IMPACT MATH CONFERENCE February 20, 2010 Improving Mathematical Practices through Algebraic Connections and Technology Presentation of the Locker Problem Connie Shannon, Carey Golden, Angela Lindsay, Marlene Vitko, Crystal Welsh, Sherri Rist

Prior Knowledge Before the actual activity was started, I had students complete a warm up that asked them to write down everything that they knew about factors and multiples. Three examples of student responses follow.

Student 1

Student 2

Student 3

Hand out and discussion of “The Locker Problem” Hand outs were given to the students, a blank sheet with locker numbers 1-20 along with the instructions for the problem. Students were asked to read instructions silently then we dissected the problem to define any terms they did not know.

Locker Problem Handouts

Starting the Locker Problem A Visual Representation Five students stood in front of the class opening and closing books to simulate the state of the lockers. A PowerPoint program showing the opening and closing of lockers was used simultaneously as the problem was introduced..

Locker Simulation using Textbooks

Opening and Closing Lockers (textbooks)

Now it’s Your Turn! There are 20 lockers in one hallway of the Godwin Middle School. In preparation for the beginning of school, the janitor closed all of the lockers and put a new coat of paint on the door, which are numbered from 1 to 20. When the 20 students from Mrs. Johnson’s class returned from summer vacation, they decided to celebrate by working off some energy. The came up with a plan: The first student ran down the row of lockers and opened every door. The second student started with locker #2 and closed every second door. The third student started with locker #3, and changed the state of every third locker door. The fourth student started with locker #4 and change the state of every fourth locker door. The fifth student started with locker #5 and changed the state of every fifth locker door, and so one, until 20 students had passed by the lockers.

The Locker Problem

Student 1, runs down the row of lockers and opens every door

Student 1, runs down the row of lockers and opens every door OPENOPEN

Student 1, runs down the row of lockers and opens every door OPENOPEN OPENOPEN

Student 1, runs down the row of lockers and opens every door OPENOPEN OPENOPEN OPENOPEN

Student 1, runs down the row of lockers and opens every door OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 1, runs down the row of lockers and opens every door OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 1, runs down the row of lockers and opens every door OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 1, runs down the row of lockers and opens every door OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 1, runs down the row of lockers and opens every door OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 1, runs down the row of lockers and opens every door OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 1, runs down the row of lockers and opens every door OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 2, closes the door of lockers 2, 4, 6, 8, and so on to the end of the line OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 2, closes the door of lockers 2, 4, 6, 8, and so on to the end of the line OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 2, closes the door of lockers 2, 4, 6, 8, and so on to the end of the line OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 2, closes the door of lockers 2, 4, 6, 8, and so on to the end of the line OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 2, closes the door of lockers 2, 4, 6, 8, and so on to the end of the line OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 2, closes the door of lockers 2, 4, 6, 8, and so on to the end of the line OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 3, changes the state of the door of lockers 3, 6, 9, 12, and so one to the end of the line. (The student opens the door if it is closed and opens it if is closed.) OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 3, changes the state of the door of lockers 3, 6, 9, 12, and so one to the end of the line. (The student opens the door if it is closed and opens it if is closed.) OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 3, changes the state of the door of lockers 3, 6, 9, 12, and so one to the end of the line. (The student opens the door if it is closed and opens it if is closed.) OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 3, changes the state of the door of lockers 3, 6, 9, 12, and so one to the end of the line. (The student opens the door if it is closed and opens it if is closed.) OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 4, changes the state of the doors of lockers 4, 8, 12, 16, and so on OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 4, changes the state of the doors of lockers 4, 8, 12, 16, and so on OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 4, changes the state of the doors of lockers 4, 8, 12, 16, and so on OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student 5 changes the state of every fifth door, student 6 changes the state of every sixth door and so on until all 20 students have had a turn OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN OPENOPEN

Student work as discussion progressed

Doing and Undoing

After Action Review

Students use cards to simulate lockers opening/closing

Connected Math Application

Website Simulation Online /index.htmlhttp://connectedmath.msu.edu/CD/Grade6/Locker /index.html

Locker Blogging on School Fusion The Locker Problem Here is a site where you can enter the student number and see which lockers were changed. This is a video of students solving the locker problem using less lockers than we used. · Carey Golden: How many times was locker 40 touched??? Sky Lynx: 1 i have no idea –Carey Golden: Think about it...anything student 5 touches will end with a 5 or 0. So 5 will touch locker 60. What are some of the lockers student 12 will touch? Sky Lynx: 6? –Carey Golden: Student 12 will start with locker 12 and then touch what? »Sky Lynx: 24? »Carey Golden: Hooray!!! You've got it!!! Now which one will both 12 and 5 touch?? »Sky Lynx: 24 and 10? »Carey Golden: You are right again! Student 12 will touch 24 and student 5 will touch locker 10 but which locker will student 12 and student 5 both touch? They will both touch the same locker...which one will that be? »Sky Lynx: 60? »Sunset Lemur: Both of the students will not touch locker 10. They will both touch locker 60 because it is a multiple of 12 and 5. »Sky Lynx: really? »Sunset Lemur: 12, 24, 36, 48, 60 · Carey Golden: Which locker is the first locker that students 5 and 12 will both change? Brick Wolverine: locker 60 i think –Carey Golden: Yes, you are right!!! Students 5 and 12 will touch locker 60! Why??? Brick Wolverine: i used lcm –Carey Golden: You've got it! Student 5 will touch multiples of 5! What will student 12 touch? · Orchid Bison: Huh? Sunset Lemur: Read the problem on this website or look at the slide show on class pages –Orchid Bison: Ok, thanks. Sunset Lemur: Do you know how to tell which lockers were touched?

Summation At the end of class student’s were asked to answer questions about what they had learned.