Journal of Mathematics Education © Education for All
December 2015, Vol. 8, No. 2, pp. 56-73
Integration of Math Jingles into
Gregory J. Crowther
University of Washington, Bothell, USA
University of Washington, Seattle, USA
Lekelia D. Jenkins
Arizona State University, USA
Jennifer L. Breckler
San Francisco State University, USA
Biology, especially physiology, includes quantitative relationships that explain
key concepts, yet many biology students have poor math skills or math anxiety
which might hinder their learning. We propose that students who are motivated
to learn but are intimidated by math may benefit from in-class activities such
as singing or listening to content-rich jingles that make the relevant math more
accessible. Here we describe a three-part process by which we have used
feedback from 231 students in four college physiology classes to develop math-
related jingles suitable for use in similar classes. In Part 1, we report three
classes’ overall reactions (>60% positive) to educational songs as reported in
standard multiple-choice surveys, while noting the limitations of this approach.
In Part 2, we mine open-ended survey comments for common themes among
students’ reactions. Among music-related comments, we repeatedly find three
main constructive suggestions: songs should be kept very short; connections
between lecture material and songs must be obvious; and songs must be heard
or sung more than once to be maximally helpful. In Part 3, we present seven
mathematical physiology jingles (with URLs for online access) whose
development was driven partly by insights from Part 2.
Key words: content-rich songs, educational music, physiology mnemonics,
The Next-Generation Science Standards and Vision and Change report
– representing current best practices for K-12 and undergraduate education,
respectively – state unequivocally that mathematical literacy is a central
component of biology education (American Association for the Advancement
of Science, 2011; NGSS Lead States, 2013). Thus, quantitative reasoning
should be included in all high school and college biology courses, yet this is
more easily said than done. Among the sciences, biology has a reputation for
harboring students and faculty who dislike or fear math (Sorgo, 2010). Indeed,
at the college level, performance on standardized multiple-choice math tests is
lower among college biology students than in their counterparts studying
Crowther et al. 57
computer science, physical science, and engineering (Wai et al., 2009). We and
others have observed that many college biology students struggle with many
aspects of simple algebraic equations: remembering them, solving them,
grasping their conceptual meaning, embracing their relevance to biology, and
so forth (Breckler et al., 2013; Watkins & Elby, 2013).
Numerous possible strategies for improving biology students’ math
skills have been noted by authors such as Madlung et al. (2011). These include
incorporating more math problems into biology classes, incorporating more
biology problems into math classes, creating more math-centric biology
textbooks, getting biologists and mathematicians to team-teach biology, and
developing new hybrid biology/math courses. These approaches could perhaps
be complemented with efforts to make math more fun and engaging for the
students, thus reducing barriers to learning. In the context of teaching statistics,
Lesser & Pearl (2008) offer a “taxonomy of fun” – including such modalities
as humor, kinesthetic activity, music, and poetry – and advice on implementing
fun activities. More specifically, the use of music to enliven mathematics and
statistics courses has been discussed cogently by others (Robertson & Lesser,
2013; Lesser, 2014) and is a primary theme of this special issue.
Here we explore a somewhat different use of music: to emphasize and
clarify important mathematical relationships found in biology courses. In
particular, we show how simple equations can be presented and sometimes
explained in the form of song lyrics, potentially promoting both memorization
and understanding. Our efforts are focused on physiology, a core component of
biomedical science taken annually by hundreds of thousands of high school,
undergraduate, and graduate/professional students in North America alone
(Human Anatomy & Physiology Society, 2006).
Our long-term goal is to determine whether content-rich STEM songs
can improve students’ academic performance, as suggested by some previous
reports (VanVoorhis, 2002; McCurdy et al., 2008; Smolinski, 2011; Lesser et
al., 2014). However, the answer to this question may depend heavily on whether
students like the songs and the pedagogy in which they are embedded. Thus, as
a preliminary step toward our ultimate goal, we have used students’ feedback
on classroom song interventions to guide the creation of short math jingles that
may be broadly useful to physiology teachers and students. The purpose of this
paper, therefore, is to report on this feedback-guided song development process
and the songs resulting from it; we plan to assess actual learning gains in future
Courses and students studied. As part of our ongoing exploration of
educational science music (Crowther, 2012a; Crowther & Davis, 2013;
58 Math Jingles in Physiology Courses
Crowther et al., 2016), we collected and examined feedback on educational
songs used by the lead author in teaching quarter-long (11-week) undergraduate
physiology courses on three different campuses during 2014 and 2015. The
courses and campuses are as follows: Biology 220 (Introductory Physiology),
University of Washington-Seattle (UWS; 145 to 573 students); Biology 241-
242 (Human Anatomy & Physiology for pre-nursing students), South Seattle
College (SSC; 18 students) and University of Washington-Bothell (UWB; 27
students); Biology 352 (Anatomy & Physiology for biology majors), UWB (30
students). The 200-level courses (220 and 241-242) are considered introductory
courses and are taken mostly by freshmen, sophomores, and juniors of various
majors; the 300-level course (351) is taken predominantly by junior and senior
biology majors. While detailed demographic information on the students was
not obtained, the proportion of “non-traditional” (older) students is relatively
low at UWS, somewhat higher at UWB, and even higher at SSC.
Different subsets of the above courses were used in the three different
parts of the study described below; Table 1 shows which courses were used in
which parts. In order to maximize narrative clarity, the study’s parts are not
presented in a strictly chronological order.
Song development. Five to nine physiology songs were incorporated
into each of the above courses; 26 different songs were used in all. Songs were
generally written specifically for the above courses by the lead author. The
songs were intended to cover material central to many physiology courses, and
to present information as well as possible within the constraints of musical
rhythms and rhymes. Seven of the songs covered mathematical relationships,
as discussed below.
Song implementation in the classroom. Songs were generally
performed live by the instructor in the classroom a cappella (without
instruments), though karaoke backing tracks were used occasionally. Each song
was performed once. Lyrics were simultaneously provided to students via
PowerPoint slides. Students were sometimes encouraged to sing along and/or
make gestures illustrating the meaning of the lyrics. Lyrics and sheet music
were also available to students outside of class via the instructor’s website, but
links to these files were not always included on the slides.
Ethical treatment of human subjects. No personally identifiable
information was collected in this study. Because this study’s surveys were
originally created and administered primarily for purposes of course
development, they were not prospectively reviewed by an Institutional Review
Board (IRB). However, the subsequent decision to publish the data was
approved by the Human Subjects Division of the University of Washington.
Researcher positionality. The lead author taught all of the students
surveyed in this study. In presiding over the classes listed above, he made no
attempt to hide his enthusiasm for science-based music, and may have given
students the impression that he expected them to enjoy it as well. Thus, it is
possible that the lead author’s position of authority over the students influenced
Crowther et al. 59
the students’ responses even though the responses were collected anonymously.
The other authors had no relationship with the lead author’s students.
Timeline of Study
Part 1: Students’ overall
reactions to physiology
Part 2: Students’ detailed
reactions to physiology
Part 3: Development of a
suite of math -themed
Part 1: Students’ Overall Reactions to Physiology Songs
All SSC and UWB students completed Likert-style survey questions of
the following format: “To what degree did [course component] help you learn
the material? (A) very helpful, (B) helpful, (C) neither helpful nor unhelpful,
(D) unhelpful, (E) very unhelpful.” Course components that we asked about
included songs as well as (depending on the quarter) in-class discussions, in-
class worksheets, kinesthetic movements, laboratory exercises, and study
guides/practice tests. These surveys were completed by >90% of enrolled
students. To simplify analysis, the categorical responses above were converted
to numbers between 0 (very unhelpful) and 4 (very helpful).
Part 2: Students’ Detailed Reactions to Physiology Songs
General UW student evaluations of teaching. UWS students
completed standard anonymous end-of-quarter course evaluations administered
by UW’s Office of Educational Assessment. These evaluations asked students
to rate many aspects of the course and the instructor on a 0-to-5 scale, and also
to answer the following open-ended questions: “Was this class intellectually
stimulating? Did it stretch your thinking? Why or why not? What aspects of this
class contributed most to your learning? What aspects of this class detracted
from your learning? What suggestions do you have for improving the class?”
These optional evaluations were completed (either online or in person,
depending on the quarter) by 59% to 76% of enrolled students (depending on
To classify students’ song-related comments, the following categories
were created post hoc. (A) Songs were a positive aspect of the course, without
specific mention of themes C, D, or E below. (B) Songs were a negative aspect
of the course, without specific mention of themes C, D, or E below. (C) Songs’
60 Math Jingles in Physiology Courses
length and/or class time devoted to discussing them were excessive. (D)
Connections between song lyrics and lectures were not always clear or strong.
(E) Songs would be more beneficial if heard or sung multiple times (i.e., more
than the one time each was presented in class).
Song-specific survey. After one quarter, Biology 220 students at UWS
were invited to complete a survey about the six specific songs used during that
quarter: “Erythropoietin,” “Fick’s Law of Diffusion,” “Meet My Threshold,”
“Surface Area-to-Volume Ratio,” “The Sodium Jeer,” and “Where Is That
Sound?” (“Fick’s Law of Diffusion” and “Surface Area-to-Volume Ratio”
focused on mathematical relationships; the others did not.) Performance and
discussion of these songs – which varied greatly in style and length –
collectively filled 24 minutes of class time, spread over 17 hours of animal
physiology lectures. (An additional 17 hours of plant physiology lectures did
not include songs and were not covered by this survey.) Students were asked to
rate each song as a very poor use of class time, poor use of class time, okay use
of class time, good use of class time, or very good use of class time. To simplify
the analysis, these categorical responses were converted to numbers between 0
(very poor use of class time) and 4 (very good use of class time). Students were
also asked whether the maximum amount of class time that should be devoted
to a content-rich song should be 0 minutes, 1-2 minutes, 3-4 minutes, 5-6
minutes, 7-10 minutes, 10-20 minutes, or more than 20 minutes. This survey
was completed by only 15% of enrolled students, probably reflecting the
delayed timing of the survey and the limited motivation of students to complete
it at that point.
Part 3: Development of a Suite of Math-Themed Physiology Jingles
In reflecting on Part 2 of this study, the lead author decided to
expand his repertoire of math-related songs beyond “Fick’s Law of Diffusion”
and “Surface Area-to-Volume Ratio.” He identified additional mathematical
relationships that seemed sufficiently important to merit inclusion in most
physiology survey courses. He then wrote out key phrases about each
relationship and tried to find rhythms and melodies that suited these phrases.
For example, the central lesson of Poiseuille’s Law is that flow is proportional
to vessel radius raised to the 4th power. The idea arose that this relationship
could be captured in the phrase “r times r times r times r,” with the repetition of
the key variable providing appropriate emphasis. The song was then built
around this phrase, with a verse to introduce the topic and a chorus to deliver
the equation itself (Figure 1).
In this manner, over several quarters, the lead author created five
additional math-themed songs for his physiology courses. Songs were intended
to be brief – and thus may be considered “jingles” rather than full songs – as
well as clear, pleasant, and easy to sing. These goals were not 100% compatible
with each other; for example, changing the above-mentioned phrase to “radius
times radius times radius times radius” would improve its clarity but would
Crowther et al. 61
compromise its musicality. Since some ambiguities are unavoidable, we use
them to spark class discussions, as exemplified by the questions provided in the
Results and Discussion
Part 1: Students’ overall reactions to physiology songs
As part of the lead author’s teaching, he routinely inserts content-rich
songs into lectures. Our initial assessments of this approach were usually
limited to multiple-choice surveys asking students whether they liked the songs
and/or whether the songs helped them learn. Data from three different classes
of physiology students are shown in Figure 2. These data suggest that most
students (60 to 67%, depending on the course) find the songs helpful or very
helpful, as opposed to neutral (18% to 37%) or unhelpful or very unhelpful (0%
Additional analysis of surveys from these courses suggested an
important caveat regarding the Figure 2 data: students’ ratings of the songs
might reflect their overall satisfaction with the course content and/or instructor
as much as or moreso than their specific opinions of the songs per se. That is,
the more the students like the course content and/or instructor, the more highly
they will tend to rate songs (and other tools), irrespective of the specific merits
of the songs (or other tools). This possibility first occurred to us when we found
strong correlations between SSC Biology 241 students’ ratings of the songs and
their ratings of other teaching tools, with R2 values of 0.88 (songs vs. kinesthetic
movements), 0.39 (songs vs. worksheets), and 0.48 (songs vs. labs). We had
previously assumed that students’ reactions to songs would be independent of
their reactions to other course tools because, for example, the songs were
written by the course instructor (G.J.C.), whereas the lab exercises were taken
from a standard mass-published lab manual.
62 Math Jingles in Physiology Courses
Figure 1. Sheet music for the jingle “Poiseuille’s Law of Laminar Flow.”
Having noticed these correlations retrospectively at SSC, we then
prospectively tested their occurrence in two physiology courses (Biology 241
and Biology 352) at a different institution (UWB). These courses used
somewhat different teaching tools and multiple lab instructors, but the same
classroom lecturer (G.J.C.). We again found highly significant correlations
between students’ perceived usefulness of the songs and their perceived
usefulness of other tools (songs vs. in-class discussions: R2 = 0.16, p=0.002;
songs vs. study guides: R2 = 0.16, p=0.003). Thus, “bleed-over” of overall
satisfaction into ratings of songs (or any specific course component) likely
biases the latter. Additional support for this interpretation comes from surveys
from the same course (Biology 220) taught by G.J.C. two consecutive quarters
in a row at the same institution (UWS). From the first quarter to the second
quarter, students’ ratings of the course content rose from 3.2 to 3.8 on a 5-point
scale, and their ratings of the instructor’s contribution to the course rose from
3.0 to 4.0; likewise, the percentage of music-related survey comments that were
positive (as opposed to mixed or negative) climbed from 33% (46 of 140) to
61% (11 of 18). While other explanations for these quarter-to-quarter changes
cannot be ruled out, the data are consistent with the idea that students’ ratings
of specific instructional features (such as music, in our case) are biased by their
overall opinion of the course content and/or instructor. This argument has
previously been advanced (though not about music in particular) by others such
as d’Apollonia & Abrami (1997) and Young (2006). A practical implication is
that any class’s ratings of songs should be interpreted in the context of its
overall “baseline” satisfaction with the course. Several previous studies of
educational STEM songs (McLachlin, 2009; Grossman & Watson, 2015;
Weinhaus & Massey, 2015; Yee Pinn Tsin, 2015), including our own (Crowther
& Davis, 2013), have omitted this important context. Future studies could
address this issue by reporting students’ ratings of music alongside their ratings
of other aspects of the course. Students who give a course’s music a 4 on a 1-
to-5 scale might be considered pro-music if they give 3s to most other parts of
a course, but perhaps not if they give 4.5s to most other parts.
Crowther et al. 63
Figure 2. Responses of three different physiology classes to the question
“To what degree did the songs used in class help you learn the class
material?” Possible answers were: very unhelpful (red), unhelpful (orange),
neither unhelpful nor helpful (yellow), helpful (light green), and very helpful
Part 2: Students’ detailed reactions to physiology songs
Having noticed the limitations of typical survey results like those
presented above, we desired more extensive and therefore more valuable
feedback to inform our development of songs as biology teaching tools.
Fortunately, such feedback was available via generic end-of-course evaluations
completed by 433 of the 573 UWS students enrolled in Biology 220 in the
spring quarter of 2014. Of these 433 students, 348 answered one or more of the
open-ended questions following the multiple-choice questions (see Methods);
of these 348, 140 students (40%) commented specifically on the songs used
during lectures despite the lack of a song-related prompt and the limited class
time devoted to the songs (2% of total lecture time). A summary of these song-
specific comments is given in Figure 3. Remarkably, despite the lack of a song-
related prompt, many students made specific suggestions for improving the
songs’ usefulness. The three most common suggestions (explicit or implied)
were (C) class time devoted to songs should be carefully limited (13% of
students commenting on the songs), (D) connections between song lyrics and
lectures should be made more obvious (10% of students), and (E) songs should
be repeated for maximum impact (6% of students). Examples of each type of
comment are given in Figure 3.
A song-specific survey completed by 85 students in this same class (see
Methods) resulted in two additional findings of note. First, 84% of these
students said that any in-class musical exercises should be limited to 6 minutes
or less, thus confirming the prevalence of theme C above (Figure 4). Second,
64 Math Jingles in Physiology Courses
among the six featured songs, the two mathematical songs (“Surface Area to
Volume Ratio” and “Fick’s Law of Diffusion”) received the highest and 3rd-
highest ratings, respectively (Figure 5).
Part 3: Development of a suite of math-themed physiology jingles
Intrigued by the students’ possible preference for math-themed songs,
and now recognizing the need for clearer songs that reinforce key course
content in more obvious ways (theme D above), the lead author developed
additional songs covering mathematical relationships central to animal
physiology (Table 2). Each song lasts less than one minute, perhaps qualifying
it as a “jingle” rather than a full song; this brevity enables concise in-class
interludes (theme C). Lyrics, sheet music, and simple online recordings are all
now publicly available (at the URLs listed in Table 2) to facilitate subsequent
recall and practice (theme E). Thus, the development of these jingles has been
informed by students’ feedback, as well as checked for accuracy and clarity by
a second physiologist (J.L.B.). Brief notes on each jingle are included below;
possible questions to ask students about each jingle are included in the
“Cardiac Output and Pulmonary Ventilation.” This jingle compares
the analogous equations for calculating cardiac output and calculating
pulmonary ventilation. Parallels in the two equations are emphasized by the
parallels in the verses, with only a few words changed between the cardiac
output verse and the pulmonary ventilation verse. For cardiac output: “Volume
moved per beat/Times number of beats per minute/Equals volume of blood per
minute;/That’s all this equation has in it!” For pulmonary ventilation: “Volume
moved per breath/Times number of breaths per minute/Equals volume of air per
minute;/That’s all this equation has in it!” Our hope is that if a student can recall
either one of the two equations, the other will be easy to retrieve.
Crowther et al. 65
Figure 3: Comments about physiology songs from UWS Biology 220
students. (140 students who commented specifically on songs used in class
were classified as shown. Percentages sum to >100% because five students fit
into two categories).
66 Math Jingles in Physiology Courses
Figure 4. Responses of UWS Biology 220 students (N=85) to a question on
the maximum amount of class time that should be devoted to a content-
Figure 5. Ratings of UWS Biology 220 students (N=82) of six songs on a 0-
to-4 scale. Error bars represent standard errors of the mean (SEM). Means
with different letters are significantly different from each other (p < 0.01)
according to paired t-tests with a Bonferroni correction for multiple
comparisons. SA/V and Fick are the two song topics on mathematical
Crowther et al. 67
Math-Themed Physiology Jingles Developed During This Study
Cardiac output (CO) and pulmonary ventilation (PV)
are both calculated in the same way: the volume pumped
(stroke volume [SV] or tidal volume [TV]) is multiplied
by the frequency of pumping (heart rate [HR] or
respiratory rate [RR]).
Diffusion rate is directly proportional to the
concentration gradient (∆P, for partial pressures of
gases) and surface area (A), and is inversely
proportional to diffusion barrier thickness (D).
For loads moved by muscles, the mechanical advantage
(MA) equals the length of the in-lever (Li) divided by
the length of the out-lever (Lo).
An ion’s equilibrium potential (Eion) can be calculated
from its concentrations outside and inside the cell
([ion]out and [ion]in) and its electrical charge (z).
In the kidney, a substance’s excretion rate (E) equals its
filtration rate (F) plus its secretion rate (S), minus its
reabsorption rate (R).
The rate of blood flow through a blood vessel (Q)
depends most strongly on the radius of the blood vessel
(r). Q also depends on the hydrostatic pressure gradient
(∆P), blood vessel length (L), and fluid viscosity (η).
Animals’ metabolic rates reflect a balance between
intake of nutrients via their surface area (SA) and the
use of these nutrients by their internal volume (V). For
a hypothetical cube-shaped animal, as body length (L)
increases, V increases more rapidly than SA.
68 Math Jingles in Physiology Courses
“Fick’s Law of Diffusion.” When equations are expressed concisely,
the meaning of the abbreviations may be forgotten (Watkins & Elby, 2013).
“Fick’s Law of Diffusion” addresses this issue by presenting the abbreviations
in the first half of the jingle, then spelling out the full terms (in the same order)
in the second half. Thus, “delta P” corresponds to “pressure difference,” “A”
corresponds to “surface area,” “k” corresponds to “the constant k,” and “D”
corresponds to “diffusion barrier.”
“In-Lever, Out-Lever.” Students generally remember that mechanical
advantage (MA) is equal to a ratio involving the in-lever (Li) and the out-lever
(Lo), but they often flip the numerator and denominator. This jingle helps them
remember that MA equals Li divided by Lo and points out that a change in either
one can improve the mechanical advantage: “Elongate the in-lever, shorten up
“The Nernst Equation.” This jingle does not present the terms of the
equation as one would write them out from left to right; rather, it starts with the
ratio of extracellular and intracellular ion concentrations because this is the core
of the equation – an ion’s equilibrium potential (E) reflects its relative
concentrations outside and inside the cell – and the other terms should not
distract from that.
“Pee Values.” In studying the kidney, many students struggle with the
terms filtration, reabsorption and secretion. In particular, they often do not
know whether each of these processes moves substances from the blood to the
pre-urine or vice versa. They will keep these straight if they understand the
jingle’s equation, which conveys that filtration and secretion move substances
into the pre-urine (for excretion) while reabsorption does the opposite.
“Poiseuille’s Law of Laminar Flow” (Figure 1). The repetition of “r
times r times r times r” emphasizes the surprising fact that blood flow rate is
proportional to vessel radius (r) raised to the 4th power. (In addition, the rhyme
with “employ” helps students pronounce the French surname “Poiseuille.”)
“Surface Area-to-Volume Ratio.” This jingle references the formulas
for the surface area and volume of a cube: 6L2 and L3, respectively, where L is
the length of a side of the cube. These formulas should be written out explicitly
to avoid confusion (e.g., “Six L to the two” might not otherwise be understood
as 6L2). Also, the alliteration of “large” and “low” in the line “If you’re large,
it’s low” reminds students to group these two adjectives together: a large body
size implies a low surface area-to-volume ratio.
Advice on Implementation
Based on the data presented in Part 2 and past experience deploying
music in the classroom, we recommend that physiology instructors who wish
to use a jingle should consider the following. Though some instructors consider
music to be a fun way of introducing new topics (Crowther et al., 2016), we
usually use songs as recaps or extensions of already-covered topics, so that
students have some context in which to interpret the song lyrics (theme D
Crowther et al. 69
above). Instructors should facilitate multiple passes through a jingle (theme E),
perhaps by using it in class and also encouraging out-of-class, web-aided
practice. Non-singing instructors may wish to recruit musically inclined
colleagues, teaching assistants, or student volunteers. In addition, rather than
assuming that a jingle “speaks for itself,” instructors should help students
unpack the highly compact lyrics (theme D). Finally, as with any other aspect
of a lecture, advance rehearsal of jingles will help ensure that valuable class
time is used efficiently (theme C).
Obviously, further evaluation will be necessary to assess the
effectiveness of the seven jingles listed in Table 2. These jingles are now being
evaluated by hundreds of students NOT taught by the lead author; we look
forward to reporting these results in a future publication.
While this study focused on college physiology courses due to our
expertise and current teaching assignments, our work may be informative to any
high school or college-level efforts (e.g., in a math class) to teach math with
content-rich music. In particular, we believe that college students’ apparently
strong preference for very short jingles is an important finding, partly because
it contrasts with the extended length of most commercially available math songs
for this age group, as catalogued at SingAboutScience.org (Crowther, 2012b).
Indeed, it is notable that two of the only studies to demonstrate a positive impact
of content-rich math music on test performance (VanVoorhis, 2002; Lesser et
al., 2014) used jingles rather than full-length songs as their intervention.
In summary, while math remains a considerable challenge for many
biology students, brief content-rich jingles may render it less dreary and more
accessible. The examples presented here may, at the very least, provide
engaging interludes that are minimally disruptive to existing curricula.
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72 Math Jingles in Physiology Courses
Sample Questions to Ask Students about the Physiology Jingles
“Cardiac Output and Pulmonary Ventilation”
Questions: (1) The terms cardiac output, stroke volume, heart rate,
pulmonary ventilation, tidal volume, and respiration rate are not included in the
jingle’s lyrics, but their definitions ARE included. Which definitions go with
which terms? (2) How do these variables change when you start performing
Answers: (1) “Volume of blood per minute” is cardiac output. “Volume
moved per beat” is stroke volume. “Number of beats per minute” is heart rate.
“Volume of air per minute” is pulmonary ventilation. “Volume moved per
breath” is tidal volume. “Number of breaths per minute” is respiration rate. (2)
All of these values increase during aerobic exercise.
“Fick’s Law of Diffusion”
Questions: (1) Which term of the equation reflects a concentration
gradient, which is necessary for diffusion? (2) What does the “constant” k
Answers: (1) Pressure difference (delta P) refers to a difference in the
partial pressures of a gas, and thus reflects a concentration gradient. (2)
“Constant” k depends on the temperature, the size of the molecule that is
diffusing, the specific medium through which it is diffusing (water? air?), etc.
Questions: (1) What units does Mechanical Advantage (MA) have? (2)
What range of values can a Mechanical Advantage have? (3) Mechanical
Advantage can also be calculated from the force in (Fi) and force out (Fo), or
from the velocity in (Vi) and velocity out (Vo). How do those formulas compare
to the one presented in the jingle?
Answers: (1) MA is unitless; the units of the numerator and denominator
cancel. (2) In theory, mechanical advantage can be anywhere from just above 0
to far above 1. (3) MA is also equal to Fo divided by Fi and to Vi divided by Vo.
“The Nernst Equation”
Questions: (1) What is ion valence? (2) What units are carried by the
equilibrium potential (E)? (3) What does the value of E mean?
Answers: (1) Ion valence is the charge carried by an ion, such as minus-
1 or plus-2. (2) E, an electrical potential, generally is reported in units of
millivolts. (3) E is the electrical gradient across the membrane needed to
perfectly counterbalance any concentration gradient, such that there is no net
movement of the given ion from one side of the membrane to the other.
Crowther et al. 73
Questions: (1) Does secretion of a solute by the kidney increase or
decrease the rate at which it is excreted? (2) Is it possible for the excretion rate
of a solute to be 0? If so, how?
Answers: (1) Secretion of a solute increases the solute’s excretion rate.
(2) Yes, this is possible. If the filtration, secretion, and reabsorption rates are all
0, then the excretion rate will be 0 as well. (This is generally true for proteins
in the blood.) Alternatively, if the reabsorption rate is equal to the sum of the
filtration rate and the secretion rate, the excretion rate will be 0. (This is
generally true for glucose in the blood.)
“Poiseuille’s Law of Laminar Flow”
Questions: (1) How does vessel radius (the “r” in the song) relate to
resistance to blood flow? (2) What is delta P here? Is this the same delta P that
is in Fick’s Law of Diffusion? (3) Can you rearrange the equation so that pi is
in the numerator?
Answers: (1) Resistance to flow (often abbreviated with a capital R) is
proportional to radius to the 4th power. (2) Here delta P refers to a difference in
hydrostatic pressure over the length of the vessel. It is not the same as the delta
P in Fick’s Law of Diffusion. (3) The equation can be rewritten as: Flow =
“Surface Area-to-Volume Ratio”
Question: (1) If we were to assume that an animal were spherical, rather
than cube-shaped, would SA/V be similarly affected by body size?
Answer: (1) Yes. The surface area of a sphere equals 4*π*r2, where r is
the radius. The volume of a sphere equals (4/3)* π*r3. The surface area-to-
volume ratio is 3/r, which decreases as r increases. Thus this ratio decreases
with increasing size, regardless of whether the object is cube-shaped or
Gregory J. Crowther
University of Washington, Bothell
Lekelia D. Jenkins
Arizona State University
University of Washington, Seattle
Jennifer L. Breckler
San Francisco State University