Download Probabilities Not Just for Mathematicians Anymore (framework

Teachable Unit: Executive Summary
Title: Probabilities. Not Just For Mathematicians Anymore!
Learning Goals
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Students will be able to understand that certain scientific processes can be described
mathematically in terms of probability.
Students will be able to recognize patterns in data and learn to define probability results in terms
of percentages, decimals, and fractions.
Depending on the field of study, students will be able to analyze a set of data probabilistically
with an aim towards utility in that class (i.e., Punnett squares in genetics, electronic states in
inorganic chemistry, Boltzmann distributions in physical chemistry, etc.)
o Note: This specific teachable unit is defined in terms of Punnett square and genetics.
Learning Objectives
Scientific Principles
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Students will be able to describe a fundamental topic in their field in terms of probability, such as
how in genetics Punnett squares give rise to probability calculations.
Students will be able to define independent and mutually exclusive events.
Students will be able to classify independent and mutually exclusive events.
Students will be able to apply appropriate probability formulas to different data.
Students will be able to, using larger data sets or case studies, use probability data to act as a
predictive tool, such as using phenotype distributions to predict genotypes.
Scientific Skills
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Students will be able to recognize patterns in data and to begin to comprehend where data may be
analyzed probabilistically.
Students will be able to draw and analyze Punnett squares of varying complexity.
Students will be able to recognize whether scientific events are independent or mutually exclusive
and be able to offer a reason why.
Students will be able to navigate a case study and recognize the importance of case studies to
broader science as a whole.
Mathematical Skills
 Using a specific Punnett square, students should be able to use the sum and product rules to arrive
at specific probability values.
 Using traditional mathematical probability examples, students should be able to analyze dice
rolling or other examples to determine specific probabilities.
 Students will be able to make connections between traditional examples and applied examples,
such as genetic probability and prediction of genotype from phenotype distributions.
General Skills
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Students will develop reading comprehension through analysis of a case study.
Students will develop spatial thinking and familiarity with symmetry through utilization of the
Punnett square technique.
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Students will gain comfort with conversion of fraction, decimal, and percentage.
Students will develop interdisciplinary critical thinking skills by applying probability formulas
across disciplines.
Teaching Challenge
Science Content:
Biology students often have trouble understanding how mathematics ties into their subject. This
unit aims to address that issue for the field of probability. In particular, students often do not make the
connection that Punnett squares reflect probabilistic predictions. Also, students often confuse the
probability terms “independent” and “mutually exclusive,” and the appropriate probabilistic formulas that
go along with these notions: namely the “product rule” for independent events and the “sum rule” for
mutually exclusive events.
Brief Description of the Teachable Unit
This module attempts to move away from traditional lecture and worked-out example based
descriptions of biological and chemical-based probability phenomena and instead attempts to strive for a
student-centered preparatory classroom where the focus is on discussion, analysis of case studies, and
prior preparation. Each class period is designed to hook the students’ interest by an applicable, real-life
topic and then will transition into class periods where students are engaged in a variety of active learning
assignments with varying degrees of assessment. Due to nature of genetics, specifically in terms of skin
color, diversity topics are easily built into this module. Again, it is important to note that this module is
interdisciplinary by nature and could easily be modified to a number of important topics for both introlevel and upper-level chemistry and physics courses. Depending on the students’ prior familiarity with
certain terms based on math experience (i.e., independent and mutually exclusive events), this unit will
require approximately 3-4 hours.
Sequence of Objectives
Objective 1: Describe how Punnett squares give rise to probability calculations.
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Hook: Presented to students at the end of the class prior to this unit. You are aware of discrete
traits from the unit on Mendelian genetics. Now let’s consider traits with a variable distribution.
For homework, do some brainstorming and research on your own. Identify some variable traits in
humans.
Introduction: Students share their ideas of continuously variable traits. Lead into skin color and
show picture of a family with notable variation in skin color. Ask students “How can this be”?
Mini-lecture: On the genetics of skin color variation. Clearly identify that there are over 100
genes that can contribute to skin pigmentation. However, there are three genes that primarily
contribute. Describe the dosage effect: with 3 genes, skin color can be a value from 0 – 6.
Clicker question: If parents both have a value of 3 for skin color pigmentation, what is the
probability that one of their children will also be a 3? Allow for discussion and re-poll.
a. 100%
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b. approximately 80%
c. approximately 50%
d. approximately 30% * correct answer
e. 0%
Small Group Activity: Drawing a Punnett square for a trihybrid cross.
o If both parents are heterozygous for all three genes, what are the possible gamete types
produced by each parent? Draw the appropriate Punnett square.
o Use your Punnett square to determine: how many offspring will be a 0; how many
offspring will be a 1; … ; how many offspring will be a 6.
o Assessment: In a large group discussion, ask students to share the results from their
Punnett squares. Revisit the clicker question.
Clicker question: if one parent is a 0 and the other parent is a 6, what is the probability that one
of their children will be a 4? Allow for discussion (ask students to think carefully about the
possible gamete types). Re-poll.
a. 100%
b. approximately 80%
c. approximately 50%
d. approximately 30%
e. 0% * correct answer
Small Group Activity: Drawing a Punnett square for a trihybrid cross.
o If one parent is a 2 and the other parent is a 4, what are the possible gamete types
produced by each parent? Draw the appropriate Punnett square.
o Use your Punnett square to determine: how many offspring will be a 0; how many
offspring will be a 1; … ; how many offspring will be a 6.
o Assessment: A representative from each small group will draw their Punnett square on
the board. There are 225 possible parental combinations that reduce down to 3 different
offspring phenotype distributions, so it is highly likely that every group will pick a
different parental combination and all 3 offspring phenotype distributions will be
represented.
If time necessitates, ask students to complete the 2 x 4 Punnett square for homework, and perform
the assessment during the beginning of the next class.
Objective 2: Students will be able to define then classify independent and mutually exclusive
events.
Teachable Tidbit
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Setup: The setup will discuss topics from the previous objective by bringing back into focus the
8x8 Punnett squares and discussion of the skin tone of parents and children. We discussed how
irritating drawing the large Punnett square can be. This is bridged into a question/assertion about
how well the average person can multitask?
Individual Activity: Of the following two lists of pairs of events, what are distinguishing
characteristics of the two lists?
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Think/Pair/Share: Students are asked to share their opinions of what the commonality of the
events is with a partner or with neighboring groups.
Definition: This group discussion will lead to working definitions of independent and mutually
exclusive in a mathematical and scientific context.
o Note: It is important to note that the specific names will not be given to the students until
after this step.
Clicker Poll: Students will be asked to determine whether columns A and B are independent and
mutually exclusive, respectively, or vice versa.
Discussion: Students will be asked to argue their points with neighboring groups to convince
them of their answers, right or wrong.
Re-poll: Students will be re-polled on the prior poll question. If the answer is still in question,
additional discussion or a mini-lecture can be introduced.
Clicker Questions: Sets of events will be introduced, and the students will click A if the event is
independent, B is the event is mutually exclusive and C if the event is neither. Examples 1 and 2
are deliberately one or the other and will encompass prior topics in biology or chemistry.
Examples 3 and 4 are less clear and are designed to drive discussion to help students shore up
their understandings of these two terms.
o Note: If Examples 1 and 2 are unclear or have mixed results, no worries! Now you get
to bring in older topics for discussion!
Brainstorming: What are some independent events? What does it mean to be mutually
exclusive? Brainstorm some usual tasks that are independent or mutually exclusive and discuss
with your neighbors to ensure that you are correct.
Homework: Develop a set of independent and a set of mutually exclusive events that are of a
scientific nature and post them on the class blog. Be ready to discuss these in the following class.
Objective 3: Students will be able to apply the appropriate probability formulas to different
data.
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Hook: Probabilities in sports or gambling. Using examples from the class blog, remind students
about the terms independent and mutually exclusive. This reminder leads into the following
small group activity.
Small Group Activity: Discuss events that you think are independent vs. mutually exclusive.
Group Activity using Punnett squares: What are independent events? What are mutually
exclusive events? Given parents of a particular genotype/phenotype, predict probability of
having a child of phenotype x or vice versa. Given parents of a particular genotype/phenotype of
skin tone, predict the probability of having a child within a given range of phenotype.
Think/Pair/Share: Baseball batting average, lottery, cards, dice
Activity: M&M or bead activity, changing probability
Clicker Question: If a couple has 4 girls, what is the probability their 5th child will be a boy?
(More likely, less likely, equally probable)
Group Discussion: Convince your neighbor of what is the correct answer.
Objective 4: Students will be able to use phenotype distributions to predict genotype.
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Homework: Worksheet given to students at the end of the prior class (see below). As part of the
homework assignment, students will complete a reading that describes the symbols used in
pedigree analysis. Students will bring the completed worksheet to class for further discussion and
work.
Pedigree Homework
The pedigree illustrated here shows individual II-2 affected with a disease (cystic fibrosis) that is
inherited in an autosomal recessive pattern. Let “A” represent the dominant allele and let “a” represent the
recessive allele.
1. What are the genotypes of individuals I-1and I-2?
2. What is the probability that if parents I-1 and I-2 had another (fourth) child, he or she would not have
the cystic fibrosis?
3. What is the probability that II-3 is heterozygous?
Individual II-3 (John) marries a woman, Tammy, whose aunt Jill (on her mother’s side) died of cystic
fibrosis. Tammy’s mother is Amy and the parents of Amy and Jill are Tom and Mary.
4. Draw this scenario into the above pedigree.
5. What is the probability that Tammy is a carrier?
6. If John and Tammy decide to have children, what is the probability they will have a child with cystic
fibrosis?
7. If John and Tammy’s second child has cystic fibrosis, what is the probability their third child will have
cystic fibrosis?
 Case Study: Presented to students during class for small group work.
A recently married couple enters your office and explains that they would like to start having
children, but are concerned. The wife has a sister who has cystic fibrosis and the husband has a
brother with CF. What do you tell the couple about the probability that they will have a baby with
CF? What would you say about their probability of having a child who was a CF carrier?
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Mini PTBL: (because everything smaller is cuter) to be completed during the remainder of class.
Students will have the opportunity to finish work outside of class and will present their disease in
an informal presentation during the following class period.
Choose a disease that is of interest to you that is considered genetic (has a strong inheritance pattern).
Using reliable online resources, provide the following information:
1. What is the mode of inheritance of this disease?
2. What is the molecular/cellular nature of the disease? What gene(s) is/are affected, and what is the
nature of the mutation(s) involved in the disease?
3. Imagine that you have become an illustrious Genetics professor. Draw a pedigree containing at
least 3 generations. Develop 3 questions, based on the pedigree and using probability that will be
used in your exam.
As a starting point, here are two online sources of information on genetic diseases:
OMIM (http://www.ncbi.nlm.nih.gov/omim)
Genetics Home Reference (http://ghr.nlm.nih.gov/).
Both of these exhaustive databases are maintained by the National Institutes of Health of the US
government. Both government and university scientists contribute to these resources. These are
highly credible and complete sources of information on genetic traits.
How does the teachable unit address the following themes?
Assessment
1) Formative Assessment
a) Clicker Questions
i) Classification of independent and mutually exclusive events
ii) Polling and re-polling
b) Writing group answers on the board
i) Examples of independent and mutually exclusive events
ii) Writing and developing Punnett squares
c) Large class discussion of small group answers
i) What is the genetic cause of variable traits?
ii) Various probability calculations
iii) Analysis of case studies
2) Summative Assessment
a) Distinct homework problems to calculate specific probabilities by analysis of Punnett square by
inspection or mathematically
b) A short quiz at the end of the week in which student must solve a probability problem in a new
context
c) An assessment section on an exam
Diversity
In this module, this synthesis of biology and mathematics is lead into by a discussion of skin color and
how it can be determined by genetics. Because of this, biology students can be interested and engaged as
long as the topic is kept sensitive. The subject is broached by introducing a picture of a set of twins in
which one is very dark and one is very light in order to instigate discussion or at least curiosity.
Examples are selected in order to engage differences in gender and ethnic roles. Specifically, a clicker
question is introduced to engage subtle bias between gender roles in the sciences in order for students to
engage these topics at an early stage. Various types of assignments are selected to attempt to engage
students of different learning abilities. Visual learners will be engaged by clicker questions, pictures, and
finished Punnett squares. Auditory learners will be engaged through discussion and think/pair/share.
Kinesthetic learners will be engaged by working on Punnett squares and active debate.
Active Learning
Historically, skin tone has been associated with race and therefore can be related to a controversial topic.
In this exercise, we demonstrate that skin tone is predominantly determined by three genes, which can be
described as a probabilistic nature. The activities within will encourage thought-provoking discussion
and lead into calculations of this nature.
Action Plan
Tiffany Frey, Michael Roberts, and Dana Wohlbach each plan on incorporating the majority of the
teachable unit into a 200-level Genetics class. Tiffany will do so in Spring 2014; Michael and Dana will
utilize the unit during Fall 2013. Scott Brothers plans on modifying the existing framework of the unit in
order to incorporate the concepts into Physical Chemistry II in the statistical thermodynamics module
during the Spring 2014 semester. Joseph Walsh plans on using project-based learning in Numerical
Analysis.