
Type of Class:
High School Geometry (or may be adapted for
middle school
geometry), all
levels
Related VA SOL:
G.13
Time Frame:
90 Minute Block Period
Objectives:
 Students will be able to
discover the equation for the surface area of a sphere and will know how to
use that equation to solve surface area problems.
 Students will be aware
of current ecological problems and know how math is used in those problems.
Materials:
 Orange for each pair of
students (the most “spherelike” oranges you can find)
 Knife to cut oranges
 Old Newspaper
 Paper towels
 Calculators
 Rulers
 Several other spherical
objects, like a kickball, baseball, ball of yarn, halfspheres
 Apple per student
 Popsicle sticks
 Caramel Apple Wrap^{â}
(made by Concord Foods)
Procedures:
 Give each pair of
students an orange, some paper towels and newspaper. Have students lay the
newspaper on tables/desks/floor (wherever you want them to do this task) and
cut their orange in half so that it does not go through the “top” of the
orange (though if you have chosen very spherical oranges then it shouldn’t
make much difference). You may want to have that predone for them, but make
sure that the groups of students have orange halves that match each other.
 Lay half of the orange
on a sheet of paper and have students trace the circumference of the
crosssection of the orange. Tell them that they will be peeling the orange
and fitting the peel into the circles the size of their traced circles. Have
them predict how many circles the peel will fill and have them trace these
many circles on their paper. Then have them peel the orange (very carefully)
and fit the peelings into their circles. This activity works best if students
break the peelings into small pieces.
 As a class, find out how
many circles it took each group. Discuss and make hypotheses for the formula
for the surface area of a sphere assuming they already know the area of a
circle.
 Have students compute
the surface area of their orange.
 Next have small groups
of students (or pairs depending on the number of objects you bring in) find
the surface area of the different spherical items you brought in. Most items
you cannot cut in half, so bring in different things a student might be able
to use to find the surface area (string, a yard stick, tape, etc.). For
example, they can roll the ball on the floor and measure the circumference and
can then compute the radius or they can use string to find the object’s
circumference. Have students organize their data in a table so that they can
clearly see the different measurements that they have made. Many students will
be creative in this endeavor to find the surface area of their object.
 Lead a discussion with
the students on the volume of a sphere. See if they can come up with a way to
find the volume of a sphere, or to estimate the volume of a sphere. After
discussing ways to find the volume, give students the formula (unless they
figure it out on their own or already know it) and have them find the volume
of their object.
 Students will now be
working individually to make caramel apples. Make sure there is an apple per
student (as they can eat their product and obviously will have a hard time
sharing). Their job is to figure out the surface area of the apple and decide
exactly how much caramel they will need to cover the apple. They need to be
as careful as possible to compute the most exact measurement as possible. The
caramel comes in sheets and students will cut the exact amount they need for
their apple. Since they only get one chance to get the right amount of
caramel, they will need to decide how the best way to get the caramel to cover
the apple (most likely this will be by breaking it into small pieces and
putting it on the apple). Make sure students create a record sheet for all of
their measurements and have them write a description of how they found the
surface area. Once students have put the caramel on their apples, they must
show their apple to you before it can be put in the microwave and eaten. You
are checking to see how much (if any) extra caramel there is, or how much
space is left on the apple (this tells you how accurate the students have
been). You can record their accuracy on each of their record sheets as they
turn them in. Remember that the apples are not perfectly spherical, so there
may be some error for each person. (The directions for making the caramel
apples are on the packaging.)
Assessment:
Students should be assessed on how well they complete the project and how
accurate their applecovering process was. Also, for homework have students
measure the surface area of two spherical objects they find in their home. Have
them record all measurements and show all work in finding the surface area.

