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## Multiple Representations and Connections with the Sierpinski Triangle

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

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## Graphing Inequalities, Connecting Meaning

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.

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## Using Covariation Reasoning to Support Mathematical Modeling

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

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## American I Do's – The More the Merrier?

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.

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## Adding It All Up: Reconceiving the Introduction of the Integral

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

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## The Back Page: My Favorite Lesson: Modeling Exponential Decay

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).

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## Connecting Algebra to Economics

Economics can be an avenue for teaching such algebra concepts as graphing curves, writing linear equations, solving systems of equations, and shifting graphs.

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## Activities for Students: Connecting Spatial Reasoning Ideas in Mathematics and Chemistry

Stereochemistry and three-dimensional analysis constitute significant parts of this student activity.

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## Extreme Numbers and Estimation Skills

Using estimation helps increase numerical fluency and gives meaning to large numbers.

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## Spaghetti Sine Curves: Virtual Environments for Reasoning and Sense Making

The Spaghetti Sine Curves activity, which uses GeoGebra applets to enhance student learning, illustrates how technology supports effective use of physical materials.