I have this written on the board:
“Gather Round, we are going to take a look at Skeleton Towers.”
The goal is to create a fun atmosphere around the patterns we see in skeleton towers. I sit at a central table and work with a few students to create the first 4 structures of the skeleton tower. Essentially we start class with a fish bowl where the class stands around us and observes the way in which we build the towers. Sometimes I give them sketches and models of the tower, but most of the time I like to build some suspense and leave the process as somewhat of a mystery.
With the class quite and curious as to what the skeleton tower actually is (the name has science fiction and fantasy connotations), I start them off and place one cube down and say “here is the first skeleton tower. What number is this?” Its obviously 1, but students think that it can’t be so simple. I encourage them, “this one is as simple as it seems, what number is this?” When we agree that it is one, I give them the general theme, “we can represent numbers with shapes and even build numbers into structures. Here is the second skeleton tower.” I stack the 5 cubes, one at a time and repeat the counting process. I encourage students to note the visual structure of the pattern. I start the third tower, but have the student next to me finish it. The always assemble the tower quickly. It’s a great opportunity to get a student involved in building and modeling with math. I often pick a struggling student here, because this is a chance to show everyone (including themselves) that they can do well in mathematics. For the final tower, I have a different student build it. When they are done, I ask the group around us, “how did they know what the fourth skeleton tower would look like?”
When students discuss how the towers are structured, I try and elicit some valuable insight into the skeleton tower. They notice the skeleton tower seems to have steps that climb up to the peak. They notice that the tower layers get a bit smaller as you climb. They mention that the height of the center stack always matches the skeleton tower number. There are many observations, but its important to validate the ideas so that they stick with the class as they investigate the general pattern. For example, the observation of the center stack is critical. I check for reasoning on this by asking, “how many blocks will stand at the center of the fifth tower? Sixth tower? How do we know?” To validate all of the observations, I have students repeat and rephrase the ideas shared and record them on the board during their investigation.
Tables get 5 minutes to work and build skeleton towers. The goal is for them to have a tactile understanding of the shape. The more they build with it, the more options they will have in constructing an algorithm. Not only are they constructing numbers here, but they are constructing a structure. This will help them to mentally deconstruct the shape and explode it into different pattern structures. After about 5 minutes, students begin to record their work and their answer for the 10th skeleton tower. They need to solve the problem in at least three different ways. I keep this general since I don’t want them to reach the algebraic solution unless they are ready for that type of thinking. Ideally they would all search for different ways to construct the formula (which is the hexagonal number function):
I give students any type of paper they need, including graph paper (to encourage an organized pace and use of graphs in their solutions), blank white paper (for people who need that extra space) and lined paper (to encourage them to write out their thinking and reasoning).
The goal of the summary is to share the different ideas and approaches from the group. There are so many fun ways to approach this problem and its important to share them all. For example, if a student drew out the four “arms” of the tower and one central stack, they would share how the central stack is always the equal in height to the number of the tower they are working on. They will always talk about how the four “arms” equal the sum of the numbers from 1 to one less than the central stack. As a teacher you can build on this with the class for the [n-1(n – 1 +1)]/2 formula for adding an arithmetic sequence and piece this together to get a general approach to the total tower. This is really fun because many students see for the first time that the formula describes the structure of the shape, where four arms equal 4*[n(n-1)]/2 or 2n(n-1) and then we add the central stack with n cubes. This gets us to a workable formula with 2n(n-1) + n. There are many similar approaches and the goal is to encourage them all. I usually end the lesson and ask them to record their approaches and list as many strategies as they can.