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BOXES, BODIES, AND OTHER CONTAINERS
(A STUDY OF SURFACE AREA AND VOLUME)
MASTER TEACHER Jeff Duffy
GRADES 7-9
OVERVIEW
In this lesson, students will relate surface area to volume.
They will use formulas to calculate area and volume of regularly and irregularly
shaped objects. An understanding of the surface area/volume ratio and living
things will be developed. Students will have the opportunity to check
measurements of objects using the displacement technique. An appreciation of
metric measurement will be reinforced.
ETV SERIES
Math Vantage #112 Containers: Surface Area
and Volume
LEARNING OBJECTIVES
Students will be able to:
* identify various geometric shapes
* use supplied formulas to calculate surface area and volume of
various objects
* list reasons why an organism’s biological needs are
influenced by its surface area/volume ratio
* use lab equipment to measure various objects
* give examples of metric system benefits
MATERIALS
One ziplock bag per student containing candy in the shape of the
forms that will be discussed. [Life saver, M&M’s, small Tootsie
Roll, Hershey kisses, Peanut butter cup, gum ball, Jolly Rancher, caramel
(cube)]
Flat trays: One per student ideal; one per two acceptable.
Rulers: One per student: metric one side and standard on the
other.
Graduated cylinder: 100 ml minimum. One per student ideal; one
per two acceptable.
Calculators: one per student ideal. Optional at teacher’s
discretion.
Regularly shaped plastic, metal, or wooden objects. A sphere,
cube, cone should be minimal. They may be obtained from a school shop, craft or
hardware store, etc.
A plastic cup or beaker, large enough to hold the objects, one
at a time.
A bottle of soap bubbles and a wire loop.
Containers filled with enough water to fill the beakers several
times.
VOCABULARY
Sphere: A three dimensional object resembling a globe.
Circle: A line where all points are equidistant from a
central point, as in a hula-hoop.
Cube: A three-dimensional, six-sided figure, all sides
having an equal area, i.e. dice.
Volume: The amount of space an object occupies,
irrespective of shape.
Area: The amount of space occupied on the surface of an
object.
Metric: A system of measurement based on the number 10
and where all measurements are interrelated.
Cone: A three-dimensional object having a circular base
at one end and coming to a point at the other.
Ellipse: A closed curve made from the non-circular
portion of a cone.
Graduated cylinder: A device to measure the volume of
various substances.
Truncated: Cut off or shortened.
Cylinder: A three dimensional object formed by two
circles connected by parallel lines.
Torus: Generally having the shape of a donut.
PREVIEWING ACTIVITIES
Say, “Today we are going to examine several different shapes
that you are all familiar with, and we are going to see how they relate to one
another. We are also going to observe how an organism’s size and shape
influence where it fits into the world of living things. You will have an
opportunity to use some of the math skills you already have, and perhaps stretch
them a bit. Before we begin, it is important that everyone understands the
vocabulary that we will be using.” At this time, query students on their
understanding of the vocabulary terms. Have them isolate the various common
geometric shapes (candies) in different parts of the ziplock bag as those shapes
are discussed. For example, the sphere (gum ball) in the upper left hand
corner, the rectangular prism (Jolly Rancher) in the center, etc. Leave the
uncommon shapes like the frustum (peanut butter cup), the ellipsoid (M&M),
and the paraboloid (gum drop) until the end of the activity. After discussing
the terms, especially area and volume, say, “If all goes well today, you may
be able to sneak into these bags and munch on the geometric shape of your
choice."
FOCUS FOR VIEWING
To give the students a specific responsibility while viewing
say, “ Now we are going to watch a video that deals with the kind of shapes we
have just talked about, and we will calculate the area and volume of some
objects. At the end of the session, you will also need to know some of the
reasons why some organisms are very large and some small.”
VIEWING ACTIVITIES
Begin the tape. Ellen is in a pool and relates a pool of
water to a human body. Both are containers. Pause. Boy on a
skateboard slides and then the series title “Math Vantage” comes to the
screen. Ask, “What do you find in containers?” (volume). Say, “Let’s see
what shape some of these containers may look like.” Resume
when Ellen describes and names several containers in the ice-cream shop. Pause.
Bubble pops in Ellen’s face. With a bottle of soap bubbles and a wire loop,
blow a few bubbles. Ask, “What shape will the bubbles form?” (round).
Reiterate that a sphere is the most efficient way to enclose space. Then ask,
“How do we know that?” Fast forward through the bearded
gentleman wagging his head back and forth until Ellen’s head is framed in a
spinning square. Resume Ellen asks how volume and surface are computed.
She gives several examples and mentions gallons and liters, inches and
centimeters. Pause. Ellen says, “millimeters”; a triangle and fish
come onto the screen. Ask the students which system, the metric system or our
system, makes the most sense when working with area and volume. If some students
insist that our system is better, allow them to take measurements using inches
later in the exercise. They will have problems. Resume as Ellen describes
various common shapes and how their area is computed. Pause. Ellen says,
“It is like opening a whole can of worms.” She opens the lid, and formulas
erupt from the container. Distribute the handout which shows formulas for
computing area and volume of basic geometric forms. Using supplied rulers and
three-dimensional forms, calculate the area and volume of a form appropriate for
the student’s ability level. A cube or rectangular prism will be the easiest,
a cone or pyramid the most difficult. Students can use calculators (teacher’s
discretion) or manually compute the answers. When the students are finished,
say, “You were able to figure the area and volume of simple shapes, but what
do you do if they are complex?” Resume as Ellen indicates how even
complex forms are usually a combination of simple ones. She also discusses the
principle of displacement to calculate volume. She dunks Chanel in a tank of
water to determine her volume. Pause: Ellen says, “every time she goes
completely under.” Chanel is in a tank of water with 30,000 cm3. At this
time, have students determine the volume of the object they used earlier, but
use the displacement technique. This may be done in one of several ways: (1)
Place a beaker in the tray. Fill the beaker to overflowing. Submerge the object
in the beaker. Remove the object. Refill the beaker using a water-filled
graduated cylinder. The amount used is the volume of the object. (2) Place the
object in an empty beaker. Fill with water until the object is covered. Mark the
level of the water (at the meniscus) with a Sharpie or china marker. Remove the
object. Using a water-filled graduated cylinder, refill to the mark. The water
used from the graduate is equal to the volume of the object. (3) Place the
object into a water filled-to-overflowing beaker. Remove the object. Pour the displaced
water from the tray into a dry graduated cylinder. This is equal to the
volume of the object. Note: Objects that are less dense than water will need to
be pushed under the surface to render accurate measurements. Allow students to
determine volume of the object, and if time permits, recalculate using a
different technique. To the students say, “How is the relationship between
surface area and volume important to living things?” Fast forward
through a van filling with cardboard boxes until Ellen exits to the right of the
screen and says, “always have to think about keeping things cool.” Resume
as Ellen explains how objects that have the same volume (a glass of ice) but
different surface areas (cubes vs. crushed) gain and loose energy at different
rates. Stop. Ask students to identify animals that are very large or very
small and discuss their energy requirements. Whales might be at one end with
hummingbirds at the other regarding surface area/volume of warm-blooded animals.
At this time, ask students if they have questions about the lesson. If
appropriate, tell the students that they may select (and eat) one of the
geometric shapes in the ziplock bag, but must be able to write down or describe
some of the characteristics of that object.
POST VIEWING ACTIVITIES
To the students say, “Now that you have had an opportunity to
measure the area and volume of one of these objects, describe how the area and
volume of the object you selected is calculated.” They may also identify
items in society or nature that have that shape. Additionally, they can
determine the density of objects with the addition of information about the mass
of objects for which they have determined volume. The volume of a fellow student
may be estimated by assigning different geometric forms to body parts,
calculating the volume of those forms, then measuring the student’s volume
using the displacement method.
ACTION PLAN
Students may contact a zoo and inquire about the feeding habits
of warm-blooded vs. cold-blooded animals, large animals vs. small animals,
etc. They may also analyze their classmates’ general body shapes/sizes to
determine which climates they are best suited for. Invite a local heating and
air-conditioning specialist to discuss the importance of surface area/volume in
area construction.
EXTENSIONS
Language arts
Describe the difficulties an animal would have if its body type
and its habitat were not compatible.
Science/math
Calculate the density of unknown objects. Investigate the
properties of water and how freezing water affects its volume and density.
Social studies: Research the different climates of the
world and the body types that humans have evolved to survive in those
environments.
Internet:
Visit these sites to learn more about geometry.
http://www.megaconverter.com/Cv_start.htm
http://atle.abc.se/~m9847/geometr.html
http://www.geom.umn.edu/
http://www.sisweb.com/math/tables.htm
Also, try this site for an awesome (and free) search engine:
http://www.ferretsoft.com/netferret/index.html
Formulas For Area and Volume
Updated: April 01, 2008
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