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## Back Page: My Favorite Lesson: Bridging Algebra and Calculus

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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## Understanding the Derivative through the Calculus Triangle

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

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## Reflections on a Technology–Rich Mathematics Classroom

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.

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## The Back Page: My Favorite Lesson – Slopes of Parallel and Perpendicular Lines: A Visual Justification

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.

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## The Popcorn Box Activity and Reasoning about Optimization

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.

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## Using Secant Parabolas to Find Roots

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.

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## Applications in Digital Image Processing

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

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## The Back Page: My Favorite Lesson: All Knotted Up

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.

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## A Natural Approach to the Number e

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.

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## Connecting Slope, Steepness, and Angles

Students must be able to relate many representations of slope to form an integrated understanding of the concept.