Manatee Math

Reprinted from The Wild Times Teacher Connection v3n3, Spring 1998.

Matemáticas del Manatí en Español: This page is also available in Spanish.

Hypothermia is a constant problem for marine mammals who live in water several degrees colder than their body temperature. As a result, marine mammals have evolved a cylindrical body shape with small limbs to reduce the amount of body heat radiated into their environment.

This set of investigations will complement your students study of manatees and other marine mammals. The principal of reduced surface area per unit volume can easily be demonstrated in the classroom with the two following activities. Depending on the age of the students, the classroom facilities, and time constraints, these activities can be performed as a demonstration or as a small group investigation. As an investigation, each of the small groups should report their results to the class for comparison. This can provide a natural opportunity to apply averaging concepts to consolidate the data.

Surface Area and Volume:

Purpose:

This lesson will help students to understand the concepts of surface area and volume, and their relationship to each other.

Materials:

unit cubes (readily available in the math materials section of any supply catalog)
graph paper
pencils
calculators (optional)

Definitions:

Surface Area: the area of the exposed surfaces of a solid object, a 2-dimensional concept, measured in square units (i.e. square inches, square centimeters, square meters, square miles, etc.). Square units are abbreviated as unit².
Volume: the amount of space occupied by an object, a 3-dimensional concept, measured in cubic units (i.e. cubic inches, cubic centimeters, cubic meters, cubic miles, etc.). Cubic units are abbreviated as unit³.
Rectangular Prism: a solid with rectangular faces (sides). A cube is a special rectangular prism with square faces.

Procedure:

Using a fixed quantity of unit cubes, rectangular prisms can be constructed with different dimensions. Each will have the same volume if constructed from the same number of cubes, but will have different surface areas. To find surface area, the exposed square faces can be counted. Volume can be found the same way, but by counting the number of cubes.

Instruct the students to find the shape with the smallest surface area. Through exploration they will find that a cube has the smallest surface area for a fixed volume.

To record their experiments, students can use graph paper to draw elevations of their prism.

Older students with an understanding of ratio can compute and compare the Surface Area to Volume ratio in each prism. Older students can also graph and chart this information using graph paper or a spreadsheet program on the computer.

Hypothermia:

Purpose:

This activity will help students to understand the relationships of surface area, volume, and loss of body heat. This same investigation/demonstration can be used to illustrate how body shape also effects the absorption of heat. This is especially important to reptiles in desert environments.

Materials:

thermometers
tea kettle or microwave
vessels with different surface areas, but equal volumes
water
graph paper
clock

Procedure:

Find two containers with equal volumes: one nearly cylindrical or cubic in shape, the other flat and spread out. The containers should be made of the same material to ensure equal conductive and insulative properties. A ceramic coffee cup and plate make good vessels. If available, the best vessels are the plastic volume materials available in most math supply catalogs. Choose the cylinders with the greatest difference in surface area.

Measure the surface area of the vessels.

Measure and record the air temperature in the room.

Heat the required volume of water to 30° C. This is not so hot that it will cause a serious burn in an accident, but will elevate the water to 10° C above the air temperature in a typical classroom.

Decant the heated water into the test vessel(s).

Immediately record the temperature of the water in the vessel and begin timing. Continue recording the temperature at regular intervals (1 to 5 min.) until the water cools to room temperature. If time constraints do not allow for this intensive monitoring, record initial and final temperatures only. Final temperatures should be taken after an elapsed time of at least 20 minutes.

Repeat with other vessels. The same volume of water must be used in each trial.

Compare results to determine the best body shape for retention of body heat.

Graph results of each experiment. The x-axis should be time, while the y-axis should be temperature. Using this graph, it can be observed that the steeper the curve, the greater the rate of heat loss. Room temperature can be identified on the graph as the point where the cooling curve slope becomes zero. Older students with an understanding of linear equations can be challenged to write equations that will construct the lines for each container.

manatee Extension: Using the scale line drawing of a manatee, estimate the surface area and volume of a manatee. Encourage students to approach this problem with as little guidance as you can provide. By leaving them to the problem and allowing them to work in small, cooperative groups, they will have an opportunity to arrive at several different creative solutions. Allow each small group to present its solution the class, and provide an opportunity for solutions to be discussed and compared.

Hint, if your students are having difficulty, assume the drawing is an approximate profile and use the "width" of the drawing as the diameter of the animal.

Other Manatee Pages:

Understanding Manatees (lesson plans)
Manatee Natural History

Submit your own curriculum plans or resource suggestions to The Wild Ones for publication.

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