The study of scaling and scale modeling is extended from general proportions and similar figures to the actual conditions that must be present in order for a scale model to function as the actual object.

Learning outcomes

The students will learn to calculate density, viscosity, and the Reynolds Number, and they will apply these characteristics to scale models.

Teacher planning

Time required

Two to three consecutive class periods on the traditional 60-65 minutes classroom schedule or two consecutive class periods on the block 90-minute schedule.

Handouts

Density review warm-up
This warm-up may be given to the students to remind them how to calculate the density of a fluid.
Open as PDF (54 KB, 1 page)
Density review warm-up answer key
Open as PDF (65 KB, 1 page)
Data sheet three
Students will record data on this sheet and use it to calculate the Reynolds Number.
Open as PDF (106 KB, 4 pages)

Materials

  • Spheres of different densities — Ex: copper BB pellets, beads, metal ball bearings, marbles
  • Karo syrup (Other liquids, such as a pearlescent shampoo, may be used, but the amounts may need to be changed.)
  • Balance — two for the classroom
  • Graduated cylinders or long tubes — one large one per group (750 mL) and several beakers to measure
  • Meter sticks — one per group
  • Stopwatches — one per group
  • Calculators — at least one per group
  • Small fish tanks — two for the classroom (more if available)
  • Tweezer — one per group
  • Long spoon — one per group
  • Calipers — one per group
  • Masking tape or colored tape if available (painter’s tape)
  • Wind-up toys — fish, shark

Technology resources

The instructor will need a computer with an LCD projector and speakers in order to watch the video clips.

Pre-activities

  1. Before the students arrive: Mass the 750 mL cylinders on a balance and record the mass in grams. The teacher should mix up a batch of Karo/Water solution for each group. Each batch should be a different solution. If there are five groups of students, the batches could be mixed according to the table below. These amounts can vary depending on the amount of Karo available and the beakers available. If a substitute liquid is used for the Karo, the amounts may change.
    GroupAmount of water (mL)Amount of Karo (mL)
    A0750
    B45705
    C93657
    D186564
    E330420
  2. Mix the batches well until few air bubbles are visible. Label each graduated cylinder with the amounts of water and Karo and the mass of the cylinder.
  3. Separate the spheres and label each sphere as V, W, X, Y, Z.
  4. During the first class, the students need to be introduced to the critical vocabulary. This can be done at the discretion of the teacher. The mathematics and science teachers could collaborate to introduce this vocabulary during the science unit that includes density. This can also be incorporated in the warm-up density review sheet that is included in the lesson.
  5. For motivation, the teacher may show clips of “King Kong” or “Honey, I Shrunk the Kids” and discuss how the movies portray unrealistic situations. For instance, if a coffee cup with coffee in it is shrunk, the liquid would actually turn into a stickier, more viscous fluid due to the reduction in size! This is discussed in the article “Hairy noses: It’s a small, sticky world out there” (see “Supplemental information” below). In “King Kong,” King Kong’s bones would not be able to support his structure. They would actually break because he would be too large and too heavy, and the density of his bones would not support his larger build. If the size of the bones change but the density does not change, the density will not be great enough to support the increase in bone size and weight.

    In real life, certain characteristics of rescaled objects would appear and function differently. As objects are scaled up or down, the density and viscosity of the objects change (viscosity of the coffee and density of King Kong’s bones). One way to accurately simulate the actual objects with scaled models is to calculate the Reynolds Number of the original object and manipulate the Reynolds Number of the scaled object to be similar.

  6. The students need to understand that the purpose of this lesson is to calculate the Reynolds Number of certain fluids and apply these characteristics to a scale model (wind-up toy). Teacher note: In order to use the same fluids, do not conduct the scale model portion with the fish tanks until day two or day three when all of the students have successful calculated the viscosity of their fluid. Also, ensure that an appropriate sphere has been chosen for each solution. The students will need to measure the time it takes the sphere to drop through the solution, so the sphere should not float and should not fall too rapidly.

Activities

Day one

  1. The warm-up may be given to remind students how to calculate the density of a fluid.
  2. Optional: Show video clips from “King Kong” or “Honey, I Shrunk the Kids” to encourage discussion on scale models and the reality of reducing and enlarging objects. Guide the students to understand that fluid has certain properties that can appear differently when scaled. Review the critical vocabulary.
  3. Assign groups and distribute the materials to each group. Try to have only four or five students in each group.
  4. Using the calipers, the students will measure the radius of the spheres and record this on data sheet three.
  5. The students will mass the filled graduated cylinder and record this on data sheet three.
  6. On the graduated cylinder, mark a beginning reference point and ending reference point with a piece of tape. The beginning reference point should be about two cm below the top of the liquid. The ending reference point should be about five cm from the bottom of the cylinder. Have the students place a dot/mark on the tape that will be used to label the reference points. This will be a visual reminder of the beginning and ending points.
  7. The students will measure the distance between the data points (reference points marked by dots on the tape) and record this on data sheet three.
  8. One student will hold the sphere in the tweezers below the surface of the liquid but above the beginning reference point.
  9. The timer will tell the student holding the sphere when to release the sphere.
  10. The timer begins when the ball passes the beginning reference point and ends when the sphere passes the ending reference point.
  11. The recorder will record the time the sphere took to pass from the first reference point to the second reference point.
  12. The students will repeat the sphere release ten times.
  13. The students will calculate the velocity for each of the ten runs and then average the velocities.
  14. The students will calculate the viscosity on data sheet three.

Day two

  1. Review the results of the viscosities from day one. Compare the results of each group and record this information on the board in order of most viscous to least viscous. Lead the students to understand that the more Karo that the solution contained, the higher the viscosity. Objects move more slowly through more viscous fluids.
  2. Discuss the importance of viscosity in our daily lives: ketchup in a bottle, engine oil, toothpaste. The following table provides a list of some common liquids and their viscosity at the given temperature. Remind the students that viscosity changes with temperature (heating up honey allows it to flow more quickly).
    LiquidTemperature (degrees Celsius)Viscosity (Pa s)
    Honey2010
    Ketchup2050
    Sour cream25100
    Peanut butter20150-250
  3. Choose two of the solutions (unless more fish tanks are available). Preferably, do not pick the pure Karo solution (A). It is best to pick the 45mL Water + 705 mL Karo (B) and the 330 mL Water + 420 mL Karo (E) solutions for variety. Pour each solution into a fish tank.
  4. Two student volunteers will place the wind-up toy in the aquarium next to the shorter wall but not touching the wall. Another volunteer will measure the distance from the front of the wind-up toy to the wall across the tank.
  5. All students will record this distance on data sheet three.
  6. The student holding the toy will release the toy (as straight as possible) and a time-keeper will time how long it takes the toy to go across the tank. Ten repetitions should be recorded. Teacher note: Only have the students record the time if the toy goes relatively straight across the tank.
  7. The students will calculate the velocity of the wind-up toy for each of the ten repetitions and then average the velocities.
  8. A student volunteer will measure the length of the wind-up toy and all students will record this information on data sheet three.
  9. Students will calculate the Reynolds (Re) Number on data sheet three. The Reynolds Number describes the “stickiness” of the fluid. It is a mathematical calculation that involves the density and viscosity of the fluid. The Reynolds Number is useful in determining whether the fluid of a scaled model is similar to the actual fluid. In order to effectively use a model, the Reynolds Number for the model must be similar to the actual object. If this is not the case, the results of research on a fluid model may not apply to the actual object because they are dissimilar.
  10. Discuss the results of the Reynolds Number for each fish tank. The teacher will explain how this is modeling a fish swimming in a lake. For a fish swimming, the Reynolds Number is greater than one. In order for a scale model of a fish swimming in a lake to be realistic, the Reynolds Number in the tank should be approximately equivalent to the Reynolds Number in the lake. The students should apply this to other situations (model airplanes vs real planes, ship scenes in movies that are filmed with models in pools, etc). Introduce the scientific importance of the Reynolds Number. Briefly, discuss the article “Hairy noses: It’s a small, sticky world out there” with the students (see “Supplemental information” below).

Assessment

The assessment will include student participation in the lab work and the results of data sheet three. The teacher should ensure that each student correctly calculated the density, viscosity, and Reynolds Number.

Modifications

This plan is intended for eighth grade Algebra I or pre-algebra students that have completed a science unit on density. This plan could be modified by providing more of the data and calculations for the student. For instance, the density could be calculated but the viscosity could be provided.

Supplemental information

Summers, A. (2002). Hairy noses: It’s a small sticky world out there. American Museum of Natural History.
This article explains the history, purpose, and use of the Reynolds Number. It discusses practical uses of the Reynolds Number, including the use of scaled models to conduct research. Researches can change the size of airplanes or tiny insects to reproduce the item at a size that is easier to work with. Researchers have used the Reynolds Number to create a larger scaled model of the mantis shrimp or stomatopods. This article is a short article that can be given to the students to read independently, or it could be used to launch a discussion about the Reynolds number on the first day of activities. The teacher could also collaborate with a Science or Language Arts teacher to incorporate the article in another discipline. The article is written at an appropriate level for eighth grade students, but some teacher guidance may be necessary.

Websites and resources

The Physics Hypertextbook. (2010).
This site has some great information on viscosity and dimensional analysis of viscosity. The site also has some tables that list the viscosities of some well-known liquids.
King Kong video clip
This clip shows the dinosaur, Ann, and King Kong. The size of King Kong is compared to Ann and the dinosaur.

Comments

This lesson requires sufficient planning in advance. The students need to have an understanding of density. The lesson could be concluded at the calculation of density if students are not familiar with density calculations.

  • Common Core State Standards
    • English Language Arts (2010)
      • Science & Technical Subjects

        • Grades 6-8
          • 6-8.LS.3 Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks.

  • North Carolina Essential Standards
    • Science (2010)
      • Grade 6

        • 6.P.2 Understand the structure, classifications and physical properties of matter. 6.P.2.1 Recognize that all matter is made up of atoms and atoms of the same element are all alike, but are different from the atoms of other atoms. 6.P.2.2 Explain the effect...
      • Grade 8

        • 8.P.1 Understand the properties of matter and changes that occur when matter interacts in an open and closed container. 8.P.1.1 Classify matter as elements, compounds, or mixtures based on how the atoms are packed together in arrangements. 8.P.1.2 Explain...

North Carolina curriculum alignment

Mathematics (2004)

Grade 8

  • Goal 2: Measurement - The learner will understand and use measurement concepts.
    • Objective 2.02: Apply and use concepts of indirect measurement.

Science (2005)

Grade 8

  • Goal 4: The learner will conduct investigations and utilize technology and information systems to build an understanding of chemistry.
    • Objective 4.05: Identify substances based on characteristic physical properties:
      • Density.
      • Boiling/Melting points.
      • Solubility.
      • Chemical reactivity.
      • Specific heat.
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