Title
Locker Problem Activity
Annotation
Imagine you are at a school that still has student lockers. There are 1000 lockers, all shut and unlocked, and 1000 students.
Here's the problem:
1. Suppose the first student goes along the row and opens every locker.
2. The second student then goes along and shuts every other locker beginning with number 2.
3. 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.)
4. 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? Why?
Primary Learning Outcome:
Students have investigated the problem using manipulative and technology. 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.
Additional Learning Outcomes:
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, and 10,000 lockers?
Assessed QCC:
13
Topic: Problem Solving, EquationsStandard: Writes and solves an equation for a given word problem.
Standards: NCTM
• - Grades 6-8: Algebra, Problem Solving, Reasoning and Proof,
Total Duration:
One class period
Materials and Equipment:
Locker boards
"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
5. Cut cardboard or other heavyweight paper into strips.
6. Print the numbers, two sets for each board.
7. Cut out the numbers.
8. Glue one set of numbers on the board, spacing them evenly.
9. Cut out "covers" for the numbers using the same cardboard.
10. Glue the second set of numbers on the tops of the "covers."
11. 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.)
Technology Connection:
Spreadsheet
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:
12. Make a new spreadsheet file. Name it lockers.
13. Select Column 1 and set the font size to 12, style to bold, and alignment to center (for better viewing).
14. 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-cell (A1) through (A36) have a 0 in them. (view example)
15. In the first cell in Column 2-(B1) type Student 1. This is where we will keep tracking of which student is opening/closing the lockers.
16. 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)
17. Change the student counter to Student 2.
18. 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)
19. Change the student counter to Student 3.
20. 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)
21. Continue this process until 36 students have followed the pattern with the lockers. (view example)
Procedures:
After the 36th student opens/closes lockers, which lockers are open? Which are closed?
After the 100th student opens/closes lockers, which lockers are open? Which are closed?
After the 10,000th lockers remain open or closed?
What pattern do you see?
Can you find a pattern for any number of lockers?
Students can make a data table using the information gathered so far. There might be columns for just the number of open lockers and the number of open lockers expressed with an exponent.
lockers open1
4
9
16
25
36
/ lockers open - exponent
1^2
2^2
3^2
4^2
5^2
6^2
Web Links:
Classic Locker Problem With A Twist - M.S. Problem of the Week
Locker Problem - Ask Dr. Math
Opening and Closing Lockers - Ask Dr. Math
Locker Problem - University of Central Florida
Assessment:
The displaying of the data will be used to assess the lesson, if your data prove correct then 100. If your data is incorrect but your process is correct then 90 but, if your data is incorrect and your process also incorrect then 80.