Teachable Unit: Executive Summary
Title: Probabilities. Not Just For Mathematicians Anymore!
Learning Goals
· 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
· 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
· 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
· 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.
· 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 intro-level 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.
· 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%
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
· 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?
· 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.
· 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.
· 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?
· 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.