# Depicting and Analyzing Data: Enriching Science and Math Curricula through Graphical Displays and Mapping

## CONTENTS OF CURRICULUM UNIT 08.06.02

## Displaying Populations: Jellybeans, Paper and People

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## Population Dynamics Unit

Population dynamics are all the factors that affect the size, shape and growth of a population over time. Population change is due to several inherent factors; resource availability, technological change, climate change and genetic variability. The dynamics of populations are the ways in which the numbers of individuals in a population grow and shrink. Populations have two inputs and two outputs determining their dimension. Simply put, inputs are either births or immigration. The two outputs are the opposite; emigration, where organisms move out of the population for a variety of reasons, the second is death. When the inputs are equal to the outputs the population is in balance, this is when a population graph’s line would have a slope of zero, and the number of organisms reaches the carrying capacity for the area the population inhabits.

Population is often described in terms of density, the number of organisms in a specified space. In terms of organisms, other than humans, a population’s carrying capacity is largely due to the resources needed for survival in a habitat. Humans, however, have developed methods to ensure the carrying capacity of their locale continues to increase, as does their life expectancy. The human population is an exception, rather than the rule in nature. Humans have endeavored to control world population. Some authors have gone so far as to state that humans have endeavored to cleanse their population in favor of what a few deem as necessary population control (Connelly, M.2008). In any case, humans adapt their environment to suit their own needs in a way no other organism in the history of the earth.

In displaying population growth, graphs are easy to use because they are made of lines, dots and simple symbols that are uncomplicated and quick for students to draw. Graphs can show relationships between variables or show the rate of change of a given variable. If a point needs to be expresssed, a change over time depicted, or comparison shown, graphs are useful tools. Edward Tufte, a Yale University professor, suggests in his book on graphs and graphing that misleading or unclear graphics, charts, and tables can sometimes have disastrous effects (Tufte, E.2001). To help students understand change in populations of the great diversity on earth the
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relationship among changes as they occur, students must learn how graphing can enhance or alter the message. Through lecture, display and discussion in the seminar “Depicting and Analyzing Data; Enriching Science and Math Curricula through Graphical Displays and Mapping,” Dr. William B. Stewart helped participants grasp a thorough understanding of the importance of each facet of graphical display. Every component of a graphic illustration may be analyzed to construct realistic population representation convey the desired message. Only people have the ability to set in place global change. By examining the relationships between organisms through data collection and graphic display, we can reach an understanding of the relationships between lifeforms, ecosystems and people for the harmony of all.

As the unit begins, the I Didn’t Know That (
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IDKT!)
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sheets are distributed to students. This quick half-sheet paper is a strategy to monitor student understanding, and will serve three purposes. In the five minutes before each lecture or activity and the five minutes at the end, student will have the opportunity to focus their attention to making connections with the subject at hand.
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IDKT!
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is based on the K-W-L teaching strategy. In K-W-L; K equals what the student knows, W equals what the students want to know, and L equals what the students learns. Using
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IDKT!
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the teacher can quickly assess what the students know early in a lesson and address any misconceptions as the activity begins. At the end of the lesson students write what they learned and ask questions. Teachers can review the answers and plan for the next lesson based on student input. The
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IDKT!
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sheets can also be used as a type of data collection throughout the unit. Teachers should set up specific objectives for student learning to collect good data. The quality of an answer should be discussed with students to obtain good data for a final assessment for the unit.
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IDKT!
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worksheets show student progress as they learn the vocabulary necessary to understand populations. The important vocabulary for this unit is demography, density, dispersion, immigration, emigration, growth rate, growth curve, simulation, and estimation.

The study of demography is important to this unit. Demography is the statistical study of populations to examine how a population may change. There are three key characteristics of a population’s demography -- size, density and dispersion. The size of a population is simply the number of organisms, counted or estimated. Density is the number of organisms in a specified area and dispersion is how the organisms are grouped with in the habitat. Sometimes organisms are concentrated in a specific space, other times they are evenly distributed throughout the area. A species can also be randomly dispersed throughout a habitat. Immigration refers to organisms moving into an area and emigration refers to organisms leaving an area.

Students can determine the growth rate of the population to understand how populations change. The growth rate is the death rate (number of people who expired) subtracted from the birth rate (number of new births) and helps demographers predict population growth. Death rates are affected by nutrition, infant mortality, public health care and environmental quality. Birth rates are affected by the fertility rate (number of children per woman), which is determined by economic, social and biological factors. Economic factors affecting the fertility rate are child labor, cost of rearing children and pension plans. Social factors include migration, economic development of a region, and education for women, employment for women, marriage age, religious beliefs and urbanization. Biological factors included infant mortality rates, birth control methods and family planning. The growth rate (r) is considered a major factor in predicting population growth.

Growth curves whether J-shaped (showing exponential growth) or S-shaped (showing logistic growth) help students see how populations change. Population growth is estimated in terms of a logistic growth or exponential growth. A typical equation for determining the logistic growth of a population may look like:

dN/dt = r
_{
max
}
N ((K - N)/ (K)) or dN/dt = r
_{
max
}
N (1 - (N/K)).

Where dN/dt is the growth rate of the population, r
_{
m a x
}
is the maximum growth rate for a specific species in a certain environment, K is the carrying capacity, and N is the total population. In a logistic growth model, the growth rate decreases as the population increases, an inversely proportional relationship. Mathematical equations for predicting population size are usually based on book published in 1798 by Thomas Malthus utilizing the equation pt
_{
+ 1
}
= r X p
_{
t
}
. In this equation p
_{
t + 1
}
(population size at the next time-period) equals a Malthusian factor that determines growth rate (r) multiplied by p
_{
t
}
(population at the time (
_{
t
}
)). This equation, known as the difference equation of exponential growth, allows you to find the population size at separate time intervals. (Bulaevsky, J.1997). In exponential growth, the larger the quantity, the faster the population grows in absolute terms. Exponential growth demonstrates a directly proportional relationship. In exponential growth, although the increase in the population size is seems slow over a short period, the growth becomes impressibly larger over a longer period as the initial quantity continues to double.

Students investigate in the last part of this lesson: how the density and dispersion of a species can limit the carrying capacity of a habitat by examining influencing factors and habits of populations. Demographers have defined two types of growth strategies in population density. Growth rate or r-strategists are organisms that have relatively short life spans, reproduce quickly producing many young and provide little parental care. Derived from the equation where growth rate equals r
_{
max
}
N (1 - (N/K)), r-strategists, are often opportunistic species. Most weeds are examples of r-strategists. K- (or carrying capacity) strategists are organisms with long life spans, they tend to reproduce slowly, have few young and strong parental care are considered the stable species. Large mammals are good examples of K-strategists. r- and K-strategists play distinct roles in ecological succession of an ecosystem. Most organisms display both r- and K- strategy characteristics. For example, trees disperse many un-nurtured seeds of the r-strategist; however, they also display longevity and stability of K-strategist.

Computer simulations attempt to model possible scenarios based on mathematical computations. Simulations provide mathematical models for observing predicted results. Simulations also help provide estimations, calculated approximate values for statistical models also based on mathematical equations. In the next lesson, students watch how a population changes and grows using a computer simulation at Shodor.org. As the size of the populations of rabbits and wolves changes students learn about probability, randomness and chaos in a simple ecosystem. Graphic displays show students how the data can be represented, and a series of exploration questions guide students to conclusions. Students work in pairs for about thirty minutes to determine why the rabbit population grows so fast and can predict probabilities for the future. Afterwards, students look at a population in a fixed habitat that explores factors that influence populations and then practice two ways that demographers use to estimate populations. After looking at the diverse world of organisms, student will then begin to investigate their own population and the influences on human population that exist today that will determine the statistics of the future.

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