Designing activities to reconcile multiple representations supports students' focus and fluency.

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### J. Vince Kirwan and Jennifer M. Tobias

A task using multiple representations helps students write explicit algebraic equations.

### J. Matt Switzer

tudents often have difficulty with graphing inequalities (see Filloy, Rojano, and Rubio 2002; Drijvers 2002), and my students were no exception. Although students can produce graphs for simple inequalities, they often struggle when the format of the inequality is unfamiliar. Even when producing a correct graph of an inequality, students may lack a deep understanding of the relationship between the inequality and its graph. Hiebert and Carpenter (1992) stated that mathematics is understood “if its mental representation is part of a network of representations” and that the “degree of understanding is determined by the number and strength of the connections” (p. 67). I therefore developed an activity that allows students to explore the graphs of inequalities not presented as lines in slope-intercept form, thereby making connections between pairs of expressions, ordered pairs, and the points on a graph representing equations and inequalities.

### Marlena Herman and Jay Schiffman

The process of prime factor splicing to generate home primes raises opportunity for conjecture and exploration.

### Michael Tamblyn

A wonderful experience occurred in a class that I was teaching recently. It was a precalculus class, the last period of the day. The local university had brought over its cadre of preservice secondary school mathematics teachers to observe my class, so there were twenty-four additional eyes on me that day.

### Erik Jacobson

Table representations of functions allow students to compare rows as well as values in the same row.

### Margaret Cibes and James Greenwood

Students analyze items from the media to answer mathematical questions related to the article. The mathematics in these clips includes interpretation of graphs, computing percentages, making conjectures, and analyzing data. The first clip concerns college admission, a relevant topic for many students.

### Raymond N. Greenwell and Daniel E. Seabold

The Gale-Shapley algorithm can be used to match partners in a variety of contexts, such as marriage and hospital residencies.

### Steven R. Jones

An applied approach to understanding the integral—using a burst pipe—involves physical quantities and helps deepen the concept for students.

### Alison L. Mall and Mike Risinger

Our favorite lesson, an interactive experiment that models exponential decay, launches with a loud dice roll. This exploration engages students in lively data collection that motivates interest in key components of the Common Core State Standards for Mathematics: functions, modeling, and statistics and probability (CCSSI 2010).