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Grades: 7‒11

In Geometric Functions activities students create, manipulate, and experience function concepts by treating geometric transformations as functions with points as their variables.

In this activity students create a new function family, the swirl, which behaves differently from any function students have previously explored. The swirl is easily constructed but surprisingly interesting and complex, prompting students to revisit and refine their concepts of function. (This activity is a lead-in to its companion activity, Animated Special Effects—Swirl a Picture.)


In this activity students will:

  • Create a rotation function using a fixed parameter as the angle.
  • Redefine the function definition to make the angle of rotation depend on the location of the independent variable.
  • Describe the covariation behavior of the modified function.
  • Investigate the function as a mapping by restricting its domain, constructing the corresponding range, and comparing their shapes.
  • Predict the behavior of other members of the same family, and change the function to test the predictions.
  • Animate the function to explore a continuum of members of the function family, and explain the resulting “special effect” in terms of the function that produces it.
  • Conduct further investigation to quantify the attributes of the swirl in terms of the relative rates of change of the variables and the shapes of the restricted domain and corresponding range.

Mathematical Practices

(1) Make sense of problems and persevere in solving them; (2) Reason abstractly and quantitatively; (3) Construct viable arguments and critique the reasoning of others; (4) Model with mathematics; (5) Use appropriate tools strategically; (6) Attend to precision; (7) Look for and make use of structure; (8) Look for and express regularity in repeated reasoning.

Content Standards

8.F1,2; 8.G1; F-IF1,2,9; G-CO2; G-SRT1

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This activity is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License: If you adapt and/or share this activity, you must attribute it to "KCP Technologies, a McGraw-Hill Education Company." You may distribute it only non-commercially under the same or similar license.

This material is based upon work supported by the National Science Foundation under KCP Technologies Award ID 0918733, with grant period September 1, 2009 through August 31, 2013. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.