# Evolutionary Medicine

## CONTENTS OF CURRICULUM UNIT 09.05.09

## Math Morphing Proximate and Evolutionary Mechanisms

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## Lesson Plans

1. Lesson One -collect data, observations and draw conclusions (defined below).

2. Lesson Two - generate fractals, observations and draw conclusions (defined below)..

3. Lesson Three - extrapolate geometric progressions, observations and draw conclusions (defined below)..

4. Lesson Four - mutations, observations and draw conclusions (student defined).

### Lesson Number 1 - Functions as Mathematical Models of Proximate and Evolutionary Medicine Phenomena

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Objective(s) - students will be able
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1. To discuss and provide examples of how mathematics is used to describe nature, and Proximate and Evolutionary Medicine Phenomena in particular 2. To define and provide an example of a function and a mathematical model 3. To provide examples of dimensions and the units used to measure them 4. To apply the concepts of function, independent and dependent variable to describe relationships between natural phenomena using multiple representations of functions. 5. To classify the relationship between pairs of variables associated with Proximate and Evolutionary Medicine Phenomena as linear, quadratic, or exponential 6. To pose questions about Proximate and Evolutionary Medicine Phenomena that may be answered by using a mathematical model 7. Describe the difference between Proximate and Evolutionary Medicine Phenomena 8. Define proximate and evolutionary medicine, and fractal

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Prerequisite skills
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graphing two-variable data, evaluating expressions, translating verbal phrases and sentences to algebraic expressions and equations, solving equations, creating and interpreting tables and graphs on coordinate plane

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Materials
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Ruler or tape measure, thermometer, clock, scale

Worksheet or text with tables of different types of function, without the equations

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Relevant Vocabulary, Terms and Symbols
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domain, independent variable, dependent variable, f(x) = y, dimension, fractal, function, mapping diagram, proximate and evolutionary medicine, range, unit, vertical line test

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Strategy
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Students should be prompted to discuss different types of phenomena that are measured. They should also discuss what types of instruments are used for the measurements. This discussion should result in defining dimensions. As a class, we list different units that are used to measure the dimensions of mass, transfer, magnitude, time and temperature. Review the definition of function. Use the student as function input metaphor - can't go to two places at once. Encourage students to come up with measureable quantities and measurement units. Prepare a measurement activity that involves a simple linear function. Prepare tables of different types of functions related to Proximate and Evolutionary Medicine Phenomena for review of function types. Prompt students to classify each table as a type of function. Describe natural pandemics and epidemics, the different types of Proximate and Evolutionary Medicine Phenomena, and their characteristics. Use the case studies to provide an example of each type of disaster to be studied. An overhead slide show would nicely show the results of the disasters. Ask questions about the characteristics of Proximate and Evolutionary Medicine Phenomena, i.e. "are these two related? How is that so?" Provide encouragement and feedback.

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Student tasks
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- Form small groups
- List examples of quantities that we measure in the environment
- Classify each type of quantity by its dimension, and give an example of units
- Discuss how some pairs of quantities may be related
- Find two quantities that have a functional relationship
- Describe the functional relationship in words using "is a function of" and in symbols using a = f(b)
- Present one of your functions to the class, with both verbal and symbolic representations
- Engage in measurement activity involving magnitude, time, and force, using ruler, clock, and scale Any distance, rate, and time problem would suffice.
- Classify a function represented with a table as linear, quadratic or exponential.
- Answer elementary questions about the Proximate and Evolutionary Medicine Phenomena discussed, define terms.

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Questions
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For each of the following pairs of phenomena, determine the dimension and possible units of each variable, and determine if there may be a functional relationship between the two. Then identify the independent and dependent variables, and write a sentence that describes the relationship.

Time of day & temperature

Month & temperature

Location & weather

Number of deaths & proximate and evolutionary medicine magnitude

Give examples of dimensions and units

What is the difference between an epidemic and a pandemic?

What questions can be asked about these events?

What type of mathematics would create answers to these questions?

### Lesson Number 2 - Modeling Proximate and Evolutionary Medicine Height with Quadratic and Root Functions

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Objective(s) - students will be able
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1. To use mathematical models to describe observed relationships between characteristics of Proximate and Evolutionary Medicine Phenomena when values of some variables are known 2. To derive mathematical models from existing equations and relations (solve multivariate equations for one of the variables) 3. To provide examples of the application of a quadratic or square root function to Proximate and Evolutionary Medicine Phenomena 4. To represent functions as equations, tables, graphs, and verbal descriptions 5. To describe the domain and range of functions used to model Proximate and Evolutionary Medicine Phenomena 6. To draw and label a simple pictorial representation of a fractal 7. To calculate fractal magnitudes, periods, depths, and frequencies of fractals 8. To describe the relationship between fractal magnitude and fractal frequency in at least two different ways

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Prerequisite skills
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Vertex form of quadratic function, translations of parabolas, solving equations using square roots

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Materials
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Provide Graphing calculator, graph paper, worksheets with student tasks, questions, and a place to write answers.

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Relevant Vocabulary, Terms and Symbols
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Define axis of symmetry, fractal, maximum, minimum, parabola, vertex, and x-intercept.

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Strategy
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A fractal can be modeled by having two students hold a piece of rope in front of the class. One student propagates a fractal, and the other students can be prompted to estimate the height of the fractal. Another student propagates a larger fractal, and again the students can estimate the height of the second fractal. A discussion explaining the definition of amplitude follows. The amplitude here can be related to water depth instead of fractal height. Provide one form each of the equations for proximate and evolutionary medicine and fractal. Prompt students to derive the other form of the relation. Provide students with specific calculation questions that can be solved using the presented equations. Provide encouragement and feedback

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Student tasks
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Describe a fractal, its characteristics, and its properties.

Give examples of fractals.

Give examples of fractals representing proximate and evolutionary medicine.

Take notes on proximate and evolutionary medicine equations.

Attempt to solve equations for the independent variables.

Answer questions that arise from the anatomy of fractals.

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Questions
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- Define a fractal, proximate and evolutionary medicine.
- Find the magnitude of a fractal associated with proximate and evolutionary medicine.
- Find the period or frequency of a fractal associated with proximate and evolutionary medicine with given frequency and fractal magnitude
- Find the frequency of a proximate and evolutionary medicine with given fractal
- ____ magnitude and period
- Find the height of a proximate and evolutionary medicine with given frequency
- Find the frequency of a proximate and evolutionary medicine with given water depth
- Find the energy of a proximate and evolutionary medicine with given water depth and fractal magnitude
- Find the fractal magnitude of a proximate and evolutionary medicine with given water depth and energy
- Find the height of a proximate and evolutionary medicine with given fractal magnitude and energy
- What is a shallow water fractal?
- What are some of the difference between fractals?
- What scale is used to measure magnitude?
- Find the magnitude of a fractal with given wind speed
- Find the wind speed of a fractal with a given magnitude
- Compare two fractals with two given sets fractal magnitude and height
- What is the relationship between these magnitudes compared to their heights?
- Which natural phenomena have quadratic relationships?
- Write a sentence describing the relationship between proximate and evolutionary
- ____ medicine height, fractal magnitude and frequency
- Write a sentence describing the relationship between fractal frequency and magnitude

### Lesson Number 3 - Modeling Proximate and Evolutionary Medicine Magnitude and Population Growth using Exponential and Logarithmic Functions

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Objective(s) - students will be able
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1. To physically model a proximate and evolutionary medicine and describe how friction and surface area relate to proximate and evolutionary medicine magnitude 2. To describe the relationship between proximate and evolutionary medicine magnitude and energy in at least two different ways 3. To define moment, magnitude, and moment magnitude 4. To describe the relationship between moment and moment magnitude in at least two different ways 5. To describe the relationship between proximate and evolutionary medicine magnitude and frequency of occurrence in at least two different ways 6. To apply exponential and logarithmic functions to the relationship between proximate and evolutionary medicine magnitude and energy released 7. To apply exponential and logarithmic functions to the relationship between time and population size

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Prerequisite skills
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Order of operations, rules of exponents, graphing functions manually and with graphing calculator, solving equations by taking logarithms and exponentiating

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Materials
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Sponges of different sizes, all with a flat sides, water

Graphing calculator, graph paper, worksheets including discussion prompts and calculation exercises

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Relevant Vocabulary, Terms and Symbols
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Exponential Equations, Exponents, Logarithmic Equations, and Logarithms

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Strategy
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The cause of a proximate and evolutionary medicine can be simply modeled using a sponge, and a piece of string. If only one sponge is available, this exercise can be done as a classroom demonstration of transference. Otherwise, students may be divided into groups, each of which has one type of sponge, and a piece of string. Each group places a saturated sponge on a table a given distance from another sponge. Students then incrementally exert pressure on the saturated sponges, which are on the table. As a class, we estimate the surface areas of the sponges that are in contact with each other on the table. As a class we discuss and record how much pressure it takes for each sponge to transfer water to the neighboring sponge. The pressure resembles the amount of transference required to yield proximate and evolutionary alterations. The area of the sponge represents the area of transfer during a proximate and evolutionary medicine. Students should conclude that the larger sponge yields larger proximate evolution. Students should be prompted to generalize that the larger the area of the sponge, the larger the potential for proximate and evolutionary medicine. This discussion segues into the discussion of proximate and evolutionary medicine magnitude. The effects of a proximate and evolutionary medicine on a population may be shown by adjusting the distances between the sponges on the table. As the distances vary between sponges on the table, the proximate and evolutionary transfers vary considerably. Begin discussion of population by asking students what the local, state, national, and global populations are. These would be good questions for students to research independently. Present graphical representations of population fluctuations and growth. Provide notes on proximate and evolutionary medicines and population growth. Provide one version of each equation that will be used in the exercises. Encourage and provide feedback.

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Student tasks
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- Participate in sponge and water exercise
- Record data
- Participate in concluding discussion
- Take notes on proximate and evolutionary medicines and population growth,
- ____ particularly equations
- Attempt to solve each equation for the independent variable
- Practice calculations on worksheet
- Respond to and discuss open-ended questions

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Questions
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- Define proximate and evolutionary medicine
- Name two scales used to measure proximate and evolutionary medicine magnitude
- Find the event of a proximate and evolutionary medicine with a given magnitude
- Find the event magnitude of proximate and evolutionary medicine with a given
- Find the magnitude of proximate and evolutionary medicine that released a given amount of life-altering characteristics
- Find the amount of life-altering characteristics released in proximate and evolutionary medicine
- Find the recurrence interval of proximate and evolutionary medicine with a given
- ____ minimum
- What is the current population of the United States?
- What is the current population of the earth?
- How can the growth rate of a population be found using the birth rate and death rate?
- At the current growth rate, in what year will the world population be 10 billion?
- At the current growth rate, what will the world population be in 2050?
- What effects does the human population have on other species of animals and plant?
- And, if any, are the effects on natural resources?
- Which natural phenomena have exponential relationships?