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## Reflecting on the Cloud Clock problem

This problem scenario explores the analog clock, a rich source of tasks associated with angles and angle measures. The Cloud Clock problem is an opportunity for students to deepen their understanding of analog clocks, angles, and time and angle measurement. To access the full-size activity sheet, go to http://www.nctm.org/tcm, All Issues. Each month, this section of the Problem Solvers department showcases students' in-depth thinking and discusses the classroom results of using problems presented in previous issues of Teaching Children Mathematics.

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## Interactive Bulletin Boards

Attractive bulletin boards enhance the classroom environment; moreover, those designed for learning tasks can also effectively promote problem solving and consolidate learning. Boards are easily adapted to levels of ability, grade, and complexity; they offer students real-world connections and engaging ways to interact with peers and with math concepts.

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Along with suggested instructional notes each month, teachers are given a problem and asked to use it in their own classrooms and report solutions, strategies, reflections, and misconceptions to the journal audience. The winter months often keep students involved in such extracurricular activities as competing in sports, dancing, or playing a musical instrument. This set of problems involves a real-life scenario about scheduling basketball games for the local gym and offers students the opportunity to show their creativity in using elapsed time.

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## Equal Pay Day

April 10, 2018, was “Equal Pay Day.” In the United States, on average, women earn less than men, which means that they must work more days to earn the same amount of pay. Equal Pay Day represents how many more days women must work to earn what men earned in the previous year.

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## Ferris Wheel Graphs

To introduce sinusoidal functions, I use an animation of a Ferris wheel rotating for 60 seconds, with one seat labeled You (see fig. 1). Students draw a graph of their height above ground as a function of time with appropriate units and scales on both axes. Next a volunteer shares his or her graph. I then ask someone to share a different graph. I choose one student with a curved graph (see fig. 2a) and another with a piece-wise linear (sawtooth) graph (see fig. 2b).

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## Summer Travel

You are excited about taking a summer vacation with your three friends except for potentially waiting in line at the airport's security checkpoint. In December 2016, the Salt Lake City International Airport expected 23,000 passengers to depart its terminals each day (Lee 2016). As a concerned passenger who wants to spend as much time on the beach as possible, you want to figure out the best way to get through security as quickly as possible.

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## Using voice recorders to capture their thinking

Teachers share success stories and ideas that stimulate thinking about the effective use of technology in K–grade 6 classrooms. This article describes how students can use available technology to communicate and share their thinking in popular media formats.

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## The ABCs of Covariance

While filling three-dimensional letters, students analyzed the relationship between the height of water level and elapsed time.

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