# Using Mathematical Reasoning to Quantify the Microscopic Scale

Every now and then, everyone should take the opportunity to look at pond water — that “wild and beautiful Lilliput world, half Disney, half Dali” (Cudmore 1977, 15). Who isn’t amazed and excited by seeing the trumpet-shaped *Stentor*, with its beautiful blue pigment, using its cilia to sweep smaller cells into its oral groove? Or seeing *Didinium *whirl around in search of prey, like a World War I flying ace engaged in a dogfight? Or *Amoeba* spreading its pseudopods in search of victims in a manner reminiscent of Steve McQueen’s nemesis in the ’50s sci-fi classic *The Blob*?

Galileo and other scientists used telescopes to transform our view of the solar system, and eventually the universe. Like Galileo’s telescope, the microscope is one of those rare technologies that opened whole new vistas. Antonie van Leeuwenhoek first saw the microscopic world in the 17^{th} century. As described in the BioInteractive video *Animated Life: Seeing the Invisible,* Leeuwenhoek was the first person to see sperm cells, red blood cells, protozoa, and bacteria. Imagine what the world would be like without the microscope. We would have no idea that cells, the fundamental units of life, even existed. Diseases would still be mysterious without a germ theory.

It’s important for students to have opportunities to peer into the microscopic world and see the tiny critters going about their business. But viewing magnified images of microorganisms, while awe-inspiring, is not sufficient in and of itself to lead students to a deeper understanding of the scale of the microscopic world. “Understanding scale requires some insight into measurement and an ability to think in terms of orders of magnitude. … At a basic level, in order to identify something as bigger or smaller than something else — and how much bigger or smaller — a student must appreciate the units used to measure it and develop a feel for quantity” (National Research Council 2012, 90). This can be challenging for students who struggle with fractions, ratios, and unit conversions. But students need practice working with these concepts and skills to develop facility with them. And having the opportunity to actually see the critters under the microscope provides a motivational context that is often missing from the more decontextualized exercises found in textbooks.

Fortunately, BioInteractive has just the resource to provide your students with the needed experience and practice. The classroom activity “What Leeuwenhoek Saw” presents them with scaled-up images of microorganisms and guides them through the calculations necessary to determine their actual sizes in micrometers. Viewing the actual microbes under a microscope and then guiding your students through the BioInteractive activity can help them make sense of the tiny critters that inhabit pond water.

Before diving into the main activity, a set of practice problems is provided for your students to review metric units and conversions. In the problem set, they measure the width of their index finger in centimeters, then convert the measurement to meters (m), millimeters (mm), and micrometers (μm). The students are then told that an average skin cell measures 30 μm, and they convert that measurement to mm, cm, and m. Finally, students are asked to calculate how many skin cells lying side-by-side would fit across the width of their index finger.

As an extension at this point, you might ask students to try to imagine how many cells make up their own bodies. The cool thing is that no one really knows the answer to that question. Scientists have made back-of-the-envelope calculations since the ’70s, and estimates for the average adult male have been highly variable, ranging from 10^{12} to 10^{14}, or 1 trillion to 100 trillion. In more recent efforts, scientists have made more accurate estimates of the numbers of different cell types and settled on a total number of 3.0 × 10^{13}, or 30 trillion (Sender *et al*. 2016). In addition, discussing the number of cells in the human body can set the stage for another discussion, perhaps at a later time, of how many bacterial cells reside on and in our bodies. Here, science journalist Ed Yong’s book *I Contain Multitudes* (2016) and its associated video series provide an engaging introduction to the microbiome that serves as an excellent resource for teachers.

It is also interesting for students to consider the fact that all their trillions of cells came from one single cell, a zygote. With a diameter of about 130 μm, a zygote could hide under the period at the end of this sentence (Cudmore, 1977). Yet that one tiny cell contains two meters of DNA and produces all the trillions of cells that compose a human body, with their varied structures and functions.

In the main part of the “What Leeuwenhoek Saw” activity, students work with an image of *Daphnia ambigua* that has been scaled up 50×. The activity guides them through the process of determining the actual size of *Daphnia* based on their measurements of the scaled image. In the final part of the activity, students choose an organism or a cell from a set of 12 cards and use the calculation skills they’ve learned to design and construct a scale model of the organism or cell they’ve chosen. The *Paramecium* card is shown below as an example.

Another possible extension, designed to have students compare the relative sizes of prokaryotes and eukaryotes, is to estimate how many prokaryotic cells would fit inside a eukaryotic cell. I use a human skin cell and an *E. coli* cell as representative cells. Students can find the dimensions of these cells using the *Cell Size and Scale* interactive from Learn.Genetics.

The skin cell has a diameter of 30 μm. Living skin cells are 14-sided polyhedrons (Yokouchi 2016; eLife 2016), but we can model them as spheres. The volume of a sphere is given by the formula *V* = 4/3π*r*^{3}, where *r *is the radius. The volume of a skin cell is

*V* = 4/3π(15 μm)^{3 }= 14,137 μm^{3}

The *E. coli* cell is sausage-shaped with dimensions of 3 ⨉ 0.6 μm and can be modeled as a cylinder. The volume of a cylinder is *V* = π*r*^{2}*h*, where* r* is the radius and *h* is the height of the cylinder. *E. coli* cells have a volume of

*V* = π(0.3 μm)^{2}(3 μm) = 0.85 μm^{3}

Comparing these two volumes shows that we could put roughly 16,632 *E. coli* cells inside a human skin cell.

As humans, we tend to concern ourselves mainly with things we can see with the naked eye. But as microbiologist Bonnie Bassler explains in the *Animated Life* video, “Everything that you can actually see with your eye is just the smallest sliver of life on this earth. Most of life is invisible. We still have this idea that we're the most central feature of Earth, and it's the humans that are the bystanders. The microbes are doing the work.”

Ed Yong (2016) noted that earth scientists have considered naming our current epoch the Anthropocene to denote the impact of anthropogenic climate change on Earth’s systems. But Yong suggests that it might be more accurate to say that Earth has been in the Microbiocene since the dawn of life and will continue to be so to its very end. Remove all multicellular life forms and there would be some shifts in the way the biosphere operates, but remove the microbes and the whole thing collapses. Lewis Thomas (1974, 3) expressed a similar sentiment about the resilience of the biosphere when he wrote, “It is illusion to think that there is anything fragile about the life of the Earth; surely this is the toughest membrane imaginable in the universe, opaque to probability, impermeable to death. We are the delicate part, transient and vulnerable as cilia.”

The activities described here support student understanding of the crosscutting concept of scale, proportion, and quantity from the Next Generation Science Standards. "These [three] concepts are the starting point for scientific understanding, whether it is of a total system or its individual components" (NGSS Lead States, 2013, 418). I’ve found that teaching students to express the small magnitudes of microscopic organisms in mathematical terms, and make relative comparisons of cell volumes, helps them develop a better understanding of the microscopic world. Viewing the critters that inhabit the Lilliput world within the microscope is always engaging for students. Combine that with performing the mathematical operations in the “What Leeuwenhoek Saw” activity and you have what students need to begin to understand life at the microscopic scale.

**References**

Cudmore, L. L. Larison. *The Center of Life: A Natural History of the Cell*. New York: Quadrangle/The New York Times Book Co., 1977.

*eLife*. “What shape is a skin cell?” last modified December 20, 2016, https://medium.com/lifes-building-blocks/what-shape-is-a-skin-cell-172e45061e6f.

Genetic Science Learning Center. (2010, September 2). Cell size and scale [Interactive]. https://learn.genetics.utah.edu/content/cells/scale/.

HHMI BioInteractive (nd). *Animated Life: Seeing the Invisible* [video file]. https://www.hhmi.org/biointeractive/seeing-the-invisible.

HHMI BioInteractive (2017). What van Leeuwenhoek saw [classroom activity]. https://www.hhmi.org/biointeractive/what-van-leeuwenhoek-saw.

National Research Council. *A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas*. Washington, DC: The National Academies Press, 2012. https://www.nap.edu/read/13165/chapter/1.

National Research Council. *Next Generation Science Standards: For States, By States*. Washington, DC: The National Academies Press, 2013. https://www.nap.edu/read/18290/chapter/13#418.

Sender, Ron, Shai Fuchs, and Ron Milo. “Revised estimates for the number of human and bacteria cells in the body.” *PLOS Biology* 14, 8 (2016): e1002533. https://doi.org/10.1371/journal.pbio.1002533.

Thomas, Lewis. *The Lives of a Cell: Notes of a Biology Watcher*. New York: The Viking Press, 1974.

Yokouchi, Mariko, Toru Atsugi, Mark van Logtestijn, Reiko J. Tanaka, Mayumi Kajimura, Makoto Suematsu, Mikio Furuse, *et al*. “Epidermal cell turnover across tight junctions based on Kelvin's tetrakaidecahedron cell shape.” *eLife* 5 (2016): e19593. https://doi.org/10.7554/elife.19593.

Yong, Ed. *I Contain Multitudes: The Microbes Within Us and a Grander View of Life*. New York: HarperCollins, 2016.

*Robert Cooper recently retired from Pennsbury High School, Fairless Hills, PA, where he taught biology (general, honors, and AP).** Currently, he is taking online courses in biology to keep up with more recent developments in the field.*