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## Conceiving Phase Shift as Time Travel

Two teachers use homemade pendulums for their students to explore phase shift in sine curves. The use of Desmos’s Activity Builder enabled students to become mathematical explorers and supported their sense of mathematical agency.

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## Technology and Median Income Data: Opening Doors

Classroom stories show how using technology to investigate the wage gap provided opportunities to develop students’ identities and agency and enabled a classroom culture of sharing and risk-taking.

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## The Hidden Beauty of Complex Numbers

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## Enacting Co-Craft Questions Using Flexible Teaching Platforms

Teachers can implement a mathematics language routine within in-person/hybrid and remote instructional contexts.

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## Point-Line Ellipses and Hyperbolas

The author alters the definitions of ellipses and hyperbolas by using a line and a point not on the line as the foci, instead of two points. He develops the resulting prototypical diagrams from both synthetic and analytic perspectives, as well as making use of technology.

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## An Alternative Approach for Defining a Quadratic Function

Defining a quadratic function through the slopes of its secant/tangent lines leads to the fundamental theorem of calculus (FTC) and an alternative way of understanding integration.

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## Rethinking Wait Time: What Can 3 Seconds Do?

A high-leverage strategy first discussed more than 50 years ago, wait time has many benefits for both teachers and students yet is not used to its full potential. See how it can enhance your students’ mathematical discourse.

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## Using Judicious Telling to Strengthen Literacy in Mathematics

Language is a conduit for communicating and understanding mathematical ideas. This article explores how we can use judicious telling to attend to students’ written and spoken literacy in mathematics.

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## Approximating Pi

This method using the area of regular polygons inscribed in circles to approximate a value for pi is similar to the method used by Archimedes using circumferences.

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## Variations on a Theme of Fibonacci 1

We introduce variations on the Fibonacci sequence such as the sequences where each term is the sum of the previous three terms, the difference of the previous two, or the product of the previous two. We consider the issue of the ratio of the successive terms in ways that reinforce key behavioral concepts of polynomials.