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**Grades:** 10-12

**Author:** Connie Hubbard, Sandra Van Natta, AGPA staff

**Source:** Original

A nanometer is one billionth (1 x 10^{-9}) of a meter which can be about 3 to 5 atoms in width. Electrospun nanofibers produced from polymer solutions are being used in unique ways by scientists. Nanotechnology allows the manipulation of matter, atom by atom at the "nanoscale." Properties of these materials are amplified due to the fact that many fibers can fit into a very small space. Scientists have found many unique ways to use such fibers from producing new materials capable of blocking moisture, removing of toxins from both water and air, delivering medicines to a specific region in or on the body, and tissue scaffolding. The possibilities are endless as scientists and inventors produce new products formed from these extremely small fibers.

In this investigation students will determine the advantages of going "small" by comparing the amount of chocolate syrup coating on a large diameter pretzel to that of an equal volume of smaller pretzels coated with chocolate. Students will learn what happens to the surface area as the diameter gets smaller and smaller. Students will determine what advantages exist in making the size (diameter) of a pretzel smaller. Students will make the comparison by massing a cup of chocolate syrup before and after dipping the pretzel(s). Students will use their data, graphs and mathematical equations to support their conclusions. The lesson contains a Powerpoint review of the metric system with pictures to help students visualize large and small number lengths. Using a second Powerpoint set of slides in the elaboration, the teacher can relate the activity to the new field of nanotechnology and discuss with students why it is advantageous for newly developed materials to be so small. This often has to do with the large surface area available on nanosized particles and fibers.

**What should students know as a result of this lesson?**

- Students will be able to explain that as the size (diameter) of an object (in this case, a pretzel) decreases, the amount of surface area available increases when compared to its volume. This is called the "surface area to volume ratio." The geometric shape of a straight pretzel is a cylinder. This is also true for fibers.
- Students should conclude that if a greater coating is desired on a cylinder (pretzel), it is better to have several smaller cylinders rather than one large cylinder for a given volume.
- Students should be able to relate this model to nanoparticles, particularly nanofibers, since their geometry can be described as a long, very thin fiber.
- Students will also explain how scientists use extrapolation of measured data to determine dimensions too small to measure with current means.

**What should the students be able to do as a result of this lesson?**

- Students will measure the circumferences and diameters of three pretzels in three size groups (small, medium and large).
- Students will construct a data table to contain the diameters, circumferences and lengths of the pretzels.
- Students will calculate the cross-sectional areas, volumes, surface areas and surface to volume ratios of each pretzel group.
- Students will create a graph of diameter v. circumference using the averages of each group and a graph of the average radius of each group v. surface-to-volume ratios.
- Students will extrapolate from the graph to obtain values for "nanopretzel."

- Large (rod) pretzel sticks
- Medium pretzel sticks
- Small pretzel sticks
- String
- Scissors
- Centimeter ruler
- Chocolate syrup
- Cups
- Vernier caliper
- Graphing program (Recommended: Vernier Graphing Program) or graph paper

**Engagement**

Begin the lesson by showing students pictures (or actual objects) of first-generation technologies and more recent examples. Pictures of a video-tape along side a DVD and a television that used cathode ray tubes along side a flat screen television are available by clicking here. Discuss with the students these examples of developments in electronics products (technologies) that have become possible due to our new ability to "do more" in ever smaller spaces. Explain to students that in this lesson they will be exploring the advantages of going small and learning about application in the new field of nanotechnology.

Some high school students have heard of nanotechnology but are unaware of unique physical properties that make nano-sized objects so desirable. One such property is the surface area-to-volume ratio. A quick demonstration of increased surface area may be done by coating pretzels in chocolate. Take an equivalent amount of smaller pretzels that have about the same volume as a large pretzel rod and pose the questions, "Which will have the greater amount of chocolate coating adhering?" Take several different answers from students and ask them to explain their thoughts. Return to this activity after students have performed the lab activity to allow students to actually mass the remainder of chocolate in the cup of chocolate after each dip.

*Assessment* A quick assessment of understanding can be done by taking a hand count of ideas. Ask how many students believe that there will be no difference; how many think the larger pretzel will contain more chocolate; and, how many think the smaller pretzels will contain more chocolate.

**Exploration**

__Procedure: Part 1 - Data Collection__

- Obtain 4 pretzels from each size of pretzels - small, medium and large. Partners should have a total of 12 pretzels.
- Using a metric ruler or a vernier caliper, measure the diameter of each pretzel and record the data in your data table. The length of each pretzel is 100mm, which will be a constant value and will be used to calculate the volume of your pretzels.
- Wrap a piece of string snugly around the outside of each section of pretzel. Mark the string so it represents the circumference of the pretzel.
- Measure the length of string and record your data under "circumference" in table 1 of the Data Page.
- Calculate average diameters and circumferences for each size-set of pretzels.

*Assessment* Have students develop their own data tables and also have them post their data in a common data table on the board. Students can then find their own mistakes with some guidance.

__Procedure: Part 2 - Calculations and Graphing__

- Using averages, graph circumference v. diameter placing the circumference on the y-axis and the diameter on the x -axis of your graph. Do this using graph paper, a computer or a graphing calculator. (You must have a hard copy of the graph.)
- Divide the circumference of any of your pretzels by its diameter. What value should you get? Have students compare values with those of others in the class. Are they about the same?
- Find the slope of the graph and compare the slope with the actual value of π. Calculate the percentage error.
- Using the group averages, calculate the cross-sectional area, volume and surface area of each pretzel rod set using the equations below. Show all work and label appropriately.
- Cross Sectional Area = π r
^{2} - Volume = area x height or π r
^{2}h (multiply the values in "a" by the length 100mm) - Surface Area of a Cylinder = (2π r h) + (2π r
^{2})

- Cross Sectional Area = π r
- Calculate the Surface Area to Volume Ratio by dividing the surface area of each group by the calculated volume of that group. Which group of pretzels has the largest value? What do you think this means relative to the size (diameter) of the pretzel?
- Graph the average radius of each group v. surface area/volume ratio (divide the surface area for each group by its volume.) Place the radius on the abscissa (x-axis)

*Assessment* Have students develop their own calculation data tables and also have them post their calculated data in a common data table on the board. Again, students can then find their own mistakes with some guidance. It is very easy to make a mathematical error when using several equations.

**Explanation**

__Graph 1:__ Graph student averages of diameter v. circumference using graphical analysis or use a graphing template for the overhead projector. All students will graph their own values. Draw a line of best fit and then ask students to find the slope of this line.

*Oral Assessment:*

Ask students "What value is this equivalent to?" Most students should recognize π. Show students that if the equation for the circumference is divided by the equation for the diameter it equals π.

Also, ask students "What would happen if we extend our fit line in each direction?" Students should see that we can use our trend line to extrapolate the diameter or circumference of cylinders we did not measure.

__Graph 2:__ Graph the average radius of each group v. surface area/volume ratio. Draw a trend line. Ask students what relationship exists between the radius (size) and the surface-to-volume ratio. Students should notice an inverse relationship and that as the radius increases the surface-to-volume ratio decreases. Ask students what they think this means in terms of nanofibers.

Review the metric system of measurement and the prefixes that when affixed to "meter" describe very large to very small units of length. Show the PowerPoint presentation entitled, What is the Meaning of "Nano" in Nanotechnology?.

*Evaluation Questions:* (Please answer in complete sentences)

- Using your graph, predict the circumference of a pretzel that has a diameter of 2.0 mm.
- Assuming your graph relationship is linear, what would be the circumference of a "nanopretzel" that has a diameter of 100 nanometers?
- From your second graph, what happens to the surface/volume ratio of a pretzel as its radius gets larger (assuming constant length?)
- Predict the surface/volume ratio for a 400nm nanofiber.
- Which group of pretzels has the largest value surface area to volume ratio? What do think this means relative to the size (diameter) of the pretzel?
- Given the same length and volume, which pretzels do you think have more salt per pretzel? Justify your answer with your data.

**Elaboration**

Use one large paper straw and a number of smaller straws equivalent in volume to help students understand surface area compared to volume. Since the volume is constant, you can focus just on the surface area of the straws. The lengths of the straws are also constant. Cut the straws open with an exacto knife, flatten them and tape them to an overhead transparency. Have student volunteers measure the length and width dimensions. The class can calculate the areas by multiplying the length times the width. Data will show an increase in surface area when the areas of the smaller straws are added together and compared with the larger straw's surface area.

Conduct a discussion of the applications of nanotechnology after showing the PowerPoint presentation, Nanotechnology and Nanofibers.

__Optional Project:__ Visit the computer lab and have students research the following: Describe how nanofibers might be used in a productive way. Be creative, give some detail and incorporate what you have learned in this lesson in your response.

Have students begin with the websites listed in the Teacher Notes Worksheet initially. As students explore, have them post any valuable sites they have encountered. Have students site their references. Use the grading rubric found in the Worksheets section.

Students should know the equations for the surface area of cylinder, the volume of a cylinder, the cross-sectional area of a cylinder (area of a circle) and the circumference of a circle. They should also be able to perform calculations with these equations. Students should know how to graph and interpret graphs. They should be able to recognize direct and inverse relationships. Students should also know how to calculate the slope of a line.

- Using analogies
- Scientific literacy
- Conceptual understanding of problem solving
- Real world applications
- Using models

**NGSS Standards:**

- HS-PS2-6 Communicate scientific and technical information about why the molecular-level structure is important in the functioning of designed materials.
- HS-LS1-2 Develop and use a model to illustrate the hierarchical organization of interacting systems that provide specific functions within multicellular organisms.
- HS-LS1-6 Consturct and revise an explanation based on evidence for how carbon, hydrogen, and oxygen from sugar molecules may combine with other elements to form amino acids and/or other large carbon-based molecules.
- HS-ESS3-1 Construct an explanation based on evidence for how the availability of natural resources, occurrence of natural hazards, and changes in climate have influenced human activity.

**Common Core Standards:**

- RST.9-10.3 Follow pecisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text.
- WHST.9-10.2 Write informative/expanatory texts, including narration of historical events, scientific procedures/experiments, or technical processes.

**National Standards:**

- Physical Science Content Standard (10-12)
- Mathematics (10-11)

**Ohio Standards:**

- Scientific Inquiry - Benchmarks A and E (11-12)
- Scientific Ways of Knowing - Benchmarks A and B (9-10)
- Benchmark A (11-12)

Metric prefixes: nano

Information on nanofibers: http://www.nanitenews.com/, http://electrospun.blogspot.com/

Metric Measurement

Graphing using Excel or Graphical Analysis

Interpreting graphs: extrapolationg and determining relationships between variables (indirect, inverse)

Constructing an appropriate data table

Geometry equations for area, volume, surface area, cross-sectional area and circumference

No special requirements.

Nanofibers and their applications.

Additional information about nanotechnology and nanofibers is found in the PowerPoint presentation, Nanotechnology and Nanofibers.

This lesson combines a performance assessment and a written assessment. Students should be able to measure correctly using metric instruments. Students must calculate the circumference, volume, surface area and cross-sectional areas of their pretzels. They must graph variables and answer questions which demonstrate understanding of the concepts.

**Grouping Suggestions:** Recommended for groups of 2 or 3 students.

**Pacing/Suggested Time:** 1-lab period of 80 minutes; Reflection and Discussion (10-15 minutes); The optional elaboration piece would require additional class time of at least 1 period with additional time given (perhaps as homework) for students to finish the essay.