Organic Evolution # 3: The Meiotic Mix

College Algebra #26: Probability

The students will learn:
· The Hardy Weinberg principle of gene frequency.
· Ratio can indicate a probability of an event.
· That the polynomial theorem explains how the Hardy Weinberg principle works.
• Two popsicle sticks per person. For every 10 people, use 20 sticks, and mark 7 with a capital R in black, 3 with a lower case r in black. Mark 7 with a capital R in red, and the remaining 3 with a lower case r in red.
• Two paper bags: one marked "boys" for the black popsicle sticks and one marked "girls" for the red popsicle sticks.
• Student sheet: one per person
• Calculators to determine square root.
• Overhead or board for teacher.
Student challenge: "How many of you can roll your tongue?" Allow students time to experiment with how many students can roll their tongue. After the activity, explain that tongue rolling is a monogenic trait: that is, there is only one gene for this trait. Explain that the ability to roll is dominant, and the inability to roll is recessive.

For a review of Mendelian genetics, show the video ORGANIC EVOLUTION #3: The Meiotic Mix from the beginning without stopping (10 min total).
Focus students viewing by directing them to listen for how the following key terms are used:
monogenic trait recessive
dominant Hardy-Weinberg Law

Second viewing
Check for student comprehension by showing the video a second time using the second viewing technique. Start about 3 minutes into the video when the animation shows a napkin which is to be written on. This section of the tape is about 6 minutes long.
Distribute the "Genes Count" question sheet. Pass the remote control to students. Allow them to pause as the questions are answered so they can answer on their sheet.
1. What is the Hardy-Weinberg Law? It says that the ratio of dominant to recessive would (remain constant.)
2. The dominant to recessive ratio ____________ (remains constant) as the population is not (disturbed).
3. The Hardy-Weinberg Law is theoretical model comes to equilibrium when strict conditions are met. They are:
a) (large population)
b) (random selection of mates)

ACTIVITY #1

Go over their answers, ask students why a) large populations must exist and b) a random selection of mates must occur. (a) Variations are less apt to occur in larger populations than in small. b) Non-random selection of mates will cause certain alleles to be chosen for or against, changing the gene frequency of those genes.
Why is the Hardy-Weinberg Law used? (To predict gene frequency, and to determine if some selection process is occurring.)

ACTIVITY #2

PREPARATION

Prior to this activity, the students should have learned about dominant and recessive traits; how to represent them as letters; and how to use a Punnett square to determine offspring genotypes and phenotypes (genotypes: the actual allele combination of a gene and phenotype: the physical appearance of a trait. Allele= form a gene can take. Gene=specific location on chromosome fully or partially responsible for a trait.)
Review elementary probability with students: that the probability of something occurring is determined by the ratio of probability of success of an event to the number of possible outcomes of an event (i.e. probability of a tossing a head = 1 way of success /2 possible outcomes, or _, or .5, or 50%)
The probability of all outcomes = 1 (the probability of tossing a head OR a tail =1).
Students should know that the probability of two dependent events happening is equal to the product of the probability of each event happening independently. (The probability of tossing two heads in a row is equal to _ times _, or 1/4).
Pose the question "How can you tell how often a trait will appear?" "Can you predict how frequently it will appear in future generations?"
Pass out student worksheet "An Investigation into Gene Frequency."
Ask students to randomly take two marked popsicle sticks out of the bags. One bag is for boys (black lettered), and one for girls (red lettered). Each bag will have 70% marked with R and 30% marked with r. Tell them that R will represent the gene for rollers, and r the gene for non-rollers.
Ask students to fill in their papers indicating their genotype (I on student sheet).
Ask students what the possible genotypes are that exist. (RR and Rr would be for rollers, rr would be for non rollers.) (II on student sheet.)
Count how many people are in the class. Enter this at III on the student sheet.
Explain that we cannot determine the probability of the RR of Rr alleles because all people with that combination cannot be distinguished one from another. However, since we know how many people cannot roll their tongues, we can determine the frequency of that allele.
Poll the class as to how many of them have rr and enter the data in the table (number IV).
Determine the ratios of non-rollers to the total number of the group and then get a percentage by dividing that fraction. (V)
Draw a Punnett square like this, using the number just calculated at V (ratio of those with rr to the total number in the class.)

<! - - THIS IS ONE OUT OF FIVE RECTANGLES. THERE ARE 4 45-DEGREE BISECTORS WITHIN THE RECTANGLE - - >

This Punnett square represents all of the possible genotypes. From what we know, we can determine the gene and allele frequency.
Remind students that the probability of two consecutive things occurring is a product of the probability of each occurring independently. If this is so, how was the probability of the rr occurring determined? (The probability of the r from one parent times the probability of the r from the other.) How would you determine the probability of the r by itself? (Take the square root of the probability of rr).

<!- - THIS IS THE SECOND RECTANGLE, ONCE AGAIN WITH 4 45-DEGREE BISECTORS WITHIN THE RECTANGLE - - >

Calculate the square root of the frequency of rr occurring and place the answer in two places: next to each r.
Ask students what the probability of R or r occurring from one parent. It must be 1, since no other outcome exists. Either R or r will always come up. Subtract the calculated square root from 1 to get the probability of the R occurring, and place that answer next to each R.

<! - - THIS IS THE THIRD RECTANGLE WITH THE SAME IDEA AS THE FIRST TWO. - - >

Point out that the probability of R plus the probability of r =1.
Calculate the probabilities of RR and Rr occurring by multiplying those individual probabilities together, then place the answers inside the Punnett square next to the corresponding genotype. Add all of those numbers inside the Punnett square together. They should total l. (.49 + .21 + .21 + .09 = 1)

<! - - SAME KIND OF RECTANGLE AGAIN.-->

If we let P = probability of R and q = probability of r, then what was just done is the same as:
(P + q)2=(P + q) (P + q)=
P2 + Pq + qP + q2 =1 or
P2 + 2Pq + q2 =1
which is the binomial theorem.
Poll the class to see how many people have the genotypes other than rr. Fill this in the table. (How may have RR and Rr popsicle sticks?) Complete the table by determining the ratios of these genotypes and enter these 4 figures into the second Punnet square. See how they compare with the calculated figures. These figures from the raw data should be the same as those calculated in the first Punnet square.
Students have learned to calculate gene frequencies using the following information:
1) Total population figure
2) Number of members with rr trait
3) Laws of probability
4) Binomial theorem
Review with students they have just learned what is known as the Hardy-Weinberg Law. The other terms to know are gene pool (set of all genes available for a given population) and gamete (sex cell that has genes).

1. a) Simulate offspring production by combining male and female alleles. Have each person face their partner and place their popsicle sticks behind their backs. Simulate the combination of an egg and sperm, by directing each person to randomly pick one popsicle stick and present it. Look at each combination and tally up the results (RR, Rr, rr). Do this 10 times, which would be the same as having 10 children.
b) Poll the class as a whole and ask them to report the genotype frequency to be tallied on a class chart.
c) Determine a ratio the same as before. This time, use class results instead of the parents' results.
d) After all data has been collected, compare the parents' genetic ratios with those of their offspring using the same Punnet square procedure. The figures for the offspring should be very close to those from the parents for each genotype. Tell them they are demonstrating the Hardy-Weinberg Law. It states that gene frequencies will not change unless some external force selects for or against a trait.
e) Discuss why the parent ratio is not exactly the same as the offspring ratio. With small samples, variations will exist and with smaller samples, variations are likely to be greater due to one anomaly making up a greater percentage of the population.
2. Have some people "mate with others" less, and others mate more. This will simulate non-random selection, changing the gene frequencies. Have students tally the genotypes of their offspring. Follow the same procedure as in 1.
The new Punnet square will provide different figures from the original parent ratios.
Ask students what happened? (Since the ratios have changed, the Hardy-Weinberg Law says that some external force has affected that population.)
3. Try this activity using actual raw data from a class instead of popsicle sticks.
4. Investigate genetic drift (natural selection, directional selection, stabilizing selection, disruptive selection, speciation, etc.)
5. Investigate other monogenic traits, such as widows peak and attached ear lobe and their frequencies in the population.

· What's New, a site on the World Wide Web, is a source of current information in the various fields of science. Features on this site include:
· Factoids
· NewsMaker Interviews
· Media Watch
1. List 3 ways in which biotechnology or its applications are currently making news. Describe how each of these will provide for career opportunities in the future. What type of educational background will be necessary to prepare for these jobs?
2. Survey the Media Watch (a month's listing) and determine which of the programs might relate to the field of biotechnology. Have the students keep track of the program listings over the course of a few months.
3. Factoids is sort of a "Trivial Pursuit of Science". Investigate this feature for some "fun" ways to integrate math & science. "Did you know that 17% of the people in the midwest consume diet ice cream?"
What's New
http://www.gene.com/ae/WN/
· America Online features several programs which can assist teachers in the preparation of lessons in biotechnology. One of the best is Access Excellence. Access Excellence is an interactive computer network which allows teachers across the country to communicate with other teachers and research scientists. Access Excellence provides up-to-the-minute reports on seminars and tours, as well as an expanding collection of classroom and laboratory activities.
Access Excellence
http://www.gene.com/ae
America Online (keyword: excellence)