The Visual / Auditory / Symbolic / Kinesthetic Approach to Algebra

Skip Manipulative Use If Your Students Already Know The Material

  • Manipulatives are designed to be outgrown.
  • Suggestion: Don't force anyone to speak a "less sophisticated language." Use it yourself. Speak it to them. Do not force them to reply using that language.
  • Once the mastery in representation and computation is achieved or it becomes more easy to complete the problem by written or mental work, Term Tiles should be sidelined.

      Trust this advise. If your students can complete integer computation or simplify algebraically expressions mentally, don't make them complete work manipulatively! They may, however, require manipulatives to write algebraic expressions and equations.

      It may be that for a given class, writing algebraic expressions and equations and solving word problems are their biggest challenges. If so, let these be the only tasks you complete manipulatively.

Think Money

      Suppose students have mastered integer addition and subtraction, manipulatively or otherwise. Suppose they've gone on to other work but someone goofed and doesn't see the SIGNED INTEGER ADDITION OR SUBTRACTION ERROR. If working with Term Tiles, address the issue with tile manipulation. If Term Tiles are not handy, THINK MONEY.

For example:
  • Add -3 and +5.
  • Add +3 and -2.
  • Simplify 3 - -4.
  • Lose $3 and win $5.
  • Win $3 and lose $2.
  • Win $3, then the opposite of lose $4.

      Thinking in terms of money permits a concrete thought process though no concrete item is present. This technique also works extremely well with simple fractions.

For example:
  • Simplify 5 - 1 1/2.
  • What's 20% of 20?
  • You have $5 and spend $1.50, what's left?
  • What's 20% of $20?

      Suppose instead they've made a SIGNED INTEGER MULTIPLICATION OR ADDITION ERROR. Whether working with tile or paper and pencil think OPPOSITES, or, think of THE GOOD GUYS AND THE BAD GUYS.

The Good Guys and The Bad Guys

      The story goes:
    If the Good Guys (+) come into town (+) , that's good (+).     
    If the Bad Guys (-) come into town (+), that's bad (-).
    If the Good Guys (+) leave town (-) , that's bad (-).
    If the Bad Guys (-) leave town (-), that's good (+).
(+)(+) is +
(-)(+) is -
(+)(-) is -
(-)(-) is +

      More to the point and mathematically accurate:

In multiplication or division:

The Signed Number Hundreds Borad

Click on the graphic to enlarge it.

      For a signed number addition and subtraction manipulative that is more sophisticated than Term Tiles use the Signed Number Hundreds Board.

      Addition and subtraction are completed through sequences of vertical and horizontal slides.

      See The Hundreds Board for more info., Unit 4   © 2008, A. Azzolino