LESSON STUDY VIGNETTE: PRIME AND COMPOSITE NUMBERS
This classic locker problem exposes students to considering the important nature of numbers including even and odd numbers, factors, multiples, prime numbers, composite numbers, and perfect squares.
Suppose you’re in a hallway lined with 100 closed lockers. You begin by opening every locker. Then you close every second locker. Then you go to every third locker and open it (if its closed) or close it (if its open). Continue this for every nth locker on pass number n. After 100 passes, where you get to locker #100, how many lockers are open?
Our teachers in the grade band fourth through sixth planned a lesson study around this locker problem. The goal of their lesson was to solve problem and model a mathematical situation that would reveal a relationship between factors and multiples.
The Common Core Math Standard in the fourth grade reinforces the concept of factors and multiples.
Gain Familiarity with factors and multiples.
4. Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite. (CCSS-M)
During the launch of this lesson, students warmed up by giving examples of factors and multiples and then determine the greatest common factor for two or more numbers and the least common multiple for two numbers. They were also asked to categorize numbers by looking for prime numbers, composite numbers, perfect squares, and even or odd numbers. The warm-up helped remind students of the difference between factors and multiples and the number of factors for all prime numbers.
The host teacher who delivered the research lesson had a student read the problem out loud and asked another student to restate the problem to be sure everyone understood the problem. The math agenda for her research lesson was to engage the students in problem solving and reasoning as students analyzed the nature of different numbers. Odd 1, 3, 5, 7, 9, 11, ... even 2, 4, 6, 8, 10, ... square 1, 4, 9, 16, 25, 36, ... prime 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... composite 4, 6, 8, 9, 10, 12, 14, 15, 16, ... triangular 1, 3, 6, 10, 15, 21 (as seen in the handshake problem), and the interesting Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21. Number shapes like the triangular numbers like 1, 3, 6, 10 can be arranged in the shape of a triangle. Just as the square numbers can be arranged in squares.
In the locker problem, students have to look for a pattern between the actions they take opening and closing the lockers and the number of factors each number has. They see that there is a relationship between the factors a locker number has and whether it is open or closed at the end. As students realize that they need to systematically keep track of the actions they are making as they open and close lockers, it affords
Figure 4.1 Figurate numbers—triangular and square number patterns.
the opportunity to encourage mathematical practices of looking for and making use of structure and looking for and expressing regularity in repeated reasoning (Standard 7 & 8 of the CCSSM, 2010).
The host teacher noted after the lesson, “I should have defined the phrase, ‘changed the state’ better for the students’ understanding.” The phrase “changed the state” links to the number of factors for each locker number. Changing the state of the lockers results in a pattern related to the number of factors of each locker number.
For all the lockers that had an even number of factors, the locker would be closed and the lockers with an odd number of factors would stay open. The only locker numbers with the odd number of factors turn out to be the square numbers since 4 has (1, 2, 4), 9 has (1, 3, 9), etc. The factors 2 x 2 are counted once, not double counted, so all square numbers will have an odd number of factors. The classic locker problem is a rich benchmark problem that is familiar in the school curriculum and which provides multiple entry points to access a variety of math content in number theory (For more details, see Seshaiyer, Suh & Freeman, 2011).