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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.

Students independently explore rotations, dilations, and translations as functions. They construct each function, label and trace its variables, and compare their motions and paths. Students restrict the domain to reveal the corresponding range, and identify similarities and differences between the members of each family.


In this activity students will:

  • Create two members each of three different function families (rotation, dilation, and translation).
  • Investigate the behavior of each function by dragging the independent variable, observing the dependent variable, and noting patterns of covariation (by observing relative rates of change of the variables) and correspondence (using traces to observe patterns of corresponding variable locations).
  • Identify the location(s) and patterns of fixed points (where the independent and dependent variables coincide) that characterize each family.
  • Restrict the domain of an independent variable to a polygon, and observe the resulting range of the dependent variable.
  • Use function notation appropriately, in written work and during discussions, to identify functions and their dependent variables.
  • Identify the behavioral characteristics shared by the members of each function family.
  • Use differences in behavior to distinguish each function family from the others.
  • Create and describe a new function family (glide reflection) and add it to Sketchpad’s Transform menu (optional).

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.