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## Perimeter and Beyond

A simple problem-solving exercise encourages teachers to “start small” to reveal how third graders understand multiple math concepts simultaneously.

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## Problem: Building picture frames

This month's problem offers students an opportunity to determine where we find math in the world, interpret it, and engage in mathematical modeling. Each month, elementary school teachers are presented with a problem along with suggested instructional notes and asked to use the problem in their own classrooms and report solutions, strategies, reflections, and misconceptions to the journal audience.

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## Hockey-stick fundraiser

New Jersey sixth graders who were participating in a school fundraiser to help fight childhood cancer could hardly wait to explore this problem from the September 2011 issue, which invites students to work strategically with combinations of numbers. Afterward, their teacher reflected that so many available activities claim to be authentic learning activities but require little in-depth problem solving. This one does.

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## Problem: Area Beyond the Formula

This variation of Simon's (1995) rectangle exploration will have students investigating area in a conceptual manner that goes beyond tiling and formulas. Each month, elementary school teachers are given a problem along with suggested instructional notes; are asked to use the problem in their own classrooms; and are encouraged to report solutions, strategies, reflections, and misconceptions to the journal audience.

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## Connecting the Threads of Area and Perimeter

See how sixth-grade students design and create quilt squares for this geometry project.

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## Butterfly gardens

The May problem scenario has students explore what happens to the area of a two-dimensional shape as the side dimensions change but the perimeter remains the same. A fourth-grade teacher from North Carolina explored the task and uncovered her students' misconceptions concerning the corner tiles of the rectangle.

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## Hockey-stick fundraiser

Each month this section of the Problem Solvers department features a new challenge for students. Readers are encouraged to submit problems to be considered for future Problem Solvers columns. Receipt of problems will not be acknowledged; however, problems selected for publication will be credited to the author. Find detailed submission guidelines for all departments at www.nctm.org/tcmdepartments.

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## Constructing Understanding in a Mathematics Methods Course

Key elements can enhance teacher candidates' understanding, interest, and confidence with learning and teaching mathematics while decreasing their math-related anxiety and fear.

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## Coaches Engage with Principles to Actions

Here is support for coaches who work in diverse contexts to integrate high-leverage teaching and coaching practices with specific attention to mathematics content.

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## In Stitches: Knitting and Knotty Area Problems

Knitting, like other traditional crafts such as quilting or weaving, is a highly mathematical activity. Knitters need to constantly coordinate different forms of measurement, including weight, length, and area. Knitting typically involves following a pattern. If you do not follow that pattern, you can find yourself dealing with some pretty complicated knot theory.