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Frogs and Worms

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With school starting, many of us are focusing on the need to support students’ engagement in the Standards for Mathematical Practice (SMP). Regardless of whether your state has adopted the Common Core State Standards, the SMP represent processes and proficiencies that we all want to develop in our students. Within these standards, decontextualize and contextualize represent two unfamiliar terms for many of us. Here, I offer two problems to help you and your students think about the processes embodied in these terms.

First, the Frog Race problem: 

FrogTwo frogs have a race. One frog makes a jump of 80 centimeters once every five seconds. The other frog makes a jump of 15 centimeters every second. The rules of the race require that the frogs must cross a line 5 meters from the start line and then return to the start line to complete the race. Which frog wins the race?  (NCTM 1994) 

This problem is appropriate for upper elementary school students. For those in the lower grades, consider the Worm problem: 

WormA worm is at the bottom of a 12-foot wall. Every day it crawls up 3 feet, but at night it slips down 2 feet. How many days does it take the worm to get to the top of the wall? (Herr and Johnson 2001)

As students work to solve either of these problems, drawing a diagram may be an appropriate initial strategy. After that, students may move toward using symbols to represent and solve the problem. These symbols will be manipulated without considering the problem. That is, students will be decontextualizing the problem.

The richness of these problems, however, comes from contextualizing—that is, pausing during the process of working with the symbols to look back at how the symbols connect to the original problem. For both the Frog Race problem and the Worm problem, this process of “keeping an eye on” the problem is key to finding the solutions.

I encourage you to solve both of these problems and consider using them with your students. And be sure to decontextualize and contextualize—the results may surprise you.

You are invited to share your thoughts and comments here or via Twitter @TCM_at_NCTM.  I’d also like to see samples of student work. I’ll be back in a couple of weeks with my reflections on the Frog and Worm tasks.


Herr, Ted, and Ken Johnson. 2001. Problem Solving Strategies: Crossing the River with Dogs and Other Mathematical Adventures. 2nd ed. Emeryville, CA: Key Curriculum Press.

National Council of Teachers of Mathematics (NCTM). 1994. “Menu of Problems.” Mathematics Teaching in the Middle School 1 (November-December): 223.

 Angela Barlow, Middle Tennessee State UniversityAngela T. Barlow is a Professor of Mathematics Education and Director of the Mathematics and Science Education Ph.D. program at Middle Tennessee State University. During the past fifteen years, she has taught content and methods courses for both elementary and secondary mathematics teachers. She has published numerous manuscripts in Teaching Children Mathematics, among other journals, and currently serves as the editor for the NCSM Journal of Mathematics Education Leadership

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