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Chapter 2
Measurement and Problem Solving
02-00-02un
Title
 Expressing a number in scientific notation
Caption
 Scientists use scientific notation as a convenient way to express very large or very small numbers.
Notes
 Students will need the definition of the decimal part (the number's significant figures, with a single nonzero digit to the left of the decimal point) and the exponent part (a sizing factor that is always a power of ten). It can be useful to relate the convenience of scientific notation to number entry on a calculator: Numbers like the one shown would be tedious, or even impossible, to enter on a calculator in regular notation, but easy to enter in scientific notation.
Keywords
 scientific notation, exponent, value
02-00-03un
Title
 Moving the decimal point left
Caption
 By manually moving a decimal point to the left, we can determine what the exponent in scientific notation should be.
Notes
 Students should learn the leftward movement for values >=10, but a rightward movement is necessary for numbers <1. Point out that many calculators will do the conversion from regular to scientific notation without the need to perform the counting steps outlined here.
Keywords
 decimal point, exponent, scientific notation
02-00-04un
Title
 Moving the decimal point to the right
Caption
 By manually moving a decimal point to the right, we can determine what the exponent in scientific notation should be.
Notes
 Students should learn the leftward movement for values >=10, but a rightward movement is necessary for numbers <1. Point out that many calculators will do the conversion from regular to scientific notation without the need to perform the counting steps outlined here.
Keywords
 decimal point, exponent, scientific notation
02-00-07un
Title
 All measurements include a degree of uncertainty
Caption
 Measurements are written so that all except the rightmost significant figure are known with certainty. The rightmost significant figure is always an estimate.
Notes
 Students may not know how uncertainty comes into measurement. It is helpful to point out that uncertainty arises from limitations in the design and craftsmanship of the instruments used to make the measurements. It is also helpful to argue the case for measurement: Despite the uncertainty in measurement, scientists generally gain a more accurate picture of reality through quantitative observations than through purely qualitative observations.
Keywords
 uncertainty, estimate, significant figures
02-04-03un
Title
 Example of addition of measurements
Caption
 When adding measurements, we round the answer so that its significant figures extend to the right only as far as they do in the measurement in which the significant figures extend least to the right.
Notes
 In this example, the significant figures in 5.74 extend to the 0.01's place; for both 0.823 and 2.651, the significant figures extend to the 0.001's place. The 5.74 therefore extends least to the right, and it will force the sum (9.214) to be rounded to the 0.01's place. By this action, we arrive at the properly rounded answer of 9.21 for this addition problem. Note that the rounding rule depends on the location of the significant figures in the measurements, not on how many significant figures are in the measurements.
Keywords
 significant figures, addition, rounding
02-04-04un
Title
 Example of subtraction of measurements
Caption
 When subtracting measurements, we round the answer so that its significant figures extend to the right only as far as they do in the measurement in which the significant figures extend least to the right.
Notes
 In this example, the significant figures in 4.8 extend to the 0.1's place; the significant figures in 3.965 extend to the 0.001's place. The 4.8 therefore extends least to the right, and it will force the difference (0.835) to be rounded to the 0.1's place. By this action, we arrive at the properly rounded answer of 0.8 for this subtraction problem. Note that the rounding rule depends on the location of the significant figures in the measurements, not on how many significant figures are in the measurements.
Keywords
 significant figures, subtraction, rounding
02-04-05un
Title
 Significant figures in addition and subtraction
Caption
Notes
Keywords
02-04-06un
Title
 Significant figures in addition and subtraction
Caption
Notes
Keywords
02-07-03h
Title
 An example of a conversion factor
Caption
 A conversion factor relates one unit to another through a numerical equivalence.
Notes
 Units tell us what has been measured, but often we have a choice of units by which to express the measurement. To facilitate communication between people making measurements, conversions have been worked out to allow us to translate measurements from one unit to another. Conversions are often employed in dimensional analysis problems, in which a measurement is multiplied by one or more conversion factors to express the measurement in different units than those originally read from the instrument's scale.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit
02-07-04i
Title
 Example of a solution map for a multistep conversion problem
Caption
 If we don't have a single conversion factor relating the original and desired units, we can still perform the conversion using more than one conversion factor, applied in a multistep sequence.
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation. Since factors are logically equivalent to unity, there is no limit to the number of factors that can be used in a multistep conversion problem. Students should be made aware that the factors themselves may contain measurements, and therefore may influence the answer's significant figures. Usually, however, the relationships expressed by the factors are exact, and as a result, will not affect the answer's significant figures.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, multistep
02-07-05un
Title
 Solution map for converting centimeters to inches
Caption
 We can convert from cm to inches by multiplying our original measurement, in cm, by the factor shown, using the dimensional analysis approach.
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for cm to inches, based on the known relationship between the units.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, centimeter, inch
02-07-06un
Title
 Solution map for converting kilometers to miles
Caption
 We can convert from km to mi by multiplying our original measurement, in km, by the factor shown, using the dimensional analysis approach.
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for km to mi, based on the known relationship between the units.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, kilometer, mile
02-07-07un
Title
 Solution map for converting meters to millimeters
Caption
 We can convert from m to mm by multiplying our original measurement, in m, by the factor shown, using the dimensional analysis approach.
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for m to mm, based on the known relationship between the units.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, meter, millimeter
02-07-08un
Title
 Multistep solution map for converting liters to cups
Caption
 We can convert from L to cu by multiplying our original measurement, in L, by the factors shown, using the dimensional analysis approach. Note that both factors must be used in the dimensional analysis setup!
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for L to cu, based on the known relationship between L and qt, and between qt and cu.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, liter, quart, cup
02-07-09un
Title
 Multistep solution map for converting kilometers to laps
Caption
 We can convert from km to laps by multiplying our original measurement, in km, by the factors shown, using the dimensional analysis approach. Note that both factors must be used in the dimensional analysis setup!
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for km to laps, based on the known relationship between km and mi, and between mi and laps.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, kilometer, mile
02-07-10un
Title
 Solution map for converting cubic centimeters (cm3) to cubic inches (in3)
Caption
 We can convert from cm3 to in3 by multiplying our original measurement, in cm3, by the factor shown, using the dimensional analysis approach. The factor was generated by cubing the relationship between cm and in: (2.54 cm)3 = (1 in)3 or 16.387 cm3 = 1 in3
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for cm3 to in3, based on the known relationship between the units.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, centimeter, inch, cubic centimeter, cubic inch, volume
02-07-11un
Title
 Solution map for converting square inches (in2) to square centimeters (cm2)
Caption
 We can convert from in2 to cm2 by multiplying our original measurement, in in2, by the factor shown, using the dimensional analysis approach. The factor was generated by squaring the relationship between in and cm: (1 in)2 = (2.54 cm)2 = or 1 in2 = 6.45 cm2.
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for in2 to cm2, based on the known relationship between the units.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, centimeter, inch, square centimeter, square inch, area
02-07-12un
Title
 Example of a multistep solution map for problems involving units raised to a power
Caption
 In this problem, we wish to convert from dm3 to in3 by multiplying our original measurement, in dm3, by the factors shown, using the dimensional analysis approach. Note that all factors must be used in the dimensional analysis setup, and all factors use powers of three.
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for dm3 to in3, based on the known relationship between the units. The factors were generated by cubing three relationships: 1) between m and dm; 2) between cm and m; 3) between cm and in.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, centimeter, inch, cubic centimeter, cubic inch, cubic meter, cubic decimeter, volume
02-07-14un
Title
 Solution map for calculating density
Caption
 Density is defined as the mass of a substance divided by its volume. If the mass and volume are known, we can calculate the density.
Notes
 The map is a flowchart designed to guide students in setting up an equation to solve for density.
Keywords
 unit, equation, mass, volume, density
02-07-15k
Title
 Clogged artery
Caption
 Too many low-density lipoproteins in the blood can lead to blocking of arteries.
Notes
 Here is an example of the relevance of density to the health sciences. Lipoproteins come in two varieties, low-density (LDL) and high-density (HDL). If the concentration of LDL is too high, and/or the concentration of HDL is too low, the person is at risk for heart disease, stroke, or other circulatory problems.
Keywords
 density, lipoprotein, HDL, LDL, artery, blood, heart disease
02-07-16un
Title
 Example of multistep solution map for converting grams to milliliters, using density as a factor
Caption
 We can convert from g to mL by multiplying our original measurement, in g, by the factors shown, using the dimensional analysis approach. In this example, the density is given as 1.32 g/cm3. Note that density appears as the first factor, and both factors must be used in the dimensional analysis setup!
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for g to mL, based on the known density of the substance, 1.32 g/cm3, and the relationship between mL and cm3. Students should be told that density works well as a factor because it has units in both the numerator and the denominator, just as a factor does.
Keywords
 unit, conversion, dimensional analysis, metric system, SI unit, centimeter, cubic centimeter, milliliter, gram, mass, volume, density
02-07-17un
Title
 Example of multistep solution map for converting kilograms to cubic centimeters, using density as a factor
Caption
 We can convert from kg to cm3 by multiplying our original measurement, in kg, by the factors shown, using the dimensional analysis approach. In this example, the density is given as 0.752 g/cm3. Note that density appears as the second factor, following a metric system conversion from kg to g.
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for kg to cm3, based on the known density of the substance, 0.752 g/cm3, and the relationship between kg and g. Students should be told that density works well as a factor because it has units in both the numerator and the denominator, just as a factor does.
Keywords
 unit, conversion, dimensional analysis, metric system, SI unit, centimeter, cubic centimeter, kilogram, gram, mass, volume, density
02-07-18un
Title
 Example of multistep solution map for converting kilograms to liters, using density as a factor (Example 2.15)
Caption
 We can convert from kg to L by multiplying our original measurement, in kg, by the factors shown, using the dimensional analysis approach. In this example, the density is given as 0.789 g/cm3. Note that density appears as the second factor, following a metric system conversion from kg to g, and preceding metric conversions from cm3 to mL, and mL to L .
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for kg to L, based on the known density of the substance, 0.789 g/cm3, and the relationships between kg and g, cm3 and mL, mL and L. Students should be told that density works well as a factor because it has units in both the numerator and the denominator, just as a factor does.
Keywords
 unit, conversion, dimensional analysis, metric system, SI unit, centimeter, cubic centimeter, milliliter, liter, kilogram, gram, mass, volume, density
02-07-19un
Title
 Example of solution map for calculating density (Example 2.16)
Caption
 Density is defined as the mass of a substance divided by its volume. If the mass and volume are known, we can calculate the density.
Notes
 The map is a flowchart designed to guide students in setting up an equation to solve for density. In this problem, the mass was given as 59.9 kg, and the volume was 57.2 L. The mass was converted to 5.59 x 104 g; the volume was converted to 5.72 x 103 cm3. Following the solution map, dividing the mass by the volume yielded a density of 0.977 g/cm3.
Keywords
 unit, equation, mass, volume, density, conversion, dimensional analysis, metric system, SI unit, cubic centimeter, milliliter, liter, kilogram, gram
02-07-20un
Title
 Moving the decimal point left (Example 2.18)
Caption
 By manually moving a decimal point to the left, we can determine what the exponent in scientific notation should be.
Notes
 Students should learn the leftward movement for values >=10, but a rightward movement is necessary for numbers <1. Point out that many calculators will do the conversion from regular to scientific notation without the need to perform the counting steps outlined here. In this example, the decimal point moves seven places to the left; the exponent is therefore +7.
Keywords
 decimal point, exponent, scientific notation
02-07-22un
Title
 Multistep solution map for converting feet to meters
Caption
 We can convert from ft to m by multiplying our original measurement, in ft, by the factors shown, using the dimensional analysis approach. Note that both factors must be used in the dimensional analysis setup!
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for ft to m, based on the known relationship between inches (in) and ft, and between m and in.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, meter, inch, foot
02-07-23un
Title
 Solution map for converting square kilometers (km2) to square meters (m2)
Caption
 We can convert from km2 to m2 by multiplying our original measurement, in km2, by the factor shown, using the dimensional analysis approach. The factor was generated by squaring the relationship between km and m: (1 km)2 = (1000 m)2 = or 1 km2 = 1 x 106 m2
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for km2 to m2, based on the known relationship between the units.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, meter, square meter, square kilometer, kilometer, area
02-07-25un
Title
 Example of multistep solution map for converting grams to liters, using density as a factor (Example 2.28)
Caption
 We can convert from g to L by multiplying our original measurement, in g, by the factors shown, using the dimensional analysis approach. In this example, the density is given as 0.84 g/mL. Note that density appears as the first factor, followed by a metric system conversion from mL to L.
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for g to L, based on the known density of the substance, 0.84 g/mL, and the relationship between mL and L. Students should be told that density works well as a factor because it has units in both the numerator and the denominator, just as a factor does.
Keywords
 unit, conversion, dimensional analysis, metric system, SI unit, milliliter, liter, gram, mass, volume, density
02-07-27n
Title
 Celsius thermometer with markings every 1oC
Caption
 A Celsius thermometer is a typical scientific instrument. With markings every degree, the user makes a measurement by reading the certain digits from the scale, then visually estimates to the nearest tenth of a degree.
Notes
 This instrument's scale can be used to show the origins of the estimate in a measurement. Students might be asked to compare the reading obtained from this thermometer with one having markings every ten degrees, or every tenth of a degree.
Keywords
 thermometer, Celsius, temperature, significant figure
02-07-28o
Title
 Graduated cylinder w/markings every 0.1 mL
Caption
 A graduated cylinder is a scientific instrument commonly used to measure volume. With markings every 0.1 mL, the user makes a measurement by reading the certain digits from the scale, then visually estimating to the nearest 0.01 mL.
Notes
 This instrument's scale can be used to show the origins of the estimate in a measurement. Students might be asked to compare the reading obtained from this cylinder with one having markings every mL, or every 0.01 mL.
Keywords
 graduated cylinder, milliliter, volume, significant figure
02-07-29p
Title
 Celsius thermometer with markings every 0.1oC
Caption
 A Celsius thermometer is a typical scientific instrument. With markings every 0.1 degree, the user makes a measurement by reading the certain digits from the scale, then visually estimates to the nearest 0.01 degree.
Notes
 This instrument's scale can be used to show the origins of the estimate in a measurement. Students might be asked to compare the reading obtained from this thermometer with one having markings every ten degrees, or every tenth of a degree, as is seen in record #30.
Keywords
 thermometer, Celsius, temperature, significant figure
02-07-30q
Title
 Graduated cylinder with markings every 10 mL
Caption
 A graduated cylinder is a scientific instrument commonly used to measure volume. With markings every 10 mL, the user makes a measurement by reading the certain digits from the scale, then visually estimating to the nearest mL.
Notes
 This instrument's scale can be used to show the origins of the estimate in a measurement. Students might be asked to compare the reading obtained from this cylinder with one having markings every mL, or every 100 mL. This is a fairly low-quality instrument, in that it yields few significant figures.
Keywords
 graduated cylinder, milliliter, volume, significant figure
02-07-31r
Title
 Burette with markings every 0.1 mL
Caption
 A burette is a scientific instrument commonly used to measure volume. With markings every 0.1 mL, the user makes a measurement by reading the certain digits from the scale, then visually estimating to the nearest 0.01 mL.
Notes
 This instrument's scale can be used to show the origins of the estimate in a measurement. Students might be asked to compare the reading obtained from this burette, and a graduated cylinder having markings every mL. A burette is commonly used to deliver a fairly precisely measured volume of liquid.
Keywords
 burette, milliliter, volume, significant figure
02-07-32s
Title
 Graduated pipette with markings every 1 mL
Caption
 A graduated pipette is a scientific instrument commonly used to measure small volumes of liquid. With markings every 1 mL, the user makes a measurement by reading the certain digits from the scale, then visually estimating to the nearest 0.1 mL.
Notes
 This instrument's scale can be used to show the origins of the estimate in a measurement. Students might be asked to compare the reading obtained from this pipette, and a graduated cylinder having markings every mL. A pipette is commonly used to deliver a small volume of liquid, typically under 10 mL.
Keywords
 pipette, milliliter, volume, significant figure
02-07-33t
Title
 Example of a digital balance
Caption
 A digital balance uses an electronic device to detect and report mass. The balance's circuits automatically display both the certain digits and the estimate when a measurement is made.
Notes
 Students might be asked to comment on why the balance pan is enclosed in a glass compartment. This discussion could develop into an exploration of the balance's sensitivity to air currents, and how this sensitivity is related to the quality of the measurements.
Keywords
 balance, mass, significant figure
02-07-38un
Title
 Solution map for converting inches to centimeters
Caption
 We can convert from inches to cm by multiplying our original measurement, in inches (in), by the factor shown, using the dimensional analysis approach.
Notes
 The map is a flowchart designed to guide students in setting up a dimensional analysis calculation for inches to cm, based on the known relationship between the units.
Keywords
 unit, conversion, dimensional analysis, English system, metric system, SI unit, centimeter, inch
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