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How does the surface area to volume ratio effect daily life?
This project is a demonstration of surface area and volume manipulation. It's purpose will be to engage both the audience and the demonstrator to apply their knowledge and critical thinking skills to gain a new comprehension of surface area, nets, volume, and scale factors, using linking cubes and card stock to manipulate a "unit dog" based upon changes in any of these factors. To do this, demonstrators and experimenters will have to manipulate the pieces of the dog (torso, legs, head) using a system of ratios to see how the volume and surface area are affected by scale factor. Ultimately, from the data and results, they will be able to formulate an expression to describe these changes and understand how the expression(s) represent these transformations.
To see how this project relates to biology see: ELA COMPONENT
Although surface area should be easy to understand, its simplicity is often the foundation for struggle during comprehension and application. This is why its important to know the fundamentals of surface area before implementing it into any situation.
The definition is self explanatory; surface area is the total area of a three-dimensional object's surface. Surface area simply refers to the sum of the areas of each two-dimensional "plane" on a 3D object. Here is an example:
Vocabulary
Area
Surface Area
Volume
Edge/Side
Face
Procedures:
The procedure for this demonstration will revolve around a series of questions that serve to allow the participants to gain understanding and comprehension of certain concepts gradually.
Materials:
Linking Cubes
Multiple copies of 1 inch square grid card-stock paper
Tape
Scissors
Poster boards
Interactive applets (optional, for demonstration):
SA/V changes with side lengths
Front, back, side, and 3D views of rectangular prisms, shows how SA/V change as sides change
Shows nets for a cube
Allows virtual construction of solids with unit cubes that can be rotated to examine front, back, side, and 3D view.
Pictures:
Have each participant make a unit dog out of 13 linking cubes, providing them a pre-made model as a guide. Allow the participants to review over basic properties of a cubes, such as face, edge, and area/volume calculations.
Have the participants determine the volume of their unit dogs.
Have the participants think about a possible method of figuring out the surface area of the unit dog besides counting (the answer will be provided later on).
Have the participants draw the unit dog from different perspectives (top, front, and side views).
Demonstrate and discuss the connection between the unit cubes and their nets, incorporating the leg, head, and torso nets and shapes as well.
Allow the participants to come up with their own versions of nets for head/legs and torso.
Now have the participants create new nets increased by a scale factor of tow, using the linking cubes to visualize the affects.
Construct a scaled unit dog to use as a model. Assign groups to build particularly scaled dogs from the paper nets (use scales from the pre-made chart provided in this packet). Make sure that they use one net for the four legs and the head, and one for the torso. Have each group calculate surface area and volume for their dog. Fill in the chart with the student's measurements and calculations.
Let the students use the expressions from the last row of the table to calculate the surface area and volume with abnormally large scale factors.
The interactive applets are optional, but they are helpful as they provide a different perspective of transformations using smooth, clear images and transitions. (The following is an image of the interactive tool; the link is available once you hover over the picture).
Safety Regulations:
There are no safety concerns in this project. However, be weary for calculations and side views.
PBL Investigation:
Elizabeth
Harmony Science Academy Grand Prairie
PBL Investigation
Understanding the fundamentals of Surface Area, will allow others to utilize its principles critically. Consequently, the procedures for this project follow a list of questions that push observers to think about the different areas this project covers.
1. After the unit dog has been constructed, in what way(s) can its volume and surface area be determined?
The volume of the dog, can be determined simply, by counting the number of cubic units (assuming that each individual cube equals 1 cubic unit), resulting with a volume of 13 cubic units in volume.
The surface area, can also be easily determined by counting, however, a function can be used to find the total quickly and efficiently. Each segment (the legs, head, and torso) has 9 exposed surfaces. Since the cube units = 1 cube unit, each segment will have a total of 9 cubic units for surface area. There are six segments in total, so we can multiply six and nine to get the result, 54 cube units, in surface area.
2. After examining the the unit dog in both, two-dimensional and three-dimensional perspectives, what connection can be made between the nets for a unit dog and unit dog made out unit cubes?
The net is a substitute for the surface of the unit dog. The cubes have depth and account for the volume of the unit dog, the net represents the surface of the unit dog.
3. Now it’s time to think of the transformations that will occur when the dimensions are changed. Using this chart as a guide (and the same fundamental equation used to determine the surface area of the unit dog with units of 1) fill in the values for the different volumes and surface areas when their corresponding change in scaled factor.
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