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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 explore reflections as functions. They construct a reflection function, label the variables using function notation, and trace the variables to compare their paths. Students restrict the domain to reveal the corresponding range, and they move the mirror to create different members of the reflection family. Finally, students solve reflection challenges by figuring out where the mirror must go to match an existing reflection.


In this activity students will:

  • Construct an independent variable point x, a mirror (which defines the function), and the reflected image of x (the dependent variable).
  • Label the dependent variable using function notation as meaningful shorthand for the action that created it.
  • Drag the independent variable while tracing both variables, and describe both the variables’ relative motion (their covariation) and the relationship between their traces (their correspondence).
  • Move the mirror to form and investigate a different member of the same family.
  • Restrict the domain of the independent variable to the border of a polygon, and observe and describe the resulting range by dragging the independent variable.
  • Identify fixed points of the function, and explicitly compare the relative motion (both speed and direction) of the independent and dependent variables.
  • Solve challenges that involve finding a hidden mirror, given an object and its reflection.

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.