the numbers.

Align the decimal points and subtract the numbers.

The second measurement (349.0 meters) has the least number of digits (one) to the right of the decimal point. So the answer must be rounded to one digit after the decimal point.

The second measurement (28.34 meters) has the least number of digits (two) to the right of the decimal point. So the answer must be rounded to two digits after the decimal point.

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ExplainSTART A CONVERSATION The rules for rounding calculated numbers can be compared with the old adage, “A chain is only as strong as its weakest link.” Explain that an answer cannot be more precise than the least precise value used to calculate the answer. Ask In addition and subtraction, what is the least precise value? (The measurement with the fewest digits to the right of the decimal point.)Ask In multiplication and division, what is the least precise value? (The measurement with the fewest significant figures.) If students wonder why addition and subtraction rules differ from multiplication and division rules, point out that in addition and subtraction of measurements, the measurements are of the same property, such as length or volume. However, in the multiplication and division of measurements, new quantities or properties are being described, such as speed (length ÷ time), area (length × length), or density (mass ÷ volume).

Misconception Alert Students may argue that making one measurement of a dimension, such as length, is adequate. Ask What possible errors may occur when making only one length measurement? (Acceptable answers include misreading the ruler or not holding the ruler parallel to the length of the object.)

Sample Practice ProblemWhat is the total mass of three rock samples that have measured masses of 20.72 grams, 24.8 grams, and 17.35 grams? (62.9 grams)

Foundations for MathADDITION AND SIGNIFICANT FIGURES Explain to students that when performing calculations that involve measurements with significant figures, the final result must be rounded to be consistent with the measurements. When adding and subtracting, if the measurements have different numbers of significant figures, the final result should be rounded to the same number of decimal places as the number with the least number of decimal places.

In Sample Problem 3.5, the first step should be to identify the measurement with the least number of decimal places. Encourage students to underline or circle this measurement. The result of the calculation should be rounded to the same number of decimal places (not digits) as this measurement.

Sample Problem 3.6T U T O R

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a. 7.55 meters 0.34 meter 2.567 (meter)2

b. 2.10 meters 0.70 meter 1.47 (meter)2

c. 2.4526 meters2 8.4 meters 0.291 976 meter

d. 0.365 meters2 0.0200 meter 18.25 meters

0.29 meter

2.6 meters2

1.5 meters2

18.3 meters


Solve each problem. Give your 10. answers to the correct number of significant figures and in scientific notation.

8.3 meters a. 2.22 meters

8432 metersb. 2 12.5 meters

c. 35.2 seconds 1 minute

60 seconds

Calculate the volume of a warehouse 11. that has measured dimensions of 22.4 meters by 11.3 meters by 5.2 meters. (Volume l w h)

Significant Figures in Multiplication and DivisionPerform the following operations. Give the answers to the correct number of significant figures.

7.55 meters a. 0.34 meter2.10 meters b. 0.70 meter2.4526 metersc. 2 8.4 meters0.365 meterd. 2 0.0200 meter

Analyze Identify the relevant concepts. Perform the specified math operation, and then round the answer to match the measurement with the least number of significant figures.


The second measurement (0.70 meter) has the least number of significant figures (two). So the answer must be rounded to two significant figures.

Both measurements have three significant figures. So the answer must be rounded to three significant figures.

The second measurement (8.4 meters2) has the least number of significant figures (two). So the answer must be rounded to two significant figures.

The second measurement (0.34 meter) has the least number of significant figures (two). So the answer must be rounded to two significant figures.

( )

In Problem 11, the measurement

with the fewest significant figures is

5.2 meters. What does this tell you?

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Answers 8. a. 79.2 m c. 11.53 m b. 7.33 m d. 17.3 m 9. 23.8 g10. a. 1.8 × 101 m2

b. 6.75 × 102 m c. 5.87 × 10–1 min11. 1.3 × 103 m3

Explain

Sample Practice ProblemA small rectangular container has measured dimensions of 4.25 inches by 8.5 inches by 1.75 inches. What is the volume of the container? (63 cubic inches)

Evaluate

Informal AssessmentWrite the following sets of measurements on the board.

(1) 70ºC, 70ºC, 80ºC

(2) 77ºC, 78ºC, 78ºC

(3) 80.25ºC, 84.50ºC, 86.00ºC

Ask The temperature of a liquid under similar conditions was measured every ten minutes by three different students. Which student had the most precise measuring instrument? (Set 3 is the most precise because the measurements have the greatest number of significant figures.) Ask What would have to be known to determine which set is the most accurate? (the accepted value of the liquid’s boiling point)

ReteachUse Figure 3.4 to reteach the method of correctly recording the number of significant figures in a measurement. Then have students convert each measurement into scientific notation. (8 × 10–1 m, 7.7 × 10–1 m, 7.72 × 10–1 m)

Focus on ELL

4 LANGUAGE PRODUCTION Have students work in pairs to complete the lab. Make sure each pair has ELLs of varied language proficiencies, so that more proficient students can help less proficient ones. Have students work according to their proficiency level.

BEGINNING: LOW/HIGH Underline the action verbs in each step of the procedure. Work with a partner to follow each step.

INTERMEDIATE: LOW/HIGH Use a chart to keep track of the different measurements needed for the lab.

ADVANCED: LOW/HIGH With the help of a partner, answer #2 of the Analyze and Conclude section of the lab.

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Lesso3.112. Review How can you express a number in

scientific notation?

13. Review How are accuracy and precision evaluated?

14. Explain Why must a given measurement always be reported to the correct number of significant figures?

Calculate 15. A technician experimentally determined the boiling point of octane to be 124.1°C. The actual boiling point of octane is 125.7°C. Calculate the error and the percent error.

Evaluate 16. Determine the number of significant figures in each of the following measurements:

11 soccer playersa. 0.070 020 meterb. 10,800 metersc.

Calculate 17. Solve the following and express each answer in scientific notation and to the correct number of significant figures.

(5.3 a. 104) (1.3 104)(7.2 b. 10−4) (1.8 103)10c. 4 10 3 106

(9.12 d. 10 1) (4.7 10 2)(5.4 e. 104) (3.5 109)

BIGIDEA QUANTIFYING MATTER

Write a brief paragraph explaining 18. the differences between the accuracy, precision, and error of a measurement.

Quick Lab

Analyze and Conclude

Identify1. How many significant figures are in your measure-ments of length and of width?

Compare 2. How do your measurements compare with those of your classmates?

Explain3. How many significant figures are in your calculated value for the area? In your calculated value for the perimeter? Do your rounded answers have as many significant figures as your classmates’ measurements?

Evaluate4. Assume that the correct (accurate) length and width of the card are 12.70 cm and 7.62 cm, respectively. Calculate the percent error for each of your two measurements.

Accuracy and Precision

ProcedureUse a metric ruler to measure in centimeters the length and width of an 1.

index card as accurately as you can. The hundredths place in your measure-ment should be estimated.

Calculate the area (2. A l w) and the perimeter [P 2 (l w)] of the index card. Write both your unrounded answers and your correctly rounded answers on the chalkboard.

Purpose To measure the dimensions of an object as accurately and precisely as possible and to apply rules for rounding answers calculated from the measurements

Materials3-inch 5-inch index cardmetric ruler

0.010 square meterd. 5.00 cubic meterse. 507 thumbtacksf.

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OBJECTIVES After completing this activity, students will be able to measure length with accuracy and precision, apply rules for rounding answers calculated from measurements, and determine experimental error and express it as percent error.SKILLS FOCUS Measuring, calculatingPREP TIME 5 minutesCLASS TIME 15 minutesMATERIALS 3 inch × 5 inch index cards, metric rulersTEACHING TIPS Emphasize that students should use an interior, marked line, such as 10.0 cm, as the initial point, instead of the end of the ruler, which may be damaged.EXPECTED OUTCOME Measured values should be similar, but not necessarily identical, for all students.ANALYZE AND CONCLUDE

1. Four for length; three for width2. See Expected Outcome. 3. Significant digits for rounded-off answers are

area, 3, and perimeter, 4. Some students may not round to the proper number of digits.

4. Errors of ± 0.03 cm are acceptable. Such errors yield percent errors of 0.2% for length and 0.4% for width.

FOR ENRICHMENT Have students devise methods of calculating the volume of one card. Point out that measuring the thickness of one card with a ruler would be very inaccurate. Ask How might the measurement of the thickness of the card be improved? (Use a more precise instrument, such as a micrometer, or measure the thickness of a stack of cards and divide by the number of cards.) Have students determine the thickness of one card and calculate its volume. Using the class average of the calculated volumes, have each student determine the percent error using the average as the accepted value.

Af

Quick Lab

Lesson Check Answers12. Write the number as a product of two

numbers: a coefficient greater than or equal to one and less than ten, and 10 raised to an integer power.

13. Accuracy compares the measured value to the correct value. Precision compares more than one measurement.

14. The significant figures in a calculated answer depend on the number of significant figures of the measurements and the mathematical operation used in the calculation.

15. error = –1.6ºC; percent error = 1.3%

16. a. unlimited d. 2 b. 5 e. 3 c. 3 f. unlimited17. a. 6.6 × 104 d. 8.65 × 10−1

b. 4.0 × 10−7 e. 1.9 × 1014

c. 107

18. BIGIDEA Accuracy compares a measured value to an accepted value of the measurement; precision compares a measured value to a set of measurements made under similar conditions; and error is the difference between the measured and accepted values.

CHEMISTRY YOU: EVERYDAY MATTERY YYY&

Take It FurtherMeasure 1. What is the measured height

of the tomato shown above? How many significant figures does your answer have?

Identify 2. What are some other activities that involve measurements done by hand? What units and measuring tools are used?

Chemistry & You 73

Watch What You MeasureJust because you live in a digital age doesn’t mean that you no longer have to do things by hand. In fact, manually measuring quantities remains an important everyday skill in a number of professions and activities. For example, chefs measure volumes of ingredients in cups (C) or liters (L). Tailors use a tape measure calibrated in inches (in. or ) to measure length, while biologists use metric rulers or calipers calibrated in centimeters (cm). A ship’s navigator uses a sextant to measure the angle between the sun and the horizon. The angle is expressed in degrees (°) and minutes ( ).

The next time you make a measurement in lab, keep in mind that lots of other measurers are rounding and noting significant figures, just like you are.

7 ”

7.92 cm

42° 31.4’

3_4

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AnswersTAKE IT FURTHER

1. 32.72 mm (four significant figures)2. Answers may vary; possibilities include

measuring length of fabric (yards and yardsticks), measuring weight of a letter (ounces and a scale), measuring weight of produce in a store (pounds and a scale).

CHEMISTRY YOU YOYY U&& Have students study the photographs closely. Point out that the most important decision these professionals make when they begin their measurement is not whether to use a manual tool or a technologically advanced tool—the most important decision is to determine the correct tool for the task. The choice of tool helps determine the overall accuracy of the measurement.

Explain to students that the flexible cloth tape measure used by the tailor is divided into increments of 1/16 of an inch. Pose the following question to students: Imagine that the only tool the tailor had available was a wooden ruler divided into 1-inch increments. How might the use of the wooden ruler affect the outcome of the custom-tailored suit the tailor is constructing? You may need to assist students in the following ways:

• Custom-tailored clothing is designed to fit the body measurements of the intended wearer.

• Many of the measurements a tailor makes involve curves or circumferences.

• A typical non-retractable cloth tape measure ranges in length from 60 to 96 inches.

Extend

Connect to ARCHITECTURE

Explain to students that one field which makes use of manual systems of measurement is architecture. Discuss how manual tools are utilized by an architect during both the design stages of a project (such as making blueprints or producing models) and by the various tradesmen during the implementation of the architect’s design and blueprints. Have students think of a specific example of an architectural project and write a brief paragraph discussing how, during each stage of construction, various manual systems of measurement might be utilized to create a finished product.

Differentiated InstructionL1 STRUGGLING STUDENTS Have students produce a list of several tools used to manually take measurements. The list should include both the name of the tool and what quantity it is used to measure.

ELL ENGLISH LANGUAGE LEARNERS Have students print out pictures of tools commonly used measure and arrange them on a poster board. Students should write a description of the tool and people who might use such a tool for their work or everyday lives.

L3 ADVANCED STUDENTS Have students choose a trade that commonly utilizes computerized measuring tools and research how the systems of measurement have evolved over time in that trade. Encourage students to be creative in what format they use to display their findings.

Documents

3 Scientific Measurement Scientific M Planning Guide