# Stair Skipping

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The goal of the "Problem Solvers" department is to foster improved communication among teachers by posing one problem each month for K-6 teachers to try with their students. Every teacher can become an author: pose the problem, reflect on your students' work, analyze the classroom dialogue, and submit the resulting insights to this department. Remember that even student misconceptions are interesting.

PROBLEM

Alexis, Hamilton, and Dallas were at the bottom of a rickety spiral staircase that was their only way out of a mineshaft. They could leave only by going out one at a time and thought that their best plan was not to step on the same steps on the way out. After a short consultation, they made a plan: each person would step with his or her left foot on the first step. After that, Alexis would walk the steps one at a time, alternating feet as usual. Hamilton and Dallas would also alternate feet as usual, but Hamilton would skip one step and Dallas would skip two steps on their way out. They pondered a few questions:

* On which step from the bottom would their right feet first land?

* On which steps would all three people land?

* On which steps would they all tread with their right feet? With their left feet?

* If the staircase had 191 steps, what would be the last step on which all three people would all land? Land with their left feet? Land with their right feet?

Please spend some time discussing this problem with your students. Talk with your students about how skipping steps is related to skip counting. Because all three children in the problem step on the first step, however, the skip counting starts at the number 1 rather than at 0. Students may also need to explore the patterns of using left feet versus using right feet. Encourage students to think about a way to determine the last step on which the three children all land, without writing out the complete pattern.

Although the students may need help understanding the problem and its constraints, please avoid giving too much guidance. Collect actual students' work, and jot down notes about discussions that occurred and the variety of students' solution processes. View this task as more than an exercise for which students are seeking a correct answer. …