My favorite lesson introduces algebra students to a key concept from calculus: instantaneous rate of change. In this lesson, I help students develop an intuitive understanding of this abstract concept by framing questions within a real-world context.

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### Eric Weber, Michael Tallman, Cameron Byerley, and Patrick W. Thompson

We know derivatives are about rates. Why don't our students?

### Thomas E. Hodges and Elizabeth Conner

Integrating technology into the mathematics classroom means more than just new teaching tools—it is an opportunity to redefine what it means to teach and learn mathematics. Yet deciding when a particular form of technology may be appropriate for a specific mathematics topic can be difficult. Such decisions center on what is commonly being referred to as TPACK (Technological Pedagogical and Content Knowledge), the intersection of technology, pedagogy, and content (Niess 2005). Making decisions about technology use influences not only students' conceptual and procedural understandings of mathematics content but also the ways in which students think about and identify with the subject.

### Patricia Wallace-Gomez

When teaching slopes of parallel and perpendicular lines, I want students to have a visual image of the lines, not just memorize a formula. A simple exercise with parallel lines can get the message across.

### Walter J. Whiteley and Ami Mamolo

Investigating rates of change in volume without calculation leads to an enriched sense of the optimization process and encourages reflection and connection among different approaches.

### Jason Silverman, Gail L. Rosen, and Steve Essinger

Use digital signal processing to capitalize on an exciting intersection of mathematics and popular culture.

### Jamie-Marie L. Wilder and Molly H. Fisher

Our favorite lesson is a hands-on activity that helps students visually “tie” (pun intended) the concepts of rate of change and *y*-intercept together in a meaningful context using strings and ropes. Students tie knots in ropes of various thicknesses and then measure the length of the rope as the number of knots increases. We provide clothesline, twine, bungee cord, and other ropes found at local crafts, sporting goods, and home stores. We avoid very thin string, such as thread or knitting yarn, because the knots are small and the string length does not change enough to explore a rate of change. A variety of thicknesses is important because this allows for variability in the rates of change.

### Helen M. Doerr, Donna J. Meehan, and AnnMarie H. O'Neil

Building on prior knowledge of slope, this approach helps students develop the ability to approximate and interpret rates of change and lays a conceptual foundation for calculus.

### Yajun Yang and Sheldon P. Gordon

Almost all root finding methods use linear functions to calculate the next approximation to a real root. The authors introduce a method based on using parabolas through three points and use one of the two roots via the quadratic formula to identify the following approximation.

### Ryan Brydges and Kevin Webster

Students analyze items from the media to answer mathematical questions related to the article. Exponential growth is the topic of both clips this month.