Students explore material properties in hands-on and visually evident ways via the Archimedes' principle. First, they design and conduct an experiment to calculate densities of various materials and present their findings to the class. Using this information, they identify an unknown material based on its density. Then, groups explore buoyant forces. They measure displacement needed for various materials to float on water and construct the equation for buoyancy. Using this equation, they calculate the numerical solution for a boat hull using given design parameters.
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- Common Core State Standards for Mathematics: Math
- 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.<sup>★</sup> (Grades 9 - 12)  ...show
- 1. 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. (Grades 9 - 12)  ...show
- 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (Grades 9 - 12)  ...show
- International Technology and Engineering Educators Association: Technology
- J. Engineering design is influenced by personal characteristics, such as creativity, resourcefulness, and the ability to visualize and think abstractly. (Grades 9 - 12)  ...show
- National Council of Teachers of Mathematics: Math
- make decisions about units and scales that are appropriate for problem situations involving measurement (Grades 9 - 12)  ...show
- recognize and apply mathematics in contexts outside of mathematics (Grades -1 - 12)  ...show
- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships (Grades -1 - 12)  ...show
- Washington: Math
- Analyze a problem situation and represent it mathematically. (Grades 9 - 12)  ...show
- Measure masses and volumes of known and unknown substances.
- Calculate density using given measurements.
- Predict physical behaviors by employing a numerical model.
- Apply predictions to an engineering design challenge.
- graduated cylinder marked in milliliters (alternatively, provide a 2-liter bottle with the top cut off and a ruler so students can measure displacement in cm and calculate volume)
- scale (can be digital), to measure mass in grams
- known objects of a variety of materials, such as wood, steel, aluminum, plastic, glass, Styrofoam, water and oil; these objects can be found (glass marbles, steel marbles, wood blocks, etc.) or cut to appropriate size; find items in your home, school shop or hardware store
- irregularly shaped "mystery object" of unknown material; make it the same material as one of the known objects, but irregular in shape and masked with paint or tape to conceal its identity
- 4 plastic 500-ml drinking bottles
- aluminum foil, 1 sheet
- Eureka - Archimedes Principle Worksheet 1, one per student
- Eureka - Archimedes Principle Worksheet 2, one per student
|weight:||How heavy an object is. What a scale reads when it is weighed in a given setting. Note that an object's weight would be different on the moon than on Earth.|
|mass:||The property of an object that gives it weight. We will be using metric unit of gram (g) as the unit of mass and equating it to the weight measured by a scale under classroom conditions.|
|volume:||The space an object takes up. We will use both the metric unit cubic centimeter (cm3) for solids and milliliters (mL) for liquids. It is convenient that 1 cm3 = 1 ml.|
|density:||The ratio of mass per volume of a material. Mass is an intensive property (as opposed to extensive), which means that it is a characteristic of the material and independent of the size of the object. Density of water is 1 g/cm3. density = mass / volume.|
|displacement:||The volume of water that is moved away or replaced by an object. This is viewed as a change in apparent volume and we measure it in milliliters (ml).|
|buoyant force:||The force exerted by water due to displacement of the water. Because water has a density of 1 g/cm3, for each cubic centimeter (or milliliter) displaced, 1 g of water has been displaced. This means that by measuring the change in volume in milliliters, we have found the mass of the object in grams.|
|volume equations:||Volume of a block = l * w * h or length * width * height; volume of a sphere = 4/3 * π * r3, where r is the radius; volume of a cylinder = h * π * r2, where h is the height.|
Before the Activity
- Gather materials and make copies of the two worksheets: Eureka - Archimedes Principle Worksheet 1 and Eureka - Archimedes Principle Worksheet 2.
- Read all materials and do the experiment and worksheets in advance to understand the activity and be aware of any challenges students might encounter.
- Instead of graduated cylinders, you can cut the tops off 2-liter bottles so students can measure displacement in cm and calculate volume.
- The known objects can be found or cut to appropriate sizes. Many are household items, or available in school shop classes or hardware stores.
- Make the mystery object of the same material as one of the known objects, but irregular in shape. Mask its identity with paint or tape.
With the Students
- Review fractions and how to calculate the volume of cubes, spheres and cylinders. Provide sample problems that include measuring using a ruler.
- Divide the class into groups of three or four student each. Give each team a set of equipment.
- The group is responsible for working together and completing the activity together. Each student is responsible to fill out his/her own worksheet to be handed in.
- The group presents its findings to the class (see below).
- Day 1: Hand out worksheet 1. Students measure the mass and dimensions of known materials and calculate the density of each.
- Each team records its findings on the board and the class discusses the findings, including sources of error and possible variations in density results for different samples.
- The teams measure and calculate the density for the mystery object and determine what material the "mystery" object is, based on a comparison of this material and the list of known densities, those already calculated.
- The class comes together again to compare each group's results for the mystery object and assess what the groups have discovered.
- Day 2: Hand out worksheet 2. Students experiment with boats (plastic bottles) filled with various materials to determine an equation for the buoyant force. They then apply this equation to a real-life engineering problem outlined in the worksheet.
- After worksheets are completed, bring the class together to discuss these findings.
- The force supporting the floating object is known as the buoyant force. When an object is floating on water, the force of gravity on that object is equal to the buoyant force of the water.
- Buoyant force of water = density * displacement, which is equal to the force (due to gravity) of the boat = weight.
- The height above or below the water may change with the boat's orientation, but the volume above and below the water does not change.
- Day 3: For the final engineering challenge, each group designs a boat hull based on what they know about densities and buoyant force.
- Each group uses a numerical model to calculate the dimensions of the hull with given design parameters, as outlined in the worksheet.
- Each group reports its findings to the class, as directed by the teacher.
- Explain how this activity relates to the engineering design process.
- The following is an example of the design project: Some of the largest oil supertankers are designed to carry 500,000 GWT (gross weight tons). This is 500,000,000 kg. It would require 500,000 liters of water to be displaced or 500,000 m3! The hull for these ships can be 400 m (1312 ft) long and 60 m (197 ft) wide, giving a draft (submerged depth) of 20.8 m (68 ft)!
Activity Embedded Assessment
- Find conversion factors and do all calculations independently.
- Analyze any geometric shape and complete the task independently.
- Apply the mathematics to the experimental results.
- What was something that was really good about this project? It could be something you were proud of accomplishing or something that went well in the activity. Explain why this was important.
- What is something you would do differently if you did this activity again or something (a skill or a process) you would like to work on after this activity? State how you would do this.
- What is something significant that you learned from this activity? This could be something you never noticed before or a light bulb (aha) moment.
Buoyancy. Last revised 26 March 2013. Wikipedia, The Free Encyclopedia. Accessed 28 March 2013. http://en.wikipedia.org/wiki/Buoyancy
Nave, C.R. Buoyancy. Hyperphysics. Department of Physics and Astronomy, Georgia State University. Accessed 28 March, 2013. http://hyperphysics.phy-astr.gsu.edu/hbase/pbuoy.html#buoy
© 2010 by Regents of the University of Colorado; original © 2010 Board of Regents, Washington State University
CREAM GK-12 Program, Engineering Education Research Center, College of Engineering and Architecture, Washington State University
Last modified: April 21, 2015