## FAMILY RELATIONSHIPS—ROTATION, DILATION, AND TRANSLATION FAMILIES

Prerequisites:

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.

OBJECTIVES

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).

COMMON CORE CONNECTIONS
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