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## The Do Nothing Machine

The Trammel of Archimedes traces an ellipse as the machine’s lever is rotated. Specific measurements of the machine are used to compare the machine’s actions on GeoGebra with the graph of the ellipse and an ellipse formed by the string method.

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## Algebraic Thinking in the Context of Spatial Visualization

This pattern-related problem, appropriate for high school students, involves spatial visualization, promotes geometric and algebraic thinking, and relies on a no-cost computer software program.

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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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## 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.

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## A Radian Angle Measure and Light Reflection Activity

During a Desmos activity, students adjust the measures of angles in radians to reposition a laser and a mirror so the beam passes through three stationary targets. The Radian Lasers activity can be extended to simulate project-based learning.

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## Promoting Exploration through Synthesis

The author discusses “synthesizing" teaching practice, which encourages students to explore patterns and its underlying mathematics structure through technology.

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Using technology to explore a rich task, students must reconcile discrepancies between graphical and analytic solutions. Technological reasons for the discrepancies are discussed.

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## Everybody Still Plays: Virtual Engagement without Webcams On

When learning is virtual and students’ webcams are turned off, the ways that we interacted in an in-person classroom fall short. These six strategies for hearing from all students during whole-group instruction and small-group work honor students’ need to keep their webcams off.

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