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. 1 of 11
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!!! 2 of 11
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 3 of 11
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. 4 of 11
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. 5 of 11
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. 6 of 11
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. 7 of 11
Novice 8 of 11
Apprentice 9 of 11
Practitioner 10 of 11
Expert 11 of 11