The Visual / Auditory / Symbolic / Kinesthetic Approach to Algebra

  Table of Contents

Contents of This Table of Contents
  • Even Before Representation
  • Manipulatives Theory
  • Term Tiles & Tokens Work Areas - READ THIS
  • Signed Number & Math Suggestions
  • REPRESENTATION & Tough Topics to Teach
  • Integer Computation
  • Algebraic Computation & Factoring
  • Solution of Equations and Inequalities
  • Advanced Topics -- Fractions, Linear Systems, Binomial Expansion
  • Appendix / Masters
  • Source Material

Even Before Representation

Manipulatives Theory
Unit 1. Term Tiles - OVERVIEW
Unit 1. Tiles, Tokens, & Toys
Unit 1. Manipulatives As A Language
Unit 1. Manipulative Use
Unit 2. Play, Work, Visualize, Debrief, Extend
Unit 2. Longer Assignments and Projects

Term Tiles & Tokens Work Areas - READ THIS
Unit 3. Storing Tiles During Use
Unit 3. Display Areas During Use
Unit 3. Moving Tiles

Signed Number & Math Suggestions
Unit 4. Skip Manipulative Use If Your Students Already Know The Material
Unit 4. Think Money
Unit 4. The Good Guys and The Bad Guys Or
Unit 4. The Signed Number Hundreds Board

Representation & Tough Topics to Teach
Unit 5. A Number
Unit 5. Zero
Unit 5. The Opposite
Unit 5. "Zero In" and "Zero Out" to Simplify.
Unit 6. Represent Expressions. Evaluate Expressions. Verify Statements.
Unit 7. More, Less, The Sum, The Difference, Double, Triple
Unit 8. "Five Decreased by a Number" Vs. "A Number Decreased by Five"
Unit 9. "The Other Number If The Sum Of Two Numbers Is Five"
Unit 10. Of: The Opposite of , Two Of , The Square Of
Unit 11. Perimeter and Area
      Unit 11. Vocabulary
      Unit 11. Tile Representation of Area
      Unit 11. Tile Representation of Perimeter
      Unit 12. Exercises

Creation & Interpretation
Unit 13. Make Experts Using Creation and Interpretation Activities.
Unit 13. Digital Manipulatives Spread Sheets
      tiles.xls     create.xls     hot.xls


Integer Computation
Unit 14. Integer Addition
Unit 14. Integer Subtraction
Unit 15. Integer Multiplication
Unit 15. Integer Division

Algebraic Computation & Factoring
Unit 16. Algebraic Addition
Unit 16. Algebraic Subtraction
      By "Adding the Opposite"
      Using "Take Away"
Unit 17.Distributive Property of Multiplication over Addition
Unit 17.Algebraic Multiplication
Unit 18.Algebraic Division
Unit 19.Factoring
      Factor - A Noun and A Verb
      Algebraic Factoring
Unit 20. Special Products
      Unit 20. Binomial Squared and Cubed
      Unit 20. Binomial Squared
      Unit 20. Binomial Cubed
      Unit 20. The Squares and Cubes of the Sums and Differences of Two Numbers
      Unit 21. Sums and Differences of Squares and Cubes
      Unit 21. Difference Two Squares
      Unit 21. Sum of Two Squares
      Unit 21. Difference of Two Cubes
      Unit 21. Sum of Two Cubes

Solution of Equations and Inequalities

Unit 22. Solve Equations and Inequalities.

Unit 22. Solve A Linear Equation By Undoing One Operation.
  • Undo addition.
  • Undo subtraction.
  • Undo division.
  • Undo multiplication.
  • Undo multiplication by -1.
  • Undo multiplication by a fraction.
Unit 23. Solve A Linear Equation By Undoing Two or More Operations.
  • Undo more than one operation.
  • Solve an equation with a binomial on each side.
  • Solve an equation with complicated expressions on each side.
Unit 24. Solve All Linear Equations.
  • Solve an equation with 1 solution.
  • Solve an equations with no solution.
  • Solve an equation in which all numbers are solutions.
Unit 25. Solve A Linear Inequality.

Unit 26. Solve A Quadratic

Unit 27. Solve A Proportion.
  • Remind Me. Why Is It Legal to Cross Multiply?
  • Check for extraneous roots.

Advanced Topics

Unit 30. Rational Expressions, Fractions
Unit 28. Fraction Representation and Computation
Unit 29. Represent Rational Expressions
Unit 29. Tilted Tile Display
Unit 29. Numerator / Denominator Display
Unit 30. Multiples, Equivalent Fractions, Reduced Fractions
Unit 30. Term Tiles Require Multiplication to Create Equivalent Fractions.
Unit 31. Reduced Fractions
Unit 10. (A-B)/(B-A) is -1.
Unit 32. Add Two Fractions
Unit 33. Subtract Two Fractions
Unit 34. Multiply Fractions and Mixed Numbers
Unit 35. Divide Two Fractions
Unit 36. Solve An Equation with Fractions.
Unit 39. Find A Common Denominator.
Unit 37. Solve A Linear System

Unit 38. Solve A Linear System by Substitution.
Unit 39. Solve A linear System by Linear Combination.
Extend the Pattern.

Unit 40. Dimensions
Unit 41. Sets of Numbers
Unit 42. Factorial, Permutations, Combinations
Unit 43. Binomial Expansion Presented Symbolically and Pascal's Triangle
Unit 44. Term Tiles and Binomial Expansions

Appendix / Masters
  • Contents
  • Basic Tile & Token Set: front back
  • Extended Tile & Token Set: front back
  • Additional Variable Tile & Token Set: front back
  • Paper Folded Solids
    x³ and a 1 cube -x³ -1 cube
    x², x, y², y, and 1         -x², -x, -y², -y, and -1
    x²y -x²y
    xy² -xy²
    y³ and -y³
  • 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

Source Material
  • A Journey with Self-Assessment as a Compass,
    and related pages 1995, 1998, 2001, A. Azzolino and related pages
  • "algebraic fractions" 2006 A. Azzolino
  • Algebraic inequalities & graphing on a number line, March 2008, A. Azzolino
  • area 2005, Agnes Azzolino
  • Common Denominator, 2006, Agnes Azzolino
  • compute.xls, 2008, A. Azzolino
  • Features of the Languages of the Math Classroom 1998, 2005 Agnes Azzolino
  • I Got Your Number, RATIONAL Computation, 2005, Agnes Azzolino
  • MATH SPOKEN HERE! - an arithmetic and algebra dictionary 2005, Agnes Azzolino
  • Multiple Strips & Fraction Bars, 2005, Agnes Azzolino
  • Multiples, 2005, Agnes Azzolino
  • "Options" 2008, Agnes Azzolino
  • perimeter 2005, Agnes Azzolino
  • prime.xls, 2003, 2008, A. Azzolino
  • "Remind Me Why "Invert and Multiply" Works When Dividing Fractions." 2008 A. Azzolino
  • So, What Is A Fraction? 2005, Agnes Azzolino
  • "Solve A Systems of Linear Equations by Substitution" 2008, A. Azzolino
  • Solve linear equation w/many operations - page 3, 2005, A. Azzolino
  • Solve linear equation w/more than one operation in the expression, pg 2, 2007, A. Azzolino
  • Solve linear equation w/paper & pencil, 2005, 2007, A. Azzolino
  • "Solving Quadratics" 2007, Agnes Azzolino
  • Spreadsheet Notes & Index 2006, Agnes Azzolino
  • "The distance formulas used for a line, a plane, or in space" 2003, 2006, 2007 A. Azzolino
  • The Hundreds Board, '98, '01, Agnes Azzolino
  • "Use Linear Combination to Solve Systems of Equations and Inequalities" 2004, A. Azzolino
  • Write 21, September 29, 2003, A. Azzolino
  • Writing one operation equations, expressions, statements, 2001, 2007, A. Azzolino
  • Written notes on factoring, 2005, A. Azzolino   © 2008, A. Azzolino