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 mathnstuff.com 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:

representing or graphing expressions,
presenting translantions, reflections, and dilations of functions,
finding "inverses",
solving equations,
explaining the composition of functions,
introducing and developing the concept of slope, domain and range.

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

"Math Exercises" -- whole body graphing
Presentation notes -- algebra thru precalc in 1 hour
coordinate plane, with radians marked
coordinate plane, without radians
$library$ Function Library

References:

$[MC,i. Home]$ $[Words]$ $Classes$

[Good Stuff -- free & valuable resources]

Good Stuff Page Math Languages, page 9