## Locker Problem Activity
## Teacher Lesson Plan
## Go to Student Page
This activity is aligned to NCTM Standards - Grades 6-8: Algebra, Problem Solving, Reasoning and Proof, and Communication and to California Mathematics Standards Grade 7: Algebra and Functions #1 and Mathematical Reasoning #1.1, 1.2, 2.2, 2.4, 2.5, 2.6, 3.2
## A Thousand Lockers*
Here's the problem: - Suppose the first student goes along the row and
*opens every locker.* - The second student then goes along and
*shuts every other locker*beginning with**number 2**. - The third student
*changes the state*of every**third**locker beginning with**number 3**. (If the locker is open the student shuts it, and if the locker is closed the student opens it.) - The fourth student
*changes the state*of every**fourth**locker beginning with**number 4**.
Imagine that this continues until the thousand students have followed the pattern with the thousand lockers. At the end, which lockers will be open and which will be closed?
Present the
"Locker boards" can be constructed using cardboard, paper, glue, and tape. If two locker boards are made per group of 4 students, two students can manipulate a board with lockers 1-10 and a second pair of students can manipulate a second board with lockers 11-20. Making locker boards- Cut cardboard or other heavyweight paper into strips.
- Print the
**numbers**, two sets for each board. - Cut out the numbers.
- Glue one set of numbers on the board, spacing them evenly.
- Cut out "covers" for the numbers using the same cardboard.
- Glue the second set of numbers on the tops of the "covers."
- Use two pieces of tape to fasten the "covers" over the numbers. (A good tape to use is the kind of "paper tape" that is sold to secure bandages.)
Give students time to work through the problem using the locker boards. One person in each group should
The Locker Problem can be simulated using a spreadsheet. Students can investigate the problem further by working individually, in pairs, in groups, or with one classroom display, following these directions:
- Make a new spreadsheet file. Name it
**lockers**. - Select
**Column 1**and set the font size to 12, style to bold, and alignment to center (for better viewing). - Select the
**first cell (A1)**and type 0 (zero) to denote that the first locker is shut. Continue typing 0 down the first column until 36 cells (A1) through (A36) have a 0 in them. (view example) - In the first cell in
**Column 2 (B1)**type**Student 1**. This is where we will keep track of which student is opening/closing the lockers. - Student 1 opens all of the lockers. Simulate this by selecting the first cell in Column 1 (A1) and typing 1 (one). Continue changing all of the 0's to 1's. (view example)
- Change the student counter to
**Student 2**. - Student 2 starts with locker 2, closes it, and then closes every other locker. Simulate this by selecting A2 (skipping the first cell), typing 0 (zero), and then changing the 1 to a 0 every other locker. (view example)
- Change the student counter to
**Student 3**. - Student 3 starts with locker 3 and since it is still open, closes it and then continues to change the state of every third locker. Simulate this by selecting A3 (skipping the first two cells), and typing 0 to signify that the locker is closed. Now at every third locker if there is a 0 replace it with a 1, and if there is a 1 replace it with 0. (view example)
- Continue this process until 36 students have followed the pattern with the lockers. (view example)
Suggestions
As the students are working, encourage them to do different tasks. For example, one student can enter the data while another student counts. Encourage them to look for a pattern. Can they predict what would happen with 100 lockers, 1000 lockers, 10,000 lockers... ?
At this point, students have investigated the problem using manipulatives (the locker boards) and technology (a spreadsheet). Some students will discover the pattern while working with the locker boards, while others might not "see" it until they have worked through the spreadsheet simulation. Still other students will not master the activity, but may gain a better understanding of the task.
Have the students sit down and think in terms of a pattern:
Students can make a data table using the information gathered so far. There might be columns for just the
Why?
Every number has as factors itself and 1. Therefore, every locker is opened on the first pass and shut on the pass where the student number equals the locker number.
1000 Lockers - Ask Dr. Math |