## DILATION CHALLENGES—THE DILATION FAMILY

Prerequisites:

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

Students explore dilations as functions. They construct a dilation 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 change the scale and move the center to create different members of the dilation family. Finally, students solve dilation challenges by figuring out the center and scale to match an existing dilation.

OBJECTIVES

In this activity students will:

• Construct an independent variable point x, a center point, a scale factor, and the dilated image of x defined by the center and scale factor.
• Label the dependent variable using function notation.
• 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).
• Change the scale factor to form and investigate different members of the dilation 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 center and unknown scale factor, given an object and its dilated image.

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