Towering Mathematics 

In a recent visit to a seventh-grade mathematics classroom, I was delighted to see students working collaboratively on the Skeleton Tower task as shown in figure 1.

Fig. 1 How many cubes are needed to build a tower like this but 12 cubes high? (Source: http://www.illustrativemathematics.org/illustrations/75)

The task is well suited for exploration with middle-grades students. The problem lends itself to multiple solution strategies and multiple representations and can be explored with a variety of tools, including manipulative cubes, spreadsheets, and design software (e.g., see NCTM’s Isometric Drawing Tool at http://illuminations.nctm.org/Activity.aspx?id=4182).

Student Approaches 

In small groups, many seventh graders broke the tower into 5 pieces. They saw a center column 12 cubes tall and 4 identical “legs,” each consisting of 1 + 2 + 3 + … + 9 + 10 + 11 = 66 cubes. Ultimately, these students calculated that a Skeleton Tower with height 12 units would require 12 + 4 ´ 66 = 276 cubes. Others saw different patterns. For instance, one seventh grader told me that the problem essentially involved “finding the area of a rectangle.” Creatively rearranging cubes, she noted that “12 times 25” cubes were needed for the tower of height 12. Following up, I asked this student how many cubes would be needed for a tower 100 cubes tall. She quickly replied, “Easy! 100 ´ 199.” Then she showed a diagram similar to the one below.

 
Do you see what she saw?

The Skeleton Tower task is well aligned with the Standards for Mathematical Practice, in particular, 7. Look for and make use of structure. Note that both solution strategies described here highlight students’ use of structure, both numerical and geometric, to solve the task. In the first approach, students sum consecutive integers to find a solution, which is an approach intimately linked to triangular numbers and Gauss. For instance, students can calculate 1 + 2 + 3 + … + 9 + 10 + 11 by pairing addends creatively.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = (0 + 11) + (1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6) = 6 ´ 11 = 66

In the second approach, students rearrange blocks to create an easier shape, a 12 ´ (11 + 11 + 1) rectangle, to analyze. Both approaches illustrate the suitability of the task for a wide range of learners. Both strategies hint at generalizations that may be explored with more advanced students. Meanwhile, the task’s concrete context makes it a meaningful problem for students to explore with hands-on materials.

Michael Todd Edwards is an associate professor of mathematics education at Miami University in Oxford, Ohio. He is the coeditor of Contemporary Issues in Technology and Mathematics Teacher Education, executive editor of the North American GeoGebra Journal, and codirector of the GeoGebra Institute of Ohio. His research interests focus on the teaching and learning of school mathematics with technology (specifically, dynamic mathematics software), ethical issues surrounding the use of free software and the free software movement, and writing as a vehicle to learn mathematics at all levels of instruction.