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## Patterns, Quantities, and Linear Functions

Students explore linear functions through patterns and the measurement of real-world quantities.

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## Commutativity of Linear Functions

Quite early in a student's mathematical study, he is introduced to the idea of commutativity. Initially the concern is primarily with the arithmetic operations, and in most instances the study of this property terminates there. In this article, we incorporate some ideas typically taught at the high school level (linear functions and composition of functions) to investigate the notion of commutativity in yet another setting. Of perhaps even greater interest than the main result of this paper is the strategy used to arrive at the original conjecture. Polya has written extensively about such strategies, and the authors believe that the material can be easily presented to senior high school students with a view to illutrating how certain s trategies are used in the cretttion of a mathematical theorem.

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## Connected Representations: From Proportion to Linear Functions

Building on mutually reinforcing representations, students can trace a path from proportion to linear functions.

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## Understanding Linear Functions and Their Representations

A linear card-sort activity helps students synthesize their understanding of linear functions while problem solving and reasoning; contains an activity sheet.

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## Introducing the Derivative through the Iteration of Linear Functions

The traditional approach to introducing the study of the derivative is through the tangent line as the limiting behavior of secant lines. This geometric discussion leads to the definition of the derivative, after which the differentiation formulas are developed and tangent lines can finally be computed. Unfortunately, this approach seldom gives the students any idea of what tangent lines actually are and why they might be of interest. Today's computer-graphics capabilities, however, allow exciting new ways to introduce the derivative. Topics from dynamical systems serve as interesting applications that are accessible even to students in precalculus courses (see, e.g., Devaney ). Coupled with modern graphing technology, the iteration of functions, especially linear functions, not only allows an instructor to illustrate ideas from an area of great current mathematical interest but also affords an excellent opportunity to introduce the derivative and tangent lines.

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## STEAMing Up Linear Functions

These four activities connect mathematics to science, technology, engineering, and art.

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## Activities: Using Linear Functions

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## Commuting linear functions and fixed points

An invitation to research, accompanied by at least one elegant proof

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## Applications: Linear Function Saves Carpenter's Time

Recently, a carpenter asked me a very interesting question. He was building roofs that were 30 feet long at the base (fig. 1). The pitch of each roof rises 4 inches for every horizontal foot. His job was to put in vertical supports every 16 inches, forcing him to climb up the ladder, measure 16 inches horizontally, and then measure the vertical height at that point. Then he climbed down the ladder, sawed the support, and climbed back up to put it in place. This procedure was repeated for each support. He asked me if a formula existed that he could use to determine the lengths of the supports in advance so that he wouldn't have to climb up the ladder each time to measure.

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## Linear Functions with Two Points of Intersection?

Of course two straight lines in Euclidean space cannot intersect in more than one point unless they are the same line and intersect everywhere—or can they? Follow this problem on the graphing calculator, however, and the surprising twist that gives this article its name will be seen. The material covered should be readily accessible to first-year-algebra students who have studied systems of equations, but it also contains valuable lessons for college mathematics professors who have been easily deceived by its apparent simplicity and familiarity.