By Angela T. Barlow, Posted August 29, 2014 – 

How did your students do with the Frog problem and the Worm problem? When I have used these problems in the past, typically students have quickly decontextualized them, representing the problems in some way and finding a solution. Below are some common responses. Both these solution processes are straight­forward and mathematically correct. In fact, the students providing these solutions have done a nice job of decontextualizing the problem.

The Frog problem

• 5 meters is 500 cm. The total race is 1,000 cm.
  
  
• Frog 1 jumps 80 cm every 5 seconds.

1000 cm ÷ 80 cm per jump = 12.5 jumps 

12.5 jumps ´ 5 seconds per jump = 62.5 seconds 

• Frog 2 jumps 15 cm every second.

1000 cm ÷ 15 cm per jump = 66.67 jumps

Each jump takes one second, so 66.67 seconds 

• Therefore, Frog 1 wins.

The Worm problem

• Each day, the worm has a net gain of 1 foot.
  
  
• If he gains 1 foot per day, he will take 12 days to get to the top of the 12-foot wall.

A Closer Look 

I will usually have one or two students who are quick to say, “Wait a minute!” These wait-a-minute students have noticed something that other students have not, and what they have noticed resulted from them having contextualized the problem. Let’s take a closer look at the Frog problem.

Do the frogs really travel 1000 cm to complete the race? The wait-a-minute students say no. They argue that if that were the case, the frogs would have to stop in mid-air at the 500 cm mark and reverse their path—an impossibility. Instead, the frogs have to complete their jump over the 500 cm and then turn around to go back to the starting line. As a result, how far does each frog actually travel? Does this change which frog wins the race?

Similarly, consider the Worm problem. The wait-a-minute students argue that at some point, the worm makes it to the top of the wall and does not continue sliding up and down.

Wait a minute! Keeping your eye on the problem, or contextualizing, seems to be important.

I hope that these two problems gave you and your students an interesting way of thinking about the importance of decontextualizing and contextualizing. In the Comments section below, please share how your students handled the problem.
 

 Angela 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.