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


Digital Manipulatives Spread Sheets

  • (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