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## Fraternal Twins

A same-area but different-perimeter problem is explored.

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## The Impact of Open-Ended Tasks

This department publishes brief news articles, announcements, and guest editorials on current mathematics education issues that stimulate the interest of TCM readers and cause them to think about an issue or consider a specific viewpoint. This month's guest editorial provides the platform for individuals to reflect on the positive impact that open-ended tasks can play in the teaching and learning of early mathematics. Classroom examples of open-ended expectations establish the immediate tie to fostering both 21st century skills and the Common Core State initiatives.

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## Solutions: Which lake is bigger?

Each month, this section of the problem solvers department showcases students' in-depth thinking and discusses classroom results of using problems from previous issues of Teaching Children Mathematics. In these solutions to the November 2015 problem, readers have a window into early elementary students' problem solving and understanding of measurement. Third graders were presented with tasks using maps of two lakes and various manipulatives to determine the bigger lake. Students discovered and were able to articulate that identifying the bigger lake depends on the attributes, area, and perimeter explored and that different attributes could result in different solutions.

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## Back Talk: Thea's thinking about area

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## Geometry and Proportional Reasoning in Agriculture

As you are enjoying your favorite icecream cone or sundae, take a moment to think about where your ice cream came from. If you have never visited a dairy farm, you might have a quaint image of a person sitting on a stool, milking a cow by hand into a metal pail. However, technological advances have automated many aspects of dairy farming, including processes for efficient milking. A rotary milking parlor is one such technology currently in use.

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## Gerrymandering: When Equivalent Is Not Equal!

Students explore the idea of equal versus equivalent, then learn about the social, political, economic, and educational implications of gerrymandering.

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## Pac-Man Takes a Bite out of Geometry

Do you recall the formula for the surface area of a cylinder? How about the surface area of a cone? Did you find one of these formulas more difficult to recall than the other? I have observed a difference in our students' understanding of these two formulas, and this has motivated me to think about a more conceptually based approach to learning geometric formulas.

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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.