The Locker Problem
Task
Introduction of the Problem/Context: In this problem, students will use what they have learned about factors and multiples to solve an interesting problem about a school with 1000 lockers.
Selected GPS:

M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will apply these concepts and associated skills in real world situations.

Apply factors and multiples.

Determine the greatest common factor (GCF) and the least common multiple (LCM) for a set of numbers.

M6A2. Students will consider relationships between varying quantities.

Analyze and describe patterns arising from mathematical rules, tables, and graphs.

M6P1. Students will solve problems (using appropriate technology).

Build new mathematical knowledge through problem solving.

Solve problems that arise in mathematics and in other contexts.

Apply and adapt a variety of appropriate strategies to solve problems.

Monitor and reflect on the process of mathematical problem solving.

M6P3. Students will communicate mathematically.

Organize and consolidate their mathematical thinking through communication.

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

Analyze and evaluate the mathematical thinking and strategies of others.

M6P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
Classroom Materials: Lesson Outlines
The Locker Problem
Grade Level/Subject: 6^{th} Grade (can be applied to grades 78), Factors and Multiples
Overview: In this problem, students will use what they have learned about factors and multiples to solve an interesting problem about a school with 1000 lockers.
Key Objectives:

To use ideas about factoring and multiples to solve problems (GPS: M6N1a, c).

To solve a problem by gathering data, using multiple strategies, and making and justifying hypotheses (GPS: M6A2a, M6P1ad, M6P5a).

To conduct peer reviews and communicate ideas to others (GPS: M6P3ac).
Learning Outcomes:
that factors and multiples can be useful in problem solving.
that there are multiple ways of solving a problem.

Students will be able to:
solve problems using knowledge about factoring and multiples.
communicate their ideas effectively to others.
Possible Materials:
Task: In a middle school, there is a row of 100 closed lockers numbered 1 to 100. A student goes through the row and opens every locker. A second student goes through the row and for every second locker if it is closed, she opens it and if it is open, she closes it. A third student does the same thing for every third, a fourth for every fourth locker and so on, all the way to the 100^{th} locker. The goal of the problem is to determine which lockers will be open at the end of the process.
Sample Questions:
In words, explain your thinking to the following problems clearly. Be sure to use appropriate mathematical language and models:

Which lockers remain open after the 100^{th} student has passed?

If there were 500 students and lockers, which lockers remain open after the 500^{th} student has passed?

If there were 1000 students and lockers, which lockers remain open after the 1000^{th} student has passed?

What is the rule for any number of students and lockers? Explain why your rule works.

Which lockers were touched by only two students? How do you know?

Which lockers were touched by only three students? How do you know?

Which students touched both lockers 36 and 48?
Assessment Ideas:

Have students write a problem of their own that can be solved using factors and multiples. Have students create an answer key and exchange their problem with other groups and then ‘grade’ their responses.

Ask students to write about their current understanding of factors, multiples, square numbers, etc. by using questions such as the following:

How are the lockers related to factors and multiples?

What kinds of patterns did you find as you worked on the problem? Were all of the patterns useful or were some more helpful than others for predicting which lockers would be left open?

What is special about square numbers in the lockers problem?

What do prime numbers do in the lockers problem?

Conduct a peer review of your answers by switching with another group. They will review your work and you will review theirs. Justify and defend your explanations to the other group and have them do the same.
Sample Question Solutions:
* Please note: This is an openended problem and thus, there are many ways to solve it.

One possible method of organizing data can be shown below:

Locker

1

2

3

4

5

6

7

8

9

10

11

12

Student














1


O

O

O

O

O

O

O

O

O

O

O

O

2


O

C

O

C

O

C

O

C

O

C

O

C

3


O

C

C

C

O

O

O

C

C

C

O

O

4


O

C

C

O

O

O

O

O

C

C

O

C

5


O

C

C

O

C

O

O

O

C

O

O

C

6


O

C

C

O

C

C

O

O

C

O

O

O

7


O

C

C

O

C

C

C

O

C

O

O

O

8


O

C

C

O

C

C

C

C

C

O

O

O

9


O

C

C

O

C

C

C

C

O

O

O

O

10


O

C

C

O

C

C

C

C

O

C

O

O

11


O

C

C

O

C

C

C

C

O

C

C

O

12


O

C

C

O

C

C

C

C

O

C

C

C


Another idea is to have students act as model lockers or to hang sample lockers up around the classroom that students can either open or close. This might allow students to see the emerging patterns.
In words, explain your thinking to the following problems clearly. Be sure to use appropriate mathematical language and models:

Which lockers remain open after the 100^{th} student has passed?
Lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, & 100 were open at the end. These are the perfect squares between 1 and 100.

If there were 500 students and lockers, which lockers remain open after the 500^{th} student has passed?
Lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 225, 256, 289, 324, 361, 400, 441, & 484 were open at the end. We know this because these are perfect squares.

If there were 1000 students and lockers, which lockers remain open after the 1000^{th} student has passed? What if there were 500 students and lockers? What about 1000 students and lockers?
Lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, & 961 were open at the end. We know this because these are perfect squares.

What is the rule for any number of students and lockers? Explain why your rule works.
Answers will vary. The rule should focus on the fact that square numbers are the only numbers that have an odd number of factors because one of the factors is repeated (e.g. 6 x 6).

Which lockers were touched by only two students? How do you know?
Possible answers include: 2, 3, 5, 7, 11, 13, 17, 19, and 23. The only lockers that were touched two times were the prime numbered lockers because those are the only lockers with only 2 factors (one and itself).

Which lockers were touched by only three students? How do you know?
Possible answers include: 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, & 961. The lockers that were touched only three times are all squares of prime numbers. So, the factors are 1, the number itself, and the square root of the number (e.g. for 9, the factors are1, 3 and 9 where three is a prime number that has been squared to yield 9).

Which students touched both lockers 36 and 48?
1, 2, 3, 4, 6, & 12. For any pair of numbers, the students that touched both are the students that represent common factors of the two numbers.

Conduct a peer review of your answers by switching with another group. They will review your work and you will review theirs. Justify and defend your explanations to the other group and have them do the same.
Answers will vary. As the teacher, you may want to provide a rubric, sample discussion questions, or a checklist to help focus students on the important aspects of the work they are reviewing.
* Lesson adapted from Connected Mathematics: Primetime: Factors and Multiples, © 2004 by Pearson/Prentice Hall.
