Modeling the motion of a speeding car or the growth of a Jactus plant, teachers can use six practical tips to help students develop quantitative reasoning.

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### Eric Weber, Amy Ellis, Torrey Kulow, and Zekiye Ozgur

### Dung Tran and Barbara J. Dougherty

The choice and context of authentic problems—such as designing a staircase or a soda can—illustrate the modeling process in several stages.

### Erik Jacobson

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

### 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).

### S. Asli Özgün-Koca, Michael Todd Edwards, and Michael Meagher

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

### Jamie-Marie L. Wilder and Molly H. Fisher

Our favorite lesson is a hands-on activity that helps students visually “tie” (pun intended) the concepts of rate of change and *y*-intercept together in a meaningful context using strings and ropes. Students tie knots in ropes of various thicknesses and then measure the length of the rope as the number of knots increases. We provide clothesline, twine, bungee cord, and other ropes found at local crafts, sporting goods, and home stores. We avoid very thin string, such as thread or knitting yarn, because the knots are small and the string length does not change enough to explore a rate of change. A variety of thicknesses is important because this allows for variability in the rates of change.

### Dan Kalman and Daniel J. Teague

Using ideas of Galileo and Gauss but avoiding calculus, students create a model that predicts whether a fly ball will clear the famous left-field wall at Fenway Park.

### Jennifer L. Jensen

Five problems—relating to gas mileage, the national debt, store sales, shipping costs, and fish population—require students to use functions to connect mathematics to the real world.

### Joel A. Bryan

During the thirteen years that I taught high school physics and mathematics, I found that my physics students typically came to class excited to learn. As in all science classes, they interacted with fellow classmates while performing laboratory investigations and other group activities requiring higher-order thinking skills. To create a similar experience for my mathematics students, I developed a laboratory investigation for my precalculus class. These students responded just as favorably as my physics students to hands-on data collection activities.

### John F. Mahoney

The author presents an activity in which the lines in students' hands are analyzed, with curves and lines fit to each one.