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Integrating Mathematics and Biology

Published Modules Now Available

IMB modules can be flexibly adapted for use in a variety of courses at a variety of grade levels in both biology and mathematics.

All 20 IMB modules are now published by COMAP and available for download.

COMAP's BioMath webpage contains module descriptions, information on mathematics and biology content, and any suggested prerequisites. Student and Teacher versions of each module can be downloaded for free. Downloadng teacher versions requires registration with COMAP.

Module descriptions and links to student versions are also provided below.

Module Descriptions

  1. Evolution by Substitution: Amino Acid Changes Over Time

    This module introduces students to the Central Dogma of Biology. In short, DNA, or deoxyribonucleic acid, carries the code for life and that code directs the making of proteins that will carry out the organism’s functions. Proteins are made from twenty different amino acids and the number and order of those amino acids will determine the properties and function of the protein. Any alterations in the sequence of amino acids may have an effect on the function of the protein. The protein may not function as well, may lose all function, or may possibly function better. It is also possible that the substitution may not affect the function of the protein at all. Mathematical analysis of similar proteins in different organisms based on the sequence of amino acids may give insight into their possible evolutionary history and perhaps even that of the organisms that contain those proteins. Such analysis may also lead to explanations of the mechanisms of evolution, which resulted in the natural selection of these proteins.

    [Download Student Materials]

  2. Imperfect Testing: Breast Cancer Case Study

    The results of a mammogram, like those of many tests, are not always correct. A false positive test result may create unnecessary anxiety, while a false negative test result may lead to a false sense of security. In this module, students examine the case of an adult female who learns her mammography test is positive. They then use real data to calculate the probabilities of receiving true (or false) test results and discuss the possible implications of a positive test result, given the properties of the test. These properties, which include sensitivity and specificity, can be used to help determine the rates of incorrect test results. Students also investigate the importance of disease prevalence.

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  3. Genetic Inversions: Relationships Among Species

    In this module students explore the basic concepts of DNA and chromosomal inversions. The module starts with a game that introduces the idea of gene rearrangements, and then gradually leads the students through a series of improved algorithms designed to rearrange one genome into another in the least number of steps.

    [ Download Student Materials]

  4. Spider Silk: Examining Biological Sequences

    This module asks students to apply knowledge of protein structure and function to pose and answer the fundamental question, “What alignment of two sequences is biologically most meaningful?” The unit introduces students to the rapidly emerging field of bioinformatics by developing the basic mathematical principles that underlie computer programs used to align amino acids. It also provides instruction on how to use one such resource, Biology Student Workbench. In effect, the students become researchers and should begin to understand how mathematical modeling, computing, and biology can work together to answer important scientific questions.

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  5. Microarrays: Arrays of Hope

    In all living organisms, DNA carries the code for life. Your DNA and the DNA of every organism is made up of four basic subunits that are arranged in different orders to provide the secret code of life. This is very similar to the way our alphabet works. Read the following two sentences and visualize in your mind what each of these looks like.


    Only one letter in our code was changed, but the meaning of the sentence has changed. The same is true of our DNA. In DNA our alphabet consists only of T, A, C, and G, which represent the four basic subunits of DNA called bases. The bases follow very specific pairing rules to form a molecule of DNA that has 2 strands and resembles a ladder.

    [Download Student Materials]

  6. Habitat: An Organism’s Environment

    As human populations increase and spread into areas shared with other organisms, more and more species are added to the endangered species list. As awareness increases about the impact of human activities on the environment, many questions are asked. Are there ways to lessen the negative impact that humans have on other organisms? Can human developments be designed to prevent the demise of other populations of organisms? If, for example, a community feels a golf course would be a nice addition, could it be built to limit its negative impact on other species or even possibly have a positive impact on the environment? In order to answer these questions, it is necessary first to understand the problem at a deeper biological level. Then, mathematical tools can be used to state the problem in precise language and to help one to arrive at a satisfactory solution.

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  7. Food Webs: Community Feeding Relationships

    Food webs are abstract representations of feeding relationships in communities and use a series of arrows from one species to another where the first is a source of food for the second. Discrete mathematics provides a model for a food web using a directed graph (digraph) whose vertices are the species and an arc goes from a to b if a is food for b. Digraphs representing food webs make understanding predator prey relationships easier and various properties of digraphs provide insight into properties of the food web and the species contained within. Overarching questions in this module include, “What effect would the removal of a species have on the associated food web?” and “Why are there so few top predators?”

    [Download Student Materials]

  8. Ecological Footprint: What’s My Impact?

    Most people think of humans as being separate from the environment and when they think about ecology it is usually the ecology of some nonhuman ecosystem. This unit helps students see themselves, and humans in general, as intimately connected to the environment. It introduces the ecology of humans as a topic in its own right. This focuses on both the human dependence on the environment and the human impact on the environment’s ongoing capacity to meet the needs of our world’s people. To better understand human impact on the environment, basic mathematics is used to quantify ecological impact. Ecological footprinting is developed as a tool for assessing human impact and as a decision-making tool. This module deepens students’ awareness of the human role in environmental crises, and enables them to make more informed decisions about their behaviors and their environmental impacts. Although the mathematics is basic, students gain profound insights from consideration of simple ratios and proportions and working with conversion factors. Finally, their work with the ecology of humans prepares them to understand similar issues in other ecologies, and for connecting humans with the environment. The central concept of footprinting is a simple but surprising one with profound implications for human society.

    [Download Student Materials]

  9. Drawing Lines: Spatial Arrangements of Biological Phenomena

    One of the fundamental needs of any organism is space in which to exist. Depending on how organisms engage in vital activities, such as finding shelter, foraging for food, courtship, and reproduction, the spatial needs of different species can vary and interact. Some animals range over many hundreds of square miles while foraging, while others never leave the small pond in which they were born. Many animals, regardless of their sizes or the scale of their habitats, are very territorial. These territories can be formed both within and between species. In addition, there are many other scales at which the biological organization of space is important. For example, at the cellular level, some cells interact with their neighboring cells, and at the ecosystem level all of the plants that comprise a forest together structure the space in which they grow. The organization of space, at scales ranging from microns to kilometers, both affects and is affected by various biological phenomena. Similar spatial patterns arise in widely diverse biological contexts at very different scales. This unit examines a single underlying principle governing the partitioning of a space in a wide range of biological contexts. In the unit, we will come to understand how the minimization of energy expenditure results in a widely applicable “nearest-neighbor” dynamic. Considering nearest-neighbors helps us model and understand biological phenomena with Voronoi diagrams. Creating mathematical models based on Voronoi diagrams helps biologists understand and interpret many different biological phenomena. We will examine the use of these diagrams in one of several different contexts. Thinking critically about how space is used and partitioned enables us, to ask and answer significant questions that depend more on common sense than on advanced biological and mathematical ideas.

    [ Download Student Materials]

  10. Home Range: Species’ Living Rooms

    How do researchers determine the home range of a particular species? What is meant by a species’ home range? How does the home range of a species connect to its habitat?
    This module explores how data is collected and analyzed to determine the home range of a number of species. Students use actual data for prairie dogs, black-footed ferrets, and pronghorn antelopes. They determine the home range of these animals, including the size and breadth of the home range, and how one would create a buffer zone for the home range. Students are encouraged to draw conclusions as they compare their data to other student’s data. They consider the usability and the effectiveness of different tracking techniques.

    [ Download Student Materials]

  11. Competition in Disease: Pass It On

    Members of a population display many variations. They may be visible traits or internal differences in genetic make up. These variations may result in either “success” or “failure” of the individuals. The concept of natural selection is developed in that those organisms with more successful traits reproduce at a greater rate than those with less successful traits. Some research uses the concept of infectious disease-causing organisms to model this concept. The ability to increase their rate of transmission from one host to another increases these organisms’ chance of being successful in an evolutionary sense; therefore, various methods of transmission are important to discuss. Mathematical analysis of these different methods of transmission provides insight into their possible evolutionary success rates and perhaps even that of the organisms that utilize these methods. This analysis illustrates connections between reproductive success and evolutionary success.

    [ Download Student Materials]

  12. Mathematical Modeling of Disease Outbreak

    While the 20th century saw a marked decline in infectious disease deaths and an impressive eradication of some infectious diseases, current populations are still faced with outbreaks of new diseases and the resurgence of some previous declining diseases. Disease control in the 20th century resulted from improved sanitation and hygiene, the discovery of antibiotics and the introduction of worldwide childhood vaccination programs. Science and technology played major roles in these improvements. In the 21st century, scientists, researchers, public health officials and governments continue efforts to control infectious diseases such as HIV, West Nile virus, various strains of influenza, severe respiratory syndrome (SARS) and encephalopathy. This module is about the study of infectious diseases — their causes, prevention, spread, and control.

    [ Download Student Materials]

  13. Genetic Inversion: Relationships Among Species

    The surface of our planet is populated by living things — curious, intricately organized chemical factories that take in matter from their surroundings and use these raw materials to generate copies of themselves. he living organisms appear extraordinarily diverse in almost every way. What could be more different than a tiger and a piece of seaweed, or a bacterium and a tree? Yet our ancestors, knowing nothing of cells or DNA, saw that all these things had something in common. They called that something “life,” marveled at it, struggled to define it, and despaired of explaining what it was or how it worked in terms that relate to nonliving matter.

    The paragraph above opens Lesson 1 and links to the module’s problem of determining a newly discovered species’ place on the evolutionary tree of life. In this module students will explore evolutionary separation of species through the mathematical process of sequence inversions. While biologically mutations occur when genes detach, rotate and reattach, in this module students will code original species with numerical sequences and then perform subsequence inversions in an effort to reach a new species represented by a target sequence. Students will develop and refine inversion algorithms to find the most efficient one. They will then represent and analyze possible evolutionary relationships among species using phylogenetic trees.

    [ Download Student Materials]

  14. CRIME: Criminal Investigations through Mathematical Examination

    This module introduces mathematical examination of fingerprints. It provides a possible method to identify individuals in a species. Tracking individuals can be an important step in learning more about the species as a whole. This module concentrates on human fingerprints. To accomplish this, we use a fictitious crime case to explore a mathematical procedure to examine and compare fingerprints. The module consists of multiple activities to investigate the characteristics of a fingerprint and the biology behind why everyone’s fingerprints differ.

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  15. Modeling Neuron Networks: The Neuroscience of Pain

    The human nervous system is made up of 100 to 200 billion neurons, supported by 1 to 2 trillion glial cells. Each neuron has an average of about 5 thousand synapses. Each of these synapses may release multiple neurotransmitters across a range of concentrations. It’s been estimated that the number of potential neural states that these synapses can produce in a single person is more than the number of molecules in the known universe!

    Every conscious experience is built from the interaction of neural activity at multiple levels within the nervous system, and the perception of pain is no exception. This summary of pain is just the tip of the iceberg. Each year we learn more and more about the nervous system and current advances are progressing at an accelerating rate. This module explains the basics in order to show how pain from the skin (somatosensory pain) is processed. Noted references allow you to go deeper than this brief overview.

    Overall, it is important to remember that pain is not a stimulus. This parallels the fact that there is no ‘red’ light. Red is a perceptual response to electromagnetic wavelengths within a narrow range. What we call pain is a collection of responses within the body, most often, to tissue damaging or potentially tissue damaging stimulation. Such stimuli are referred to as ‘noxious’. The receptors that have evolved to detect tissue damage are called ‘nociceptors’ and they send their information through ‘nociceptive’ pathways from the skin, through several neural relays, to the highest levels of the brain. Pain is a hierarchy of responses organized at multiple levels of the nervous system.

    Pain is a sensation. It’s a complex perceptual and motivational state. It’s an interpretation of an event. Our first exercise in this unit will be to try to capture some of pain’s complexity by analyzing words that humans use to describe the pain experience. Some of them relate to the sensory experience — how you would describe your pain. Some relate to how pain motivates — how much you want it to stop. Some relate to the interpretation of the pain — how it affects your life.

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  16. Tomography: Looking Inside With Outside Tools

    Tomography is a form of imaging using sections. Tomography truly brings together mathematics, biology and technology to solve problems in many areas. The module starts with an introduction to tomography and then introduces some basic mathematics used in computed tomography (CT scans). The module looks at the biology of the human body, what anatomical conditions can be observed using various kinds of scans, and the differences between CT scans and positron emission tomography (PET scans). Many of you have heard of (or even have had) an MRI or CT scan, or even a PET scan, thus you may find the topic of tomography very interesting.

    The module provides real-life examples that use skills from algebra II and geometry, including:

    • modeling using e and exponential functions
    • solving systems of equations, and
    • visualization in 3-D,

    to understand one of the important tools available to physicians in understanding the status of the human body.

    In addition to medical imaging, applications of tomography include autopsies, structural soundness, archeology, and others involving imaging that are increasingly important today. Throughout the module, there are many activities, worked examples, homework problems, and various assessment tools.

    [ Download Student Materials]

  17. Biostatistics in Practice: Using Statistics To Discover Links Between Eating Behaviors And Overweight Children

    Biostatistics (a contraction of the words biology and statistics) is the application of statistics to topics in biology. This module deals with two biological contexts. In Lesson 1 students look for factors that help them estimate a person’s age. They use dotplots and numeric summaries for center and spread to analyze data from team age estimates. Lessons 2 – 4 are based on the Infant Growth Study that sought connections between eating behaviors in children and being overweight. Students grapple with determining which factors are the best indicators that a child is overweight. Using scatterplots and regression lines, students study relationships between various factors (BMI at age 4, rate of eating during test meal, calories consumed during test meal) and BMI at age six. After creating binary variables (overweight (yes/no), exceeded recommended dinner Calories at test meal (yes/no)), students again search for relationships between factors and the variable overweight. This time the statistical analysis is based on two-way tables and calculating percentages.

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  18. Quorum Sensing: Organisms Communicating and Coordinating

    We live in an ever-changing world. Many people crave information about those changes. As a result, new means of communicating are continually evolving. People originally relied on word of mouth. With the invention of the printer, newspapers spread information to more people, more quickly. Later, telephones, television, cell phones and the Internet increased both the rate at which people could communicate and the number of people that the information reached. For many young people, it is hard to imagine a world without online social networks, cell phone texting, emails and television.

    The cells in the human body and the cells in all living things also crave information. Cells are designed to both receive and send messages from their environment and others in their environment. From dusk to dawn, environmental parameters like temperature, pressure, and oxygen fluctuate in a periodic or arbitrary manner. Since the survival and well-being of organisms depends on how they respond to environmental change, it is important to be able to capture environmental cues and respond accordingly. For example, when the temperature drops, humans sense the temperature change and protect themselves by wearing warmer clothes. In addition, humans have developed means of communication, as simple as shouting to as complex as satellites, to alert as many people as possible of an upcoming event. Doing so has helped humans to survive life-threatening events, coordinate actions, and ultimately be more fit in uncertain environments.

    Until recently, scientists thought that such means of communication and coordinated action was the characteristic of animals only. Recent discoveries, however, reveal that even the simplest organisms, bacteria, have a similar capacity to communicate with each other by secreting specialized, small molecules that are then received by other bacteria and interpreted accordingly. Bacteria are mainly using this form of communication to find out what the population density is in their neighborhood in order to decide whether or not to act, and, if they act, in what way. This ability is called quorum sensing, and it is the focus of this module.

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  19. Evolutionary Game Theory: The Game of Life

    This module examines the role that behavior plays in evolutionary fitness. Through studying and playing games, students develop an understanding of natural selection as organisms compete for limiting resources (e.g. food, water, space, mates, safety, etc.). Traditionally, this idea is developed in the high school curriculum by focusing on adaptations due to the physical phenotype of the organisms. This module looks specifically at behavioral choices made to obtain the resources organisms need for survival and for reproduction. The possible choices in this unit have been termed “Moderate” or “Greedy”. Moderate means that the player will take only the resources that can be attained without confronting the other player for control of the resources. Greedy indicates the player will confront the other player for control of the resource even if the resource is not immediately needed. By playing the game for M & M’s, students will develop different strategies under different conditions and examine the effects of these strategies on survival. If this behavioral trait is due to a genetically inherited variation, then these genes may be passed to the next generation. Those that are more successful will survive to live another day and perhaps even go on to reproduce. Unsuccessful individuals will not reproduce and will die before passing their genes on to the next generation. The successful individuals will pass their genes on to their descendants.

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  20. DNA Sequencing and Sorting: Identifying Genetic Variations

    Each of the cells in your body contains a copy of your genetic inheritance, the DNA that has been passed down to you, one half from your biological mother and one half from your biological father. This DNA, mixed and mutated in your ancestors over the generations, determines in large part, obvious physical features like eye color and hair color, and, combined with environmental factors like diet and exposure to toxins, can determine susceptibility to medical conditions like hypertension, heart disease, diabetes, and cancer. In this module you will learn about the joint laboratory and computer technique called genome sequencing, which is being used to study DNA and identify the variations that exist among people. Variation detection is important for understanding how our genomes differ, which will lead to a better understanding of human health and disease. The module introduces background material for the following concepts in biology and genetics: DNA, chromosomes, mutations, and polymorphisms. Mathematical and algorithmic concepts include binary numbers and sorting. You will identify where short DNA sequences fit in a reference genome. This is intended to illustrate the mapping process used in genome sequencing and the types of genomic variations that can be discovered as a result of mapping. You will appreciate the need to develop an efficient procedure for locating where short DNA sequences match a reference genome. Sorting the genome is suggested as an approach, and you will investigate the concept of sorting DNA sequences using Radix Sort.

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    [Module Teaser]