Archimedes Takes a Bath (or, How
Can You Find the Density of Two
Different Objects?)
As the story goes, King Argo was given 3 pounds of gold. The
king decided to use this gold to have a new crown made. He
sent for his master crownmaker who took the gold and made a
beautiful crown for the king. The king was very happy when he
saw the crown, but when he put it on he was not sure if
everything was right. He sent for his famous mathematician,
Archimedes, and said, “Listen, Arch, I’ve got this problem. I
gave the crownmaker 3 pounds of pure gold to make me a
crown, but this thing feels a little strange. I also noticed that he
just bought himself 200 new goats, and I’m a little suspicious
that he may have stolen some of the gold. I want you to figure
out if he did or not. If you can figure it out, I will give you
however much gold is left out of the crown.” Archimedes was
puzzled by this problem. The first thing he did upon returning
home was to have a large bath drawn, because he loved to
ponder problems in the bathtub. As he slipped into the bathtub
he came up with a wonderful solution and yelled “EUREKA!!”
as he ran to his laboratory to work out the problem. After an
hour in the lab he had his answer ready for the king. He found
out that although the crown weighed 3 pounds, it was not
entirely made of gold. He realized that the crown was made of
materials other than gold. Archimedes was able to do all of this
without harming the crown.
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In this activity, you will play the part of Archimedes and will
figure out the solution to this problem. Using the available
materials - 25 grams of gold (clay), Styrofoam, corks, a metal
washer and other materials, a balance beam and graduated
cylinders - create a demonstration for the king that will illustrate
how to tell the difference between a crown made of pure gold
and a crown made with other materials.
The king liked your demonstration and would like you to write
up your results in a memo for his brother, who recently had a
similar problem with his crownmaker. In your memo, explain
step-by-step on the back of your sheet how you can determine
if his crown is pure gold or not. Use illustrations when possible.
GOOD LUCK!!!
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Archimedes Takes a Bath (or, How Can You Find the
Density of Two Different Objects?)
Suggested Grade Span
6–8
Task
As the story goes, King Argo was given 3 pounds of gold. The king decided to use this gold to
have a new crown made. He sent for his master crownmaker who took the gold and made a
beautiful crown for the king. The king was very happy when he saw the crown, but when he put
it on he was not sure if everything was right. He sent for his famous mathematician,
Archimedes, and said, “Listen, Arch, I’ve got this problem. I gave the crownmaker 3 pounds of
pure gold to make me a crown, but this thing feels a little strange. I also noticed that he just
bought himself 200 new goats, and I’m a little suspicious that he may have stolen some of the
gold. I want you to figure out if he did or not. If you can figure it out, I will give you however
much gold is left out of the crown.” Archimedes was puzzled by this problem. The first thing he
did upon returning home was to have a large bath drawn, because he loved to ponder problems
in the bathtub. As he slipped into the bathtub he came up with a wonderful solution and yelled
“EUREKA!!” as he ran to his laboratory to work out the problem. After an hour in the lab he had
his answer ready for the king. He found out that although the crown weighed 3 pounds, it was
not entirely made of gold. He realized that the crown was made of materials other than gold.
Archimedes was able to do all of this without harming the crown.
In this activity, you will play the part of Archimedes and will figure out the solution to this
problem. Using the available materials - 25 grams of gold (clay), Styrofoam, corks, a metal
washer and other materials, a balance beam and graduated cylinders - create a demonstration
for the king that will illustrate how to tell the difference between a crown made of pure gold and
a crown made with other materials.
The king liked your demonstration and would like you to write up your results in a memo for his
brother, who recently had a similar problem with his crownmaker. In your memo, explain stepby-step on the back of your sheet how you can determine if his crown is pure gold or not. Use
illustrations when possible. GOOD LUCK!!!
Big Idea and Unifying Concept
Cause and effect
Physical Science Concept
Properties of matter
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Mathematics Concepts
Algebra
Data collection, organization and analysis
Graphs, tables and representations
Measurement
Time Required for the Task
One to two class periods.
Context
Throughout this unit on matter, students have been investigating atoms and molecules,
chemical and physical changes, and phases and properties of matter. This problem solving
activity builds on density experiments that involve layering different liquids. Students have
developed a basic understanding of the formula for density, and this task is designed to
illustrate this formula.
What the Task Accomplishes
"Archimedes Takes a Bath" is designed so that students can develop and implement a strategy
to determine the difference in density between two objects. This activity will require proficiency
in determining mass and volume to reach the value of density. This task will cover Vermont
Framework Standards: 7.10 cc, “Create and use a variety of approaches” and “determine how
to break down a complex problem into simpler parts”; and 7.12, “Observe and measure
characteristic properties of matter (in this case mass), volume and density, and use them to
distinguish one substance from another.”
How the Student Will Investigate
The experiment begins with the background story about King Argo’s crown and the
commissioning of Archimedes to decipher how much of the crown was made of gold. After
reading the story, students set up their labs. They use the materials to fashion two crowns and
conduct their tests.
Interdisciplinary Links and Extensions
Language Arts
Students may wish to write their own riddles and problem-solving stories for classmates to
solve. This could be done individually or in teams as a culminating activity to review concepts
learned in this unit of study.
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Science
Determining the purity of a substance, amalgamations of metals, use of basic laboratory
equipment. Other density and displacement activities are natural extensions of this task.
Social Studies
Science throughout history: Explore how original theories and solutions were discovered.
Teaching Tips and Guiding Questions
In learning how to measure volume by displacement, students often forget to record the starting
point of the water in a graduated cylinder before they add the object to be measured. As a
result, they record the volume as the total volume in the graduated cylinder rather than the
amount the water in the cylinder moved up. Students sometimes spend too much time
fashioning a nice-looking crown using the clay, rather than spending time making two crowns of
the same mass (one with just clay and one with some non-clay material added). Encourage
them to make both crowns at the same time. Ask the students: "How will you make sure that the
king can't tell by looking or by mass that the false crown is not make of pure clay?"
Concepts to be Assessed
(Unifying concepts/big ideas and science concepts to be assessed using the Exemplars
Science Rubric under the criterion: Science Concepts and Related Content)
Physical Science – Properties of Matter: Students understand that density is a property of
matter, that different substances have different densities, and that combining different
substances will change the density.
Scientific Method: Students understand the terms mass, density, volume, and displacement
appropriately. They describe cause-effect relationships with some justification, using data and
prior knowledge, and observe and explain reactions when variables are controlled.
Mathematics: Students represent and analyze data appropriately. Students identify trends and
patterns and use numerical data and precise measurements in describing events, answering
questions, providing evidence for scientific explanations and challenging misconceptions.
Skills to be Developed
(Science process skills to be assessed using the Exemplars Science Rubric under the criteria:
Scientific Procedures and Reasoning Strategies and Scientific Communication Using Data)
Scientific Method: Observing, predicting/hypothesizing, collecting and recording data,
manipulating of materials, measuring mass and volume, using scientific tools, drawing
conclusions, communicating findings, challenging misconceptions and raising new questions.
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Other Science Standards and Concepts Addressed
Scientific Method: Students describe, predict, investigate and explain phenomena. Students
control variables.
Scientific Theory: Students look for evidence that explains why things happen and modify
explanations when new observations are made.
Physical Science – Properties of Matter: Students describe and sort objects and materials
according to observations of similarities and differences of physical properties.
Scientific Tools:Students use computers to organize, analyze and interpret data.
Suggested Materials
•
•
•
•
•
25 grams of gold (clay)
Styrofoam
Corks
Metal washers
Balance beam and graduated cylinders
Possible Solutions
In the first part of the task, students create two crowns (one of pure clay and one of clay mixed
with other materials) that both mass 15 grams. Teachers can verify that the two crowns look
similar and mass the same. The second part of the task asks students to write a step-by-step
procedure with illustrations. The procedure needs to be complete and accurate, detailing how to
determine the density of each crown by measuring mass and volume. The use of displacement
in order to measure volume should be clearly described. It is possible to solve this problem
without calculating density, because equal masses are being used. Simply comparing the
volumes will help determine if the crown is a fake.
Task-Specific Assessment Notes
Novice
The student appears to have some grasp of the elements needed to complete the problem but
ius not able to successfully document the procedure for carrying out the task. No mention is
made of how to determine mass or density. No drawing is made.
Apprentice
This student seems to have a basic understanding of the concepts but fails to fully articulate the
procedure and results in writing. There is some confusion regarding how much material is to be
included (one or three pounds). However, there is some description of how to use displacement
to determine volume. The student’s writing does not include all steps necessary to complete the
task – no mention is made of how to determine density. Drawings are unclear.
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Practitioner
This student exhibits understanding of the concepts in the task and articulates them
successfully in writing. The given solution has all the necessary steps and procedures written
out clearly and completely. The drawings are clear and well-labeled.
Expert
This student articulates a solution in a clear and concise way. General principles that arise from
the specific task are mentioned: "Because metals have a different density, their volumes
change." Equal weights are mentioned subsequently. The illustrations could stand out on their
own as a means of articulating the task's results. The student also entertained the possibility
that the volume of the fake crown could be either greater or less than the volume of the real
crown depending on what was in the fake crown.
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Novice
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Apprentice
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Practitioner
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Expert
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