Lesson Plan Presentation
Subject- Mathematics
Teacher- Hemant Mishra, Mathematics Teacher, Strawberry Mansion High School,
Venue- Computer and Design Laboratory, Drexel University, Philadelphia.
Date- Thursday, July 31, 2008
Objective- To study one-dimensional, two-dimensional and three-dimensional measurements of nanoparticles at nano scale.
Purpose- This lesson is designed to study the 1-D, 2-D and 3-D measurements of the various types of nanoparticles at nanoscale produced while toughening a Petroleum product (Vinyl Ester ResinVE-Derakane 411-350) using Soy oil ( Bio-based Rubber). By investigating the 1-D Length, Circumference, 2-D Area, Surface Area, Total Surface Area, 3-D Volume , to understand how various shapes and size may affect the bonding behavior of the nano particles in order to toughen the polymer and increase its mechanical properties like Flexural Strength, Flexural Modulus, Fracture Toughness, Tensile Strength, Viscosity and other characteristics regarding toughness.
Grade Level- 10^{th} to 12^{th}
Duration- 1 period of Block Roster.
Pennsylvania State Academic Standards:
2.3.11A: Select and use appropriate units and tools to measure to the degree of accuracy required in particular measurement situations.
Background information:
Sphere shaped nanoparticles: Hemisphere shaped nanoparticles:
Cylinder shaped nanoparticles: Rectangular Prism & Cubical shaped nanoparticles:
Conical shaped nanoparticles Pyramidal shaped nanoparticles
1. Sphere is a 3-D geometrical structure having uniform radius.
Surface area of sphere= 4π(r^2)
Volume of sphere=(4/3)π(r^3)
2. Hemisphere is 3-D geometrical structure, it is exactly half of a sphere.
Surface area of Hemisphere= 1/2{4π(r^2)}
Volume of Hemisphere=1/2 {(4/3)π(r^3)}
3. Cylinder is 3-D geometrical and curvilinear geometric structure its surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder having two planes perpendicular to the axis.
Surface area of a Cylinder= 2π(r^2) +2 πr(r+h)
Volume of a Cylinder=π(r^2)h
4. Rectangular prism is a 3-D solid geometrical object having six rectangular surfaces.
Surface area of Rectangular prism=2(l.w+w.h+h.l)
Volume of Rectangular prism=l.w.h
5. Cube is a3-D solid geometrical object having six square surfaces.
Surface area of Cube= 6(side)^2
Volume of Cube= (side)^3
6. Cone is a 3-D solid geometrical object that tapers smoothly from a flat, round base to a point called the vertex.
Surface area of Cone=πr(r+l)
Volume of Cone= 1/3{π(r^2) h}
7. Pyramid is a 3-D solid geometrical objecthaving polyhedron faces with polygon base and its faces converge on one point, called the vertex. Surface area of Pyramid=1/2(perimeter of base)(side length) + (base area)
Volume of Pyramid=1/3(base area)(height)
Maximizing surface area is vital to making Polymers & catalysts work. Any of the above 3-D shapes can be ideal for this purpose.
Illustration: If we shrink the volume of a cylinder by a factor, say 1000 then the surface area goes down by a factor of 100. The aspect is by making cylinders small enough, some polymers can do amazing things, i.e. limiting the material exposed to an environment or building the largest enclosure with given amount of material. It is one of the promises of nanoparticles, minuscule bits of material that have properties the same material in larger sizes does not. Nanoparticles, the first real commercial breakthrough in nanotechnology are now found in everything from polymers to paint to tennis balls etc.
Materials:
· 8.5inches x 11inches sheet of waxed paper
· Modeling clay in the shapes of Cylinders, Spheres, Hemispheres, Cones and Rectangular Prisms, Pyramids, etc.
· Metric Ruler,
· Vernier Calipers with standard graduations,
· Geometrical Instruments, pencil, etc.
· Calculator
Note: - No special safety instructions for this laboratory.
Introduction and Brainstorming: Answer the following questions.
Ques- What do you mean by 1-D, 2-D and 3-D objects, give 2 illustrations of each?
Ques- Illustrate 5 objects having Spherical shape in the playground.
Ques- Illustrate 5 objects having Hemispherical shape in your house.
Ques- Illustrate 5 objects having Cylindrical shape in your class.
Ques- Illustrate 5 objects having Rectangular Prism shape on the street.
Ques- Illustrate 5 objects having Cubical shape in the shopping mall.
Ques- Illustrate 5 objects having Conical shape in your neighborhood.
Ques- Illustrate 5 objects having Pyramidal shape you know.
It is important to know the meaning of Surface Area and Volume of given object before calculating them at nanoscale.
Surface Area of an object is the Area of the total surface enclosed by the object.
Volume of an object is the amount of total Space the object takes up.
In order to find the Surface Area and Volume of given samples of nanoparticles at nano scale we must assume a scale for study.
Scale to be used: 1 centimeter = 1 nanometer
Questions asked to the students.
Ques- What are the 2 dimensions in the Surface Area of a Sphere?
(Possible answers) radius and radius of the sphere.
Ques- What are the 3 dimensions in the Volume of a Hemisphere?
(Possible answers) radius, radius and radius of the hemisphere.
Ques- What are the 2 dimensions in the Surface Area of a Rectangular Prism?
(Possible answers) length, width and height of the rectangular prism.
Ques- What are the 3 dimensions in the Volume of a Pyramid?
(Possible answers) length and width of the base, height of the pyramid.
Remind Student to measure the radius, length, width and height of each object using Vernier Calipers and write into the columns of the assigned table and then in the last two columns calculate the Surface Area and the Volume of each object using the corresponding values.
Table for Calculation of Surface Area and Volume of the sample nanoparticles
Shape of nanoparticle | Radius(r) (in nm) | Length(l) (in nm) | Width(w) (in nm) | Height(h) (in nm) | Surface Area (in sq nm ) | Volume (in cu nm) |
Sphere | | | | | | |
Hemisphere | | | | | | |
Cylinder | | | | | | |
Rectangular Prism | | | | | | |
Cube | | | | | | |
Cone | | | | | | |
Pyramid | | | | | | |
Surface area of the objects will be measured in square nanometers (sq nm)
Volume of the objects will be measured in cubic nanometers (cubic nm)
Student Worksheet
In Nanoparticles as the size increases, say 10 unit then the increases in its Volume, say 1000 cubic unit, is more than the increase in its Area, say 100 square unit. In order to investigate this fact, we can find the “Surface Area to Volume Ratio” of each type of Nanoparticle in the given shape.
Analysis of “Surface Area to Volume Ratio” of the following shaped Nanoparticles
Use the above table to calculate the following “Surface Area to Volume ratios.
- For Sphere shaped nanoparticle
Surface Area = _____________ per nm
Volume
- Hemisphere shaped nanoparticle
Surface Area = _____________ per nm Volume
- Cylinder shaped nanoparticle
Surface Area = _____________ per nm
Volume
- Rectangular Prism and Cubical shaped nanoparticle
Surface Area = _____________ per nm
Volume
- Conical shaped nanoparticle
Surface Area = _____________ per nm
Volume
- Pyramidal shaped nanoparticle
Surface Area = _____________ per nm
Volume
Conclusion and Prediction
Ques- Which shape had the smallest Surface Area-to-Volume ratio?
Answer-
Ques- Which shape had the largest Surface Area-to-Volume ratio?
Answer-
Ques- Among the given shapes you tested, which shape would you recommend as the most reactive catalyst? Explain.
Answer-
Ques- Why it is important to understand the shapes & other characteristics of the nanoparticles?
Answer-
Ques- What shaped we should use for the Bio-based Rubber nanoparticles to make them TOUGHER & less BRITTLE.
Answer-
Ques- Which shape would be more Tough & less Brittle among the all given shapes.
Answer-