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 “sphere-like” 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, half-spheres
• Apple per student
• Popsicle sticks
• Caramel Apple Wrap^â (made by Concord Foods)

Procedures:

1. 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 pre-done for them, but make sure that the groups of students have orange halves that match each other.

2. Lay half of the orange on a sheet of paper and have students trace the circumference of the cross-section 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.

3. 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. 

4. Have students compute the surface area of their orange.

5. 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.

6. 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.

7. 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 apple-covering 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.