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Examining student understanding of geometric scaling

David Smith SmilingAuthors:

  • David Smith, he/him, Associate Teaching Professor, Physics, University of Washington, Seattle campus
  • Peter Shaffer, Professor, Physics, University of Washington, Seattle campus




Project Question

When considering animal size and scale, it is helpful to understand how the quantities of surface area and volume scale with the linear dimensions of the animal. We are thus interested in examining student understanding of geometric scaling and its applications to biological contexts. Based on the outcomes of the research, we aim to design curriculum that addresses student difficulties with power law functions commonly used to explain scaling relationships.


Tutorials in Introductory Physics is a set of teaching materials that has been implemented in the small quiz sections of the calculus-based physics sequence for approximately 30 years and many results from this innovation have been published by the Physics Education Group at the UW. Considering the positive impact of these materials, the teaching faculty decided to modify this material such that it can be delivered in a large-lecture setting in the algebra-based sequence.

This project is conducted in an introductory physics course that meets for three lectures and one lecture-tutorial per week. In the lecture-tutorial, research-validated curriculum that is designed for a small quiz section is delivered in a large lecture hall setting and taught by an instructor and multiple teaching assistants.  Students are typically in their sophomore or junior year and pursuing careers in the life-sciences. The enrollment for the course is ~400 students per quarter.


To measure student understanding of scaling relationships prior to instruction, we administer an online pre-test that includes concept-focused questions that aim to elicit student difficulties with the material. Student performance on these questions inform the curriculum development of a 1-hour lecture-tutorial such that the curriculum addresses the most pertinent student difficulties. The efficacy of the lecture-tutorial curriculum is assessed by comparing student performance on exam questions on both midterm and final exams to their responses on the pre-test.


Early results from this research highlight student difficulties with several ideas related to isometric scaling. These difficulties are typically elicited from questions that ask students to compare the surface area and/or volume of one object to that of an isometrically scaled version of that object. A series of questions have also asked students to assess changes in quantities that are dependent on surface area and/or volume as the object changes in size (e.g., mass dependence on volume for an object with uniform density).

We first noticed that a significant portion of students were unable to recognize that surface area and volume scale with the square and cube of the isometric scaling factor, respectively. Our results show that these students tend to assume that both surface area and volume scale linearly with the isometric scaling factor.

A second difficulty can be seen in an inability to reconcile the differences between the surface area and volume of a three-dimensional (3-D) object. There seems to be a tendency for students to think that the surface area of a 3-D object scales with the cube of the isometric scaling factor. We have also noticed learning challenges when considering the isometric scaling of complex shapes such as an animal’s body.

We are editing our teaching material to target these difficulties and early results show learning gains between pre- and post-test questions. With continuing development we believe these learning gains can be increased.


Over the last decade, there has been an increased interest in modifying the introductory physics course taken by life science majors (IPLS). Course redesign projects have focused on deleting content not deemed biologically relevant (e.g., planetary motion) and adding content important in the life sciences (e.g., diffusion). As an introduction to mechanics and measurement, we believe that an investigation of scaling in biological contexts strongly aligns with the goals of the redevelopment of IPLS courses.

The scaling of an animal’s body is strongly related to thermoregulation and the surface area/volume ratio is critical for animals who absorb oxygen through their skin. We believe that our results would be of interest to biology instructors who teach these concepts.

It is our intention to eventually publish the results of this research in a peer-reviewed journal and to share the curriculum with the Living Physics Portal for physics instructors who are involved in similar IPLS course redevelopment projects.

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