Foundation Lessons
Numbers in Science
Exploring Measurements, Significant Digits, and Dimensional Analysis
About this Lesson
This lesson is an introductory activity for proper measuring techniques, the correct use of significant digits, and dimensional analysis. Students are asked to gather data on a cube and a sphere
using proper metric measuring techniques and significant 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
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 document, 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 Standards for Mathematical Content as well as the Common Core Literacy Standards for Science and
Technical Subjects.
Code
Standard
(LITERACY)
RST.9-10.3
Follow precisely a multistep procedure when
carrying out experiments, taking measurements,
or performing technical tasks, attending to special cases or exceptions defined in the text.
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.
(MATH)
A-CED.4
Level of
Thinking
Apply
Depth of
Knowledge
II
Apply
II
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i
T E A C H E R
Students will:
• Be introduced to proper measurement techniques, the correct use of significant digits,
and dimensional analysis
• Take dimensions of and identify significant 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
Teacher Overview – Numbers in Science
Code
(MATH)
N-Q.1
Standard
Use units as a way to understand problems and
to guide the solution of multi-step problems;
choose and interpret units consistently in formulas; choose and interpret the scale and the
origin in graphs and data displays.
Level of
Thinking
Apply
Depth of
Knowledge
II
Connections to AP*
Students are expected to report measurements and perform calculations with the correct number
of significant 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:
paper towels
die
marble
ruler, clear metric
string
T E A C H E R
aprons
balance
beaker, 250 mL
goggles
graduated cylinder, 100 mL, plastic
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
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ii
Teacher Overview – Numbers in Science
Teaching Suggestions
This lesson is designed to introduce or reinforce accurate measurement techniques, the correct use of significant 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 significant 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 significant digits. Emphasize that points will be deducted for answers containing too many or too few
significant digits. The correct number of significant 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 find
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 floats, forceps
can be used to gently submerge the object just under the surface of the water.
T E A C H E R
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 flexible 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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iii
Teacher Overview – Numbers in Science
Answer Key
Data and Observations
Mass (g)
Table 1. Measurements and Significant Digits
Cube Data
15.05 (4 sd)
Dimensions (cm)
Length
3.68 (3 sd)
Width
3.65 (3 sd)
Height
3.67 (3 sd)
Beaker
Initial
100 (1 sd)
Final
150 (2 sd)
Graduated cylinder
175.0 (4 sd)
225.1 (4 sd)
Volume (mL)
Mass (g)
T E A C H E R
Sphere Data
19.38 (4 sd)
Circumference
7.62 (3 sd)
Dimensions (cm)
Beaker
Initial
100 (1 sd)
Final
110 (2 sd)
Graduated cylinder
175.0 (4 sd)
182.3 (4 sd)
Volume (mL)
Volume of a cube
Formulae for Calculating…
V = length × width × height
Circumference of a circle
C = πd
Diameter of a circle
d = 2r
Volume of a sphere
4
V   r3
3
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iv
Teacher Overview – Numbers in Science
Answer Key (continued)
Exercise 1
Left:
5.75 mL
Middle:
3.0 mL
Right:
0.33 mL
Analysis
1. The number of significant digits will be determined by the equipment you are using.
2. a. 15.05 g 
b. 15.05 g 
1000 mg
 15, 050 mg
1g
1 lb 16 oz

 0.5304 oz
454 g 1 lb
3
4. 49.3 cm 
T E A C H E R
3. V = l × w × h = 3.68 cm × 3.65 cm × 3.67 cm = 49.3 cm3
1m
1m
1m


 4.93 10 –5 m3
100 cm 100 cm 100 cm
5. V = Vfinal – Vinitial = 150 mL – 100 mL = 50 mL = 50 cm3
6. V = Vfinal – Vinitial = 225.1 mL – 175.0 mL = 50.1 mL = 50.1 cm3
7. a. D 
b. D 
c.
D
15.05 g
 0.300 g/cm3
3
50.1 cm
15.05 g
 0.3 g/cm3
3
50 cm
15.05 g
 0.305 g/cm3
49.3 cm3
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v
Teacher Overview – Numbers in Science
Answer Key (continued)
8. a. 0.305
b. 0.3
g
1 kg 100 cm 100 cm 100 cm




 305 kg/m3
3
cm 1000 g
1m
1m
1m
g
1 kg 100 cm 100 cm 100 cm




 300 kg/m3
3
cm 1000 g
1m
1m
1m
c. 0.300
g
1 kg 100 cm 100 cm 100 cm




 300 kg/m3
3
cm 1000 g
1m
1m
1m
The bar above the last zero of the number 300 communicates it is a significant zero,
transforming the recorded answer from one significant digit to three.
It is equally appropriate to teach your students to use scientific notation to effectively
communicate three significant digits. The number could be correctly written as
3.00 × 102.
9. a. 19.38 g 
1 kg
 0.01938 kg
1000 g
b. 19.38 g 
1 lb
 0.04269 lbs
454 g
10. C = πd
d
C


7.62 cm
 2.43 cm
3.14
11. d = 2r
r
d 2.43 cm

 1.22 cm
2
2
4
4
12. V   r 3  ( )(1.22)3  7.61 cm3
3
3
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vi
T E A C H E R
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.
Teacher Overview – Numbers in Science
Answer Key (continued)
13. V = Vfinal – Vinitial = 110 mL – 100 mL = 10 mL = 10 cm3
14. V = Vfinal – Vinitial = 182.3 mL – 175.0 mL = 7.3 mL = 7.3 cm3
15. a. D 
b. D 
c.
D
19.38 g
 2.55 g/cm3
3
7.60 cm
19.38 g
 2 g/cm3
10 cm3
19.38 g
 2.7 g/cm3
3
7.3 cm
3
3
3
3
3
T E A C H E R
16. a.
3
2.55 g 1 lb  2.54 cm   12 in 
3


 
  159 lbs/ft
3
1 cm
454 g  1 in   1 ft 
2g
1 lb  2.54 cm   12 in 
3


b.
 
  100 lbs/ft
3
1 cm 454 g  1 in   1 ft 
2.7 g 1 lb  2.54 cm   12 in 
3


c.
 
  170 lbs/ft
1 cm3 454 g  1 in   1 ft 
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vii
Teacher Overview – Numbers in Science
Answer Key (continued)
Conclusion Questions
1. The density of the cube has three significant digits when measured with the ruler. After
subtracting to find the difference between the initial and final water levels in the graduated
cylinder and beaker, there are two significant digits when measured with the graduated
cylinder but only one significant 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 significant 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
mass
volume
D 
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,
d
 r  r
2
the volume will thus be reported as too small. If the volume is reported as too small,
4
V   r3 V
3
the density will thus be reported as too great,
D
m
 D
V
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viii
T E A C H E R
the density value reported will be too great.
Foundation Lessons
Numbers in Science
Exploring Measurements, Significant Digits, and Dimensional Analysis
Centimeters
Centimeters
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.
Figure 1. Measuring a steel pellet
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 confidently 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 centimeters? 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°.
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1
Student Activity – Numbers in Science
Figure 2. Reading a thermometer
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 3. Reading graduated cylinders
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2
Student Activity – Numbers in Science
Significant 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
definition. Counting a stack of 12 pennies is an exact number. Defining a day as 24 hours is an
exact number. Exact numbers have an infinite number of significant digits.
As we have seen previously, measured numbers involve some estimation. Significant 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 significant digits represented in a measurement, follow some basic rules:
1. If the digit is not a zero, it is significant.
2. If the digit is a zero, it is significant only if:
a. It is sandwiched between two other significant digits; or
b. It terminates a number containing a decimal place.
Examples:
•
•
•
•
•
•
•
•
3.57 mL has three significant digits (Rule 1)
288 mL has three significant digits (Rule 1)
20.8 mL has three significant digits (Rule 1, 2a)
20.80 mL has four significant digits (Rules 1, 2a, 2b)
0.01 mL has only one significant digit (Rule 1)
0.010 mL has two significant digits (Rule 1, 2b)
0.0100 mL has three significant digits (Rule 1, 2a, 2b)
3.20 × 104 kg has three significant digits (Rule 1, 2b)
Significant Digits in Calculations
A calculated number can never contain more significant digits than the measurements used to
calculate it.
Calculation rules for significant 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 significant digits.
Example: 1.23 cm × 12.34 cm = 15.1782 cm2 (calculator value), report as 15.2 cm2
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3
Student Activity – Numbers in Science
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 equivalent 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 defined numbers, such as 1 foot or 12 inches, are exact numbers and
thus will not affect the number of significant 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 significant digits based on the
number with the least number of significant digits in your original problem.
Example 1
How many inches are in 1.25 miles?
1ft  12in. 
1ft
12in.
or
12in.
1ft
5280 ft  1mile 
1.25 mile 
5280 ft 1mile
or
1mile
5280 ft
5280 ft 12in.

 79,200 in.
1mile
1ft
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4
Student Activity – Numbers in Science
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 fill your tank?
From a reference table, we find 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 find
23gal. 
4 qt
1L
33.5¢ $1



 $29
1gal. 1.06 qt 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 significant 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 find that 1 in. = 2.54 cm.
Setting up the factors, we find
1.0 L 
1000 (cm  cm  cm)
1in.
1in.
1in.



 61 in.3
1L
2.54 cm 2.54 cm 2.54 cm
(The answer must have two significant digits because our given value 1.0 L contains two significant digits.)
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5
Student Activity – Numbers in Science
As you become more comfortable with the concept of unit cancellation, you will find 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
V
1.10 atm
The answer is reported to two significant digits because our least accurate measurement (1.5 mol)
has only two significant digits.
Note: Never rely on the number of significant digits in a constant to determine the number of significant digits for reporting your answer. Consider only the number of significant digits in given
or measured quantities.
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6
Student Activity – Numbers in Science
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:
aprons
balance
beaker, 250 mL
goggles
graduated cylinder, 100 mL, plastic
paper towels
die
marble
ruler, clear metric
string
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, final 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 final 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, final 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 final water volume.
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7
Student Activity – Numbers in Science
Data and Observations
Table 1. Measurements and Significant Digits
Cube Data
Mass (g)
Dimensions (cm)
Length
Volume (mL)
Width
Height
Initial
Final
Beaker
Graduated cylinder
Sphere Data
Mass (g)
Circumference
Dimensions (cm)
Initial
Volume (mL)
Final
Beaker
Graduated cylinder
Formulae for Calculating…
Volume of a cube
Circumference of a circle
Diameter of a circle
Volume of a sphere
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8
Student Activity – Numbers in Science
Analysis
Show your organized work on a separate piece of paper. Transfer your final 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 significant digits.
Length
1 in. = 2.54 cm
1 ft = 12 in.
1 mile = 5280 ft
1 mile = 1.61 km
1 m = 1.09 yds
1 m = 100 cm
1 m = 1000 mm
1 km = 1000 m
Table 2. Common Conversions
Mass
Standard
1 lb = 16 oz
Volume
1 gal. = 4 qts
1 qt = 2 pints
1 pint = 2 cups
Standard to Metric
1 lb = 454 g
1 kg = 2.21 lbs
1 L = 1.06 qts
1 tsp = 5 mL
Metric
1 g = 1000 mg
1 kg = 1000 g
1 cm3 = 1 mL
1 L = 1000 mL
1. For each of the measurements recorded in Table 1, indicate the number of significant 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)
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9
Student Activity – Numbers in Science
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.
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10
Student Activity – Numbers in Science
Analysis (continued)
7. Using the density formula
D
mass
volume
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.
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11
Student Activity – Numbers in Science
Analysis (continued)
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.
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
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Student Activity – Numbers in Science
Analysis (continued)
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.
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
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Student Activity – Numbers in Science
Analysis (continued)
15. Using the density formula
D
mass
volume
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.
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
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Student Activity – Numbers in Science
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 significant digits to explain your answer.
2. A student first 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 justified 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
justified and should state clearly whether the calculated density will increase, decrease, or
remain the same.
Copyright © 2012 Laying the Foundation®, Inc., Dallas, Texas. All rights reserved. Visit us online at www.ltftraining.org.
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