Graphing w/Manipulatives & Exploring Functions through the Use of Manipulatives

© 1991, 1998, 2010 Agnes Azzolino

The Most Valuable Manipulative in Secondary Mathematics!

      The materials listed below come from the publication EXPLORING FUNCTION THROUGH THE USE OF MANIPULATIVES, 1994, Agnes Azzolino, ISBN: 0-9623593-3-5. It, as other material at is provided for not-for-profit purposes at no cost to the user. Please remember to site the source.
      A sample of manipulatives is shown below. Links to additional material are also listed.

Click on the image to see an enlargement.
      Start with a coordinate plane printed on paper (1/2 page works well).
      Provide students with waxpaper and have them place it on the paper coordinate plane (positioned for the function at hand).
      Have them point-plot a function ON THE WAX PAPER. They have made their first function manipulative. Repeat the process with more wax paper and more functions, as desired.
      The function I use first is 1/x, x-1, the reciprocal function. The absolute value, and square root are also important, but, not as important as the big 3 -- the identity function, the opposite function, and the reciprocal.
      The Absolute Value Dilator at the left is made of printed transparency film with the 2 arms hinged by a clothing snap. Once placed on a coordinate plane printed on paper, functions such as 2|x| and |x|/2 may be graphed as well as the more traditional |x|, |x+2|, and -|x-1|+3.

Click on the image to see an enlargement.
      The Slopemeter is also printed on transparency film and a tangent arm is hinged with a clothing snap at the origin. Placed on a point on a function and positioned so the x-axes and y-axes are parallel, the slope may be read by placing the arm tangent to the curve and determining where the tangent line crosses a grid mark on the coordinate plane. The (x,y) coordinates provide the y/x slope.
      Function manipulatives may be used in:
      On the next page, animations on Representing Expressions Kinesthetically, provides a taste of the power of manipulative / kinesthetic language. It uses aminmated gifs to simulate their use in:
  • graphing a function,
  • graphing translations and reflections,
  • solving equations.

Additional Resources

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© 2005, 2010 Agnes Azzolino