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November 2012, Volume 106, Issue 4

FEATURES

Using Multiple Representations to Teach Composition of Functions
Scott Steketee and Daniel Scher
Experience with multiple representations fosters students’ robust understanding of what functions are, how they behave, and how they can be composed.

Thinking Like a Mathematician
Michael K. Weiss and Deborah Moore-Russo
The moves that mathematicians use to generate new questions can also be used by teachers and students to tie content together and spur exploration.
Second Look:
Think Like a Mathematician: Connections though Representations

Understanding the Derivative through the Calculus Triangle
Eric Weber, Michael Tallman, Cameron Byerley, and Patrick W. Thompson
We know derivatives are about rates. Why don’t our students?

Exploring Quadrilaterals through Flexible Approaches- FREE PREVIEW!
Kerri Richardson, Anne Reynolds, and Catherine S. Schwartz
Rich mathematical tasks—here, finding and categorizing the quadrilaterals that can be made with vertices on a 4×4 grid—can promote adaptability in any classroom.

Looking for Connections between Linear and Exponential Functions
Jane-Jane Lo and James L. Kratky
Students compare and contrast linear and exponential relationships from tabular, graphical, and algebraic representations.

Spanish Word Problem

Tasks That Make Connections through Representations
Joe Garofalo and Christine P. Trinter
Students think resiliently about using the quadratic formula, analyzing factors graphically, finding the shortest distance between two points, and finding margin of error.

Second Look - Think Like a Mathematician: Connections though Representations

Activities for Students: Becoming Mathematicians: Exploring an Unsolved Arithmetic Problem
The Collatz conjecture is an open mathematical question easily accessible to prealgebra students; the authors present some activities based on the conjecture to show students how mathematicians really work.

Encouraging Preservice Mathematics Teachers as Mathematicians
An assignment that asks preservice secondary mathematics teachers to make connections between the mathematics they know and the mathematics they will teach. It describes how one preservice teacher's project resulted in a physical representation of the statement and proof that the sum of cubes of the first n natural numbers is equal to the square of their sum.

Discovering Relationships Involving Baravelle Spirals
This article details a classrooms exploration of Baravelle spirals as visual representations of infinite geometric series, focusing on a variety of strategies used by preservice teachers in discovering patterns and investigating relationships of variables. Student discussion and activities are discussed.

Illuminations Unit: Trout Pond
This investigation illustrates the use of iteration, recursion and algebra to model and analyze a changing fish population. Graphs, equations, tables, and technological tools are used to investigate the effect of varying parameters on the long-term population.

Connections Standard for Grades 9-12
Students in grades 9-12 should develop an increased capacity to link mathematical ideas and a deeper understanding of how more than one approach to the same problem can lead to equivalent results, even though the approaches might look quite different. Students can use insights gained in one context to prove or disprove conjectures generated in another, and by linking mathematical ideas, they can develop robust understandings of problems.

Representation Standard for Grades 9-12
In grades 9–12, students' knowledge and use of representations should expand in scope and complexity. As they study new content, for example, students will encounter many new representations for mathematical concepts. They will need to be able to convert flexibly among these representations. Much of the power of mathematics comes from being able to view and operate on objects from different perspectives.