**Title:** Variations on a Human Face Lab

**Author:** Andrew Renaud

**Grade Level:** 9

**Subject/Content:** Integrated/Applied Mathematics

**Summary of Lesson:** Through the application of math concepts, the purpose of the investigation is to examine the application of Theoretical and Experimental Probability, reflections, and ratios and rates that exists in one's everyday environment, in order to develop an understanding of how these concepts apply to the human genetics.

**Focus Question:** How are Theoretical and Experimental Probability, reflections, and ratios and rates related to human genetics?

**Databases(s):** *Student Resource Center*, *WEB FEET K-12*

**Procedures:**

- Materials needed: Variations on a Human Face handout, loose leaf paper, graph paper (five divisions to the inch), coins, colored pencils or crayons, glue, pencils
- After our general review of the prerequisites: Theoretical and Experimental Probability, reflections, ratios and rates, genetic traits, and Punnett squares students are randomly placed into pairs
- Students will need to review and discuss the following on the Gale Web sites:
*Student Resource Center*: Keyword search "Probability.*Math & Mathematics. The History of Math Discoveries Around the World*, "Probability Theory*U·X·L Encyclopedia of Science*and, under the Multimedia tab,*Probability Theory*and*Snake eyes*. "Ratio*Gale Encyclopedia of Science,*Ratio, proportion, percent*Math & Mathematicians: The History of Math Discoveries Around the World,*and*Gale Encyclopedia of Science.*Under the Magazines and Journals tab: (Book Review)*Science.*Symmetry "*Gale Encyclopedia of Science,*under the Magazine and Journal Article tab, Symmetry and the Beautiful Universe (Brief Article)(Book Review)*Science News.**WEB FEET*K-12: Keyword search "Probability. Web sites such as 2. http://www.cut-the-knot.org/curriculum/ should be reviewed. Each person in the pairing is given the handout*Science Resource Center*: "Ratio, Rate and Proportion*Mathematics*

- After the students complete the data table and the analysis section of the handout, students complete the analysis questions
- Students create genetic offspring front-faced and side view portraits using graph paper

**Steps/Activities by student(s):**

- Anticipatory Set: Students view the video "Black or White" by Michael Jackson. The students are told that the Austrian monk Gregor Mendel determined the basic laws of genetics, the rules governing how traits are handed down from parent to offspring. A trait is passed down through genes, the basic unit of genetic information, with at least two genes, one from each parent, controlling its inheritance. Some traits are dominant, that is they prevent other traits from appearing. Some are recessive; these do not appear when a dominant gene is present
- Key terms/skills: Students will review the prerequisites including these math ideas: Theoretical and Experimental Probability, reflections, and solving linear equalities and inequalities
- Modeling: Students listen to teacher demonstrate a brief section of the lab with results posted on the overhead projector. Students take notes when necessary
- Student Test for Understanding: Students will follow this list of procedures with teacher support
- The student and his/her partner will flip a coin to determine the facial characteristics of one of their offspring. One of the students will represent the father and the other will represent the mother. In each coin toss, heads represents a dominant gene and tails represents a recessive gene. In some cases a hybrid result will look like a mixture of the two traits. This is called incomplete dominance
- First flip to determine the sex of the offspring. Only the father flips, as the father determines the child's gender. Heads will be a boy. Tails will be a girl. Record your data in the Data Table. See the following Web site for an example of how to model the remainder of the start of this lab
- When creating the genetic front and side view face representations, students should adhere to general concepts related to symmetry. Students may make a rectangular coordinate grid for this. Students may fold the graph paper both horizontally and vertically down the center for which the fold represent the dependent and control axes
- Recap – What did we learn? Explain verbally the important math skills that you utilized. What are important group skills (communication, cooperation, collaboration) used?
Students should also answer each of the following questions:

- How did you determine which chromosome (with its associated gene) each parent would contribute to the child?
- If each coin represents a homologous pair of chromosomes, how does the flip represent the behavior of chromosomes in meiosis?
- What is the difference between experimental and theoretical probability relative to this lab? How are these two concepts related (tied to this lab)?
- Using an example from this activity, explain your understanding of the words genotype and phenotype
- What is the experimental probability that both recessive genes will be contributed to the child for a specific trait? Make sure that you write out the probability formula used in this determination
- Use examples from this activity to explain the difference between complete and incomplete dominance
- Tabulate the ratios of dominant to recessive phenotypes in each of the traits. Explain the difference between a ratio and a unit ratio here relative to this question

- Home link – Students answer these basic questions: Why is a sound understanding of the difference between Experimental and Theoretical Probability important for this lab? What is the significance of ratios in this lab? What does symmetry play a major role in human genetics? How is symmetry not an exact science relative to this subject? What does Human Genetics mean in the context of math? Why is this stuff important for you and your life?
- Follow-up – Class presentation of the results. Students type a three paragraph summary in proper (Type III format – proper grammar, punctuation, spelling, 5-7 sentences per paragraph, at least ten word per sentence, no sentence starts with the same word in a paragraph.) Paragraph One explains what the lab was about. Paragraph Two describes what the student learned. Paragraph Three explains how this applied to the individual's own current or future life. The highlights of paragraph three are verbally shared with the class by each student

**Outcome:** Through the application of math concepts (listed above), the students gain at least an application level understanding of Probability, symmetry and ratios and rates that exists in one's everyday environment relative to human genetics.

**Related Activities**: H.O.T.S. (Higher Order Thinking Skills) -- Stresses Bloom's Syntheses and Evaluation levels. Students participate in The "Human Body Project. Students can complete an up-scaled version of this experiment in order to create a life sized human offspring. Large, rolled paper will be needed for this task. Science teachers could have students go into detail regarding the biological systems and the labeling of these during this process.

**Standard Date**: October, 1998

**Content Standard(s)**:

- 1.1 Understanding numbers, ways of representing numbers
- 1.2 Understanding the meaning of operations and how they relate to each other
- 1.3 Use computational tools and strategies fluently and estimate appropriately
- 2.1 Understand various types of patterns and functional relationships
- 2.2 Use symbolic forms to represent and analyze mathematical situations and structures
- 2.3 Use mathematical models and analyze change in both real and abstract contexts
- 3.3 Recognize the usefulness of transformations and symmetry in analyzing mathematical situations
- 5.1 Pose questions and collect, organize, and represent data to answer those questions
- 5.2 Interpret data using methods of exploratory data analysis
- 5.3 Develop and evaluate inferences, predictions, and arguments that are based on data
- 5.4 Understand and apply basic notions of chance and probability
- 6.1 Build new mathematical knowledge through their work with problems
- 6.2 Develop a disposition to formulate, represent, abstract, and generalize in situations within and outside mathematics
- 6.3 Apply a wide variety of strategies to solve problems and adapt the strategies to new situations
- 6.4 Monitor and reflect on their mathematical thinking in solving problems
- 7.2 Make and investigate mathematical conjectures
- 8.1 Organize and consolidate their mathematical thinking to communicate with others
- 8.2 Express mathematical ideas coherently and clearly to peers, teachers and others
- 8.3 Extend their mathematical knowledge by considering the thinking and strategies of others
- 8.4 Use the language of mathematics as a precise means of mathematical expression
- 9.1 recognize and use connections among different mathematical ideas
- 9.2 Understand how mathematical ideas build on one another to build a coherent whole
- 9.3 Recognize, use, and learn about mathematics in contexts outside mathematics
- 10.1 Create and use representations to organize, record, and communicate mathematical ideas
- 10.2 Develop a repertoire of mathematical representations that can be used purposefully, flexibly, and appropriately
- 10.3 Use representations to model and interpret physical, social and mathematical phenomena

**Learning Expectation:** Students will apply understanding of concepts related to Theoretical and Experimental Probability, reflections, and ratios and rates within the context of Genetics.

**Performance Indicators**:

At Level 1, the student is able to:

- Write (show basic knowledge for) about the meaning of these: Theoretical and Experimental Probability, reflections, and ratios and rates

At Level 2, the student is able to:

- Comprehend and Apply these concepts: Theoretical and Experimental Probability, reflections, and ratios and rates.

At Level 3, the student is able to:

- Synthesize new ideas related to and Evaluate concepts: Theoretical and Experimental Probability, reflections, and ratios and rates

**Computer Literacy and Usage Standards 9-12:**

- Students demonstrate a sound understanding of the nature and operation of technology systems
- Students are proficient in the use of technology
- Students understand the ethical, cultural, and societal issues related to technology
- Students practice responsible use of technology systems, information, and software
- Students develop positive attitudes toward technology uses that support lifelong learning, collaboration, personal pursuits, and productivity
- Students use technology tools to enhance learning, increase productivity, and promote creativity
- Students use productivity tools to collaborate in constructing technology-enhanced models, prepare publications, and produce other creative works
- Students use technology tools to process data and report results
- Students use technology resources for solving problems and making informed decisions
- Students employ technology in the development of strategies for solving problems in the real world

**ISTE NETS for Students**

- Identify capabilities and limitations of contemporary and emerging technology resources and assess the potential of these systems and services to address personal lifelong learning, and workplace needs
- Make informed choices among technology systems, resources, and services
- Select and apply technology tools for research, information analysis, problem solving, and decision making in content learning
- Collaborate with peers, experts, and others to contribute to a content-related knowledge base by using technology to compile, synthesize, produce, and disseminate information, models, and other creative works

**Information Power; Information Literacy Standards 1-4:**

- The student who is information literate accesses information efficiently and effectively
- The student who is information literate evaluates information critically and competently
- The student who is information literate uses information accurately and creatively
- The student who is an independent learner is information literate and pursues information related to personal interests
- The student who is and independent learner is information literate and strives for excellence in information seeking and knowledge generation
- The student who contributes positively to the learning community and to society is information literate and practices ethical behavior in regard to information and information technology

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