For a while now, I’ve been intrigued by the ways in which the study of geometric transformations can provide students with a very effective introduction to function concepts. Daniel and I have written a couple of articles about this topic, and we created a number of activities to take advantage of what can arguably be the most visual and tactile functions that students ever have the opportunity to experience. (The activities are in the Geometric Functions collection on the dynamicnumber.org website, .)

In our presentation at last April’s NCTM Annual Meeting, we took this approach one step further, by proposing a sequence of student experiences that would connect the commonly-studied geometric transformations directly to linear functions in the form *f*(*x*) = *mx* + *b*, and by extension to a variety of other algebraically-expressed functions.

Though our proposed student experiences are not yet fully fleshed out, we’re eager to bring these ideas to our blog, and in this post I’m taking the first step by setting the stage: on each page of today’s dilation challenges you’ll be asked to figure out which member of the dilation family is responsible for the function behavior you see and experience on the screen.

(Click the arrow at the bottom right of the sketch to go to the next challenge.)

These challenges are taken from our Dilation Function Family activity, which is designed as an introduction to dilations from a function point of view. You can download the full activity here including the sketch, student worksheet, and extensive teacher notes. (The downloaded version requires Sketchpad 5; if you don’t already have Sketchpad, you can download the free preview here: http://info.mheducation.com/sketchpad.trial.html.)

We’re also beginning to experiment with putting some of these activities on the web, with both the interactive sketch and the student directions on the same web page. You can try out the web version of the Dilation Function Family activity here: http://geometricfunctions.org/wsp/tegf/dilation-family/. (The main difference between the downloaded activity and the web activity is that on the web, students can’t actually construct the dilation functions themselves, so they need to press buttons in the websketch instead.)