The Visual / Auditory / Symbolic / Kinesthetic Approach to Algebra

 Appendix Contents
 Tiles & Tokens

 Basic Tile & Token Set: Constants, Linears, Squares, Cubes x³, x², x, 1, and their backs/opposites       front back PRINT THESE SHEETS BACK TO BACK.   Extended Tile & Token Set: Constants through Quartics     After printing and cutting, punch a hole through the quartic, x4 tokens so they are different from the cubic, x3 tokens and so they do not suggest being solid. x4, x³, x², x, 1, and their backs/opposites       front back PRINT THESE SHEETS BACK TO BACK.   Additional Variable Tile & Token Set: Constants through Cubic y³, y², xy, y, 1, 5, 10 and their backs/opposites       front back PRINT THESE SHEETS BACK TO BACK.

 Paper Folded Solids
 x³ and a 1 cube -x³ and a -1 cube x², x, y², y, and 1 -x², -x, -y², -y, and -1 y³ and -y³ x²y -x²y xy² -xy²

 Multiplying, Dividing, Factoring

 (x+y)³ = x³ + 3x²y + 3xy² + y³ (x-y)³ = x³ - 3x²y + 3xy² - y³ (y-x)³ = y³ - 3y²x + 3yx² -x³   (x+1)³ = x³ + 3x² + 3x + 1 (x-1)³ = x³ - 3x² + 3x -1 (y+1)³ = y³ +3y² + 3y +1 (y-1)³ = y³ - 3y² + 3y -1   Use these sheets to produce (x+y)³ and (x-y)³ and (y-x)³ Print 1 copy of each x³, -x³, and y³ (which also contains -y³)             x³, 1 -x³ -1 y³, -y³ Print 3 copies of each -xy² and xy², -x²y and x²y             x²y -x²y xy² -xy²   Use these sheets to produce (x+1)³, (x-1)³, (y-1)³, and (y+1)³ In addition to the cubes listed above, print 3 copies of each             x², x, y², y, and 1             -x², -x, -y², -y, and -1

• (for students, basic tiles only) tiles.xls
• (for students, basic + hot cells) create.xls
• (for teacher and to supplement students', all tiles, tokens, hot cells) hot.xls
• Eliminates the use of an overhead projector or opaque projector and the transporting of overhead tiles for anyone using Excel®, computer, and projector.

• Permits taking and storing notes on a lesson involving Term Tiles & Tokens.

• Permits the creation of tests, quizzes, worksheets, answer sheets using Term Tiles graphics.

• Permits students to create and store Term Tiles pictures and text for homework or projects.

• With restrictions, permits web based, or on site, Term Tile & Token staff developement.

• With restrictions, permits creation of texts, pamphlets, and papers by publishers, professionals, and parents.

• After introductory concrete use, eliminates the need for concrete tiles in a totally digital classroom since each student could work on her / his own spread sheet.

• Joins other digital spread sheets and spread sheet resource materials available at mathnstuff.com.

• Permits the user the full range of Term Tiles & Tokens (the visual / auditory / symbolic / kinesthetic) modes, permitting one to speak in pictorial, verbal, symbolic, and manipulative ways, permitting one to work with technology as well as text and tile to compute and communicate mathematically.
 www.termtiles.com, Appendix   © 2008, A. Azzolino