# 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*

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?

 Introducing the Activity

Present the Locker Problem page to the class using one of the following methods:
1. View the page.
2. Print the page and project it as a transparency.
3. Give each student a copy of the page.

 Using a Manipulative

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
1. Cut cardboard or other heavyweight paper into strips.
2. Print the numbers, two sets for each board.
3. Cut out the numbers.
4. Glue one set of numbers on the board, spacing them evenly.
5. Cut out "covers" for the numbers using the same cardboard.
6. Glue the second set of numbers on the tops of the "covers."
7. 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.)

Simulation

Give students time to work through the problem using the locker boards. One person in each group should record the process. The others should open and close the lockers. Here is the locker open/close sequence.

 Using Technology

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:

1. Make a new spreadsheet file. Name it lockers.

2. Select Column 1 and set the font size to 12, style to bold, and alignment to center (for better viewing).

3. 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)

4. 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.

5. 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)

6. Change the student counter to Student 2.

7. 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)

8. Change the student counter to Student 3.

9. 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)

10. 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... ?

 Displaying the Data

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.

Before making a graph of the spreadsheet data, discuss the entries in cells A1 through A36 with the class. Provide directions for Graphing Data from a Spreadsheet to help students display the data.

 Extending the activity - looking for patterns

Have the students sit down and think in terms of a pattern:

1. After the 36th student opens/closes lockers, which lockers are open? Which are closed?
2. After the 100th student opens/closes lockers, which lockers are open? Which are closed?
3. What pattern do you see?
4. 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 open 1 4 9 16 25 36 lockers open - exponent 1^2 2^2 3^2 4^2 5^2 6^2

Ask students: What pattern do you see? Are there any relationships among the numbers? What will be the next locker that will remain open?

Will locker 50 be open?
Will locker 100 be open?
Will locker 1000 be open?

The completed table might look like this:

 lockers open 1 4 9 16 25 36 49 64 81 100 ... n x n lockers open - factored 1 X 1 2 X 2 3 X 3 4 X 4 5 X 5 6 X 6 7 X 7 8 X 8 9 X 9 10 X 10 ... n X n lockers open - exponent 1^2 2^2 3^2 4^2 5^2 6^2 7^2 8^2 9^2 10^2 ... n^2
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.

In addition, all numbers (lockers) except perfect squares have factors that occur in pairs, so that every locker except those whose number is a perfect square has its state changed an even number of times: it gets changed and then changed back, or opened and shut again.

Only perfect squares have a duplicate factor pair like 3x3 = 9, so that the state of these lockers is changed an odd number of times or opened and left open.

 Other "Locker Problems" on the Web

1000 Lockers - Ask Dr. Math
Classic Locker Problem With A Twist - M.S. Problem of the Week
Illustrating the Locker Problem - Wolfram Demonstration Project
Locker Problem - Ask Dr. Math