A giant stopped by my class and left several of his giant pencils. As my students observed those pencils, we brainstormed a list of things we could figure out about the giant by knowing the size of his pencil. They knew that they could use ratios and proportions to estimate the height of the giant, the size of his shoes, the size of a pencil sharpener for his giant pencils, and the height of his desk and chair. They were enthusiastic as they joined their collaborative working groups to get to work on this problem of giant proportions!
Students collected the tools they needed, including meter sticks, yard sticks, calculators, typical pencils, and “giant” pencils. Their discussion was self-directed and rich with Math Practice Standards as they discussed how to approach this problem. Should they use centimeters or feet and inches? How would they record their information? What information will they need to find the ratios and proportions to determine the height of the giant?
As they worked, they agreed on the height of an average fifth grader by searching for the information on the internet. They measured each other to find out if the information was reasonable. From that, they extrapolated the foot size of an average fifth grader and once again discussed if that made sense. When they realized that shoe sizes are different for boys’ and girls’ shoes, they wondered if this “giant” is a girl or boy. All this while, I was almost a spectator.
This open-ended “giant” problem invites many different interpretations and ideas in the process. Students quickly realized that their answers to the giant’s height were not all the same. Their discussion led me to understand that this problem is all about the process. Do my students know how to approach an open-ended problem and how to make realistic estimates to solve it? To me, that is the most important part of solving these kinds of problem.
I must give credit to the NCTM publication “Teaching Mathematics in the Middle School” in March 2009: Giant Pencils: Developing Proportional Reasoning. If you are not familiar with the NCTM publications or website, take a look. That is a valuable resource.
What kinds of open-ended problem do you design for your students? Do you have any you can share?