One of the most interesting and important unifying concepts in school mathematics is the function concept. One often hears such expressions as “the function of the brain is to enable a person to make rational decisions” or “the function of a lamp is to give light.” However, the mathematical use of the word “function” is linked to the historical use of the word by mathematicians and scientists. Historically, mathematicians and scientists have used the word “function” to illustrate how one condition or “state” affects another. For example, they use such phrases as “distance is a function of time,” “water pressure is a function of depth,” and “the area of a circle is a function of its radius.” You will note that these expressions suggest pairs—distance and time, pressure and depth, area and radius. In the following discussion we will first review the mathematical definition of function, and then present some practical uses of functions. Part 2 of this article give examples of functions that can be used to describe some real life situations.

### Edward Esty

Some of the recent elementary school arithmetic textbooks introduce functions, a topic formerly appearing no earlier than in high school. The University of Illinois Arithmetic Project has long used functions (called “jumping rules” by the Project) in classes for elementary school children.

### Jennifer N. Lovett, Allison W. McCulloch, Lara K. Dick and Charity Cayton

the task of study in this article—which is focused on the function concept—and to report research considering PSMTs’ reflections on their learning as a result of engaging with the task. Technological Pedagogical Function Knowledge Building off

### Jennifer N. Lovett, Allison W. McCulloch, Blain A. Patterson and Patrick S. Martin

released? As a matter of fact, vending machines are programed with functions that assign to each button pressed a particular item to dispense. Given that vending machines are a common experience for students in today's world, they provide a wonderful

### Judith Macks

Contexts from the fields of geography and history emphasize functions as situations in which each input has exactly one output.

### Aaron Bakst

The development of *hyperbolic functions* in the traditional trigonometry courses (if this is ever reached during a one-semester instruction) is usually confined to purely algebraic methods. However effective the latter procedures may be, it is doubtful that a student realizes the import of the properties of hyperbolic functions. The student is never offered the opportunity to realize the fact that, essentially, the properties of hyperbolic functions are analogous to the properties of circular functions. It is possible, however, to develop the properties of hyperbolic functions in a manner which is analogous to the processes which are employed in the development of circular functions. Thus, it is proposed to examine and to develop hyperbolic functions by means of a geometric approach.

### Tim Erickson

height.” If you then plot each bounce height against its bounce number, an exponential function can model the data; that is, the function will approximate the points in your plot. A quick online search yields a wealth of lessons on this topic from

### Charles A. (Andy) Reeves

Examples of elementary teachers using physical models of function machines with their K-5 students. Teachers will learn how to make function machines and use them in the classroom to introduce students to algebra functions

### Robert C. Yates

Herein is presented a definition of trigonometric functions not usually found in elementary texts. Continuing in the same unusual way, properties of these functions are derived in a manner simple enough to use in an early calculus course.

### G. N. Wollan

The function concept, a concept with many connections to other concepts, needs careful study by teacher and student.