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## (Counter)Productive Practices for Using Student Thinking

Learn why collecting, clarifying, and revoicing—often great teaching moves—do not always work.

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## Recalibrating Beliefs and Teaching Practices

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## Exploratory Writing to Support Mathematical Sense Making

A teacher implements this type of personal prose in the classroom to help students make sense of fractions and communicate ideas.

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## Area of a Changing Triangle: Piecing It Together

Examining the covariation of triangle dimensions and area offers a geometric context that makes analyzing a piecewise function easier for students.

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## Beyond the Sign Rules

Are your students negative about integers? Help them experience positivity and joy doing integer arithmetic!

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## Trigonometry and the World Water Crisis

This full unit in trigonometry introduces the world water crisis. Students engage in real-world problem-solving activities that access 21st-century skills while learning mathematics.

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## The Importance of Play in Middle School Mathematics

Using question 28 from the May Problems to Ponder in volume 114, the author and her seventh- and eighth-grade students launched into a discussion of creativity, linearity, piecewise, and recursive definitions of functions. This pattern to ponder provided rich mathematical opportunities for all students in my middle school classroom.

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## Problems to Ponder

Problems to Ponder provides 28 varying, classroom-ready mathematics problems that collectively span PK–12, arranged in the order of the grade level. Answers to the problems are available online. Individuals are encouraged to submit a problem or a collection of problems directly to mtlt@nctm.org. If published, the authors of problems will be acknowledged.

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## Problems to Ponder

Problems to Ponder provides 28 varying, classroom-ready mathematics problems that collectively span PK–12, arranged in the order of the grade level. Answers to the problems are available online. Individuals are encouraged to submit a problem or a collection of problems directly to mtlt@nctm.org. If published, the authors of problems will be acknowledged.

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## Using TileFarm to Support Emerging Multiplication

Symmetric dot patterns are a particularly powerful object for investigation, providing opportunities for foundational learning across PK–5. We found that second-grade students naturally used repeated addends to count symmetric dot patterns created using the new software TileFarm.