Making Sense of the Surface Area of a Sphere
Introduction: This lesson reinforces mathematical concepts and formulas for calculating surface area (SA). Using the story of Louis Zamperini’s travels while lost at sea in the Pacific Ocean, students will gain an understanding of how far Louie traveled while ‘lost at sea’ and how much surface area would have had to have been covered by the search and rescue planes.
Content Area & Grade Level: Algebra and Geometry, High School Grades 9 and 10
Original Author: Victoria Miles, Chair, MHS Mathematics Department. Updated with permission for Technology Integration: Carrie Haner, Student, BSU Instructional Technology Program. Credit Source: AIMS Activity: A-Peeling Spheres
Objectives: After group interaction and individual reviewing, students will be able to:
- Know and understand the formula for calculating surface area of a sphere.
- Demonstrated estimation and compatible numbers.
- Apply these concepts to solve real world problems involving spheres.
Standards: Common Core State Standards Initiative www.corestandards.org
Mathematics • Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. • 7.G.b.4 “Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.”
Mathematics • Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. • 7.G.b.6 “Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.”
Timeline : One 86-minute block or two 45-minute classes.
Students – List the continents and oceans of the earth. Sort the continents and oceans from smallest to largest.
Instructor – Play video “Continents and Oceans”
on smartboard screen. On the board, list the continents with surface area and list oceans without surface area, allowing room for writing these later.
Instructor – Display online document of the first two paragraphs of the Preface of Unbroken on smartboard screen, ask for volunteer to read aloud. GOAL: Find surface area of all the earth’s ocean, derive formula for surface area of sphere.
Display Interactive Map of Louie’s Journey on the Pacific Ocean, ask
What does the “sixty-four millions of ocean” refers to? [Answer: the area of the Pacific Ocean.]
“How can the area of all the earth’s oceans be measured?” [Answer/Activity: Subtract total area of 7 continents from total earth surface area.]
Instructor – Launch Activity: Hold up orange, ask
What part of the orange best represents surface area? [Answer: the peel.]
What part represents volume? [Answer: the inside edible part.]
If they sliced through the orange, what would the cross-section be? [A circle.]
Where would you slice that would result in the greatest possible circular cross-section? [The middle of the orange, between the two navels.]
Instructor – Demonstrate the activity using a document camera. Cut the orange exactly in half. Show the cross-section. Let students know this part of a sphere is known as the ‘GREAT CIRCLE’. Put the orange flat side down on a paper plate. Trace the great circle with a pen. Lift the orange off the plate. Ask
If we were to peel the orange using little pieces so they are relatively flat, and if we were to put the small, flat pieces of peel into the circle just traced, how many ‘Great Circles’ would the entire peel cover? Solicit guesses, but do not verify the answer [Four.]
Group Explore – Each group receives an orange, a plastic knife, paper plates.
Students work together solving the problem:
How many great circles are needed to fit the peel of an orange inside?
How many Great Circles were needed to hold the small pieces of peel? [Answer: four.]
How can we use this information to write a formula for the surface area (SA) of a sphere?
Surface Area = 4 Great Circles; the area of each great circle is A= πr^2;
Group Explore – Focus Problems – Provide earth’s circumference: 24,901 miles. Discourage the use of calculators; suggest students estimate and use compatible numbers. In groups, students work to solve the following problems:
a) Estimate the surface area of the earth.
b) Estimate the surface area of all the earth’s oceans.
c) What percent of all the earth’s oceans does the Pacific Ocean represent?
d) Estimate how many square miles Louis drifted on the life raft and/or the area a plane would have had to cover in the search. (Based on possible crash site to the Marshall Islands.)
Summarize Lesson: Invite groups of students to present their solutions to the whole class.
Exit Slip: One Question Quiz:
If the circumference of a large beach ball is 31 inches, estimate the surface area of the ball. Show reasoning.
Online Quiz by MathGuides.com: Surface Area of a Sphere
Unbroken by Laura Hillenbrand
Fresh oranges, plastic knife with serrated edge, paper plates, paper & pen
PC for access to Youtube, PowerPoint, Office, Websites
Grouping Strategies: Groups of 2-4
Calculators discouraged, exceptions for students accompanied by SPED Aide.
For advanced students or those seeking more involved geometry, suggest watching video Surface Area of a Sphere, deriving the formula by mathematicsonline.