This is the fourth Investigation of the LabLearner CELL Density. In this Investigation, students will determine the densities of several different solids.  They will use both mathematic formulas and volume displacement methods to determine volume.

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A. Begin the Investigation by encouraging students to recall what they discovered about density in the previous three Investigations.  Pose the following questions to prompt student discussion:

1. Ask students: Which property of solids and liquids have you been investigating? Students should indicate that they have been investigating density.

2. Ask students: What are some of the ways you have determined the density of solids and liquids? In Investigation One, students determined the density of liquids by combining two different liquids together.  They found that the less dense liquid floated on the surface of the liquid with the greater density.  Students also combined solids and liquids together in this Investigation.  They found that if a solid had a greater density than the liquid, the solid sank to the bottom of the liquid.  If the solid was less dense than the liquid, it floated.  In Investigation Two, a volume of water was measured and the mass of the volume was determined.  The density was then calculated by dividing the mass by the volume.  In Investigation Three, students determined the density of rubbing alcohol and vegetable oil.  The volume of each liquid was measured and the mass of the volume was determined.  The density was then calculated by dividing the mass by the volume.

3. Ask students: Which two factors affect density? Students should indicate that mass and volume affect density.  As the volume of a substance is increased or decreased, the mass of the substance increases or decreases by the same magnitude.  The ratio of mass to volume, the density, is unchanged.

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B. Continue this Investigation by broadening the discussion to include the three states of matter: gases, liquids, and solids.  To facilitate discussion, divide the class into groups, and allow students to brainstorm as each question is asked.  As each state is discussed, encourage students to predict what each particle diagram will look like for gases, liquids, and solids.

1. Initiate a discussion with students about some of the characteristics, or properties, of each of the three states of matter.

a. Ask students:  What are some of the characteristics of gases? Student answers may vary.  Students may describe gases as being light, transparent, or weightless.

b. Ask students:  What are some of the characteristics of liquids? Student answers may vary.  Students may describe liquids as wet, runny, and being able to spread and flow without a definite shape.

c. Ask students:  What are some of the characteristics of solids? Student answers may vary.  Students may describe solids as hard with a definite shape.

d. Ask students:  Do you think gases have density? Student answers may vary. 

Encourage students to realize that gases have density because all matter has the property of density. You can also talk about the different densities of air at different temperatures, like a hot air balloon or cool air sinking to the basement and warm air floating upstairs or to the attic.

e. Ask students:  Do you think liquids have density? Student answers may vary.  Encourage students to realize that, as shown in Investigations Two and Three, liquids have density.

f. Ask students: Do you think solids have density? Student answers may vary.  Encourage students to realize that, as shown in Investigation One, solids have density (metal cube in water).

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2. Explain to students that the amount of space between the particles in the three states of matter determines each of their densities.  Ask students to predict the arrangement of the particles that compose all three states of matter.

a. Ask students:  Of the three states of matter, which do you think has the highest density?  The lowest density? Student answers may vary.

b. Ask students:  Think about the characteristics of gases.  Do you think that the particles that compose gases are close together or far apart? Student answers may vary. Ask students to draw what the particle diagram for a gas would look like. 

Encourage students to draw their diagram in Problem 1a of their Student Data Record.

c. Ask students:  Think about the characteristics of liquids.  Do you think that the particles that compose gases are close together or far apart? Student answers may vary.

Ask students to draw what the particle diagram for a liquid would look like and encourage students to draw their diagram in Problem 1b of their Student Data Record.

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3. Ask students to begin to think about how they would determine the densities of gases, liquids, and solids.

a. Ask students: In Investigations Two and Three, how did you measure the density of liquids? Student answers may vary.  Students should indicate that they poured the liquids into a graduated cylinder to measure volume and then determined the mass of the liquid in the graduated cylinder.  Density was calculated by dividing the mass by the volume.

b. Ask students: Do you think that you can measure the density of a gas?  How do you think you could do it?  Student answers may vary.  Encourage students to think about how the mass and volume of a gas could be calculated.  An example would be as follows:  A balloon of known mass is filled with the gas.  The mass of the balloon and the gas is determined and the mass of the empty balloon is subtracted to yield the mass of the gas.  The density is then calculated by dividing the mass by the volume of the balloon.

c. Ask students: Do you think that the density of a solid can be measured?  How do you think you could do it? Student answers may vary.  Students should indicate that they would need to measure the mass and volume of the solid.

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Note: The length, width and height of the rectangular solid book on the slide may be different than the one measured in class. Use these dimensions as practice.

C. Begin a discussion of how the volume of a solid can be determined.  Tell students that measuring the volume of a solid can be done in two different ways.

1. Explain to students that the first method to measure the volume of a solid uses a tool from the Procedural Toolbox called Determination of Volume Using a Mathematical Formula.  The formula uses three dimensions of the solid object.  Since solid objects may have different shapes (rectangular, spherical, triangular), their dimensions will be different.  This may require that each differently shaped solid object have a different formula.

a. Show students a textbook.  Explain that an object, such as a textbook, can be identified as a rectangular solid because of its shape.  The shape of a rectangular solid or similarly shaped object is determined by sides that are parallel to each other.

• Ask students:  What three dimensions determine the shape of a rectangular solid? Students should indicate that a rectangular solid is determined by length, height, width.  Encourage students to realize that in the metric system, each of the three dimensions can be measured in centimeters.

• Ask students:  How would you find the volume of a rectangular solid? Student answers may vary.  Students should indicate that the volume of a rectangular solid or similarly shaped object can be determined by multiplying the three dimensions together: length (cm) × width (cm) × height (cm).

• Ask a student volunteer to use a metric ruler to measure the three dimensions of a textbook in centimeters. Write the three dimensions on the board including the units of centimeter (cm).  

Encourage students to copy the dimensions in Problem 2a of their Student Data Record. 

• Ask students to multiply the three dimensions in Problem 2a.  Tell students that the units used to derive the volume of the book are cm³.

• Ask students:  Do you think you could use this information to help you calculate the density of this textbook?  Do you need any additional information to calculate the density? Students should indicate that the volume that they calculated could be applied to the formula for density, density = mass ÷ volume.  However, in order to calculate the density, they would need to know the mass of the textbook.  For the purpose of this problem, tell students that the textbook has a mass of 1.9 kg or 1900 g.

• Encourage students to complete Problem 2a of their Student Data Record.

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Note: The radius of the sphere on the slide may be different than the one measured in class. Use these dimensions as practice.

b. Tell students that they will have to determine the volume of a sphere in this investigation.  Use a ball as an example of a sphere.

• Explain to students that in order to calculate the volume of a sphere, they will have to use the formula:  

Volume of sphere = 4.2 × radius × radius × radius.

• Draw a circle on the board. Place a dot in the center of the circle. Ask a student volunteer to come to the board and indicate how to find the radius of the circle.

Encourage students to draw the radius on the circle in Problem 2b of their Student Data Record.

• Ask students:  How would you measure the radius of a sphere? Student answers may vary.  Students should indicate that they could use a ruler to measure from one edge of the sphere into the center.

• Ask a student volunteer to use a metric ruler to measure the radius of the ball. Write the measurement on the board including the units of centimeter (cm). Encourage students to copy the dimensions in Problem 2b of their Student Data Record.

• Ask students to use the radius of the ball to calculate the volume of the ball in Problem 2b. Tell students that the units used to derive the volume of the ball are cm³.

• Ask students:  Do you think you could use this information to help you calculate the density of this ball? Do you need any additional information to calculate density? Students should indicate that the volume that they calculated could be applied to the formula for density, density = mass ÷ volume. However, in order to calculate the density, they would need to know the mass of the ball.  For the purpose of this problem, tell students that the mass of the ball is 30.9 g.

• Encourage students to complete Problem 2b of their Student Data Record.

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2. Tell students that the second method of determining the volume of a solid object uses a tool from the Procedural Toolbox called Determination of Volume Using the Volume Displacement Method.

a. Explain to students that in this method, a solid object, such as a steel marble, is added to a graduated cylinder that contains a known volume of water.

b. Ask students: What do you think will happen to the level of the water after the steel marble is added?

Student answers may vary.  Encourage students to realize that the steel marble displaces the water and raises the level of the water in the graduated cylinder so it appears that the volume of the water has increased.

c. The increase in the water volume is equal to the volume of the solid object.

d. Draw a diagram of a graduated cylinder on the board to show this concept.

e.  Tell students to locate the Table in Problem 2c of their Student Data Record.  This Table contains data for the volume of a steel marble using the volume displacement method. Encourage students to calculate the volume of the steel marble.

f. Ask students:  After you determine the volume of the steel marble, do you think you could use this information to help you calculate the density of the steel marble? Students should indicate that the volume they calculate could be applied to the formula for density, density = mass ÷ volume.  However, in order to calculate the density, they would need to know the mass of the steel marble.  For the purpose of this problem, tell students that the mass of the steel marble is 28.2 g.

g. Encourage students to complete Problem 2c of their Student Data Record.

h.  Explain to students that when using the volume displacement method, volume is expressed as milliliter (ml).  When using the formula for calculating volume, volume is expressed as cubic centimeter (cm³).  In the metric system, ml and cm³ are the same when measuring volume.

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Archimedes

Archimedes was a scientist who lived in Greece around 300BC. A story goes that he was assigned a problem to solve by the King. Archimedes had to determine the volume of an irregularly-shaped solid object (the King’s crown). 

One day, his bath was filled too full – right to the very brim. When Archimedes got into the tub, water overflowed and he immediately realized that the water that overflowed water was equal to his body’s volume. He discovered volume displacement! 

It is also told that upon making this discovery, he was so excited he shouted aloud “Eureka, Eureka!“. Since that time, many people, not just scientists, will say the word “Eureka” when they figure out something important.

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D. Explain to students that in this Investigation, they will determine the density of several solid objects using both methods of determining volume.  Their results will answer the following questions:

• Can the volume of all solids be measured using both the formula for measuring volume and the volume displacement method?
• Do all solids possess the same density?
• Does the density of solids depend on the size of the sample?