The Visual / Auditory / Symbolic / Kinesthetic Approach to Algebra

Multiples, Equivalent Fractions, Reduced Fractions

      Fraction Bars are a better manipulative than Term Tiles for introducing multiples, equivalent fractions, and addition or subtraction of arithmetic fractions.

      Term Tiles are a better manipulatives than Fraction Bars when it comes to representing or computing with algebraic fractions.

      To write an equivalent fraction:

  • using pencil and paper,
          multiply the fraction by 1, this means,
          multiply the numerator and denominator by the same number.
  • using Multiple Strips and Fraction Bars,
          choose the desired equivalent fraction already displayed in the Fraction Bar,
          visually created by the Multiple Strips display of the times tables.
  • using Term Tiles The Short Way using Tilted Tile Format,
          make multiple identical rows.
  • using Term Tiles The Long Way using Numerator / Denominator Format,
          multiply the fraction by 1, this means,
          multiply the numerator and denominator by the same number,
          exactly as one does with pencil and paper.
Fraction Bars Made from Multiple Strips Create Equivalent Fractions.

      Below, what looks like a piece of a times table, a 3-strip or multiples-of-3-strip is pictured. In boxes the three times tables is displayed from left to right, smallest to largest. It is not an infinite strip so only the smallest multiples of 3 are printed in the boxes.

      Below, a 3-strip is paired with a 4-strip to create, in Numerator / Denominator Display, a 3/4-strip. It shows the smallest multiples of three-fourths and in this three-fourths-stip, one can see that the fractions 6/8, 9/12, 12/16, all reduce to 3/4.

Term Tiles Require Multiplication to Create Equivalent Fractions.

      Writing equivalent fractions looks a lot like reducing fractions.

      Writing equivalent fractions and reducing fractions are inverses. One involves making multiples in the numerator and denominator, the other involves factoring out the common multiple in both the numerator and denominator.

      Identical rows are used to both reduce and write equivalent fractions.

      In the Tilted Tile Display of 6/10, and 9/15, note the identical row of tiles. Each fraction may be reduced to the 3/5, the ratio depicted in one row.

      The fraction 3/5 may be rewritten as 6/10, or 9/15, or 12/20, or 15/25, If the desired denominator is 12, the required numeration is 9 1/5, and, this can not be represented manipulatively without knowledge of proportions. But, it isn't usually addressed by pencil and paper computation without proportions either.

      Below in the Numerator / Denominator Display of 2/(x+1) and (2(x+2))/[(x+2)(x+1)] and (2x)/(x(x+1)), it is necessary to write numerator and denominator in factored form in order to see the common factors.

Write Equivalent Fractions in Tilted Tile Display.
1st:Place in ONE row (or column) EVERY tile in the display area which represents the fraction.
2nd:Place one (or more) EXACT DUPLICATES of this row (or column) in the SAME DISPLAY AREA so as to obtain a fraction with the desired denominator.
3rd:Recognize that if the desired denominator can not be obtained through this procedure, an equivalent manipulative fraction with that denominator is not possible.

Write Equivalent Fractions in Numerator / Denominator Display.
1st:Place the numerator of the original fraction in the STORAGE AREA ABOVE the numerator display area and leave the numerator display area empty.
2nd:Place the denominator of the original fraction in the STORAGE AREA BELOW the denominator display area and leave the denominator display area empty.
3rd:In the VERTICAL storage areas to the left of the numerator display area and the denominator display area, place the desired multiplier (factor).
4th:Complete the multiplication in both numerator and denominator display areas.
5th:Remove tiles from the storage areas., Unit 30   © 2008, A. Azzolino