exponential growth
https://teacherswithguts.org/
enPapercatchers
https://teacherswithguts.org/resources/papercatchers
<span>Papercatchers </span>
<p>RUNNING THE ACTIVITY—PART 1</p>
<p>In the first stage of this Activity, ask everyone to crumple up a piece of (scrap) paper. Pick one person to come up to the front of the room and be the initial member of the population. Members of this population should follow these rules:</p>
<ul><li>If you are standing at the front of the room, then throw your paper high in the air (at least several feet above your head) when the facilitator gives the “next generation” command.</li>
<li>If you catch your paper, then you survive and may “reproduce” by calling up another member of the audience to join the population.</li>
<li>If you don’t catch your paper, then you “die” and must sit down. Lead participants through several “generations.”</li>
</ul><p>If the population crashes or becomes extinct (because all of the population members drop their pieces of paper), begin again, noting that sometimes populations will crash by chance or become extinct when their numbers are small. Record the size of the population over time. What is the maximum population?</p>
<p>Once the members of the audience are all standing at the front of the room, take a look at the graph of the population over time. Ask the participants to reflect on some of the following issues:</p>
<ul><li>What type of population growth is modeled, and why does it occur?</li>
<li>The growth curve changes shape dramatically when all potential members of the group are already in the population. What does this limitation add to (or take away from) the model?</li>
<li>How realistic is the implementation of “death” in this model?</li>
<li>How does chance play a role in this model? How does the inclusion of randomness make this model different from the penny growth model?</li>
</ul><p>RUNNING THE ACTIVITY—PART 2</p>
<p>Now place a newspaper-sized piece of paper on the ground. Follow the same rules you did earlier, adding one new rule:</p>
<p>Each member of the population must have part of one foot on the paper at all times. Removing your foot from the paper (even for an instant) results in “death” (sitting back down).</p>
<p>Proceed, as in the last experiment, by calling up one initial population member and asking that person to throw the paper in the air. Continue propagating new generations, noting the number of people in the population over time. What happens to this population? Is everyone able to stand on the paper? When the population size stabilizes, discuss the implications of this new feature. </p>
<ul><li>Hold up the paper and show participants what they did to the “environment.”</li>
<li>What kind of growth does this model demonstrate?</li>
<li>What kinds of death processes are modeled by implementing “dying” when you take your foot off the paper? How does this relate to limited resources in nature?</li>
<li>Calculate a carrying capacity (maximum number of individuals supported) for this environment.</li>
<li>Is there a newspaper size that leads to oscillating populations (many people die when the paper is crowded, which frees up a lot of space for growth, which fills up the paper, and so on)?</li>
</ul><p>Compare the birth and death processes in this model and the penny growth model. What are the advantages and disadvantages of the two models? When would it be more appropriate to use one or the other?</p>
<p>FACTS FOR FACILITATORS</p>
<p>It is likely that people were cooperating in order to avoid collisions when they were standing close together on the newspaper. How do social factors like this one influence population dynamics? Depending on the paper-catching skills of the people in the group, you might need to adjust how high the population members throw their papers. You might also use throwing height as a variable in the model. For a description of a similar activity and additional explorations, see Byington (1997). Modeling a growing population is one avenue for exploration. You and your students can consider the many different ways to model a single phenomenon and weigh the advantages and disadvantages of each approach.</p>
<span><span lang="" about="https://teacherswithguts.org/members/turtle" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">turtle</span></span>
<span>Fri, 07/27/2018 - 15:39</span>
<img src="https://teacherswithguts.org/sites/default/files/styles/resources/public/2016-09/Screen%20Shot%202016-09-01%20at%201.38.32%20PM.png?h=459c57f2" width="270" height="300" alt="Participants in Paper Catchers from Adventures in Modeling" typeof="foaf:Image" class="img-responsive" /><h4 class="label field-label">Original author</h4>
<div class="field field-field-original-author field-single">Adventures in Modeling (By Colella, Klopfer, and Resnick)</div>
<h4 class="label field-label">Resource type</h4>
<ul class="field field-field-content-type field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/resources/search/content_type/25-0" hreflang="en">Activity</a></li>
</ul><h4 class="label field-label">Content area</h4>
<ul class="field field-field-curricular-area field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/resources/search/curricular_area/16-0" hreflang="en">Science- Life Sciences</a></li>
<li><a href="https://teacherswithguts.org/resources/search/curricular_area/15-0" hreflang="en">Mathematics</a></li>
</ul><h4 class="label field-label">Use type</h4>
<ul class="field field-field-education-level field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/resources/search/education_level/21-0" hreflang="en">Middle School</a></li>
<li><a href="https://teacherswithguts.org/resources/search/education_level/379" hreflang="en">Just-in-time resources</a></li>
</ul><h4 class="label field-label">Tools</h4>
<ul class="field field-field-tools field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/taxonomy_redirect/term/201" hreflang="en">Hands-on / Offline</a></li>
</ul><h4 class="label field-label">Tags</h4>
<ul class="field field-field-tag field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/taxonomy_redirect/term/105" hreflang="en">carrying capacity</a></li>
<li><a href="https://teacherswithguts.org/taxonomy_redirect/term/111" hreflang="en">limited resources</a></li>
<li><a href="https://teacherswithguts.org/taxonomy_redirect/term/112" hreflang="en">exponential growth</a></li>
</ul><h4 class="label field-label">Related links</h4>
<ul class="field field-field-links field-multiple list-unstyled"><li><a href="http://projectguts.org/files/Fileshare_Area493/2008-9_Curriculum/unit_2-_Shared_Resources/Papercatchers.pdf">Instructions for Papercatchers</a></li>
</ul>Fri, 27 Jul 2018 19:39:07 +0000turtle212 at https://teacherswithguts.orgPenny Growth
https://teacherswithguts.org/resources/penny-growth
<span>Penny Growth</span>
<p>Did you ever count bacterial colonies growing in a petri dish? Or track the accumulation of money in your savings account? Or watch as mold spread over an old piece of bread? If so, you have observed population growth in action. As populations grow, the change in the number of individuals in those populations over time can be classified in different ways. Exponential growth and logistic growth are terms applied to specific patterns of population expansion. Both exponential and logistic growth are central to many processes and are the basis for many models. Perhaps because of their prevalence, or perhaps because of their “observable” quality, these types of growth are commonly described in biology, mathematics, and even economics courses. However, most textbooks present them in terms of somewhat sophisticated mathematical equations. Unfortunately, those equations do not provide a handy structure for thinking about growing systems from the “bottom up.”</p>
<p>In this Activity, you will get a chance to explore exponential and logistic growth models, relying on simple rules and observable aggregate behavior rather than on specialized mathematical techniques. This Activity approaches system modeling from the perspective of one individual or one creature—much like StarLogo does. The penny model presented in this Activity is an idea model. You might think about the advantages and limitations that this type of model provides for understanding exponential and logistic growth. You might also consider what modifications would be required to create a penny-based minimal model for a system. </p>
<p>ACTIVITY MODELING CONCEPTS </p>
<ul><li>Discover how simple rules can define behavior that is typically described by complex mathematical equations. </li>
<li>Learn about the concepts of exponential and logistic growth. </li>
<li>Analyze the elements of a model that make it appropriate or inappropriate for thinking about specific systems.</li>
</ul><p>MATERIALS</p>
<ul><li>String</li>
<li>Approximately 250 pennies per group </li>
<li>Note paper and graph paper </li>
</ul><p>RUNNING THE ACTIVITY</p>
<p>In this Activity, each group will “grow” a population of pennies. Start by dividing the participants into three or more small groups. Give one group a small piece of string (about a foot long), another group a medium-sized piece of string (about two feet long), and the final group a long piece of string (a few feet long). If you have more than three groups, you can give out varying string lengths or give two groups the same string length. Have each group form their string into a circle on a large flat surface. Give each group three pennies and have them place the pennies “randomly” inside the string circle. Then distribute the rest of the pennies. Explain the following rules:</p>
<ol><li>The pennies must all lie flat on the surface of the table (or floor).</li>
<li>Pennies can only “live” inside the string circle.</li>
<li>Every time step, reproduction is simulated by placing the maximum number of “offspring” pennies around the perimeter of every “parent” penny.</li>
<li>Pennies reproduce synchronously. This process means that a generation happens in a single time step, during which every “parent” penny reproduces to the full extent that it can.</li>
</ol><p>Ask each group to begin penny reproduction and to note the number of pennies at the end of each generation. Groups can record either the total number of pennies or the number of new offspring pennies. Later you might discuss the relationship between these two measures. Let each group proceed at its own pace. Groups with shorter strings usually finish before the other groups. As groups finish, ask them to graph the numbers of pennies over time. After some time, stop the remaining groups and have them construct the same graphs.</p>
<p>Compare all of the graphs. Most likely the short-string graphs will illustrate logistic (S-shaped) growth, while the long-string graphs will show exponential (continually accelerating) growth. Ask the groups to discuss the following issues:</p>
<ul><li>What effect does the length of string have on the population growth? </li>
<li>What effect does the initial placement of pennies have? Very few groups will actually place their pennies in truly random initial positions, so you may want to compare groups’ methods for placing their pennies. </li>
<li>Are there some animals whose growth might resemble the penny growth? If so, do all of the graphs represent growth patterns of the same animals? </li>
<li>Would these models be appropriate ways to describe a gazelle population on the open plains of Africa? How about fish in a pond? Why or why not? </li>
<li>What kinds of modifications might enable this model to better represent other systems?</li>
</ul><p>Consider running the model again with one or more of the following modifications: </p>
<ul><li>Have the pennies move around after each generation is done reproducing. </li>
<li>Invent a mechanism for simulating sexual reproduction (e.g., require two pennies to be less than an inch apart for them to give birth to a new penny). </li>
<li>Add ways for the pennies to die.</li>
</ul><p>This alteration might be a random method that selects a few pennies to die each generation or it might be a predictable mechanism, such as one that causes pennies in crowded areas to die. Do those modifications change the answers to some of the questions above?</p>
<p>FACTS FOR FACILITATORS</p>
<p>Encourage participants to think of ways that this model might be more or less appropriate for certain systems, but also make sure that they evaluate the model critically and understand its limitations. This Activity is an idea model meant to convey the concepts of exponential and logistic growth. It is not designed to be an accurate representation of any given system. It does, however, demonstrate one way that you can convey the ideas of exponential and logistic growth using simple rules for individuals, instead of using mathematical equations to describe the system level behavior. </p>
<p>This Activity provides a good entry into creating simple StarLogo models. Try programming a model of exponential growth. How is the StarLogo TNG or StarLogo Nova model different from the penny model? A logistic growth model is a bit more complicated to create, but a basic logistic growth model can be built around the concepts of turtles-here and hatch. (Hint: You can limit the number of turtles on any given patch to one and hatch (or create) newborn turtles only if there are adjacent empty patches for the new offspring to inhabit.)</p>
<p>You may be surprised to learn that many researchers have used coins to augment thought experiments. A good place to read more about these kinds of experiments is Schelling (1978).</p>
<span><span lang="" about="https://teacherswithguts.org/members/turtle" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">turtle</span></span>
<span>Wed, 08/31/2016 - 13:40</span>
<img src="https://teacherswithguts.org/sites/default/files/styles/resources/public/2016-08/Screen%20Shot%202016-08-31%20at%2011.40.06%20AM.png?h=db4311ad" width="270" height="300" alt="Image of penny growth from Adventures in Modeling" typeof="foaf:Image" class="img-responsive" /><h4 class="label field-label">Original author</h4>
<div class="field field-field-original-author field-single">Project GUTS (excerpted from Adventures in Modeling (By Colella, Klopfer, and Resnick)</div>
<h4 class="label field-label">Resource type</h4>
<ul class="field field-field-content-type field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/resources/search/content_type/25-0" hreflang="en">Activity</a></li>
</ul><h4 class="label field-label">Content area</h4>
<ul class="field field-field-curricular-area field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/resources/search/curricular_area/13-0" hreflang="en">Computer Science</a></li>
<li><a href="https://teacherswithguts.org/resources/search/curricular_area/15-0" hreflang="en">Mathematics</a></li>
</ul><h4 class="label field-label">Use type</h4>
<ul class="field field-field-education-level field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/resources/search/education_level/21-0" hreflang="en">Middle School</a></li>
<li><a href="https://teacherswithguts.org/resources/search/education_level/379" hreflang="en">Just-in-time resources</a></li>
</ul><h4 class="label field-label">Tools</h4>
<ul class="field field-field-tools field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/taxonomy_redirect/term/201" hreflang="en">Hands-on / Offline</a></li>
</ul><h4 class="label field-label">Tags</h4>
<ul class="field field-field-tag field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/taxonomy_redirect/term/112" hreflang="en">exponential growth</a></li>
<li><a href="https://teacherswithguts.org/taxonomy_redirect/term/117" hreflang="en">logistic growth</a></li>
</ul><h4 class="label field-label">Related links</h4>
<ul class="field field-field-links field-multiple list-unstyled"><li><a href="http://projectguts.org/files/PennyGrowth-AIM.pdf">Instructions for Penny Growth as off-line activity</a></li>
<li><a href="http://projectguts.org/files/Penny%20Growth%20data%20sheet.pdf">Data sheet for use with Penny Growth</a></li>
</ul><h4 class="label field-label">Project name</h4>
<ul class="field field-field-project-name field-multiple list-unstyled"><li><a href="https://teacherswithguts.org/taxonomy_redirect/term/380" hreflang="en">Project GUTS afterschool</a></li>
</ul>Wed, 31 Aug 2016 17:40:36 +0000turtle198 at https://teacherswithguts.org