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RFoundation Lessons

Numbers in ScienceExploring Measurements, Signifi cant Digits, and Dimensional Analysis

About this Lesson

This lesson is an introductory activity for proper measuring techniques, the correct use of signifi -cant digits, and dimensional analysis. Students are asked to gather data on a cube and a sphere using proper metric measuring techniques and signifi cant digits. The students use the data to calculate volume, circumference, diameter, and density.

This lesson is included in the LTF Middle Grades Module 2.

Objectives

Students will:• Be introduced to proper measurement techniques, the correct use of signifi cant digits,

and dimensional analysis• Take dimensions of and identify signifi cant digits for a cube and a sphere• Calculate the volume and density of a cube and a sphere• Calculate the circumference and diameter of a sphere• Use dimensional analysis to make conversions

Level

All

Common Core State Standards for Science Content

LTF Science lessons will be aligned with the next generation of multi-state science standards that are currently in development. These standards are said to be developed around the anchor docu-ment, A Framework for K–12 Science Education, which was produced by the National Research Council. Where applicable, the LTF Science lessons are also aligned to the Common Core Stan-dards for Mathematical Content as well as the Common Core Literacy Standards for Science and Technical Subjects.

Code Standard Level of Thinking

Depth of Knowledge

(LITERACY)RST.9-10.3

Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to spe-cial cases or exceptions defi ned in the text.

Apply II

(MATH)A-CED.4

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Apply II

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RTeacher Overview – Numbers in Science

Code Standard Level of Thinking

Depth of Knowledge

(MATH)N-Q.1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in for-mulas; choose and interpret the scale and the origin in graphs and data displays.

Apply II

Connections to AP*

Students are expected to report measurements and perform calculations with the correct number of signifi cant digits. *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product.

Materials and Resources

Each lab group will need the following:

Assessments

The following types of formative assessments are embedded in this lesson:• Visual assessment of measuring techniques used within the lesson

The following assessments are located on the LTF website:• Short Lesson Assessment: Numbers in Science• Introduction to the Science Classroom Assessment• 2008 6th Grade Posttest, Free Response Question 1

apronsbalancebeaker, 250 mLgogglesgraduated cylinder, 100 mL, plastic

paper towelsdiemarbleruler, clear metricstring

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Teaching Suggestions

This lesson is designed to introduce or reinforce accurate measurement techniques, the cor-rect use of signifi cant digits, and dimensional analysis. Dimensional analysis is also called the “Factor-Label” method or “Unit-Label” method, and is a technique for setting up problems based on unit cancellations. Lecture as well as guided and independent practice of these topics should precede this activity. Students should be provided with reference tables containing metric and standard conversion factors.

The purpose of signifi cant digits is to communicate the accuracy of a measurement as well as the measuring capacity of the instrument used. Remind students repeatedly to take measurements including an estimated digit and to perform their calculations with the correct number of signifi -cant digits. Emphasize that points will be deducted for answers containing too many or too few signifi cant digits. The correct number of signifi cant digits to be reported by your students will depend entirely upon your equipment.

Small wooden alphabet blocks or dice should be inexpensive and easy to obtain. Be sure to fi nd a cube/graduated cylinder combination that ensures total submersion of the cube because its volume will be determined by water displacement. If the chosen cube or sphere fl oats, forceps can be used to gently submerge the o bject just under the surface of the water.

Spherical objects could be a marble or small rubber ball. Again, be sure to check the sphere/cylinder size to ensure that total submersion of the sphere is possible.

Provide students with a length of string and metric ruler or a fl exible tape measure. The string can be wrapped around the sphere, marked, and then removed and measured.

v. 2.0, 2.0, 2.0

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Answer Key

Data and Observations

Table 1. Measurements and Signifi cant DigitsCube Data

Mass (g) 15.05 (4 sd)

Dimensions (cm) Length Width Height3.68 (3 sd) 3.65 (3 sd) 3.67 (3 sd)

Volume (mL) Initial FinalBeaker 100 (1 sd) 150 (2 sd)

Graduated cylinder 175.0 (4 sd) 225.1 (4 sd)

Sphere DataMass (g) 19.38 (4 sd)

Dimensions (cm) Circumference7.62 (3 sd)

Volume (mL) Initial FinalBeaker 100 (1 sd) 110 (2 sd)

Graduated cylinder 175.0 (4 sd) 182.3 (4 sd)

Formulae for Calculating…Volume of a cube V = length × width × height

Circumference of a circle C = πd

Diameter of a circle d = 2r

Volume of a sphere34

3V r

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Answer Key (continued)

Exercise 1

Left: 5.75 mL

Middle: 3.0 mL

Right: 0.33 mL

Analysis

1. The number of signifi cant digits will be determined by the equipment you are using.

2. a. 1000 mg15.05 g 15,050 mg1g

b. 1 lb 16 oz15.05 g 0.5304 oz

454 g 1 lb

3. V = l × w × h = 3.68 cm × 3.65 cm × 3.67 cm = 49.3 cm3

4. 3 –5 31 m 1 m 1 m49.3 cm 4.93 10 m100 cm 100 cm 100 cm

5. V = Vfi nal – Vinitial = 150 mL – 100 mL = 50 mL = 50 cm3

6. V = Vfi nal – Vinitial = 225.1 mL – 175.0 mL = 50.1 mL = 50.1 cm3

7. a. 33

15.05 g 0.300 g/cm50.1cm

D

b. 33

15.05 g 0.3 g/cm50 cm

D

c. 33

15.05 g 0.305 g/cm49.3 cm

D

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8. a. 33

g 1 kg 100 cm 100 cm 100 cm0.305 305 kg/mcm 1000 g 1 m 1 m 1 m

b. 33


c. 33


The bar above the last zero of the number 300 communicates it is a signifi cant zero, transforming the recorded answer from one signifi cant digit to three.

It is equally appropriate to teach your students to use scientifi c notation to effectively communicate three signifi cant digits. The number could be correctly written as 3.00 × 102.

Another way to communicate a number accurate to the one’s position is to use a decimal at the end of the number. The number could be written as 300., representing that this measurement is accurate to the last digit.

9. a. 1 kg19.38 g 0.01938 kg

1000 g

b. 1 lb19.38 g 0.04269 lbs

454 g

10. C = πd7.62 cm 2.43 cm

3.14Cd

11. d = 2r2.43 cm 1.22 cm

2 2dr

12. 3 3 34 4 ( )(1.22) 7.61cm3 3

V r

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13. V = Vfi nal – Vinitial = 110 mL – 100 mL = 10 mL = 10 cm3

14. V = Vfi nal – Vinitial = 182.3 mL – 175.0 mL = 7.3 mL = 7.3 cm3

15. a. 33

19.38 g 2.55 g/cm7.60 cm

D

b. 33

19.38 g 2 g/cm10 cm

D

c. 33

19.38 g 2.7 g/cm7.3 cm

D

16. a. 3 3

33

2.55 g 1 lb 2.54 cm 12 in 159 lbs/ft1cm 454 g 1in 1 ft

b. 3 3

33

2 g 1 lb 2.54 cm 12 in 100 lbs/ft1cm 454 g 1in 1 ft

c. 3 3

33

2.7 g 1 lb 2.54 cm 12 in 170 lbs/ft1cm 454 g 1in 1 ft

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Conclusion Questions

1. The density of the cube has three signifi cant digits when measured with the ruler. After subtracting to fi nd the difference between the initial and fi nal water levels in the graduated cylinder and beaker, there are two signifi cant digits when measured with the graduated cylinder but only one signifi cant digit when measured with the beaker.

The ruler is the more accurate measure of the volume when compared to the volume obtained by water displacement using the graduated cylinder. Any instrument used to submerge the cube will contribute a small amount to the volume recorded because it contributes to the total amount of water displaced. See if your students can discover this concept.

Student answers may vary in signifi cant digits depending on the equipment used.

2. The calculated density of the cube would increase. Measuring a wet block will make the mass appear greater. Because mass is in the numerator of the equation

massvolume

D

the density value reported will be too great.

3. The density of the sphere would increase. If the student measured the circumference at any point other than the center, the circumference would be reported as too small. If the diameter is reported as too small,

C

d d

the radius will thus be reported as too small. If the radius is reported as too small,

2d

r r

the volume will thus be reported as too small. If the volume is reported as too small,

343

V r V

the density will thus b e reported as too great,

mD DV

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Foundation Lessons

Numbers in ScienceExploring Measurements, Signifi cant Digits, and Dimensional Analysis

The accuracy of a measurement depends on two factors: the skill of the individual taking the measurement and the capacity of the measuring instrument. When taking measurements, you should always read to the smallest mark on the instrument and then estimate another digit beyond that.

For example, if you are reading the length of the steel pellet pictured in Figure 1 using only the ruler shown to the left of the pellet, you can confi dently say that the measurement is between 1 and 2 centimeters. However, you must also include one additional digit estimating the distance between the 1 and 2 centimeter marks. The correct measurement for this ruler should be reported as 1.4 or 1.5 centimeters. It would be incorrect to report this measurement as 1 centimeter or even 1.45 centimeters given the scale of this ruler.

What if you are using the ruler shown on the right of the pellet? What is the correct measurement of the steel pellet using this ruler: 1.4 centimeters, 1.5 centimeters, 1.40 centimeters, or 1.45 cen-timeters? The correct answer would be 1.45 centimeters. Because the smallest markings on this ruler are in the tenths place, convention states we carry our measurement out to the hundredths place.

If the measured value falls exactly on a scale marking, the estimated digit should be zero. The temperature on the thermometer shown in Figure 2 should read 30.0°C. A value of 30°C would imply this measurement had been taken on a thermometer with markings that were 10° apart, not 1° apart.

The value 30°C represents anything that will round to the value 30. This means a value could fall between 29.5°C to 30.4°C, or a full 1° of possible error. Yet by including an additional digit, the number 30.0°C indicates a value between 29.95°C to 30.04°C, or a possible error of only 0.1°.

Centimeters

Centimeters

Figure 1. Measuring a steel pellet


Student Activity – Numbers in Science

Accuracy is important, so remember to always report measurements one decimal place past the accuracy of the measuring device. When using instruments with digital readouts, you should record all the digits shown. The instrument has done the estimating for you.

When measuring liquids in narrow glass graduated cylinders, most liquids form a slight dip in the middle. This dip is called a meniscus. Your measurement should be read from the bottom of the meniscus. Plastic graduated cylinders do not usually have a meniscus. In this case, you should read the cylinder from the top of the liquid surface.

Practice reading the volume contained in the three cylinders shown in Figure 3. Record your values in the space provided.

Left: __________________

Middle: __________________

Right: __________________

Figure 2. Reading a thermometer

Figure 3. Reading graduated cylinders



Signifi cant Digits

There are two types of numbers you will encounter in science, exact numbers and measured numbers. Exact numbers are known to be absolutely correct, and are obtained by counting or by defi nition. Counting a stack of 12 pennies is an exact number. Defi ning a day as 24 hours is an exact number. Exact numbers have an infi nite number of signifi cant digits.

As we have seen previously, measured numbers involve some estimation. Signifi cant digits are digits believed to be correct by the person making and recording a measurement. (We assume that the person is competent in their use of the measuring device.)

To count the number of signifi cant digits represented in a measurement, follow some basic rules:

1. If the digit is not a zero, it is signifi cant.

2. If the digit is a zero, it is signifi cant only if:

a. It is sandwiched between two other signifi cant digits; or

b. It terminates a number containing a decimal place.

Examples:

• 3.57 mL has three signifi cant digits (Rule 1)• 288 mL has three signifi cant digits (Rule 1)• 20.8 mL has three signifi cant digits (Rule 1, 2a)• 20.80 mL has four signifi cant digits (Rules 1, 2a, 2b)• 0.01 mL has only one signifi cant digit (Rule 1)• 0.010 mL has two signifi cant digits (Rule 1, 2b)• 0.0100 mL has three signifi cant digits (Rule 1, 2a, 2b)• 3.20 × 104 kg has three signifi cant digits (Rule 1, 2b)

Signifi cant Digits in Calculations

A calculated number can never contain more signifi cant digits than the measurements used to calculate it.

Calculation rules for signifi cant digits fall into two categories:

1. Addition and Subtraction: Answers must be rounded up or down to match the measurement with the least number of decimal places.

Example: 37.24 mL + 10.3 mL = 47.54 mL (calculator value), report as 47.5 mL

2. Multiplication and Division: Answers must be rounded up or down to match the measurement with the least number of signifi cant digits.

Example: 1.23 cm × 12.34 cm = 15.1782 cm2 (calculator value), report as 15.2 cm2



Dimensional Analysis

Throughout your study of science, it is important that a unit accompanies all measurements. Keeping track of the units in problems can help you convert one measured quantity into its equiv-alent quantity of a different unit, or help set up a calculation without the need for a formula.

In conversion problems, equality statements such as “1 foot = 12 inches” are made into fractions and then strung together in such a way that all units except the one desired are canceled out of the expression. Remember that defi ned numbers, such as 1 foot or 12 inches, are exact numbers and thus will not affect the number of signifi cant digits in your answer. This method is also known as the “Factor-Label” method or the “Unit-Label” method.

To set up a conversion problem, follow these steps:

1. Think about and write down all the “=” statements you know that will help you get from your current unit to the new unit.

2. Make fractions out of your “=” statements. There should be two fractions for each “=” and they will be reciprocals of each other.

3. Begin solving the problem by writing the given amount with units on the left and then choose the fractions that will let a numerator unit be canceled with a denominator unit, and vice versa.

4. Using your calculator, read from left to right and enter the numerator and denominator numbers in order. Precede each numerator number with a multiplication sign and each denominator number with a division sign. Alternatively, you could enter all of the numerators, each separated by a multiplication sign, and then all of the denominators, each separated by a division sign.

5. Round your calculator’s answer to the correct number of signifi cant digits based on the number with the least number of signifi cant digits in your original problem.

Example 1

How many inches are in 1.25 miles?

1ft 12in.1ft 12in. or 12in. 1ft

5280ft 1mile5280ft 1mile or1mile 5280ft

5280ft 12in.1.25mile 79,200 in.1mile 1ft



As problems get more complex, the measurements may contain fractional units or exponential units. To handle these situations, treat each unit independently. Structure your conversion factors to ensure that all the given units cancel out with a numerator or denominator as appropriate and that your answer ends with the appropriate unit. Sometimes information given in the problem is an equality that will be used as a conversion factor.

Squared and cubed units are potentially tricky. Remember that a square centimeter (cm2) is really cm × cm. If we need to convert square centimeters to square millimeters (mm2), we need to use the conversion factor of 1 cm = 10 mm twice so that both centimeter units cancel out.

Example 2

Suppose your automobile tank holds 23 gallons and the price of gasoline is 33.5¢ per liter. How many dollars will it cost you to fi ll your tank?

From a reference table, we fi nd 1 L = 1.06 qt and 4 qt = 1 gal. We should recognize from the problem that the price is also an equality (33.5¢ = 1 L) and we should know that 100¢ = $1.

Setting up the factors, we fi nd

4qt 1L 33.5¢ $123gal. $291gal. 1.06qt 1L 100¢

In your calculator, enter

23 × 4 ÷ 1.06 × 33.5 ÷ 100 = 29.0754717

However, because the given value of 23 gallons has only two signifi cant digits, your answer should be rounded to $29.

Example 3

One liter is exactly 1000 cm3. How many cubic inches are there in 1.0 liters?

We should know that 1000 cm3 = 1 L, and from a reference table we fi nd that 1 in. = 2.54 cm. Setting up the factors, we fi nd

31000(cm cm cm) 1in. 1in. 1in.1.0L 61 in.1L 2.54cm 2.54cm 2.54cm

(The answer must have two signifi cant digits because our given value 1.0 L contains two signifi -cant digits.)



As you become more comfortable with the concept of unit cancellation, you will fi nd that it is a very handy tool for solving problems. By knowing the units of your given measurements and by focusing on the units of the desired answer, you can derive a formula and correctly calculate an answer. This is especially useful when you have forgotten (or never knew) the formula.

Even though you may not know the exact formula for solving this problem, you should be able to match the units up in such a way that only your desired unit does not cancel out.

Example 4

What is the volume in liters of 1.5 moles of gas at 293 K and 1.10 atm of pressure? The ideal gas constant is 0.0821L atm

mol K

.

It is not necessary to know the formula for the ideal gas law to solve this problem correctly. Working from the constant (because it sets the units), we must cancel out every unit except liters. Doing this shows us that moles and Kelvin must be in the numerator and atmospheres in the denominator:

atm L(1.5 mol) 0.0821 (293 K)mol K 32.8 L, or 33 L

1.10 atmV

The answer is reported to two signifi cant digits because our least accurate measurement (1.5 mol) has only two signifi cant digits.

Note: Never rely on the number of signifi cant digits in a constant to determine the number of sig-nifi cant digits for reporting your answer. Consider only the number of signifi cant digits in given or measured quantities.



Purpose

In this activity, you will review some important aspects of numbers in science and then apply those number handling skills to your own measurements and calculations.

Materials

Each lab group will need the following:

Procedure

Remember that when taking measurements, it is your responsibility to estimate a digit between the two smallest marks on the measuring instrument.

1. Determine the mass of the small cube on a balance and record your measurement in Table 1 on your student answer page.

2. Measure dimensions (the length, width, and height) of the small cube in centimeters, being careful to use the full measuring capacity of your ruler. Record the lengths of each dimension.

3. Fill the 250 mL beaker with water to the 100 mL line. Carefully place the cube in the beaker. Record the new, fi nal volume of water. Remove and dry the cube.

4. Fill the large graduated cylinder three fourths full with water and record this initial water volume. While holding the graduated cylinder at an angle, gently slide the cube down the length of the graduated cylinder to submerge the cube. Record the fi nal water volume.

5. Measure the mass of the spherical object on a balance and record your measurement in Table 1.

6. Use the string to measure the widest part, or circumference, of the sphere. Mark the circumference on the string with a pen and the use the ruler to determine the value of the circumference in centimeters. Be careful to use the full measuring capacity of the ruler.

7. Fill the 250 mL beaker with water to the 100 mL line. Carefully place the spherical object in the beaker. Record the new, fi nal volume of water. Remove and dry the spherical object.

8. Fill the large graduated cylinder three fourths full with water. Record this initial water volume. While holding the graduated cylinder at an angle, gently roll the sphere down the length of the graduated cylinder to submerge the sphere. Record the fi nal water volume.

apronsbalancebeaker, 250 mLgogglesgraduated cylinder, 100 mL, plastic

paper towelsdiemarbleruler, clear metricstring



Data and Observations

Table 1. Measurements and Signifi cant DigitsCube Data

Mass (g)

Dimensions (cm) Length Width Height

Volume (mL) Initial FinalBeaker

Graduated cylinder

Sphere DataMass (g)

Dimensions (cm) Circumference

Volume (mL) Initial FinalBeaker

Graduated cylinder

Formulae for Calculating…Volume of a cube

Circumference of a circle

Diameter of a circle

Volume of a sphere



Analysis

Show your organized work on a separate piece of paper. Transfer your fi nal answers to the blanks beside each question. Staple your work to your answer sheet before turning it in to your teacher.

Remember to follow the rules for reporting all data and calculated answers with the correct number of signifi cant digits.

Table 2. Common ConversionsLength Mass Volume

Standard1 in. = 2.54 cm 1 lb = 16 oz 1 gal. = 4 qts

1 ft = 12 in. 1 qt = 2 pints1 mile = 5280 ft 1 pint = 2 cups

Standard to Metric1 mile = 1.61 km 1 lb = 454 g 1 L = 1.06 qts1 m = 1.09 yds 1 kg = 2.21 lbs 1 tsp = 5 mL

Metric1 m = 100 cm 1 g = 1000 mg 1 cm3 = 1 mL

1 m = 1000 mm 1 kg = 1000 g 1 L = 1000 mL1 km = 1000 m

1. For each of the measurements recorded in Table 1, indicate the number of signifi cant digits in parentheses after the measurement. For example, 15.7 cm (3 sd).

2. Use dimensional analysis to convert the mass of the cube to:

a. Milligrams (mg)

b. Ounces (oz)



Analysis (continued)

3. Calculate the volume of the cube in cubic centimeters (cm3).

4. Use dimensional analysis to convert the volume of the cube found in Question 3 from cubic centimeters (cm3) to cubic meters (m3).

5. Calculate the volume of the cube in mL as measured in the beaker. Convert the volume to cubic centimeters (cm3) using 1 cm3 = 1 mL.

6. Calculate the volume of the cube in mL as measured in the graduated cylinder. Convert the volume to cubic centimeters (cm3) using 1 cm3 = 1 mL.




7. Using the density formula

massvolume

D

calculate the density of the cube as determined by the following instruments:

a. Ruler

b. Beaker

c. Graduated cylinder

8. Use dimensional analysis to convert the three densities found in Question 7 into kg/m3.




9. Convert the mass of the sphere to the following units:

a. Kilograms (kg)

b. Pounds (lbs)

10. Using the measured circumference, calculate the diameter of the sphere in centimeters.

11. Calculate the radius of the sphere in centimeters.




12. Calculate the volume of the sphere in cubic centimeters (cm3) from its calculated radius.

13. Calculate the volume of the sphere in mL as measured in the beaker. Convert this volume to cubic centimeters (cm3) using 1 cm3 = 1 mL.

14. Calculate the volume of the sphere in mL as measured in the graduated cylinder. Convert this volume to cubic centimeters (cm3) using 1 cm3 = 1 mL.




15. Using the density formula

massvolume

D

calculate the density of the sphere as determined by the following instruments:

a. Tape measure

b. Beaker

c. Graduated cylinder

16. Use dimensional analysis to convert the three densities found in Question 15 into lbs/ft3.



Conclusion Questions

1. Compare the densities of the cube when the volume is measured by the ruler, beaker, and graduated cylinder. Which of these instruments gave the most accurate density value? Use the concept of signifi cant digits to explain your answer.

2. A student fi rst measures the volume of the cube by water displacement using the graduated cylinder. Next, the student measures the mass of the cube before drying it. How will this error affect the calculated density of the cube? Your answer must be justifi ed and should state clearly whether the calculated density will increase, decrease, or remain the same.

3. A student measures the circumference of a sphere at a point slightly above the middle of the sphere. How will this error affect the calculated density of the sphere? Your answer must be justifi ed and should state clearly whether the calculated density will increase, decrease, o r remain the same.

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