Covering the Sphere: A Pi-Day Activity NOYCE, June 2007

1. Covering the Sphere:A Pi-Day ActivityNOYCE, June 2007 Presented by : Kimberly Halsey Matthew Wells Department of Mathematics Department of Mathematics Bath County High School (KY) University of Kentucky

2. Kimberly Halsey: High School teacher at Bath County HS Teaches classes ranging from basic Algebra to Pre-Calc Has been a teacher for 15 years Matthew Wells Mathematics Graduate doctoral student Teaches two classes per semester at University of Kentucky Going into 5th year in Graduate School Background of Presenters

3. Philosophy to the Mathematics • To get students to understand the mathematics in a conceptual way • Build a foundation that will be beneficial for later mathematics • Challenge students to think abstractly and problem solve

4. Mathematics Focus • Geometry and Algebra • Area of a circle • Surface Area of a sphere • Specific focus on PI (good for a Pi-Day activity) • The main activity involved the formula for Surface area of a sphere

5. Area of a Circle • Formula for area of a circle: A =πr2 • Challenge : Present a way to understand “WHY” the formula is the way it is • Idea : Break the circle up into triangles

6. Subdivide into 4 Triangles

7. Subdivide into 6 Triangles

8. Subdivide into 8 Triangles

9. Subdivide into 12 Triangles

10. Conceptualize Area of a Circle • Area of circle is approximately areas of all the triangles • Area of Triangle is ½ x Base x Height

11. Ideas Observed • Students understand areas of triangles and squares easier than areas of round objects • There is a connection between triangles and circles • The students can measure one triangle easily and find the areas of all the triangles

12. Formula for Area of a Circle • Area of Circle = ∑(Area of Triangle i ) = ∑( ½ x Base x Height)i =∑( ½ x Base x radius) i =(½ r) ∑(Base) i = ( ½ r )(2πr) = πr2

13. Ideas Observed (part II) • Students understand areas of triangles and squares easier than areas of round objects • There is a connection between triangles and circles • The Circumference of a circle is used (how might this apply to Spheres???)

14. Surface Area • Question: How do we find the surface area of a sphere? • Question: Can we get the students to understand the concept of Surface Area?

15. Balloons and CD’s

16. Observations • Students can connect the idea of Surface Area to 2-dimensional area • Idea of using triangles is easily transferable to higher dimension • Question: Are the triangles really triangles? • ANSWER: NO!!!

17. An INCORRECT Proof • Surface Area = ∑(Area of Triangle i ) • = ∑( ½ x Base x Height) i • =∑( ½ x Base x ½πr ) • = ¼ πr ∑(Base)I • = ¼ πr (2 πr ) • = ½ π2r2 SURFACE AREA OF SPHERE WOULD BE: π2r2

18. Comments about the Proof • The “triangles” are not actually triangles • When the “triangles” are made thinner, the error in the formula remains • Can we assimilate what we did for the area of a circle? • SOLUTION: Use triangles that shrink from all sides

19. Pi-Day Activity • Materials: One foam Hemisphere Pair of scissors Box of pins Paper triangles

20. Pi-Day Activity • Cut out the triangles and attach them to the round part of the hemisphere with three pins:

21. Pi-Day Activity • Try to cover the Hemisphere with as many NON-OVERLAPPING triangles as possible: CORRECT INCORRECT

22. Pi-Day Activity • Count up the number of triangles • Measure the radius of the ball • Measure the length of a side of the triangle • Collect and Compare

25. Students doing this Activity

26. More Students

27. Even More Students

28. Further Activities HINT: Volume of a Pyramid is : (1/3) x Base x Height • Can you find the volume of a sphere using the ideas presented today?

29. Contact Information • Matthew Wells mwells@ms.uky.edu (859) 257-7216 • Kimberly Halsey Kimberly.Halsey@Bath.kyschools.us