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February 2012, Volume 105, Issue 6


The Popcorn Box Activity and Reasoning about Optimization
Walter J. Whiteley and Ami Mamolo
Investigating rates of change in volume without calculation leads to an enriched sense of the optimization process and encourages reflection and connection among different approaches.

The Popcorn Box: GSP File for creating animations of the activity
Second Look:
Volume of a Rectangular Prism

Geometry in Medias Res
Mimi Cukier, Tony Asdourian, and Anand Thakker
Plunking students down in the middle of a geometry course—in medias res—helps them appreciate the axiomatic structure of geometry.

A Natural Approach to the Number e
Helen M. Doerr, Donna J. Meehan, and AnnMarie H. O’Neil
Building on prior knowledge of slope, this approach helps students develop the ability to approximate and interpret rates of change and lays a conceptual foundation for calculus.

Geometry + Technology = Proof
Irina Lyublinskaya and Dan Funsch
Symbolic geometry software, such as Geometry Expressions, can guide students as they develop strategies for proofs.

Teaching through Problem Solving
Cos D. Fi and Katherine M. Degner
TtPS is an approach to teaching mathematics that provides students with a way to learn mathematics with understanding.

Teaching moves for TtPS

Second Look - Volume of a Rectangular Prism

Thinking out of the Box . . . Problem
Student explorations and mathematical generalizations of a classic pre-calculus and calculus problem "The box problem."

What Else Can You Do with an Open Box?
Starting with the classic Open Box problem, we present extensions of this problem that can be used in high school mathematics classes. We also challenge high school teachers to use this process of problem analysis in their own practice as a way to enrich the content of their lessons and as a means of individualized professional development.

Sharing Teaching Ideas: Optimization: A Project in Mathematics and Communication
The mathematics in the problem addresses many of the NCTM's Standards, as well as reform-calculus initiatives. The opportunity for meaningful communication about the mathematics is a vital component that was met well in the context of the assignment.

Illuminations Lesson: Popcorn Anyone?

This lesson can be used for students to discover the relationship between dimension and volume. Students create two rectangular prisms and two cylinders to determine which holds more popcorn. Students then justify their observation by analyzing the formulas and identifying the dimension(s) with the largest impact on the volume.

Measurement Standard for Grades 9-12

Instructional programs from prekindergarten through grade 12 should enable all students to—

Understand measurable attributes of objects and the units, systems, and processes of measurement

Expections: In grades 9–12 all students should—

  • make decisions about units and scales that are appropriate for problem situations involving measurement.

Apply appropriate techniques , tools, and formulas to determine measurements

Expections: In grades 9–12 all students should—

  • analyze precision, accuracy, and approximate error in measurement situations;
  • understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders;
  • apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations;
  • use unit analysis to check measurement computations.