# SKILL BUILDER: Exponential and logistic growth

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## Objective

Students will be able to 1) explain the assumptions of an exponential and logistic growth model; 2) accurately predict how a population will grow based on initial characteristics of the population; 3) model the growth of houseflies and yeast with exponential or logistic growth curves.

#### Big Idea

Under ideal conditions, every population has a particular maximum growth potential in the shape of a J-curve, but under normal conditions population growth levels off. How might we model exponential and logistic growth using real-world populations?

## FRAME: Application and evaluation

How do we model the growth of real populations? Students have developed an understanding of key concepts in population ecology and learned to model population growth rates as exponential or logistic. This SKILL BUILDER supports' students ability to apply these growth models to real-world scenarios.  As such, the two activities available to students--the mystery of the houseflies and the complex reality of yeast--are meant to be personalized and self-paced. Students come into this lesson with different levels of proficiency and require differentiated tasks.

As such, the teacher's role during this lesson will be to help students identify a "best fit" learning activity and to collect formative assessment data. What can subgroups of students already do? What is an appropriate learning activity that builds students' skills at the right level of complexity?  Can students explain why exponential growth becomes logistic?  Can students use population ecology vocabulary to describe population growth? Can students identify mistakes in the graphing of exponential and logistic growth curves? Can students identify the limiting factors that influence a population's growth curve? Teachers will need to rapidly develop answers to such questions in real time so that all students are working on activities that develop proficiency.

This lesson begins with students considering the role of a growing yeast population in beer production. Students then choose a personalized learning pathway; a virtual housefly or yeast lab provide a basic framework. When students have complete this work they check their understanding by evaluating a short film. By the end of this lesson successful student will have met the following learning objectives:

1. explain the assumptions of an exponential and logistic growth model
2. accurately predict how a population will grow based on initial characteristics of the population
3. model the growth of houseflies and yeast with exponential or logistic growth curves.

RESOURCE NOTE: The attached PROTOTYPE ACTIVITY GUIDE might be modified by educators for classroom use.

## ENGAGE: Beer goggles

8 minutes

What is the purpose of this section?

Students learn about the role of exponential growth in the production of beer as a hook for the "virtual labs."  Students are familiar with yeast from the "African Lions" activity in the previous lesson; the hook revisits yeast from the perspective of the species' utility to human beings. By the end of this activity students should be to use the language of population ecology, especially the language of growth models, to explain why yeast is an excellent species to use for beer production.

What will students do?

Students engage with a visual presentation of the process of brewing beer.  The focus questions for these presentations are:

1. How do yeast populations grow? (Hint: Check your notes from the African Lions exercise/)
2. What does yeast produce that is essential for beer production?
3. What is the connection between answers to the first and second question?

HINT: Yeast are able to metabolize sugar through fermentation.  Basically, yeast "eat" sugar and create something that is important for beer production.

What will teachers do?

At the end of this presentation, I will elicit ideas about the role of yeast in the beer making process.  Yeast produces alcohol!  I will also elicit an understanding of why yeast is such a perfect fit for food production based on our work from the previous lessons.  The population rapidly reproduces and therefore creates alcohol quickly.  Finally, I will transition into the virtual lab exercises by explaining that population growth models have real-world applications and that the ability to work with growth curves is a skill that cuts across disciplines. For students that I anticipate will have trouble connecting yeast population growth with alcohol production, I may introduce this section with a review of how yeast populations growth using this or a similar graph. It is important to elicit understanding of this graph from students; otherwise it is impossible to understand if students need a more comprehensive review or a different introductory activity. Teachers should have evidence that students can be successful with this activity before attempting it.

What resources might be used during class time?

Options 1 and 2: Short animations

Option #3: Longer, but with an actual human.

## EXPLORE: Virtual Labs-Housefly or Yeast

35 minutes

What is the purpose of this section?

Students demonstrate current understanding of growth models by applying skills to two lab scenarios.  The teacher gathers formative assessment data to identify lagging students' skills and develops effective feedback strategies based on this data. By the end of this section students should be able to describe the growth of real populations as either logistic or exponential, draw conclusions from this analysis, and explain the growth factors that could create changes in the growth rate of a given population.

DIFFERENTIATION NOTE:  This section describes two virtual labs.  This experience could easily extend to another 55 minute class period depending on students needs.  The description below is for an ideal scenario--all students are proficient in these skills, all students efficiently complete tasks, all students are able conduct complex tasks, and all students are able to effectively self-assess and ask for targeted feedback.  Anticipate that this process will likely take more than one class period, especially for English Language Learners, students with IEPs, or students with an underdeveloped mastery of population growth principles from previous science courses.

What will students do?

Students will choose one of two mastery pathways: 1) the mystery of the missing housefly or 2) the complex reality of yeast.  See the PROTOTYPE ACTIVITY GUIDE for the student versions of these labs.  I use a blended learning system to push unique assignments that match the specific learning objectives of various student subgroups within a class.  (I will do this through direct email, or the Google add-on Doctopus.)

The housefly lab may be a better fit for students with more advanced mathematical thinking skills or students that are comfortable with complex, open-ended tasks that require higher order thinking.  Students that choose this option will be asked to grapple with mathematical expressions of population growth; as such, teachers will need to carefully curate learning exercises that match students' abilities. Students choosing the housefly lab should have demonstrated proficient use of population ecology vocabulary and an ability to accurately model population growth trough graphs.

The yeast lab may be a better fit for students that are more comfortable with prescriptive, more close-ended directions.  Additionally, while students will have to grapple with some math, pattern recognition and data extrapolation are the most advanced skills that students will need to be successful. These students may still struggle with appropriate use of population ecology vocabulary and likely have not yet demonstrated consistent proficiency with the skill of creating a graph of population growth.

What will teachers do?

For this assignment I carefully worked through both labs and identified strengths and weaknesses based on the needs of student subgroups in all of my classes.  I think created differentiated versions of the attached document that I believed represented a "best fit" learning pathway for students.

Regardless of the lab, while circulating around the room, I collected mastery data by assessing students ability to demonstrate appropriate discipline-specific thinking in responses to the following questions:

• Is logistic growth ever exponential?
• Why can't yeast/housefly populations grow to an infinitely large size?
• What factors cause logistic growth?
• Why are some populations able to grow exponentially?

RESOURCES NOTE: The attached STUDENT WORK SAMPLES are from a student's submitted work for the yeast pathway. Each demonstrated a proficient level of understanding of the differences between the exponential and logistic growth model as well as the ability to use these models to analyze real populations.

## EVALUATE: The incident at tower 37

12 minutes

What is the purpose of this activity?

Students are able to demonstrate successful mastery of learning objectives ("level 3" work) through evaluation of an ecosystem portrayed in the short film The Incident at Tower 37. By the end of this section students should be able to explain the growth of an observed population and accurately use all learned population ecology vocabulary and concepts to do so.

What will students do?

As students view this short film, they answer the following questions on loose leaf:

• What is the non-human population in this short film?
• Which model of population growth does the non-human population fit?
• What shape would the graph of population growth be for the non-human population?
• What are the limiting factors that this population faces?  Is this factor density-dependent or density-independent?  How do you know?

What resource will a class need for this activity?