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The BioMath Connection (BMC) & Integrating Mathematics and Biology (IMB) Projects

BMC/IMB was a pioneering project linking biology and mathematics in the high schools. It provided an opportunity for high school teachers, writers, researchers, and others to get in on the ground floor of developing innovative classroom materials. The materials consist of 20 modules that can be flexibly adapted for use in a variety of courses at a variety of grade levels in both biology and mathematics. The project was run by DIMACS at Rutgers University in collaboration with the Consortium for Mathematics and its Applications (COMAP) and Colorado State University (CSU).

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Modules

Click here to view a list of BioMath authors.


Student Material


Teacher Material

Evolution By Substitution: Amino Acid Changes Over Time
This unit 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.

Topics
Biology:
Changes in DNA may result in amino acid substitutions in proteins that may then be used to study evolutionary relationships.
Mathematics:  Single and multi-stage probability events, disjoint and independent events, matrices, matrix multiplication, powers of matrices.

Prerequisites
Biology:
The unit deals primarily with evolution as the result of amino acid changes over time. It does not assume any prior knowledge about biology.
Mathematics:
Decimal multiplication and percentage calculations. Other mathematics topics in this unit are not assumed as prior knowledge, and are developed in such a way that students can learn it for the first time or refresh their prior knowledge of those topics. The materials in the unit give teachers options about how much emphasis to place on the development of the mathematical content.

Length
This unit consists of 4 lessons and a project and will take 5-7 class periods (45 minutes each) if the majority of the work is done during class. Length will vary depending on the number of additional practice and extension problems done.

 


Student Material


Teacher Material

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 unit, 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.
Following a positive mammogram the woman has a genetic test through which she learns she has the BRCA gene mutations associated with breast cancer. The students investigate what it means to have this mutation and how scientists are working on medical treatments that can be tailored to a particular genetic profile.
Finally, knowing their mother is positive for the BRCA mutation leads to a dilemma for her daughters. They must decide if they will be tested for this BRCA allele, since results from testing for this allele still do not definitively determine whether or not a woman will develop breast cancer.

Topics
Biology:
Imperfect testing, genetic testing, genetic variation, pharmacogenetics, ethical choices, decision making based on data interpretation, taking perspectives, gold standards.
Mathematics: Probability, conditional probability, Bayes’ Rule.

Prerequisites
Biology:
Basic understanding of DNA structure and function; what a gene is and that an allele is a version of a gene.
Mathematics: Ratios and proportions; calculating percentage.

Length
This unit consists of 6 lessons and will take 4-6 class periods (45 minutes each).

 


Student Material


Teacher Material

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.

Topics
Biology:
Genetic mutations and phylogenetic trees.
Mathematics: Mathematical algorithms and optimization.

Prerequisites
Biology:
Relationships among DNA, genes, and chromosomes.
Mathematics: Number patterns, sequences, algorithms.

Length
This unit consists of 4 lessons and will take 5-7 class periods (45 minutes each).

 


Student Material


Teacher Material

Spider Silk: Examining Biological Sequences
This unit asks students to apply knowledge of protein structure and function to pose and answer the fundamental question investigated in this unit, “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.

Topics
Biology:
Comparison of amino acid and nucleotide sequences between species, alignment of protein and DNA sequences, use of computers with databases.
Mathematics: Edge-vertex graphs and tree diagrams as models for sequential decisions, numerical scores as measures of similarity between letter sequences, dynamic programming, minimal- and maximal-path algorithm, path-counting algorithm, and combinatorics.

Prerequisites
Biology:
Basic ideas behind DNA, RNA, protein composition and synthesis, amino acids, and classification.
Mathematics: Integer arithmetic and ability to select the minimum (or maximum) number from a list of Integers.

Length
This unit consists of 5 lessons and will take 5-6 class periods (45 minutes each. The length may vary depending upon the time you allocate for student explorations on Biology Student Workbench.

 


Student Material


Teacher Material

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.
THE BIG CAT ATE THE FAT RAT.
THE BIG BAT ATE THE FAT RAT.
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.

Topics
Biology:
Microarrays are used to explore the genes that could be associated with genetic conditions. The production of cDNA from mRNA and the hybridization of 2 strands of DNA are covered. Gene expression and the level of expression are explored.
Mathematics: The mathematics of this unit is primarily statistical in nature. The key idea in statistics is variability, so much of this unit deals with how to display, describe, and measure variation in various data settings. Color is introduced as a quantifiable variable, useful in measuring a biologically important phenomenon. Dotplots provide easy-to-make and easy-to-interpret displays for univariate data. The unit introduces four summary statistics (mean, median, standard deviation, and, optionally, interquartile range) to help quantify the characteristics of location and spread within univariate data. Clustering of points within a scatter plot helps identify patients with similar characteristics, and Euclidean distance provides a tool for measuring that similarity.

Prerequisites
Biology:
This unit is appropriate for an introductory biology class of 9th or 10th graders, a genetics class or human physiology class that deals with genetic disorders, or an AP biology class. The unit deals primarily with the genetic cause of disease as researched through the use of microarray technology. Background knowledge of DNA, replication, transcription, and genes is helpful but not required.
Mathematics:Any high school math level, but most appropriate at Algebra 1, geometry, or Introductory Statistics levels. The mathematics of this unit includes introductions to exploratory data analysis in one and two dimensions.  Students should be familiar with representing numbers on number lines and plotting ordered pairs in a Cartesian coordinate system.  Knowledge of the Pythagorean theorem or distance formula is also assumed.

Length
This unit consists of 5 lessons and an assessment. It will take 7-8 class periods (45 minutes each) if the majority of work is done during class time.

 


Student Material


Teacher Material

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.

Topics
Biology:
This unit discusses issues of habitat suitability and ecological niche. Students will develop an understanding of one method for the evaluation of the influence of specific contributory habitat characteristics on species presence (e.g., availability of food resources, nesting sites, conspecific attraction, low predator density, etc.). This method will involve discussion of biotic and abiotic environmental factors, consideration of both numerical and categorical data, isolation of independent variables governing the dependent variable of habitat selection, and how to approximate those relationships by trend lines for use in either description of the observed species preferences, or to predict relative suitability of designed habitats for efforts in conservation and management.
Mathematics: This unit discusses quantitative analysis of observational data. Students will develop an understanding of the appropriateness of numerical versus categorical interpretations of functions and the potential effects of viewing data via both representations. Further, they will explore lines of best fit as a way to quantify the relationship between dependent and independent variables in a system. By creating scatter graphs, fitting lines to the observed data, and then formalizing the relationship into an equation for a line, they will understand the use of mathematical representations of causal/correlated outcomes and be able to use these model representations to describe current systems succinctly and, further, to predict the outcome of hypothetical scenarios.

Prerequisites
Biology:
An understanding of the concept of an organism and a species is useful.
Mathematics: Basic algebra to include solving linear equations and using both the point-slope and slope-intercept representations of a line.

Length
This unit consists of 5 lessons, a lab, a project and an assessment. It will take 6-8 class periods (45-minutes each) if the majority of work is done during class.

 


Student Material


Teacher Material

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?”

Topics
Biology:
Food webs, predator/prey dynamics, energy transfer.
Mathematics: Discrete mathematics, graph theory, mathematical modeling.

Prerequisites
Biology:
Basic understanding of species requirements for survival.
Mathematics: Basic understanding of algebra and flow charts.

Length
This unit consists of 5 lessons and will take 4-6 class periods (45-minutes each) if the majority of work is done during class.

 


Student Material


Teacher Material

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 unit 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.

Topics
Biology:
Ecology of humans and ecological footprinting, ecosystems, the biosphere, resources, and carrying capacity, interactions among individual, population, community, and ecosystem levels.
Mathematics: Dimensional analysis, units of measure, ratios and proportions, elementary mathematical modeling, interpreting models.

Prerequisites
Biology:
Basic ideas about the environment, animals’ survival needs, population growth, and natural resources; awareness of the carbon cycle and photosynthesis.
Mathematics: Pre-algebra coursework including the use of variables and formulas; unit analysis.

Length
This unit consists of 5 lessons and 1 optional extension lesson and will take 6-7 class periods (45 minutes each).

 


Student Material


Teacher Material

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.

Topics
Biology:
Ecology, to include habitats and biological organization of space.
Mathematics: Geometry, to include polygons and concept of perpendicular bisectors, and creating mathematical models based on Voronoi diagrams.

Prerequisites
Biology:
Students need an understanding of basic animal needs, the notion of territorialism, what a cell is, and a readiness to critically examine the implications of difference uses of space.
Mathematics: Students need a basic understanding of basic algebra and geometry, of how to calculate averages, and a readiness to work with coordinate geometry; in particular the concept of perpendicular bisectors and equations for them will be developed in some detail.

Length
This unit consists of 5 lessons and an assessment. It will take 5 to 7 class periods (45 minutes each) if the majority of work is done during class.

 


Student Material


Teacher Material

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 unit 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.

Topics
Biology:
This unit discusses home range, habitat, trophic levels, buffers, human impact, corridors, tracking methods and other areas of conservation biology.
Mathematics: This unit includes concepts of unit conversions, graphing, estimation, area and perimeter, polygons and similar polygons.

Prerequisites
Biology:
Basic understanding of habitat and differentiation of animal species.
Mathematics: Basic Algebra 1 and Geometry concepts.

Length
This unit consists of 5 lessons and will take 5-7 class periods (45-minutes each) if the majority of the work is done during class. Extension activities provide further development of the modeling if time allows.

 


Student Material


Teacher Material

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.


Topics
Biology:
Biological success of disease-causing agents due to success at reproduction and survival, adverse reactions of infectious diseases in their host organisms, means of transmission of infectious diseases, susceptibility to diseases
Mathematics: Multiplying fractions, decimals, and percentages; single- and multi-stage probability events, disjoint and independent events.

Prerequisites
Biology:
This unit deals primarily with disease transmission and successful survival strategies of the disease-causing organism. It does not assume any prior knowledge about biology, but a basic understanding that "biological success" depends only on the two mutually linked characteristics of survival and reproduction would provide a greater biological context for the explicit examination of evolution of disease.
Mathematics: The unit uses decimal multiplication and percentage calculations to introduce basic concepts in probability calculation. Concepts beyond decimal multiplication and percentages are developed in such a way that students can learn them for the first time or refresh their prior knowledge of those topics. The materials in the unit give teachers options about how much emphasis to place on the development of the mathematical content.

Length
This unit consists of 4 lessons, one simulation and a multi-part assessment. It will take 4-5 class periods (45 minutes each) if the majority of work is done during class.

 


Student Material


Teacher Material

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 unit is about the study of infectious diseases — their causes, prevention, spread, and control.

Topics
Biology:
The biology content relates to those microorganisms whose names are generally referred to as “germs”: bacteria and viruses. Students should have already learned some basic information about bacteria and viruses so they can apply the principals of epidemiology to what they already know. In addition, students will simulate the spread of a disease with a hands-on investigation — an activity that dramatically convinces students how a disease can spread rapidly.
Mathematics: The mathematical knowledge and skills used in this unit are probabilistic in nature, beginning with the calculation of probabilities and rates (as in a germ’s basic reproduction number, called R0). Students also learn to think symbolically and numerically by working with relationships among the number of cases (patients) who are initially in a “susceptible” state and move to the “infected” state, and eventually to the “recovered” state.

Prerequisites
Biology:
Know the parts of a cell, basic differences among bacteria, viruses and eukaryotic cells, and elementary ideas of epidemiology.
Mathematics: Be familiar with the concept of probability and use of simple formulas.

Length
This unit consists of 6 lessons and an assessment. It will take 7-8 class periods (45-minutes each) if the majority of work is done during class.

 


Student Material


Teacher Material

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. The 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.

This paragraph above opens Lesson 1 and links to the unit’s problem of determining a newly discovered species’ place on the evolutionary tree of life. In this unit 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 unit 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.

Topics
Biology:
Genetic mutations and phylogenetic trees.
Mathematics: Mathematical algorithms and optimization.

Prerequisites
Biology:
Relationships among DNA, genes, and chromosomes.
Mathematics: Number patterns, sequences, algorithms.

Length
This unit consists of 4 lessons and will take 5-7 class periods (45 minutes each).

 


Student Material


Teacher Material

CRIME: Criminal Investigation through Mathematical Examination
Welcome to our unit on the mathematical examination of fingerprints! This unit is 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 unit concentrates on human fingerprints. To accomplish this, we use a fictitious crime case to explore a mathematical procedure to examine and compare fingerprints. The unit consists of multiple activities to investigate the characteristics of a fingerprint and the biology behind why everyone’s fingerprints differ.

Topics
Biology:
The biology of this unit involves skin anatomy, fingerprints, and the genetics associated with individuality, at least in the form of an expressed phenotype. The genetics include the concepts of penetrance, expressivity and epigenetics. Biological diversity is an important issue in evolution and sexual reproduction allows for the creation of individuals who are different from their parents and from any other individuals previously born. The unit also allows a convenient opportunity for the teacher to explore the physiology of the skin in more detail, if they wish. Teachers may add their own material for that purpose.
Mathematics: The mathematics of this unit is primarily the mathematics of graph theory. In graph theory, a primary concern is the connections (edges) between given vertices (points). This unit introduces basic graph theoretic concepts (networks, vertex, edge, connectedness). The process of formulating questions and answering those questions is introduced through data collection and organization, useful in identifying biological patterns of behavior within species. Constructing ordered pairs to label specific characteristics of the data provides easy-to-make and easy-to-interpret displays for the dataset. Matrices (rectangular arrays of numbers) are introduced to provide a more mathematically efficient representation of the graphs.

Prerequisites
Biology:
Students should be familiar with basic genetics (e.g. DNA, genes, alleles).
Mathematics: The mathematics of this unit includes introductions to sets (databases), collection and organization of data, basic graph theory, and matrices. Students should be familiar with representing quantities as ordered pairs, ratio and proportion, percent, sets and subsets.

Length
This unit consists of 4 lessons and will take 4-6 class periods (45 –minutes each) if the majority of the work is done during class.

 


Student Material


Teacher Material

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 unit 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.

Topics
Biology:
The nervous system, cellular biology, and neural networks..
Mathematics: Descriptive and inferential statistics, to include histograms, boxplots, t-tests, and analysis of variance.

Prerequisites
Biology:
Students need a basic understanding of the basic structure of the nervous system, gross neuroanatomy and neural function.
Mathematics: Students need a basic understanding of descriptive statistics, inferential hypothesis testing, t-tests, basic algebraic properties, operations and functions.

Length
This unit consists of 4 lessons. It will take 9 to 11 class periods (45 minutes each) if the majority of work is done during class.

 


Student Material


Teacher Material

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 unit starts with an introduction to tomography and then introduces some basic mathematics used in computed tomography (CT scans). The unit 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.

This unit 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 unit, there are many activities, worked examples, homework problems, and various assessment tools.

Topics
Biology:
Investigate medical imaging, consider tissue properties (attenuation coefficient), and examine applications of imaging methods.
Mathematics: Apply of Beer’s law and visualize shapes and surface are.

Prerequisites
Biology:
Basic ideas of anatomy, organs and tissues, and a basic understanding of radiography in terms of x-rays.
Mathematics: nderstand function as an input-output model. Understand concept of exponential functions, the concept of a 3-dimensional grid, and be able to determine surface area.

Length
This unit consists of 4 lessons, an optional unit project lesson and an assessment. It will take 5 – 7 class periods (45 minutes each) if the majority of work is done during class.

 


Student Material


Teacher Material

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 unit 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.

Topics
Biology:
Scientific method, national health problem of increasing rates of childhood obesity, identification of being overweight (use of weight, BMI, CDC Growth Charts for boys and girls), waist circumference, analysis of factors that might contribute to being overweight.
Mathematics: Statistical tools for analyzing data on one and two variables. Tools for one-variable analysis include numeric summaries for center and spread (mean, median, range, interquartile range, percentiles), five-number summaries, dotplots, boxplots, 1.5×iqr rule for identifying outliers. Tools for two-variable analysis include scatterplots, regression lines, residual error, R2, and using the regression equation to make predictions. Tools for analyzing two categorical variables (such as binary variables) include two-way tables and percentages.

Prerequisites
Biology:
Some exposure to the scientific method.
Mathematics: Plotting ordered pairs; graphing, evaluating, and determining slope of linear equations; percentage calculations.

Length
This unit consists of 5 lessons and will take 6 – 8 class periods (45 minutes each) if the majority of work is done during class. Length will vary depending on whether students complete the project and/or the optional activity on regression.

 


Student Material


Teacher Material

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 will be the focus of this unit.

Topics
Biology:
This unit discusses bacterial growth and communication. Students will develop an understanding of the basic structure of a bacterium and how they grow through binary fission.  The majority of the unit develops the idea of how, through chemical communication, a single-celled organism can gain, in a sense, multicellularity. Examples of this “multicellularity” are discussed and include the plaque on teeth, disease-causing bacteria, industrial examples, etc.  To understand how the simplest of cells can achieve cooperation, a mathematical model is developed that takes into consideration the production and degradation of a communication molecule (the inducer), the production and degradation of a receptor molecule, and the production and degradation of the final product.  For students that have studied DNA and protein production, this provides an example of how transcription and translation are regulated.
Mathematics: This unit presents a mathematical model that uses precalculus mathematics to explain the phenomenon of quorum sensing. After a review of exponential functions and an introduction to rate of change, students will be introduced to a set of equations that describe the rates of change for various components of cellular communication. Several of these equations express a rate of concentration change as a function of the concentration itself and will require students to solve linear equations to find desired concentrations. All of the equations employ the use of parameter values, provided to be representative of unique values for specific organisms. The development of a simulation of cellular concentrations over time will introduce students to iterative computations, particularly with numbers expressed in scientific notation. The notion of a density coefficient is also one of the mathematical concepts of the unit.

Prerequisites
Biology:
A basic understanding of cell structure and functions along with the Central Dogma would be helpful, but is not necessary.
Mathematics: Exponents, scientific notation, function notation, solving linear equations, using a graphing calculator’s symbolic-table-graph features, spreadsheet construction.

Length
This unit consists of 5 lessons and will take 4-6 class periods (45 minutes each) if the majority of the work is done during class. Length will vary depending on whether the activities and questions are each done individually, in groups or as a class. An extension provides further development of the mathematical model if time allows.

 


Student Material


Teacher Material

Evolutionary Game Theory:  The Game of Life
This unit examines the role that behavior plays in evolutionary fitness. Through studying and playing games, students will 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 unit will look 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; therefore, having only the genes of.

Topics
Biology:
Evolution in terms of genetics, competition and natural selection.
Mathematics: Application of game theory, to include understanding strategies and maximizing outcomes, to analyze behavior in terms of evolutionary fitness.

Prerequisites
Biology:
An understanding of the basic ideas of evolution, competition and survival of the fittest.
Mathematics: Basic mathematics operations and an understanding of ratio and percent.

Length
This unit consists of 5 lessons and will take 5-7 class periods (45 minutes each) if the majority of the work is done during class. Length will vary depending on whether the activities and questions are each done individually, in groups or as a class.

 


Student Material


Teacher Material

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 unit 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 unit 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.

Topics
Biology:
Genetics, DNA, chromosomes, sequencing, mutations, and polymorphisms.
Mathematics: Binary numbers and sorting, proportional reasoning and use of algorithms.

Prerequisites
Biology:
A basic understanding of DNA structure and function.
Mathematics: An understanding of arithmetic, proportional reasoning, and algorithmic thinking. A basic understanding of numbers in bases other than 10.

Length
This unit consists of 6 lessons and an assessment. It will take 6 or 7 class periods (45 minutes each) if the majority of work is done during class.

 

This material is based in part upon work supported by the National Science Foundation under Grant Number NSF DRL-1020166. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.