In this section a standard and "basic" topic -- lines and the linearity property which only certain functions have -- in increasingly greater depth.

In Counting Surface Boxes teachers explore Cuisenaire rods with an eye towards looking for functional relationships. A link is made with linear functions. This much can be used or modified for use right in the middle school or other K-12 classroom. There is also a twist at the end challenging teachers to think carefully in terms of both geometry and algebraic (functional) relationships [Hint: The type of function one arrives at when looking at surface area, for example, depends, crucially, on what the independent variable is]

The activities Some Properties a Function can Have and the followup, Linearities: Two properties a Function Can Have. challenge teachers to delve into the heart of how function work, as well as the modern notation, which itself requires care; for example 2·f(3) is twice f(3), but for most functions, this is very different from f(2·3). For some special functions, one does get the same result.

This and other special properties which work only for certain types of functions, are explored in detail. Plenty of in-class guidance is given which by its very nature cannot be reproduced on the web -- do not be intimidated, rather, take the course and find out for yourself! In addition to plenty of supportive prodding, there is also plenty of time given, as well as opportunity to interact with fellow teachers in one's group. It is a challenging mountain climb for some, particularly to elementary-certified teachers with less mathematical background, but such teachers have successfully completed both activities, reaching a new level of understanding of functions, f-of-x notation, and where lines and linear functions fit in.

Cousin Phil: Linear Models for Financial Analysis is a nice application of linear functions to a very concrete situation. Even though the humorously named "Phil Bates" isn't quite as obvious a reference to Bill Gates as it was when this activity was first created circa 1999, nevertheless K-12 students and teachers find this an enjoyable activity.