VIRTUAL CONFERENCE ictcm.com | #ICTCM 32 nd International Conference - PowerPoint PPT Presentation

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE ictcm.com | #ICTCM

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Exploring Volumes with GeoGebra Dissection Models Dr. Thomas (Tom) Cooper Professor, University of North Georgia Department of Mathematics Dahlonega Campus tom.cooper@ung.edu http://faculty.ung.edu/tecooper/

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM BIG IDEA • Software such as Geogebra allows us to build, view, and manipulate 3D objects. • We can explore volume formulas intuitively.

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Two Interesting Solids Frustum of a Square Pyramid Burr Buzzle & Pieces GeoGebra Model

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM https://www.gathering4gardner.org/g4g13gift/math/ BanchoffThomas-GiftExchange-Foxtrot-G4G13.pdf

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Both can be decomposed into square pyramids and “semi - orthocentric” tetrahedra.

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Some Volumes We Can Explore Intuitively Volume of any Prism = height × (area of the base) Volume of any Pyramid 𝟐 = 𝟒 × height × (area of the base) = 𝟐 𝟒 × (volume of the prism it fits in)

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Some Volumes We Can Explore Intuitively Volume of any tetrahedron 𝟐 = 𝟕 × (volume of the parallelepiped it fits in) Volume of a “semi - orthocentric” tetrahedron with a pair of perpendicular opposite edges = 𝟐 𝟕 × "height" × (𝐪𝐬𝐩𝐞𝐯𝐝𝐮 𝐩𝐠 𝐮𝐢𝐩𝐭𝐟 𝐟𝐞𝐡𝐟 𝐦𝐟𝐨𝐡𝐮𝐢𝐭)

Now to Geogebra …

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM The volume of a right rectangular prism is LWH = height × (area of the base). GeoGebra Model of Prisms

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM • By Cavalieri’s Principle , the volume of any prism is height × (area of the base). Geogebra Model of Cavalieri for Prisms GeoGebra Model of Cavalieri for Pyramids

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Pyramids (Volume = 1/3 Prism) • Geometric Argument by decomposition • Any pyramid (including cone) by Cavalieri’s Principle These were called “ Yangma ” by Liu Hui in The Nine Chapters on the Mathematical Art , 263 AD GeoGebra Model for a Cube More General Case (GGB) using a pyramid with a right triangle base

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Tetrahedron 1 6 𝒃 × 𝒄 ∙ 𝒅 Volume= • By decomposition, Volume of tetrahedron = 1/3 x (Volume of Triangular Prism) = 1/6 × (Volume of Parallelepiped) GeoGebra Construction

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Semi-Orthocentric Tetrahedron When a pair of opposite edges are “perpendicular”, Volume • = 1/6 × h × product of these side lengths. = = = + a a ,0,0 , b x y h , , , c x y , b h , i j k 1  = 6 𝒃 × 𝒄 ∙ 𝒅 Volume= a b a 0 0 x y h = − + (0) i ( ah ) j ( ay ) k = − 0, ah ay , 1 1 0,   = −  + a b c ah ay , x y , b h , 6 6 1 0 = + − + + ( ah y )( b ) ayh 6 1 = abh 6 Geogebra Model

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Frustums (with Square Bases) 𝑊 = 𝑐 2 ℎ + 1 3 ℎ(𝑏 − 𝑐) 2 +4 × 1 𝑏 − 𝑐 𝑐ℎ 2 2 1 3 ℎ𝑏 2 − 2 3 ℎ𝑐 2 + 𝑏𝑐ℎ − 𝑐 2 ℎ 1 = 𝑐 2 ℎ + 3 ℎ𝑏𝑐 + 1 3 ℎ𝑏 2 + 1 1 3 ℎ𝑐 2 3 ℎ𝑏𝑐 + = 1 3 ℎ(𝑏 2 +𝑏𝑐 + 𝑐 2 ) = GeoGebra Model Liu Hui’s (263 AD)decomposition into a square prism, 4 “ qiandus ” (triangular prisms), and 4 “ yangmas ” (square pyramids)

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Dissection Idea of Tom Banchoff (Professor Emeritus Brown University) Divide the bottom into n by n squares, and divide the top into ( n +1) by ( n +1) squares. The frustum can be divided in: 2 1 𝑏 n 2 pyramids pointing up, each with V = 3 ℎ 𝑜 2 1 𝑐 ( n +1) 2 pyramids pointing down, each with 𝑊 = 3 ℎ 𝑜+1 1 𝑏 𝑐 2 n ( n+1) semi-orthocentric tetrahedra, with 𝑊 = 6 ℎ 𝑜 𝑜+1 This give a total volume of 2 2 𝑊 = 𝑜 2 1 3 ℎ 𝑏 1 6 ℎ 𝑏 𝑐 + (𝑜 + 1) 2 1 𝑐 + 2𝑜 𝑜 + 1 3 ℎ 𝑜 𝑜 𝑜 + 1 𝑜 + 1 = 1 3 ℎ𝑏 2 + 1 3 ℎ𝑏𝑐 + 1 3 ℎ𝑐 2 = 1 3 ℎ 𝑏 2 + 𝑏𝑐 + 𝑐 2 GeoGebra Model

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Four Piece Dissection GeoGebra Model Consider the frustum with the top being an 𝑏 × 𝑏 square and the bottom being a 𝑐 × 𝑐 square. This can be dissected into pyramids with volumes 1 1 3 ℎ𝑏 2 and 𝑊 3 ℎ𝑐 2 and semi-orthocentric tetrahedra with volumes 𝑊 1 = 2 = 1 1 𝑊 3 = 6 ℎ𝑏𝑐 and 𝑊 4 = 6 ℎ𝑏𝑐 . 1 3 ℎ 𝑏 2 + 𝑏𝑐 + 𝑐 2 So the frustum has volume 𝑊 =

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM   h 1 s 1 Burr Puzzle = = 2 2 3   2 Pyramids, each with: = 3 V s s s   3 2 6   1 1 s 1 = = = 2 2 3   4 Tetrahedra, each with: V hs s s   6 6 2 12 2 4 2 = + = 3 3 3 Puzzle Piece Volume: V s s s 6 12 3   2 = = 3 3   Total Volume: V 6 s 4 s   3 GeoGebra Model Note the total volume is half the volume of a cube with sides of length 2s.

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Yoshimoto Cube GeoGebra Demonstration

32 nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE #ICTCM Contact Information Dr. Tom Cooper Professor University of North Georgia tecooper@ung.edu http://faculty.ung.edu/tecooper