Intent

This POW, like others, is designed to engage students in exploring a larger problem over time. Writing about the problem—discussing the process of their investigation as well as justifying their solutions—is an important component of the problem-solving activity.

Mathematics

Ideas associated with constant rate are in play in this activity. In addition, students often demonstrate creativity and ingenuity in employing spatial and relational skills to model the proposed situation.

Progression

Students will be introduced to the problem and will begin to explore it in class. After a few days, if time allows, let them share ideas and generate methods for further exploration at home. After approximately one week, have a few students present the POW as a concluding activity.

Approximate Time

10 minutes for introduction

1 to 3 hours for activity (at home)

20 minutes for presentations

Classroom Organization

Individuals, followed by whole-class presentations

Doing the Activity

Have students read the problem statement,and give them some class time to begin working on it. The problem may seem deceptively simple at first. You may need to offer suggestions for how students might begin their investigation. Acting out the situation can be helpful. The actual physical movement of passing another person “midmonth” often generates significant insight for students. You might also suggest creating diagrams of or using objects to represent the traveling ships.

Questions will probably arise about whether to count a ship that arrives in New York as your ship leaves or a ship that leaves San Francisco as yours arrives. The time difference between New York and San Francisco may also lead to a need for clarification. The manner in which these questions are resolved is not as important as students recognizing the need for making decisions of interpretation and clarifying their assumptions. Emphasize that students should make their assumptions clear in their write-ups.

This is a good POW on which to spend a few minutes of class time midway between the day it is introduced and the due date, encouraging students to share methods and findings. Students will often find that their peers have a variety of ideas, which can stimulate further thinking. If you don’t have class time for this, at least check in with students to remind them to be working. Get a sense of how much progress they have made, and remind them of the importance of stating their assumptions in their write-ups.

Select three presenters for the discussion of the POW.

Discussing and Debriefing the Activity

Have the three selected students present their ideas about the POW.

There may be disagreement about whether to count the ships that are arriving as a student leaves and those that are leaving as the student arrives. Encourage classmates to ask the presenters to state their assumptions.

If any students created extensions of this POW, such as having ships leave once a day instead of once a month, you might have them present their work as well.

Key Question

What assumptions did you make?