## DENSITY DERBY Grades 8-10

In this lesson, the students will explore the relationship between mass, volume and density. They will use model boats to discover how changing mass and volume affect density. Previous knowledge should include an understanding of mass and how to measure mass on a triple beam balance. Some previous experience calculating volume of regular solids and irregular solids by displacement would be helpful for the extension activities. The students should have the math skills to multiply and divide using decimals. This investigation will take 2-3 days to complete. Pre-viewing activities, the video and making the boats can take one class period. Testing the boats and discussing results can take another class period or two depending on how may trials the students perform with their boats.
Eureka #25: Volume and Density

Students will be able to:
1. identify two methods for changing density.
2. demonstrate how the density of an object
affects floatation.
3. predict and demonstrate how changes in mass and volume affect density and floatation.

Texas Assessment of Academic Skills (TAAS) Objectives
Science Objectives, Grade 8:
#1: The student will demonstrate the ability to acquire scientific data and/or information.
#3: The student will demonstrate the ability to communicate scientific data and/or information.
#5: The student will demonstrate the ability to make inferences, form generalized statements, and/or make predictions, using scientific data and/
or information.
Math Objectives, Exit Level:
#1: The student will demonstrate an understanding of number concepts.
#2: The students will demonstrate an understanding of mathematical relations, functions, and other algebraic concepts.
#4: The student will demonstrate an understanding of measurement concepts using metric and customary units.
#6-9: The student will use the operations of addition, subtraction, multiplication, and division to solve problems
#11: The student will determine solution strategies and will analyze or solve problems.

NCTM Standards for Grades 9-12:
Standard 1: Mathematics as Problem Solving
Standard 2: Mathematics as Communications
Standard 3: Mathematics as Reasoning

Per group of 4 students:
• 2 pieces of EXTRA HEAVY aluminum foil, 22 cm x 22 cm
• 2 metric rulers
• 1 triple beam balance
• approx. 50 washers, marbles, pennies or any other equal weights
• 1 small tub filled with water for floating boats
• 1 calculator

Per student:
• data sheet
For teacher demonstration:
• 1 aquarium filled with water
• 1 ball of clay
• 1 Coke
• 1 Diet Coke
• 1 rectangular container smaller than the aquarium

• mass - the amount of matter in an object; measured in this experiment with grams (g)
• volume - the amount of space an object takes up; measured in this experiment with cubic
• centimeters (cm3).
• density - the ratio between mass and volume; measured in this experiment with grams per cubic centimeter (g/cm3).

Before the students walk into the classroom, have the aquarium set up in front with a Coke and Diet Coke floating in it. (Actually, the Diet Coke will float but the Coke will sink). Initiate a short discussion about this discrepant event, encouraging the students to ask questions and make hypotheses. Do not acknowledge whether their hypotheses are right or wrong or actually conduct any experiments yet. After about five minutes, ask the students to write down their hypotheses as to why the Diet Coke floats but the regular Coke sinks.
Next, take a ball of clay and ask the students to predict whether the clay will float or sink. Put the clay in the water. Students will note that it sank. Remove the clay from the water and shape it into a small boat. Ask the students to predict whether this will float or sink. Place the clay boat in the water. Students will note that it now floats. Ask the students to write an explanation as to why the ball of clay sank but the clay boat floated.
Allow the students to share some of their explanations. The wording of their explanations will influence how you proceed from this point, but it is very likely that at least some students will use phrases such as "it weighed more" or "it was bigger" while some may actually use terms such as mass, volume or density. It is not necessary here to acknowledge whether each explanation is right or wrong. Synthesize from these explanations something like the following statement: "So we've determined that the reason an object floats has something to do with the amount of space it takes up and possibly how much mass (or weight if the term mass was not mentioned by the students) it has. We will watch a video called Eureka to learn more about how space and mass affect a property of matter called density. During the video, you will need to take several notes. First, there will be a space in the video notes section of your data sheet to define the terms volume and density. In addition, you will need to describe on the video notes section two ways the density of an object can be increased."
Pass out the data sheets to each student. Ask the students how they would begin a complete sentence to answer this question. Write on the board and have students begin their sentence in the video notes section like this:
The density of an object can be increased by:
1.
or 2.
Tell the students that they will also need to write the mathematical formula for calculating density during the video. Ask the students how that formula would start out and have them write it on their paper. (density = )
Say, "After the video we will be conducting some experiments with boats to learn more about density. We will also test your theories about the Coke cans. This video will last only about five minutes and I will be pausing periodically to discuss certain segments so pay close attention."

BEGIN the Eureka video at the beginning. PAUSE the video when the narrator asks, "How small should you crush them so that they'll fit exactly in to the container?" Ask the students what it is they have to know before figuring out how small these eight cars have to be crushed to fit exactly into the container. (how big the container is)

RESUME the video until the narrator says, "That's all that the word volume means - how much space something envelops." PAUSE the video. Ask a volunteer to repeat this definition and write it on the board. Ask, "How do you figure out how much space that cube envelops?" This question will help you assess how familiar the students are with the concept of volume. At this age level, most of the students should know that volume can be calculated by multiplying length, height, and width. The video answers this question in the very next sentence, so the students' answers are validated. Inquiries like this also help to keep the students focused when the video is resumed.

RESUME the video.
PAUSE when the screen shows "Volume = 64 m3" and the narrator says, "The container has a volume of 64 cubic centimeters." Ask the students where the little 3 comes from next to the "m" in 64 m3. Demonstrate that the superscript 3 comes from multiplying 3 different quantities measured in meters together. "If you have eight cars but only 64 cubic meters to stuff these cars into, how do you figure out how big each car can be? (Divide 64 by 8.)

RESUME the video.
PAUSE when the narrator says, "In other words, you'll have to crush each old wreck into a cube measuring two meters by two meters by two meters." Ask the students if there is another shape the cars could be crushed into besides 2 x 2 x 2 and still all fit into the box. (4 x 2 x 1)

RESUME the video.
PAUSE when the narrator says, "A density machine takes a car with a mass of say 2000 kilograms and squeezes the kilograms into a much smaller volume." Let the picture of the car with the formula Density = Mass/Volume appear but pause before anything is said on this screen. Tell the students the following:
"They have not actually given us a verbal definition of density, but they have given us a mathematical one. [Point to the mass/volume part of the picture on the screen.] What is this part the equation called? [You might have to give other examples like m/v or _ to get the students to understand that this is a fraction.] A fraction is a way of comparing two numbers. In density we are comparing an object's mass to its volume, or how much matter there is to how much space this matter takes up. Another way of stating this is to say we are making a ratio of mass compared to volume. So we can define density mathematically using the formula, and verbally as the ratio of mass to volume in an object." [Write this definition on the board.]
REWIND back to the picture of the density machine and RESUME the video. PAUSE after the narrator says, "In this case you increase density by keeping the volume the same but increasing the mass." Ask, "So which was more dense, the box with two cars or the same box with eight cars? (8) What was changed in the system? (mass) How was mass changed? (increased) Which would be more dense, 2000 kg in this container (point to aquarium) or 2000 kg in this container? (hold up a smaller box) What is the difference between these two systems? (different volumes) Now you should be able to write two ways to increase density if you haven't already."

RESUME the video as students complete their statements. STOP at the end.

Discuss the students' statements on how to increase density. Show examples on the board to demonstrate how increasing the numerator (mass) will always increase the density if volume remains the same. Show how increasing the denominator (volume) will always decrease density if mass stays the same. Demonstrate how doubling the volume will halve the density but doubling the mass will double the density.
Ask the student to now make a hypothesis about the Coke cans using the following three words: mass, volume, and density. Have a volunteer measure the mass of each can and calculate density. The density of the Diet Coke should be less than one because it floats while the density of the Coke should be greater than one. [Note: Make sure you test your soda cans first. Not all brands of diet and regular sodas follow this rule. In general, the diet drinks float because they contain aspartame (nutrasweet) rather than sugar and much less aspartame is needed to sweeten the soda compared to sugar.] Emphasize that when the mass to volume ratio is greater than one (the density of water at room temperature), objects will sink. If an object has a mass to volume ratio that is less than one, however, it is less dense than water and will float on the water.
Tell the students that they will be constructing and experimenting with two boats to explore the concept of density further. They will need to begin with two identical pieces of aluminum foil which they will fold into the shape of a boat according to the diagram. Because these two pieces of foil are the same size, they should also be the same mass. Before students begin folding they should mass each piece of foil to make sure they are within at least 0.5 grams of each other. The students should then follow the instructions on the student activity sheet and complete the questions at the end of the activity with their group. [Note: Due to the precision necessary for this activity the teacher should allow at least 50 minutes for the boat construction, testing and discussion. It might be most efficient to construct each boat and find the average mass of one washer directly after the video but wait to do the data collection and analysis until the next class period.]
To find the average mass of one washer, have each team mass one washer and then use these masses to find an average the entire class can use in
their calculations.
After the activity is complete, discuss the results with the class putting special emphasis on the idea of mass to volume ratio and how this was changed with each boat. The students should note that the boat with the smaller volume had a higher mass to volume ratio and, therefore, needed less mass to make it sink. The sinking boats should all have had mass to volume ratios greater than one. There may likely be teams whose results did not seem to follow this general rule. Use this opportunity to initiate a discussion on experimental error and control of variables. Ask the students to identify any aspect of the experiment that might have led to inaccurate data collection. It is important to emphasize that this is the type of analysis that scientists must conduct any time they perform an experiment. After students identify any inconsistencies in experimental technique, have them discuss how this could be improved upon in a future experiment. Emphasize that this is why scientists often use repeated trials in an experiment to control error or bias. It would be helpful to put together the data from all of the teams to calculate class averages upon which they can base their conclusions.

Students can choose one of the following fruits to test at home: banana, orange, apple. Will the fruit float in water? How does peeling the fruit change its mass to volume ratio (or does it)?
Invite a scuba diver to class to discuss how divers control their mass to volume ratio.
Have students conduct research on how submarines dive and return to the surface.

Science: Allow students to conduct tests of various regular and diet sodas to determine if they float or sink. What about other types of canned drinks? Have students create their own data chart comparing various brands of soda.
Science: Challenge students to explain (using the terms density, mass and volume) why helium balloons float
in air.
Math: Use this activity to explore other common ratios such as speed, acceleration, blood alcohol content, and in baseball, earned run average.
Creative Writing: The density of ice is less than the density of liquid water. Imagine and describe how different the world would be if ice behaved like most substances and became more dense than liquid water when it solidified. Would lakes freeze from the bottom up? How would this affect the sea level?

1995-1996 National Teacher Training Institute / Austin

### Master Teacher: Carol Fletcher

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