Task
The 20 students in Mr. Wolf's 4th grade class are playing a game in a hallway that is lined with 20 lockers in a row.
 The first student starts with the first locker and goes down the hallway and opens all the lockers.
 The second student starts with the second locker and goes down the hallway and shuts every other locker.
 The third student stops at every third locker and opens the locker if it is closed or closes the locker if it is open.
 The fourth student stops at every fourth locker and opens the locker if it is closed or closes the locker if it is open.
This process continues until all 20 students in the class have passed through the hallway.

Which lockers are still open at the end of the game? Explain your reasoning.

Which lockers were touched by only two students? Explain your reasoning.

Which lockers were touched by only three students? Explain your reasoning.

Which lockers were touched the most?
IM Commentary
The purpose of this instructional task is for students to deepen their understanding of factors and multiples of whole numbers. This is a classic mathematical puzzle; often it is stated in terms of 100 lockers, so students have to make the connection to factors and multiples to solve it. In this version, students can just go through all the rounds of opening and closing locker doors and observe after the fact that there is a relationship between the factors a locker number has and whether it is open or closed at the end.
This task provides students with an excellent opportunity to engage in MP7, Look for and make use of structure (if they see early on that there is a relationship with factors and multiples) or MP8, Look for and express regularity in repeated reasoning (if they start to see and describe the pattern as they imagine students opening and closing the lockers).
Because the total number of lockers is only 20, students might answer the questions without thinking about the underlying reasons for their answers. In the first question for example, a student might say "1, 4, 9, 16 are all still open because I tried it out and those were the ones that were left open." If a student goes this route, the teacher can steer the conversation back to factors by asking the students if they notice anything special about the numbers of the lockers which are still open. Asking students to try a larger number of lockers (say 50 or 100) and repeating the game can also help students look for a pattern since going through all the rounds of the game becomes less and less feasible as the number of lockers increases. Once students see the pattern, they should be pressed to explain why it works out this way.
Comments
Log in to commentNader says:
over 1 yearWhat do you call these types of problems,please?
Kristi says:
over 4 yearsVery challenging question but a great one to do with students if you have room to get them up and out of their seats. The combination of math learning with movement provides students multiple ways to engage their brains (ie. Howard gardner's Multiple Intelligences) and to cement the learning. A recent National Math Panel report encourages more practice of basic math skills. "Few curricula in the United States provide sufficient practice to ensure fast and efficient solving of basic fact combinations and execution of the standard algorithms." (National Mathematics Advisory PanelFinal Report, page 26).
Good luck and have fun!
Cam says:
over 4 yearsExcellent addition to the commentary  thanks for this!