# The locker problem algebra

For example, 4 has three factors: 1, 2, and 4 The lockers which are touched the most are those lockers whose number has the most factors. Locker 3 is opened by the 1st person, left alone by the 2nd and then closed by the 3rd. Now note that the square numbers are the only numbers with an odd number of factors all other numbers have factors which come in pairs.

Which lockers were touched by only three students? 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.

Now consider the factors of 16 which are 1, 2, 4, 8, and On this student work, for exampleyou'll see two additional "hints" both of which came by way to student questions: "Can we do this with real lockers?

Lets use a table. I say that when everyone is seated with their group, I'll explain the task.

## Locker problem simulation

The easiest way to see why is to consider what happens to a single locker. The lockers that were touched by only two students are the prime numbers since prime numbers only have two factors. Wolf's 4th grade class are playing a game in a hallway that is lined with 20 lockers in a row. Like I said, I'll work hard to stay quiet. Because the total number of lockers is only 20, students might answer the questions without thinking about the underlying reasons for their answers. Lets use a table. Look for and make use of structure. Spend a couple of minutes with this group's work to see the great ideas they're starting to have. See here. To see why this is true, first note that: The 1st student changes all the lockers with a locker number that is a multiple of 1.

Asking students to try a larger number of lockers say 50 or 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.

Notice that if the locker number, 24, is divisible by the person number, then the state changes: The number 24 has 8 factors, that is, 8 numbers that divide evenly into it.

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