Finding Your Niche: Microbial Growth Functions for Native ...
Finding Your Niche: Microbial Growth Functions for Native and Extreme Bacteria
|Sherree Komp | |Jonathan Soule |
|Math Teacher | |Pre-Service Teacher |
|Pullman High School | |Clarkston High School |
|Pullman, WA 99163 | |Clarkston, WA 99403 |
Under the mentorship of
Dr. Brent Peyton
Assistant Professor
Department of Chemical Engineering
Washington State University
Pullman, WA 99164-2710
Summer, 2004
A project funded by the Nation Science Foundation grant #EEC-0338868
[add infor in table of contents for the resource you provide in pp 3-10. In the Table of Contents just incliude the key divisions] The T of C must refelect exactly what follows.Check the pagination carefully.] {you have two teaching modules that are not even listed.. Check those of insertion in T of C.]
Table of Contents
Project Summary 1
Lesson 1: Microbial Review [there is much info pp 3-8. Sidehaeadings? ] 10
Lesson 2: Habitat and limiting factors………………………………………….....8
Lesson 3: Video on habitat destruction……………………………………….…10
Lesson 4: Energy and nutrients in communities…………………………………12
Lesson 5: Sources of your energy. Includes food chains and food web. ………..14
Lesson 6: Accumulation of pesticides in the food web………………………….16
Lesson 7: Ecological niches……………………………………………………...18
Lesson 8: Symbiosis and interdependence of ecological systems……………….20
Lesson 9: Populations……………………………………………………………22
Lesson 10: Wildlife’s social, historical, and political value…………………...…24
Lesson 11: Unit Review…………………………………………………………..26
Unit Calendar……………………………………………………………………..28
Quizzes and Worksheets……….…………………………………………………32
Assessments
List of Propositions…………………………………………………..………..…..41
Table of Specifications……………………………………………………………42
Unit Exam...……………………………………………………………………….43
Unit Poster……………………………………………..…………………………..49
Answer Keys……………………………………………..………………………..52
Project Summary
This teaching module is designed to provide students with the opportunity to explore problem-solving, engineering-oriented exercises. Students will be given a problem requiring collaborative efforts to design, construct, and test an engineering application needed to answer a biological question. Since the design of a spectrophotometer is essential to complete this module, students will need to work together to produce a machine that can distinguish light intensity using the set of components provided.
Students will need to understand microbiological concepts to be covered previously or reviewed at the beginning of this section. Students will first need a functional understanding of microbial life. The biological component of this module focuses on ecological diversity within a local environment and questions that can be posed. By culturing soil microbes and examining the range of colors and colony morphology, students will discover that simple soil is a habitat in itself. This mini-habitat can then be used to examine how these organisms interact by simulating a competition experiment to conclude how these organisms exist with one another in a fixed environment.
This module is aimed at connecting mathematics and science together through exploration and simulation. The mathematics lessons require students to investigate exponential and logarithmic functions and apply the properties associated with them to solve real-life problems. The use of graphing calculators and data collection units will be used to enhance student learning. Many of the math activities require the use of CBL units and sensors in order to conduct the experiments. For information about these resources or to request a catalog, visit .
TEACHING MODULE (SCIENCE)
Objectives
( Students will examine biological diversity by culturing soil microbes using minimal media.
( Students will construct a spectrophotometer to measure and analyze bacterial growth rates.
( Students will test competitive features between natural bacteria by devising and conducting comparative experiments.
( Students will graph growth curves and describe why populations in general mimic this growth response.
Rationale
It is assumed that students have previous materials covering microbial biology. This unit will expand on those concepts by placing the environment and aspects of behavior together with their previous knowledge. Students need to have background knowledge to confront environmental issues. They need to be able to grasp the global importance of loss of habitat and how it affects behaviors of plants and animals. An understanding of these principles enables students to consider how their actions impact the environment. As humans continue to expand and alter more of the natural environment, their role has become that of an intruder, not as fellow resident and neighbor.
In addition, this module intends to introduce principles of engineering to the students. Open-ended design problems assigned will require students to use science principles to create a machine to answer further biological questions. It is hoped this experience will encourage students to look at engineering as a potential career.
Goals
Students will gain an ecology background to make informed, conscientious decisions and to develop and exercise good moral values in making these decisions. While doing this, students will engage in problem solving activities designed to encourage engineering principles. The teacher will provide many student centered activities to create engaged lessons. Students will be continuously assessed through written and performance exercises.
Background
Microbiology
Bacteriology, the study of bacteria, is a useful science to study the many of the simplest, but most persistent forms of life on earth. Bacteria inhabit every conceivable niche in the environment from the surfaces of living animals to the soil and the deep seas. What is impressing scientists lately is the ability of the organisms to expand to extreme habitats thought to be uninhabitable to life. Microbes have been discovered in deep-sea vents where temperature exceeded 110(C and on frozen landscapes where temperatures are often well below freezing. These bacteria offer scientist opportunities to discover chemicals and proteins that are adept at conquering these environments.
In addition, bacteria’s simple cellular structure and quick generation time allow them to be used to address biological questions as model systems. Bacteria can be used to demonstrate exponential growth pattern, which can be used to model more complex ecological population dynamics (Fig. 1). These dynamics include the affect of other community members as well as limitations on energy resources. This module utilizes bacteria to demonstrate biological diversity, inter-specific competition between two species, and the basic mathematical function of population growth.
Fig.1 Population Growth Curves
[pic]
Spectrophotometry
A spectrophotometer is a device that measures the amount of light passing through a media. The extent to which a sample absorbs light depends strongly upon the wavelength of the light. For this reason, highly precise spectrophotometers use monochromatic light to ensure only one wavelength is being tested. A spectrophotometer can also be used to empirically collect data on the growth of microorganisms. Light emitted from a light source can be measured to determine the ratio that is passing through a substrate. In a culture media, the cloudy appearance of growing bacteria is able to absorb or deflect source light, providing the photoreceptor with a different value. This relationship was recognized the German mathematician August Beer and written into the mathematical equation known as Beer’s Law (Appendix A).
Competition
Literature on competition theory abounds. The basic tenet is that similar, but not identical, species compete or have competed in the past. That is, species with similar requirements, such as habitat and food requirements are in competition with each other for those requirements. Competition theory states that perfect competitors cannot coexist and that in order to coexist, species must utilize resources differently and have different competitive abilities. Species may utilize different food sources, habitats, or times of activity to avoid competition. Species may become specialized to be the best at utilizing a particular resource, or they may be generalists and be only marginally good at exploiting a wide variety of resources to avoid competition. Competitive displacement occurs when competition leads to the evolution of different resource utilizations or competitive abilities of species exploiting similar resources.
According to competition theory, many of the niches that species currently occupy are the result of competition between species that occurred in the past. Current niche diversification that is driven by competition may be difficult to observe. Invasive species may offer an opportunity to observe the effects of competition as they occur. Invasive species enter a habitat where they have not previously existed and may be better competitors than the native species that already live in that habitat. Invasive species often drive native species to extinction through their superior competitive ability. This is an example in nature where one may witness competition leading to either extinction or competitive displacement.
Creating a Spectrophotometer
In the absence of a quality monochromatic light, a simple spectrophotometer can be created that depends on a photodiode to create an electrical current in the presence of photons (light). Using such a design, a simple and wholly effective measuring devise can be engineered.
Fig. 2. Function of a Spectrophotometer
[pic]
In clear liquids, few particles are present to absorb or deflect light passing through. However, when bacteria start growing in culture, bacteria absorb light to a greater extent than distilled water (Fig. 2). This reduction in transmittance prevents the light from reaching the detector, thus producing a reduced reading. If the detector is a photoreceptor, the reduced light will decrease the electrical current produced, causing a lower value to be read on the multi-meter. As the bacteria continue to grow, more and more light is absorbed and a reduced milliamp reading is observed.
National Mathematics Standards
In 1989, the National Council of Teachers of Mathematics (NCTM) released their Principles and Standards for School Mathematics. Subsequently, many states, including Washington State, have developed their own curriculum guidelines that are based on the NCTM standards. The Washington Essential Academic Learning Requirements (EALR’s) parallel these national standards and this teaching module addresses several of them.
NCTM Algebra Standard for Grades 9–12
• Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;
• Use symbolic algebra to represent and explain mathematical relationships;
• Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;
• Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
In addition to these algebra standards, the math lessons in this module emphasize these other strands:
NCTM Connections Standard for Grades 9–12
|Instructional programs from prekindergarten through grade 12 should enable all students to— |
|recognize and use connections among mathematical ideas; |
|understand how mathematical ideas interconnect and build on one another to produce a coherent whole; |
|recognize and apply mathematics in contexts outside of mathematics. |
NCTM Representation Standard for Grades 9–12
|Instructional programs from prekindergarten through grade 12 should enable all students to— |
|create and use representations to organize, record, and communicate mathematical ideas; |
|select, apply, and translate among mathematical representations to solve problems; |
|use representations to model and interpret physical, social, and mathematical phenomena. |
| | |
| |NCTM Communication Standard for Grades 9–12 |
| |Instructional programs from prekindergarten through grade 12 should enable all students to— |
| |organize and consolidate their mathematical thinking through communication; |
| |communicate their mathematical thinking coherently and clearly to peers, teachers, and others; |
| |analyze and evaluate the mathematical thinking and strategies of others; |
| |use the language of mathematics to express mathematical ideas precisely. |
| | |
[note box added to be parallel with others above.]
Teaching Module - Mathematics
Background
The lessons in this module are aimed at students in a second year algebra course or an applied mathematics course. The module can be used in conjunction with the science module or it can be used as a stand alone unit. The unit will consist of several hands-on activities for students to explore exponential and logarithmic functions. Many applications will be made to connect the mathematics with science, finance, and the social sciences.
The lessons that are included in the module will require ten class periods, not including any assessment time.
The students should already have knowledge and competency with the following mathematics concepts and skills:
• solving and graphing linear functions;
• applying the properties of exponents to simplify expressions;
• solving and graphing polynomial functions;
• finding the inverse of a given function and understand that the graphs of a given function and its inverse are reflections over the line y = x.
Goals
Students will be able to:
• evaluate and graph exponential growth and decay functions;
• evaluate and graph logarithmic functions;
• use properties of logarithms to evaluate, expand, and condense expressions that involves logarithms;
• simulate and model exponential growth and decay through the use of manipulatives and technology;
• use exponential growth and decay functions to model real-life situations such as compounding interest and depreciating value of goods;
• write exponential and power functions, using exponential and power regression, to model real-life problems;
• determine whether a given set of data is best represented as an exponential functions or a power function;
Science Lesson Plans
Time Required: Ten to twelve 55 minute class periods will be needed. This may be variable depending on growth of bacteria and the difficulty in the design of the spectrophotometer.
Intended Grade Level: Grades 10 –12
Prerequisite Skills: Students should have a background in microbiology before beginning this module. Alternatively, a background in ecology can exist and lesson written to teach microbial concepts.
Lesson 1
Rationale
While this section primarily deals with ecological relationships, bacteria will be used as the model organism for the investigations of the extended project. Therefore, students need to review how bacteria survive and function to their specific biology can be examined.
Objectives
• Students will explain the bacteria’s general cellular structure.
• Students will define the respiration needs for growth and reproduction.
• Students will extract and plate soil microbes to observe diversity.
Assessment
The class as a whole will be assessed for their prior knowledge of microbiology. What are bacteria? What places can they inhabit? How do they grow? The class activity will require several groups to contribute to class discussion by presenting information.
Learning Activity
1. Introduction [10 min]
Students will be given an overview of the spectrophotometer activity and their role in the exercise. Students should have soil samples with them in the pre-assigned groups for extraction. The teacher should collect extra samples prior to class in case some students forget.
2. Learning Activity [40 min]
A. Think-pair-share activity on bacterial physiology. Students need to write down all the cellular components of bacteria and their nutritional requirements. After a few moments, have students pair and compare answers. Students should discuss differences if possible. Finally, several groups will be asked to share their lists with the class. Discuss any misconceptions that arise. Ask other groups for additional examples the first groups overlooked.
B. Move class to the laboratory with their soil samples. For each group of 4-5 students, have them select on soil sample to extract microbes. Follow Appendix B for procedure.
3. Conclusion [5 min]
Students need to store their plates in a safe location and clean their bench space.
Materials
Paper and pencil
Soil extraction materials (Appendix B)
Lesson 2
Rationale
Students need to utilize problems solving skills to deal with complex problems. This activity forces students to consider a problem and work through it to design a solution. In this project, this design answer becomes a useful tool they can then use to answer more basic biological problems.
Objectives
• Students will explain the concept of transmittance, absorption, and Beer’s Law.
• Students will work cooperatively to design and construct a spectrophotometer
• Students will use the spectrophotometer to take readings and convert them to meaningful values.
Assessment
Each group will be visually assessed to see if a working spectrophotometer has been constructed.
Learning Activity
1. Introduction [15 min]
Students will be given a short lecture on the uses of spectrophotometers. The concept that the change of light intensity passing through a substance can be measured and given empirical values will be explained. This will lead into the differences in transmittance, absorbance, and Beer’s law, which relates the absorbance of a material to its concentration.
2. Learning Activity [35 min]
A. Each group will be given a box with the needed supplies listed for a spectrophotometer in Appendix C. Groups will need to use the concepts discussed to design a machine that will provide a useful measure of a chance in light intensity.
B. A written design needs to be created and approved before the construction. Once constructed, the machine needs to be verified as working.
C. Check plates from previous day for growth.
3. Conclusion [5 min]
Students need to store their plates and construction in a safe location and clean their bench space.
Materials
Spectrophotometer construction materials (Appendix C)
[As one alternative, do you want to note that a teacher may borrow a “spec.” from the equipment loan program. Or if a school has a “spec” it can use that instead.]
Lesson 3
Rationale
The purpose of this activity is to present students with the topics of ecology, populations, communities, habitat, niche, mutualism, parasitism, commensalism, and symbiosis. Students will need to know this material for activities later in the chapter as well as the chapter test.
Objectives
• Students will explain how ecology is important to the survival of life on earth.
• Students will define the major terms presented in lecture.
• Students will complete the construction of their group spectrophotometer.
Assessment
A short quiz will be given to the students the next class (see Quiz 1 in Appendix G).
Learning Activity
1. Introduction [10 min]
The ecology unit will be introduced using preconception statements to judge student understanding. Student responses will be used as a preassessment to gauge where student understanding is and what information will be needed as supplement:
Examples:
Plants and animals need a place to live.
There can be many populations in one community.
Some animals eat plants, some eat other animals, and still others eat both.
We get all our energy from what we eat.
We do not need things like parasites and funguses to survive. [Not or Do?]
Plants and animals are only useful to each other and us as food.
An organism’s environment is its living surroundings.
Both predators and prey are necessary to sustain a balance.
Plants and animals may have only one type of habitat.
2. Learning Activity [40 min]
A. Jigsaw activity on ecology definitions. Students will get into groups of five and look up one term and find examples of it. After 5 minutes, one partner goes to a new group to teach their term. Students rotate groups until each group has all definitions and examples.
Major Concepts
Ecology Ecologist
Populations Communities
Habitat Niche
Mutualism Parasitism
Commensalism Symbiosis
B. Students will get into their groups and continue construction of their spectrophotometer. Those groups done will have time to design their machine creatively. All groups should have a working machine at the end of the period.
C. All groups should check their plates. Some colonies may be visible at this point.
3. Conclusion [5 min]
Make sure students have all terms copied down and inform them there will be a quiz over the vocabulary to start the next class. Students need to store their plates and construction in a safe location and clean their bench space
Materials
Paper and pencil
Textbook
Lesson 4
Rationale
Populations of plants and animals tie together concepts of environmental science and traditional biology. Understanding how populations operate based on ecological criteria will provide students background information for application of complex topics such as genetics and evolution.
Objectives
• Students will observe and record the populations in a measured area.
• Students will infer what populations live in an area over an extended period of time.
• Students will collect and interpret data to determine what consumer populations eat.
• Students will streak soil bacteria for isolation using a three-phase streaking technique.
• Students will inoculate soil bacteria to media culture.
Assessment
Collect population introduction activity at end of class. Assign activity questions (Population Worksheet- Appendix G) as homework to be collected the next class period.
Learning Activity
1. Introduction [10 Min]
Take quiz 1. Ask students to provide examples of populations. List them on the board as they name them. Have students copy the list on a sheet of paper and use a graphic organizer format to connect populations listed that interact and how.
2. Activity [20 Min] (Includes moving time)
A. Students will select a one square meter area outdoors. Mark the square with popsicle sticks and string. Students will identify populations of producers in your square area. Use a hand lens to observe small plants. Record each type of producer by drawing a picture of it. Count how many of each type of producer and record it. Do the same for possible consumers.
B. Return to classroom and observe student plates. Have groups select two different bacterial colonies to streak on a plate for isolation. In addition, have students inoculate a test tube to be used as a starter culture. Plates and tube should be kept at room temp and the tube shaken at (100rpm.
3. Conclusion [10 Min]
Ask students: Did your areas have many of the same producers or different producers? Does this tell you something about the habitat or not? Collect introductory sheets from students. Inform students the Population Worksheet will be collected at the beginning of the following class. Relate the activity to the collection of soil samples and the different populations they may represent.
Materials
Meter stick or metric tape
4 popsicle sticks
String
Scissors
Hand lens
Paper and pencil
Population Worksheets (1/student)
Petri plates with media (2/group)
Inoculating loops (1/group)
Test tubes containing culture media (2/group)
Lesson 5
Rationale
All organisms are more specialized or adapted to living in a certain environment. Even within these environments, extreme specialization gives organisms survival advantages over competitors. This extreme specialization is referred to as an organism’s niche. By understanding the concepts of niches, students will better understand the interconnected roles a habitat contains.
Objectives
• Students will define ecological niche from group brainstorming in activity one.
• Students will give one example of an animal and its ecological niche.
• Students will discuss extreme environments and the bacteria that can inhabit them.
Assessment
Students will turn in Ecological Niche Worksheet (Appendix G) as groups at the end of class period.
Learning Activity
1. Introduction [10 min]
Collect Population Worksheet. Ask students: What is an ecological niche? Have students provide definitions and examples until the concept is clear.
2. Learning Activity [40 min].
A. Have student teams brainstorm a variety of animals living in one of four communities the teacher gives them (forest, stream, desert, or tundra). After a lengthy list is created, ask the groups to consider the possibility of the members of their list existing in another habitat.
B. Have group members change groups and discuss their group’s findings. Students must provide reasons why their organisms may or may not exist in the other environments.
C. Introduce the concept of extreme organisms. Examples of thermal, psychro-, halo-, and other chemical waste organism exist and are being explored by chemical engineers as tools for clean-ups.
D. Have groups check their streaked plates and culture. Plates and dilution tubes need to be prepared for upcoming experiments. Have groups prepare their own materials.
3. Conclusion [5 min]
Collect Worksheet. Clean up lab spaces.
Materials
Chalkboard
Writing materials
Pictures of forest, stream, desert, and tundra communities.
Ecological Niche Worksheet (1/student)
Media (300ml/ group)
Petri plates (15/group)
Test tubes (15/group)
Lesson 6
Rationale
This is the initial exercise to perform the bacterial growth and competition experiments. Students need to carefully start and record the measurements of their cultures to know where their experiments begin.
Objectives
• Students will aseptically inoculate bacterial cultures to the larger culture flasks.
• Students will record the initial absorbance of each culture using their group’s spectrophotometer
• Students will use dilution-plating techniques to determine the cell numbers in their cultures.
Assessment
Assessment will be indirectly observed through the results of their dilution plating and spectrophotometer readings.
Learning Activity
1. Introduction [15 min]
Remind student that the purpose of streaking the original cultures was to generate single, pure colonies representing a single bacterium. Show the class examples of cleanly streaked plates and those that did not provide single colonies. Have the groups evaluate their own plates. Review the entire procedure for dilution plating and taking spectrophotometer readings.
2. Activity [30 min]
A) Allow students to inoculate 250-ml flasks. They should collect spectrophotometer readings for samples and controls once the starter cultures have been added.
B) Groups should collect an initial dilution plating (Appendix D).
3. Conclusion [10 min]
Have groups clean areas and start large flasks shaking. New plates should be stored appropriately.
Materials
See appendix D.
Lesson 7
Rationale
The purpose of this activity is for students to learn that good habitat is the key to wildlife survival. A population will continue to increase in size until some limiting factors are imposed. Limiting factors contribute to fluctuations in wildlife populations. Nature is never in balance, but constantly changing.
Objectives
• Students will identify essential components of habitat.
• Students will describe the importance of good habitat for animals through discussion and activity.
• Students will define limiting factors and give examples on quiz.
Assessment
The concepts of habitat and limiting factors will be quizzed the next class (see quiz 2).
Learning Activity
1. Introduction [10 min]
Introduce food, water and shelter as three essential components needed for good habitat. Also mention that area is important as well. Point out that wildlife populations are not static, that is, they constantly fluctuate depending on their environment. Ask students to define limiting factors. Food, water and shelter are very important limiting factors.
2. Activity [30 min]
Begin by reviewing the essential components of habitat (food, water, shelter).
-Then ask your students to count off in fours. All the ones go to one area, and the rest go to another area.
-Mark two parallel lines on the ground about 15 yards apart.
-The 'ones' form one group and everyone else forms another. The ones become the “deer”; they need food, water, and shelter to survive.
The deer:
a) hold their hands on their stomach when looking for food
b) put their hands over their head when looking for shelter
c) put their hands on their mouth and when looking for water.
-The deer must choose one of the three components to look for each round, but cannot change what they are looking for in the middle of the round.
-The rest of the class gets to choose before each round if they want to be food, water, or shelter. They depict what they are, the same way the deer show what they are looking for.
The activity starts with both sets of students lined up on the lines with their backs to each other.
-The teacher starts the round by asking everyone to display his or her sign.
-At the count of three the deer and habitat components turn to face each other.
-When deer see the habitat component it needs it can run to it and bring it back to its side of the line. Any deer that fails to find what it was looking for dies and becomes a habitat component the next round.
-Any habitat component that is chosen by a deer becomes a deer the next round.
-The teacher keeps track of how many deer there are at the beginning of the activity, and how many are left at the end of each round.
-At the end of about 15 rounds, gather the students and discuss the activity. The teacher can make a graph having each round count as one year. Typically, in the first two years, the deer herd will increase before it depletes the habitat. It will then level off. The graph will show periods of peak, decline, and rebuilding.
3. Conclusion [15 min]
A. Pass out Limiting Factor Worksheet (Appendix G). Allow students to fill in answers from class discussion. Discussion is based on questions on the worksheet. Graphing question will be assigned as homework. The teacher will copy the activity data onto the board and ask students to graph the change in population based on resources available through the 15 rounds. Worksheet will be due beginning of next class. There will also be a short quiz to start the following class.
B. Take spectrophotometer readings and aliquot for dilution plating.
Materials
Area large enough for students to run (gym or outdoors).
Clothes and shoes that are good for the outdoors.
Writing materials to collect data
Watch
Limiting Factor Worksheet (1/student)
Media Plates
Test Tubes
Lesson 8
Rationale
Students need to become familiar with the concept of symbiosis as one example of interdependence in ecological systems. From previous lesson, interactions in habitats, food chains, and niches can be built upon and add a layer of ecological complexity.
Objectives
SWBAT define symbiosis, commensalism, mutualism, and parasitism.
SWBAT identify animals that live in each type of symbiotic relationship through class presentations.
Assessment
Students will be assessed using short personal communication demonstrations. Grading criteria will be attached as a rubric. There will also be a quiz the following class (see attached quiz 5).
Learning Activity
1. Introduction [15 min]
Take quiz 2. Introduce the terms symbiosis, commensalism, mutualism, and parasitism. Ask students: Can cooperation and competition both exist in nature? Initiate class discussion on student answers.
2. Activity [30 min]
A) Split class into pairs and equally assign the three symbiotic classification as well as competition. Groups will discuss an example for their designation and will need to share their choice and reasoning with the class.
B) Elaborate on the competition design used in the bacterial experiment. Is this a true representation of natural competition? Why?
C) Take spectrophotometer readings and perform dilution plating.
3. Conclusion [10 min]
Discuss activity with the class. Have students turn in buddy card exercise they completed for presentation. Due to lack of time and long experiment next class, learning logs will be taken home for en entry this evening.
Materials
Media plates
Test tubes
Lesson 9
Rationale
More debate involving environmental issues are in the news, and the issue touches many lives. Students must be aware of controversies and understand how emotions and politics affect how society relates to our natural environment. By looking at how issues shape opinion and public policy, students will be aware how people respond to challenges from many viewpoints.
Objectives
• Students will give examples of ways in which wildlife has influenced the development of human societies.
• Students will describe wildlife as having important social and political value for people through article summaries.
Assessment
Students will write a one-page article summary based on an assigned research area. Paper will be turned in the following class period.
Learning Activity
1. Introduction [20 Min]
A. Take quiz 3.
B. Take spectrophotometer readings and perform dilution plating.
C. Discuss with students’ ways in which wildlife has influenced human societies. Brainstorm for ideas.
D. Refine the list and make sure these are included:
1. Historical influence of the availability of wildlife on the size and location of human communities.
2. Examples of issues and conflicts related to wildlife, historically and in the present.
3. Treaties and alliances within and between people and nations involving wildlife.
4. Creative portrayal of wildlife through art, literature, dance, music and drama as historic as well as contemporary means of expressing human relationships with wildlife.
2. Activity [30 Min]
A. Each student must select one of the research areas and find a newspaper or magazine article dealing with the topic. Students are to write a one-page summary of the article, including author’s viewpoint and possible sources of bias omitted from the article (if student is aware of any).
B. Take the class to the library and allow the rest of this hour to work on it. Article summary is due next class period. While students are working, check student progress on unit posters by looking at their notes and materials.
3. Conclusion [3 Min]
Check students on way out to make sure they have an article to take home.
Materials
Media plates
Test tubes
Writing Materials
Access to Library Resources
Suggested Module Calendar
| |Monday |Tuesday |Wednesday |Thursday |Friday |
|Week 1 |Lesson 1 |Lesson 2 |Lesson 3 |Lesson 4 |Lesson 5 |
| |1. Students bring in soil |1. Construction of |1. Terminology Jigsaw Activity |1. Quiz 1 |1. Population Worksheet Due |
| |samples. |spectrophotometer |2. Continue construction of |2. Populations activity |2. Ecological Niche Activity/ |
| |2.Review microbial biology and | |spectrophotometer. |3. Streak and inoculate media |Worksheet |
| |physiology | | | |3. Media Prep |
|Week 2 |Lesson 6 |Lesson 7 |Lesson 8 |Lesson 9 |Lesson 10 |
| |1. Start Cultures for Growth/ |1. Limiting Factor Activity/ |1. Quiz 2 |1. Quiz 3 |1. Collect Article Summary |
| |Competition Experiment |Worksheet |2. Symbiotic Discussion |2. Collect Spectrophotometer |2. Collect Spectrophotometer |
| |2. Collect Spectrophotometer |2. Collect Spectrophotometer |3. Student Presentations |readings |readings |
| |readings and initial colony |readings |4. Collect Spectrophotometer readings |3. Perform Dilution plating |3. Perform Dilution plating |
| |count |3. Perform Dilution plating |5. Perform Dilution plating |4. Article Research |4. Begin Data Analysis |
|Week 3 |Lesson 11 |Lesson 12 |Unit Exam |Lesson 13 |Experimental Write-up. |
| |1 Collect Final photometer | | |1. Collect Final Dilution Plate | |
| |readings |Unit Review | |Data. | |
| |2 Perform Final Dilution Plating| | |2. Perform Data Analysis. | |
| |3 Continue Data Analysis | | | | |
Day 1
Subject: Algebra II/Trig
Topic: Exponential Growth and Decay
Objectives: Students will be able to:
1. simulate exponential growth and decay by repeatedly folding a piece of paper;
2. collect and record data;
3. graph their data and describe the relationship between the dependent and independent variables using words and/or symbols;
4. make predictions using their graphs.
5. read and analyze an example of exponential growth to determine specific points during the growth;
6. analyze the cause and effect relationship related to a specific example of exponential growth.
Materials: 8.5 x 11 inch sheets of paper, Student Activity Sheets (Appendix H), Paper Folding Table (Appendix I)
Set/Motivation: Ask the students how many times they think they can fold a sheet of paper in half. Have them write down their prediction on a piece of scrap paper. Next, have students work with a partner to complete The Paper Folding Investigation. In the first activity the students fold the paper in half and record the number of sections formed by the fold. They repeat this procedure of folding the paper in half and recording the total number of sections formed until they can no longer fold the paper. After they have finished collecting the data, students will graph their data. In the second activity, the students will repeat the paper folding process as they did in the first activity, but the data they record is slightly different. The initial area of the paper, before folding it, will be designated as 1 square unit. After making each fold, the students will record the area of the resulting smaller sections. Again, after collecting data until the paper can no longer be folded, the students will graph their data. Student Activity Sheets are in Appendix H.
Procedures:
1. A discussion about exponential growth and decay will follow these paper-folding activities. Ask the students to share the mathematical models that they came up with for each part of the activity. In the first part of the paper folding activity, the number of sections doubled after each fold. The equation that models this relationship is y = 2x. For the second part of the activity, the area of the paper got cut in half after each fold. The equation that models the data is y = ( ½ )x. Ask the students to describe what each variable represents for each equation.
2. An extension of this activity is to share with the students how the thickness of the paper after folding it repeatedly is also a model of exponential growth. Ask the students how many folds they were able to make with their paper and compare to the predictions they made earlier. Most students were probably unable to make any more folds after the seventh time, yet their predictions were most likely more than that. The paper just got too thick to continue folding. You could have the students repeat the folding procedure one more time and record how thick the paper is after each fold. You could even just give the students the fact that the thickness of the sheet of paper without any folds is about 0.1 mm thick, so after one fold, the thickness would be .2 mm, and after three folds the thickness would be .4 mm, and so on. By the time the paper was folded seven times, the thickness would be 12.8 mm. Ask the students how many folds it would take before the thickness were as tall as themselves, or as tall as the room. Share with the students the table that compares the number of folds with the thickness of the paper (Appendix I). You can either make copies of this to hand out to each student or make it into a transparency and put in on the overhead. The students will likely be shocked to see how quickly the thickness would be that of their own height or the height of a house.
3. Not all functions of exponential growth are doubled and not all functions of exponential decay are halved. As we continue through this unit, the students will identify the growth/decay rates from the functions that are given, as well as determine those rates from data they have been given or that they have collected.
Closure/Evaluation: The Lily Pond Parable
1. If a pond lily doubles everyday and it takes 30 days to completely cover a pond, on what day will the pond be 1/4 covered?
2. 1/2 covered?
3. Does the size of the pond make a difference?
4. What kind of environmental consequences can be expected as the 30th day approaches?
5. What will begin to happen at one minute past the 30th day?
6. At what point (what day) would preventative action become necessary to prevent unpleasant events?
Answers to The Lily Pond Parable
If a pond lily doubles everyday and it takes 30 days to completely cover a pond, on what day will the pond be 1/4 covered?
Answer: Day 28. Growth will be barely visible until the final few days. (On the 25th day, the lilies cover 1/32nd of the pond; on the 21st day, the lilies cover 1/512th of the pond).
1/2 covered?
Answer: Day 29.
Does the size of the pond make a difference?
Answer: No. The doubling time is still the same. Even if you could magically double the size of the pond on day 30, it would still hold only one day's worth of growth!
What kind of environmental consequences can be expected as the 30th day approaches?
Answer: The pond will become visibly more crowded each day, and this crowding will begin to exhaust the resources of the pond.
What will begin to happen at one minute past the 30th day?
Answer: The pond will be completely covered. Even though the lilies will be reproducing, there will be no more room for additional lilies, and the excess population will die off. In fact, since the resources of the pond have been exhausted, a significant proportion of the original population may die off as well.
At what point (what day) would preventative action become necessary to prevent unpleasant events?
Answer: It depends on how long it takes to implement the action and how full you want the lily pond to be. If it takes two days to complete a project to reduce lily reproductive rates, that action must be started on day 28, when the pond is only 25% full -- and that will still produce a completely full pond. Of course, if the action is started earlier, the results will be much more dramatic.
pop/facts/exponential70.htm
Day 2
Subject: Algebra II/Trig
Topic: Exponential Growth and Decay
Objectives: Students will be able to:
1. simulate exponential growth and decay using M&M’s;
2. use graphing calculators to enter data and determine a regression equation to model the data;
3. use the interest formulas to calculate the balances of accounts after a given period of time;
4. compare investments with varying terms to determine which will yield a higher return.
Materials Needed: M & M’s, paper cups, paper plates or paper towels, Student Activity Sheets (see Appendix K ), graphing calculators (TI-83 Plus), Directions for using the TI-83 Plus (Appendix J)
Procedures:
1. Go through a quick example of how to use the graphing calculators to enter data, plot a scatter plot, find an exponential regression equation and graph the equation. Distribute the directions (Appendix J) so that the students can follow along as you demonstrate the steps using the graphing calculator and the overhead view screen. The table below give the mean weight y (in kilograms) and age x (in years).
|Age |Weight |
|1 |0.751 |
|2 |1.079 |
|3 |1.702 |
|4 |2.198 |
|5 |3.438 |
|6 |4.347 |
|7 |7.071 |
|8 |11.518 |
Once an equation has been found, have the students estimate the weight of a cod that is 9 years old. This can be done with the graphing calculator by plugging in 9 for x into the equation.
Another question to ask is if a cod was caught that weighed 25 pounds, what might we guess the age of the cod to be? This can be found by tracing the exponential curve until we get a y value close to 25. It can also be done by entering a second equation, y = 25, and locating the point of intersection.
2. M & M’s Activity: This activity simulates exponential growth and decay. Students will use graphing calculators to find an exponential function to model their experiment. Through classroom discussion and teacher questioning, students will be able to make the connection between the general exponential model y = abx, their own model, and their data from the M&M’s activity. They should see that the value of a is close to the number of M&M’s they started with in each part of the activity. The value for b is close to 1.5 for the growth activity and 0.5 for the decay activity.
3. General growth/decay models: Introduce the general growth and decay models
Growth Model: y = a(1 + r)t
Decay Model: y = a(1 – r)t
a is the initial amount
r is the rate of growth/decay expressed as a decimal
t is the time
1 + r and 1 – r are called the growth and decay factors
4. Ask the student if they can identify the growth rate and the rate of decay from the M&M’s activity. Hopefully they will see that the relationship between the 1.5 and 0.5 and the probability of getting the M facing up being ½.
5. Go through several examples using these formulas. Start by having the students use the simple interest formula to calculate how much they would have after one year if they invested $50 at 4% interest. What about two years? Five years? With a calculator, all students should be able to determine the amount of money in the account after each year by successively applying the formula I = prt and keeping a running total of the balance in the account.
Year 1: I = (50)(.04)(1) = 2, so the balance is $52
Year 2: I = (52)(.04)(1) = 2.08, so the balance is $54.08 Year 3: I = (54.08)(.04)(1) = 2.16, so the balance is $56.24 Year 4: I = (56.24)(.04)(1) = 2.25, so the balance is $58.49 Year 5: I = (58.49)(.04)(1) = 2.34. so the balance is $60.83
6. The idea is to lead students to the pattern that the balance of the account after t years is modeled by (50)(1.04)t, so if we wanted to know the balance after 10 years:
Year 10: Balance = (50)(1.04)10 = $74.01
7. Note that this formula gives us the total balance as opposed to just the amount of the interest.
8. Now introduce the idea of compounding the interest more frequently, such as quarterly. The interest rate then needs to be divided by the number of times in a year the interest will be compounded. So if we were to repeat this example and calculate how much would be earned if the interest is to be compounded quarterly, we would divide 0.04 by 4 to get 0.01. The number of times the interest gets compounded gets multiplied by 4 as well. This leads to the compound interest formula:
[pic]
A is the amount in the account after t years
P is the principal, or the initial investment
r is the interest rate expressed as a decimal
n is the number of times the interest is compounded in a year
9. Have students determine how long it would take for an investment to double if $500 were invested in an account that paid 5.4% interest compounded monthly. It would be between 12 and 13 years (12.865). Ask the students if doubling the interest rate would mean that it would take half as much time to double the amount.
Closure: Explain the rule of 70 and have students determine if this rule is consistent with what we found in the previous example.
The Rule of 70
The Rule of 70 is useful for financial as well as demographic analysis. It states that to find the doubling time of a quantity growing at a given annual percentage rate, divide the percentage number into 70 to obtain the approximate number of years required to double. For example, at a 10% annual growth rate, doubling time is 70 / 10 = 7 years.
Similarly, to get the annual growth rate, divide 70 by the doubling time. For example, 70 / 14 years doubling time = 5, or a 5% annual growth rate.
The following table shows some common doubling times:
|Growth Rate |Doubling Time |
|(% per Year) |in Years |
|0.1 |700 |
|0.5 |140 |
|1 |70 |
|2 |35 |
|3 |23 |
|4 |18 |
|5 |14 |
|6 |12 |
|7 |10 |
|10 |7 |
pop/facts/exponential70.htm
Day 3
Subject: Algebra II/Trig
Topic: Exponential Growth and Decay
Objectives: Students will be able to:
1. set up and conduct and experiment to simulate Newton’s Law of Cooling;
2. collect and graph data from the experiment using a graphing calculator and a CBL unit;
3. identify and explain each variable in Newton’s Law of Cooling;
4. make predictions about the water’s temperature using the exponential function derived from the data;
5. describe why the thermometer’s temperature will never actually reach room temperature and why.
Materials Needed: CBL system, TI-83 Plus graphing calculator with unit-to-unit link cable, TI temperature probe, coffee cup, hot water, TI-GRAPHLINK (if available), Student Lab Activity Sheet (Appendix L)
Procedure:
1. Have students work in groups of 3 on the experiment that simulates Newton’s Law of Cooling. Although the students will be working in groups to conduct the experiment, each student should complete the Student Activity Sheet (Appendix L) and answer all of the questions. This activity will likely take most of the class period, as question #8 instructs the students to repeat the experiment several times. It might be helpful to specify that each group repeat the experiment a minimum of three times. If students are unable to answer a question after the first trial of the experiment, encourage them to skip the question and try it again after the next trial.
2. Once all of the groups have finished, divide the students into new groups of three so that the new groups have only one member from each of the original groups. Within this second group, have the students share their results and conclusions from the experiment.
3. As an extension, students could try the experiment several times using different types of cups (ceramic, Styrofoam, etc.) to compare the cooling rates and determine which type of cup they would recommend using to maintain the liquid’s temperature.
Day 4
Subject: Algebra II/Trig
Topic: Exponential Functions and the Natural Base e
Objectives: Students will able to:
1. define the number e and explain how it is derived;
2. simplify expressions involving the number e;
3. compare the formula for computing investments that are compounded under different frequencies to the formula for computing investments that are compounded continuously;
4. use the natural base e in real-life applications.
Materials Needed: scientific calculators
Set/Motivation: Ask the students what the number Pi represents. Most students will respond with 3.14, but hopefully someone will remember that it has something to do with the ratio between a circle’s circumference and its diameter. In fact, Pi is an approximation for this ratio, and since it is a decimal number that continues on without ever repeating or terminating, it is classified as an irrational number. But point out that the number was not merely made up by some mathematician. It was discovered as the ratio between a circle’s circumference and its diameter.
Procedure:
1. Have students simplify the following expressions:
[pic]= ?
[pic]= ?
[pic]= ?
[pic]=?
[pic]=?
Does there appear to be a number that these values are approaching? The values should be getting closer and closer to 2.718281828459….yes, another irrational number. It is the value that the expression [pic]approaches as n gets larger and larger (or as it approaches + (). We call this the number e, which was discovered by Leonard Euler.
2. There are many applications that involve using the number e as the base of an exponential function. We call it the natural base. One of the applications is with respect to investing money. Recall our formula for investing money in a bank that compounds the interest monthly or quarterly. Sometimes, though, interest is compounded continuously. The formula that is used in this case is [pic].
3. There are many applications that involve using the number e as the base of an exponential function. We call it the natural base. One of the applications is with respect to investing money. Recall our formula for investing money in a bank that compounds the interest monthly or quarterly. Sometimes, though, interest is compounded continuously. The formula that is used in this case is [pic].
4. Have students compare the following investments, keeping the principal and interest rates the same, but changing how often the interest gets compounded. $500 is invested at 4.5% interest for 5 years compounded: annually, quarterly, monthly, daily, and then continuously. The students should see that the more often the interest gets compounded, the more interest the investor will accumulate.
5. There may be some question as to exactly what continuously compounded interest means. Students tend to want to know exactly how often that is. Remind them of the definition of the number e and that n is approaching infinity.
Days 5 and 6
Subject: Algebra II/Trig
Topic: Logarithmic Functions
Objectives: Students will able to:
1. evaluate logarithmic functions using the definition of logarithms;
2. write the inverse functions of a given logarithmic or exponential function;
3. graph logarithmic functions by hand and with a graphing calculator;
4. use properties of logarithms to expand and condense logarithmic expressions;
5. use the change of base formula to evaluate logarithms that do not use base 10 or the natural base e;
6. solve real-life problems that are modeled by logarithmic functions.
Materials Needed: graphing calculator with overhead projector, scientific calculators
Set/Motivation: Have the students find the value for x in each of the following:
a.) [pic]
b.) [pic]
c.) [pic]
d.) [pic]
e.) [pic]
The students should be able to use their knowledge about exponents to guess and check the values for x that will work in each example. However, when they get to the last problem they might have a little trouble. They should notice that [pic] and[pic], so for[pic], the value for x must be somewhere between 4 and 5. With their calculators, they can then try values like 4.1, 4.2, 4.3, etc. until they find a value that is close to 25.
Procedures:
1. Definition of logarithm with base b: [pic] if and only if [pic]. Remind students that “if and only if” means that they converse is also true. This simply means that these equations are equivalent, and the definition allows us to rewrite logarithms into exponential form as well as rewrite exponential equations into logarithmic form. For this definition, point out that the variables b and y must be positive numbers and that b[pic]1. Why is there a restriction for b? What would happen if the value for b was equal to 1?
2. Logarithmic functions are inverses of exponential functions. Use the graphing calculator and the overhead projector to demonstrate that the equations [pic]and y = [pic]are inverses. Graph the line y = x at the same time so that the students can see that the two curves are merely reflections of each other over the line.
3. The calculator has two keys on it that involve logarithms: log and ln. The log key uses a base of 10, and this is referred to as the common logarithm. The ln key uses the natural base e and is called the natural logarithm. So our calculators can only evaluate common and natural logarithms.
4. Change of Base Formula: This formula allows us to use our calculators to evaluate logarithms that have bases other than 10 or the natural base.
[pic]
Have the students evaluate the following problems with their calculators:
1.) [pic]
2.) [pic]
3.) [pic]
4.) [pic](Does this look familiar?)
5.) [pic]
5. Finding Inverses: Remind students that when we found the inverse of polynomial functions, we started by interchanging the x and y variables and then solved the resulting equations for y. We are essentially doing the same thing here, except that we will use the definition of logarithm to help us. Here are two examples:
1.) Find the inverse of [pic]
[pic] switch x and y
[pic] use the definition of logarithm to write in
exponential form
2.) Find the inverse of [pic]
x = ln (y + 3) switch x and y
[pic] use the definition of logarithm to write in
exponential form (remember that the base
of the natural logarithm is e)
[pic] solve for y
6. Graphing logarithmic functions: As we saw earlier, the graphs of logarithmic functions are curves since they are reflections of their exponential function counterpart. To graph these functions by hand, it is helpful to know some important characteristics of the graphs of logarithmic functions.
The graph of [pic]has the following characteristics:
• The line x = h is a vertical asymptote. Remember that exponential functions have a horizontal asymptote, so it makes sense that the inverse function would have an asymptote that is just the opposite.
• The domain is x ( h, and the range is all real numbers.
• If b ( 1, the graph moves up to the right. If 0 ( b ( 1, the graph moves down to the right.
7. In order to graph a logarithmic function with the graphing calculator, you will need to rewrite the function using the change of base formula if the function does not use the common or natural logarithm.
8. Properties of Logarithms:
Product Property [pic]
Quotient Property [pic]
Power Property [pic]
These properties allow us to expand and condense logarithmic expressions.
Here are some examples:
1. Expand [pic]
[pic]= [pic] Quotient Property
= [pic] Product Property
= [pic] Power Property
Condense log 6 + 2 log 2 – log 3
log 6 + 2 log 2 – log 3 = log 6 + log 22 – log 3 Power Property
= log (6∙22) – log 3 Product Property
= [pic] Quotient Property
= log 8 Simplify
Days 7 and 8
Subject: Algebra II/Trig
Topic: Solving Logarithmic and Exponential Equations
Objectives: Students will able to:
1. set up an experiment to measure the color intensity of a solution;
2. collect and graph data using a graphing calculator and CBL unit;
3. determine a logarithmic function to model the data.
Materials Needed: CBL system, TI-83 Plus graphing calculator containing the program MATHSCI and its seven subroutines, Vernier colorimeter and accompanying plastic cuvettes, adapter cable to connect the colorimeter with the CBL system, graduated cylinder, 250-mL or 500-mL capacity, green food coloring, medicine droppers or plastic Beral pipettes for transferring the green liquid, tissues or low-lint towels to dry off the cuvettes during testing, Probing Intensity Worksheet (Appendix M)
Procedure:
1. Put students into groups of three and distribute the necessary materials. The students should have enough time during the first class period to set up the experiment and collect their data.
2. The second day can be used for discussion and for an extension exercise. In the teaching notes for this activity (Appendix J), there are three suggestions for extension activities. One or more of these can be available for students to participate in on the second day after a discussion of the original activity has taken place.
Days 9 and 10
Subject: Algebra II/Trig
Topic: Modeling with Exponential and Power Functions
Objectives: Students will able to:
1. write an exponential function for a graph that passes through two given points;
2. write a power function for a graph that passes through two given points;
3. determine whether a given set of data is best modeled by an exponential function or a power function;
4. find the function that would best model a given set of data using a graphing calculator;
5. make predictions about future events using the best-fit function for a given set of data;
6. search and collect data from a reliable source and determine an appropriate model for the data.
Materials Needed: Modeling Exponential and Power Functions Examples (Appendix N) as a transparency, Modeling Exponential and Power Functions Worksheet (Appendix O), graph paper, scientific and/or graphing calculators, computer lab with Internet access, library, overhead projector
Set/Motivation: Have students describe the graphs of the following types of functions:
1. linear functions
2. quadratic functions
3. cubic functions
4. polynomial functions in general
5. exponential functions
Encourage students to explain as much detail as they can about each type of graph. What are the similarities/differences between these graphs?
Procedure:
1. Make sure each student has graph paper and a scientific calculator.
|x |y |
|1 |1.6 |
|2 |2.7 |
|3 |4.4 |
|4 |6.4 |
|5 |8.9 |
|6 |13.1 |
|7 |19.3 |
|8 |28.2 |
|9 |38.2 |
|10 |48.7 |
2. Modeling Exponential and Power Functions Example #1: Put the transparency of Appendix N on the overhead so that students can read the information as they hear it being read. The table gives the number of cell-phone subscribers (in millions) from 1988 to 1997 where x is the number of years since 1987. Have the students create a scatter plot of the data on a piece of graph paper.
3. Notice that the data appears to follow some type of curve. Since we have studied several types of graphs that are curves, how do we know which type of function would be model this data? We are going to learn how to determine whether a given set of data is best modeled by an exponential function or a power function (ie. polynomial function).
4. Have the students complete the tables for (x, ln y) and (ln x, ln y). The completed tables should look like the following:
|ln x |ln y |
|0 |0.4700 |
|0.6931 |0.9933 |
|1.0986 |1.4816 |
|1.3863 |1.8563 |
|1.6094 |2.1861 |
|1.7918 |2.5726 |
|1.9459 |2.9601 |
|2.0794 |303393 |
|2.1972 |3.6428 |
|2.3026 |3.8857 |
|x |ln y |
|1 |0.4700 |
|2 |0.9933 |
|3 |1.4816 |
|4 |1.8563 |
|5 |2.1861 |
|6 |2.5726 |
|7 |2.9601 |
|8 |303393 |
|9 |3.6428 |
|10 |3.8857 |
Once the tables have been completed, have the students graph these on their graph paper and determine which of the two graphs appear to be more linear.
5. If the graph of (x, ln y) is more linear, then the best model for the original data is an exponential function. If the graph of (ln x, ln y) is more linear, then the best model for the original data is a power function. In this case, the graph of (x, ln y) will be more linear, so we will now try to write an exponential function to model the data.
6. Writing the exponential function for this data will involve solving a system of equations. First remember that the general form of an exponential function is [pic]. Choose two points from the original data. Technically, we can choose any two points, but it may be helpful to specify which two points the students choose so that there is some consistency in the functions they come up with. Choosing the second and next-to-last set of points tends to generate a fairly accurate model. So the points we will use are (2, 2.7) and (9, 38.2). Use these two sets of data and plug the values into the function [pic].
[pic] [pic]
Solve the first equation for a: [pic]
Substitute this expression in for a in the second equation and solve for b: [pic]
[pic]
[pic]
[pic]
Now plug this value for b back into either of the original equations to find the value for a:
[pic]
[pic]
[pic]
So our exponential function that models this data is [pic]
If this pattern of growth continued, how many subscribers could we expect there to be in the year 2005?
In what year would you expect there to be 50 million subscribers?
7. Have the students try the second example on their own. This example would best be modeled by a power function, so remind the students that the general form for a power function is[pic]. Notice that this looks similar to our general form of the exponential model. The difference is that the x variable switches places with the constant b. The process is similar to the first example, however the students might have a little more trouble solving the system of equations this time. After they have found an equation to model the data, ask them to estimate the period of Neptune, which has a mean distance from the sun of 30.043 a.u.
8. Appendix N is a worksheet for students to work on as independent practice.
9. On the second day of this lesson, students will work with a partner to collect data about a particular Olympic sport. Each group will need to choose a sport in which both men and women compete. They need to choose a timed event, such as swimming, running, or an event where distance is used to measure performance, such as high jump. Resources such as the Internet, sports almanacs, etc. will be necessary for the students to conduct their searches. Each group will need to write a report summarizing their results. The report should include the following:
1. a table and scatter plot of the data sets;
2. graphs of the exponential or power functions;
3. a comparison of future performances based on the models;
4. an evaluation of the limitations of each model.
Students will take the data that was collected from the biology class regarding the spectrophotometer readings of the bacteria and find a regression equation to model the growth of the bacteria. The students should find regressions equations for both of the competing strains of bacteria and compare the equations. Graphing both functions together simultaneously can help them make predictions about when one strain out competes the other.
Appendix A
Beer's Law
Absorbance varies linearly with both the cell path length and the analyte concentration. These two relationships can be combined to yield a general equation called Beer's Law.
A = ε l c
The quantity ε is the molar absorptivity. The molar absorptivity varies with the wavelength of light used in the measurement. The quantity l refers to the length the light must pass through, and c is the concentration of the solution.
Conceptually, the transmittance is an easier quantity to understand than the absorbance. If T = 30%, then 30% of the photons passing through the sample reach the detector and the other 70% are absorbed by the analyte. The absorbance is a slightly less intuitive quantity. If A = 0, then no photons are absorbed. If A = 1.00, then 90% of the photons are absorbed; only 10% reach the detector. If A = 2.00, then 99% of the photons are absorbed; only 1% reach the detector. It is the absorbance, however, that displays a simple dependence on the concentration and cell path length (Beer's Law), and thus most chemists choose to report data in terms of absorbance rather than transmittance.
Appendix B
Sample Collection, Isolation, and Growth Measurement Procedures
Day 1
1. Students in groups of 3-4 collect a sandwich bag sized soil samples in a plastic bag by digging down approx. 10cm. Have the samples as fresh as possible as time and temperature changes can affect the number of organisms in the sample.
2. Measure 10g of soil material and mix it with 90ml of high quality water in a 250ml flask. Mix thoroughly by swirling for 2 minutes, and then allow soil to settle on bottom of flask.
3. Using a pipette, serially dilute a portion of the supernatant 1:3 two times (ie. 1:3 and 1:6 final dilutions) in sterile water, plate, and spread 200(l undiluted, 1:3, and 1:6 dilutions onto petri dish containing bacterial media (Appendix C).
4. Incubate the plates for 48 to 72 hours at room temperature (or 22-25(C if incubator is available).
Days 2-4
5. Observe the bacterial growth on plates and select two colony types to use in the competition analysis (Note: if several groups get similar colony types, have groups choose different pairs to get a wider examination of bacterial competition).
6. Students need to streak these colonies onto media for colony isolation. Have one group streak a Halomonas sp. (provided by Dr. Peyton or isolated from a salt flat environment) onto a plate containing 12.5% salt. Again incubate the plates for 48 to 72 hours at room temperature.
7. Once single colonies have grown, inoculate single colonies into flasks containing minimal media with (halophilic) and without (soil bacteria) 12.5% salt. Sterile control media should also be run parallel to ensure media used is not the source of bacterial growth.
8. Take a 2.5-ml sample for each culture and place it in a clear test tube. Grow these cultures at room temperature with shaking (~100rpm).
9. Measure the electrical conductance for each sample using the spectrophotometer. Label this set of reading as “Time Zero.” Also measure the conductance using highly pure water.
Days 5-14
10. Measure the conductance daily for each sample using a 2.5-ml aliquot. Record measurements for each successive day. Also remember to take a pure water reading with each day’s data (see spectrophotometer notes).
11. Enter collected data into a spreadsheet file or inputted into a graphing calculator for analysis.
Appendix C
Plans for Low Cost Spectrophotometer
Materials Needed for Spectrophotometer
|Glue, Glue gun or epoxy |D cell battery holder (Radio Shack #270-403 $.99) |
|¼ inch plywood or shoebox |6 alligator clips (Radio Shack #270-380a $2.39) |
|2”x 2” lumber under 6in long |1 high-output infrared LED (Radio Shack #276-143A $1.49) |
|D cell Battery |1 infrared phototransistor (Radio Shack #276-145 $.99) |
|4 feet #18 gauge wire |Small test tube or vial – 47-mm tall 12-mm diameter |
|6-volt lantern battery ($30) |Multi-meter that reads 20-200 milliamps |
(Prices may vary)
Construction of Spectrophotometer
The Case and Lid (See diagram below)
1. The case should be a least 20cm tall, 10cm deep and 20cm wide. The case that was built is about 12cm tall, 20cm wide and 12cm deep. A shoebox could be used instead, but this would not be very durable. We also found the height should be increased so larger test tubes can be used. Our 12cm version required the lid to be open to take readings, allowing outside light to affect our results.
2. Cut the sides out of ¼ inch plywood and glue to form a box.
3. Two pieces of ¼ plywood are needed for the top, one piece the same dimensions as the bottom of the box and the second piece having a length and width 1.5cm smaller. Glue the smaller piece of wood centered on the larger piece. This will make a good, nearly light-tight seal with the sides of the case. For the shoebox, ensure all holes are completely covered using duct tape.
4. Finally, drill four holes in the front of the box. These need to be large enough for the wires to run through to outside battery sources.
[pic]
The Sensor Block (See diagram below)
1. The block is first drilled from the top, in the center approx. ¾ the way through using a 5/8 inch drill bit (this size fits standard culture tubes snugly).
2. The block is next drilled horizontally from the side. The hole should be the diameter of the LED (The LED and photodiode are the same diameter). Drill completely through the block. It is important to also go through the center of the hole drilled for the vial.
3. Slide the LED in the horizontal hole on the left side. The LED goes as close to the vial hole as possible without hitting the vial. Secure the LED using glue or caulk.
4. Repeat the same process for the transistor but on the right side of the horizontal hole.
5. With the sensor completed, glue the sensor block in the center of the back wall with the leads exposed to the sides. The LED should be on the left and the transistor on the right.
[pic]
Instructions for Operation the Spectrophotometer (See diagram on next page)
1. Connect the wires on the right (LED side) to the battery holder for the 1.5-volt. If the multi-meter shows no reading or negative reading, the connections to the 1.5-volt battery holder may need to be reversed.
2. On the left side, connect one transistor wire to the 6-volt battery or power source. Connect the other left side wire (transistor) to the one multi-meter probe. Connect the wire with the alligator clips from the battery to the other multi-meter probe.
3. Turn the multi-meter to the milliamp settings. The writers used a 200-milliamp setting. You need to check that you are getting a positive reading. If the multi-meter reading is negative reverse the alligator clips connected to the 6-volt source.
4. If more than one low cost spectrophotometer is built, care must be taken to install the LED and transistor within the hole at the same angle (Perpendicular to the vertical hole). Different angles will result in each spectrophotometer giving different milliamp readings. One way to adjust the additional spectrophotometer units is to connect the circuit and glue the LED and transistor when adjusted to yield the same readings as the first spectrophotometer completed.
[pic]
Adjustment of Readings for Spectrophotometer
The spectrophotometer readings are in milliamps of current based on the light being transmitted through the media. As the bacteria grow, the amount of light being transmitted decreases. This will create data that decreases in value as the light transmittance is interfered with. To graph exponential growth of the bacteria, the data must be converted to calculate 1/n, where “n” is the reading taken from the spectrophotometer. The batteries also lose power, which affects the data analysis. To compensate, multiply the inverse of the multi-meter readings by the value of the water reading. The actual value calculated would be Water Reading/Sample Reading. An example would be:
On Monday your water reading was 43.2 milliamps and your media reading was 36.3 milliamps. The number you would graph would be: 43.2/36.3 = 1.19. On Tuesday your water reading was 42.6 and your media reading was 33.2. The number you would graph would be: 42.6/33.2 = 1.28. Even though the actual media reading was lower the second day, the amount you graph shows an increase – indicating bacteria growth.
The “Coug” Spectrophotometer
[pic]
The spectrophotometer is now ready to use. Be sure to place the top on the case when getting readings. Don’t rush when obtaining readings. The reading will oscillate when first used if the system has not been allowed to warm up. This system was found to work best when connected 5-10 minutes prior to use. This can easily be managed by having the students make the electrical connections before samples are collected. To prevent unnecessary battery loss, disconnect a wire from each battery and turn off the multi-meter when finished.
Appendix D
Media Preparation
Introductory Notes on Media
Pour about 100-ml media into 250-ml flasks and seal each flask with a piece of aluminum foil. The seal should allow air to enter in order for the bacteria to breath but should not allow airborne particles to enter for contaminating growth.
Materials needed (per student group)
|3 250-ml Beakers | |Sharpie Pens |
|Sandwich Bag for Obtaining Soil | |Inoculating Loops |
|18 Test Tubes (Over 10-ml Capacity) | |Burner |
|75 Petri Dishes | |Aluminum Foil |
|Plastic Pipettes | |Plate Spreader |
|Beaker w/ Ethanol | |Test Tube Rack |
|350 ml Culture Media | | |
Growth Media Ingredients: (Enough for 4 classes with 6-8 student groups)
Table Sugar – 1 LB bag
Table Salt (non-iodized) – 3 containers
Miracle Grow( – Granulated, small box
Borax( Soap – Regular sized box
Distilled water – 2 Gallons per class
Yeast Extract – 100 grams (Science Vendor)
Media Recipes (Makes 1000ml)
|Luria Broth (LB) |Media #1 |Media #2 |
|5 g. NaCl |5 g. Salt |5 g. Salt |
|10g Bacto( Peptone (Difco) |1.5 g. Miracle Grow( (Replacing the |1.5 g. Miracle Grow( |
| |KH2PO4 and NH4Cl) | |
| | | |
|5g Yeast Extract |1 g. Yeast Extract (Science vendor) |1 g brewers yeast boiled for 15 minutes – Cool to |
| | |room temperature |
| |4 g. Borax( (Replacing the Na2B4O7 x 10 |4 g. Borax( |
| |H2O) | |
| |10 g. Table Sugar (Replacing the Glucose)|10 ml Corn Syrup |
|Adjust pH to 7.0 with 5N |Adjust pH to 7.0 with 5N NaOH |Adjust pH to 7.0 with 5N NaOH |
|NaOH | | |
Notes:
Add 1.5% (w/v) agar when making plates.
The brewers yeast will produce some precipitate in media 2. Only use the top part of the media. Do not use the precipitate portion.
1. When using extreme halophilic bacteria, such as samples used in this SWEET module, salt concentrations need to exceed 12.5% and contain a pH of 9.0.
Appendix E
Dilution Plating
For samples that are to be plated to count cell viability, students need to understand the principle of serial dilutions before they start. While they can understand that by taking less sample along each time they will plate fewer bacteria, they may find it difficult to understand that exact measurements will allow them to calculate the number of colonies growing in a flask at any time point.
[pic]Looking at the dilution values above, bacterial counts can be calculated by counting the total number of bacteria growing on a plate multiplying it with the dilution exponent. If plating the entire volume of the final tube yielded 500 colonies, the original culture contained:
500 colonies X 105 = 500 X 105 or 5.00 X 107 cells
Note: The dilution factor changed from a negative exponent to a positive one. The sign needs to be changed since the measurement was increasing from the small value to that of the entire culture.
1. Take 5 test tubes and label them 10-1, 10-2, 10-3, 10-4, 10-5, respectively. Place these test tubes in your test tube rack. Add 9 ml of pure water to each.
2. Place 1ml from the culture and place it in the test tube marked 10-1.
3. Swirl the contents of this test tube by hand. Using a new pipette, take 1ml this tube and put it in the test tube marked 10-2.
4. Repeat step three for the remaining test tubes. Always use a clean pipette to prevent contamination from previous dilutions.
5. Obtain 5 petri dishes with bacterial media in them. Label them 10-1, 10-2, 10-3, 10-4, and 10-5. Also put your group name and date on each one.
6. On petri dish number 10-1, place 100(l of liquid from test tube 10-1. Spread this using the spreader. Do the same for the other five plates.
7. Incubate at room temperature for 48-72 hours.
8. Count colonies for each plate and determine culture value. This can now be used to correspond with the spectrophotometer reading collected for the same samples.
Appendix F
Competition Experiment
Bacterial species compete for resources within their defined ecosystems. These competitions lead organisms to find niches where they can develop specialized mechanism to succeed better than others in a defined habitat. However, when two species are placed in direct competition, one species can drive the second to extinction in that environment though the process of competitive exclusion. The following experiment will utilize the skills developed in designing a spectrophotometer and performing dilution plating.
Preparing cultures.
1. Cultures for the bacterial strains to be tested need to be grown separately prior to testing competition. Inoculate from the prepared petri plates streaked in the isolation protocol. Grow culture in 100ml of culture media (either homemade version or nutrient broth from science vendor) for 48 hours at room temperature.
2. Once cultures are observed being cloudy, use the spectrophotometer to create two approx. equal cultures through dilutions with water. This will make the population numbers approximately equal.
3. Once the two cultures are equal, inoculate a 100ml culture with 5ml of each starter culture. Incubate at room temp while shaking. After swirling the culture, remove a sample to obtain spectrophotometer readings for the initial time point.
4. Each class period, check the growth rates of the cultures by sampling the culture for spectrophotometer readings. This should be done for each sample.
5. By day 3-4 the cultures should show signs of slight cloudiness. When this is observed, in addition to the daily spectrophotometer reading, an aliquot needs to be collected to be used for dilution plating (see Appendix E).
Appendix G
Assessments
Quiz 1
Name_____________
Date______________
Define ecology.
What determines where species live?
Quiz 2
Name_____________
Date______________
Name three essential components of habitat.
Define limiting factors, and give three examples.
Limiting Factors Worksheet
15 Points
Name____________
Date_____________
Directions: Using the class discussion and your experience with the population activity, answer each of the following questions. Each question is worth 2 points. The graph is worth 3 points.
1. What do animals need to survive?
2. What are some of the limiting factors that affect survival in our activity?
3. Do wildlife populations stay the same or do they fluctuate?
4. Are ecological systems balanced or in constant change?
5. What would happen if most of the habitat components decide to be shelter? How does this affect the herd?
6. What factors may have contributed to population changes on the graph. Use your activity graph to analyze the population changes. (Graph will be turned in you’re your worksheet)
Quiz 3
Name_____________
Date______________
1. a) A tick stuck on your arm is an example of a _____________ relationship.
b) An oxpecker picking mites of the back of an elephant is an example of a _________________ relationship.
c) A bird nesting in a tree branch is an example of a ______________ relationship.
2. Humans are involved in several symbiotic relationships. List one example for each category and specify your reasoning.
Ecological Niche Worksheet
Name____________
Date_____________
Instructions: Using the habitat given by the teacher, answer the questions below.
(3 points each)
1. List the organisms inhabiting your assigned habitat (at least 10).
2. What resources do these organisms require?
3. What resources do these organisms provide to the community?
4. Label each species as “capable, not likely, or not possible” as able to live in one of the other three listed habitats. What does this tell you about species interaction?
Population Worksheet
Name_________________
Date__________________
Instructions: Fill in answers based on observations of your area and class discussion.
Draw four different producers in your area. Identify them if you can.
Name as many consumers as you can identify in your area. If there are none present, look for indications that they were there (i.e. dead insects, bones, chewed leaves, etc.).
Population Worksheet Cont.
What other populations probably visit or live in your square at other times?
From your list of consumers describe what each population probably eats.
Draw a food web for the community in your square.
Unit Test: Ecology
Part I
64pts Total
Name:_______________
Date:________________
Section:______________
Part 1: Matching (14 Points) Match the word from column (I) with its definition from column (II) by placing the assigned letter in the space provided. Answers can be used only once, but some may not be used at all. [2 points each]
Column I Column II
A. Biomass ____ 1. Animals that hunt and eat other animals
B. Community ____ 2. The relationship of all the food chains in a community
C. Consumers ____ 3. Organisms that eat both plants and animals
D. Ecology ____ 4. A group of populations that live together and depend
E. Food Web upon each other for food
F. Habitat ____ 5. Many members of the same species living together
G. Mutualism ____ 6. The total of all the food material in a community
H. Niche ____ 7. A relationship between two organisms where one benefits
I. Omnivore and the other is harmed
J. Parasitism
K. Population
L. Predators
M. Limiting Factor
Part 2: Multiple Choice (20 Points). Circle the letter of the best answer. [2 points each]
8. What causes wildlife populations to fluctuate?
(a) Wildfires
(b) Poachers
(c) Limiting Factors
(d) Parasites
(e) Geology
9. What is an animal that eats only plants?
(a) Producer
(b) Herbivore
(c) Photosynthetic
(d) Carnivore
(e) Parasite
10. What role does a species occupy in its community?
(a) Habitat
(b) Environment
(c) Niche
(d) Migration
(e) Food Web
11. Which symbiotic relationship is helpful to both organisms?
(a) Hibernation
(b) Partnership
(c) Commensalisms
(d) Mutualism
(e) Parasitism.
12. What are animals hunted by predators called?
(a) Dead
(b) Niches
(c) Prey
(d) Algae
(e) Scavengers
13. What do ecologists study?
(a) Interactions of People
(b) Interactions of Organisms and Their Environments
(c) Interactions of Pesticides and Water
(d) Interaction of Land and Water
(d) Echoes
14. What does the food chain transfer?
(a) Predators and Prey
(b) Energy and Nutrients
(c) Biomass of Consumers
(d) Consumers and Producers
(e) Biomass and Producers
15. What makes up food webs?
(a) Food Chains
(b) Communities
(c) Energy
(d) Niches
(e) Symbiosis
16. The number of individuals at each level of a food pyramid __________ as you move upwards.
a) Increases
b) Decreases
c) Remains Constant
17. Suppose a particular food chain looks like this:
hawk
|
rabbit
|
clover
Which of the following would be true?
a) The hawk is a prey species.
b) The rabbit population would increase if poachers killed all of the hawks.
c) If the clover were killed by an early frost, the rabbit population would decrease.
d) b and c are correct
e) a and c are correct
Part 3. Multiple Choice (12 Points). Use the Food Web below to answer questions 18 through 21. Select the best answer for each question. [3 points each]
Consider the community represented by the following food web:
humans coyote
weasel
wolf snake red-tailed hawk
white-tailed deer cottontail rabbit squirrel
corn crabgrass oak tree
18. Which species in this food web would cause the most serious disruption to the entire system if all of its members suddenly disappeared?
(a) Corn
(b) Squirrel
(c) Weasel
(d) Wolf
(e) Human
19. What would happen if two additional packs of wolves moved into the habitat where this food web existed?
a) The Oak Trees Die Out
b) Humans Produce Less Corn
c) Crabgrass Gets Infected
d) Red-Tailed Hawks Decrease
e) Coyotes Invade Cities
20. Which population will decrease directly from the added wolf packs?
(a) Humans
(b) Squirrels
(c) Corn
(d) White-Tailed Deer
(e) Snakes
21. Which would not help the new packs survive in the current food web?
(a) Heavy Rain Produce More Corn
(b) Wolf Pack Expand Hunting Territory
(c) Wolf Pack Find Alternative Prey
(d) Hawks Are Poached
(e) Human Hunting Seasons Extended
Part 4: Fill in the blank (10 Points). Write in the word or words that complete the sentence correctly. [2 points each]
22. To survive in its niche, an organism must be _________ to the demands of that environment.
23. Ecologists use a _________ to show the loss of energy at different levels of a food chain.
24. The organism that is harmed in a parasitic relationship is the _______________.
25. Green plants use ____________ to convert light energy to chemical energy.
26. A ___________ is a group of populations that live together and depend upon each other for food.
27. A population contains many members of the ____________ species living together
Part 5: Matching (6 Points). Read the paragraph below and answer questions 28 through 30 below based on the reading. [2 points each]
An elephant on the savannah eats the fruit of the marula tree. The seeds of this plant need to pass through the digestive system of the elephant before the seeds will germinate. While the elephant eats the fruit, ticks cover its back, imbedded in the skin sucking the elephant’s blood. Fortunately for the elephant, an oxpecker lands on its back and begins eating the engorged ticks.
(a) Elephant & Tick (c) Marula Tree & Tick (e) Elephant & Oxpecker
(b) Elephant & Marula Tree (d) Marula Tree & Oxpecker (f) Oxpecker & Tick
28. Which pair demonstrates a commensal relationship? _______
29. Which pair demonstrates a parasitic relationship? _______
30. Which pair demonstrates a mutualistic relationship? _______
Unit Test: Ecology
Part II
36pts Total
Name:_______________
Date:________________
Part 6. Essay. Answer each of the following questions using ecology terminology and proper English. Answers should be based on class discussion, activities, quizzes and worksheets.
1. An ecologist studying reduced trout populations has discovered that algae have overgrown a large stretch of a stream, directly adjacent to a fertilizer farm. Discuss what is happening to the environment, trout, and algae. [8 points]
2. Human and mosquitoes have an interesting ecological relationship. Explain how they relate to one another within their habitat. [8 points]
3. As a member of a community encompassing a sensitive environmental wetland, the town has called a meeting to discuss ways to deal with the economic needs of the commercial farmers who provide 75% of the town’s financial well being, the rare wetland, and its natural inhabitants. The farmers’ use of pesticides is vital for economic success, but the pesticides wash into the wetlands, affecting insects, fish, and birds. Consider the implications of the problem and argue for both sides at the town hall meeting. [10 points for each side of debate, 20 points total]
Answer Key
I. Selected Response
1. L 16. a
2. E 17. b
3. I 18. a
4. B 19. d
5. K 20. d
6. A 21. e
7. J 22. adapted
8. c 23. energy pyramid
9. b 24. host
10. c 25. photosynthesis
11. d 26. habitat
12. c 27. same
13. b 28. b
14. b 29. a
15. a 30. e
II. Essay
1. The environment is being affected by human development. The fertilizer farm next to the stream contributes rain run-off into the stream, enriching the stream section. This rich environment will allow natural algae to increase their population, growing everywhere they can obtain sunlight. By changing the algae population, the habitat is being altered as nutrients and spaces are being changed. This changed habitat affect trout as they are unable to swim through thick algae, and their own food sources are being affected. This places a limiting factor on the trout, causing their population to decrease.
2. Humans and mosquitoes are bonded in a parasitic relationship. The mosquito benefits from human by taking their blood at a cost to the host. Mosquitoes carry many blood borne illnesses including malaria and yellow fever. Mosquitoes are one of the few animals to be higher on the food chain than humans. Since humans are preyed on by the insects, this is a reversal for humans as top predators. This is however, cyclical as mosquitoes can also be below humans on a food web. Spiders that eat mosquitoes may move the nutrients and food energy through higher predators until they reach humans. Thus we are both predator and prey to one another.
3.
For protecting wetlands:
The rare natural wetland is a treasure for all to see. While it may not be obvious to people, these ecological systems also interact with our lives and damaging them affects us. Insects that help eat pests live and breed in these waters, but have been slowly dying. Without the wetlands, the our pest problems will be worse than if we had used no chemicals.
These are also stopping locations for migrating birds. Without this natural sanctuary, migrating birds will no longer make their ancestral route, leaving us without their spectacle.
What is most frightening is the pollution getting into our drinking sources. If they are killing off fish and birds, what will this water do to your children? We must cut back the use of chemicals and focus on natural methods. Instead of pesticides, we can use natural predators and create the same effects. Once we lose the wetland, it can never come back.
For Farmers:
Farming is a way of life for our town. Not only are jobs at stake, but the money those jobs create is the foundation of the community. By pulling the plug on a tool for growing crops, the town is asking farmer to handicap themselves and risk financial ruin. In addition, although it has been reported the use of chemicals has damaged the wetlands, no scientific study has been done to support this claim. Most of the chemicals are diluted and washed away, never reaching the wetlands. It is imperative that current farming methods continue to be used, or the community risk losing its economic base.
|Points Possible|Criteria Question 1 |Points Awarded |
|2 |Uses proper English with complete sentences. Writing indicates clear thinking. | |
|6 |Answer contains ecology (or environment), habitat, and limiting factors and how the farm| |
| |affects the river, the river affects the algae, the algae affect the trout. One point | |
| |for each | |
|5 |Student mentions 5 of 6 topics | |
|4 |Student mentions 4 of 6 topics | |
|3 |Student mentions 3 of 6 topics | |
|2 |Student mentions 2 of 6 topics | |
|1 |Student mentions 1 of 6 topics | |
|0 |Student mentions 0 of 6 topics | |
| |Total | |
|Points Possible|Criteria Question 2 |Points Awarded |
|2 |Uses proper English with complete sentences. Writing indicates clear thinking. | |
|1 |Answer mentions the parasitic relationship | |
|2 |Answer mentions mosquitoes higher in food chain. One point for statement and one point | |
| |for description. | |
|1 |Mentions humans can also be higher on the food chain. | |
|1 |Example of alternate web with humans at top (may vary). | |
|1 |Mentions predator/ prey in discussion. | |
| |Total | |
|Points Possible|Criteria Question 3 |Farmer Argument Points |Wetland Argument Points |Total |
| | |Earned |Earned | |
|3 |Uses proper English with complete | | | |
| |sentences. Writing indicates clear | | | |
| |thinking. | | | |
|2 |Mentions habitat destruction. 1 point | | | |
| |for example | | | |
|2 |Mentions pesticide pollution. 1 point | | | |
| |for example. | | | |
|1 |Mentions counter argument | | | |
|1 |Shows signs of persuasive writing. | | | |
|1 |Strongly written statement. Convincing.| | | |
|10 X 2 |Total | | | |
Appendix H
The Paper Folding Activity
Part 1: Number of Sections
|Number of Folds |Number of Sections |
|0 | |
|1 | |
|2 | |
|3 | |
|4 | |
|5 | |
|6 | |
|7 | |
|8 | |
|9 | |
1. Fold a sheet of paper (8.5 x 11 inch) in half and determine the number of sections the paper has after you made the fold.
2. Record this data in the table above and continue in the same manner until you can no longer fold the paper in half.
3. Make a scatter plot of your data.
[pic]
4. What happens to the number of sections as the number of folds increases?
5. Write a mathematical model that represents the data by examining the patterns in the table.
6. What might be different if you tried this experiment with a different type of paper, such as wax paper or tissue paper?
Part 2: Area of Smallest Section
|Number of Folds |Area of Smallest Section |
|0 |1 |
|1 | |
|2 | |
|3 | |
|4 | |
|5 | |
|6 | |
|7 | |
|8 | |
|9 | |
1. Fold a sheet of paper (8.5 x 11 inch) in half and determine the area of the smallest section after you have made the fold. Let the area of the original sheet of paper be 1 square unit.
2. Record this data in the table above and continue in the same manner until you can no longer fold the paper in half.
3. Make a scatter plot of your data.
[pic]
4. Compare this graph to the graph you drew in part one. What are the similarities? What are the differences?
5. Write a mathematical model that represents the data by examining the patterns in the table.
6. Will the area of the smallest section every equal zero? Why or why not?
Appendix I
Paper Folding Activity
|# of folds |# of sections |thickness | | | | | |
|n |2n |km | | | | | |
|0 |1 |0.0000001 | | | | | |
|1 |2 |0.0000002 | | | | | |
|2 |4 |0.0000004 | | | | | |
|3 |8 |0.0000008 |thickness of fingernail | | |
|4 |16 |0.0000016 | | | | | |
|5 |32 |0.0000032 | | | | | |
|6 |64 |0.0000064 | | | | | |
|7 |128 |0.0000128 |thickness of a notebook | | |
|8 |256 |0.0000256 | | | | | |
|9 |512 |0.0000512 | | | | | |
|10 |1024 |0.0001024 |width of hand (including thumb) | | |
|11 |2048 |0.0002048 | | | | | |
|12 |4096 |0.0004096 |height of a stool | | | |
|13 |8192 |0.0008192 | | | | | |
|14 |16384 |0.0016384 |1.6 m: an average person's height | |
|15 |32768 |0.0032768 | | | | | |
|16 |65536 |0.0065536 | | | | | |
|17 |131072 |0.0131072 |13 m: height of a two story house | |
|18 |262144 |0.0262144 | | | | | |
|19 |524288 |0.0524288 | | | | | |
|20 |1048576 |0.1048576 |105 m: quarter of the Sears tower (440 m) | |
|25 |33554432 |3.3554432 |past the matterhorn | | | |
|30 |1073741824 |107.3741824 |outer limits of the atmosphere | | |
|35 |34359738368 |3435.973837 | | | | | |
|40 |1.09951E+12 |109951.1628 | | | | | |
|45 |3.51844E+13 |3518437.209 | | | | | |
|50 |1.1259E+15 |112589990.7 |approx. distance to the sun (95 million miles) |
|55 |3.60288E+16 |3602879702 | | | | | |
|60 |1.15292E+18 |1.15292E+11 |size of the solar system? | | |
|65 |3.68935E+19 |3.68935E+12 |one-third of a light year | | |
|70 |1.18059E+21 |1.18059E+14 |11 light years | | | |
|75 |3.77789E+22 |3.77789E+15 |377 light years | | | |
|80 |1.20893E+24 |1.20893E+17 |12,000 light years | | | |
|85 |3.86856E+25 |3.86856E+18 |4 times the diameter of our galaxy | |
|90 |1.23794E+27 |1.23794E+20 |12 million light years | | | |
|95 |3.96141E+28 |3.96141E+21 | | | | | |
|100 |1.26765E+30 |1.26765E+23 |12 billion light years | | | |
Appendix J
Directions for Using the TI-83 Plus to: Graph a Scatter Plot,
Determine an Exponential Model,
And Graph the Model
To enter data:
STAT 1:EDIT ENTER
If there are entries in any of the lists, you can clear them by using the up arrow button to go to the top and then push DEL until all entries have been deleted. If there are multiple entries and/or if there are several lists occupied, you can delete all entries at once by pressing 2nd + (MEM) 4:ClrAllLists ENTER.
Enter data with L1 as the independent variable and L2 as the dependent variable.
To plot scatter graph:
2nd Y=(STAT PLOT) 1:PLOT1 ENTER
Plot on; type scatter (1st option)
X list: L1; Y list: L2; Mark; 1st symbol
To choose window and graph:
ZOOM 9:ZoomStat ENTER
YOU SHOULD SEE A SCATTER PLOT.
To find the exponential regression:
STAT CALC 0:ExpReg ENTER
THIS WILL GIVE YOU THE VALUES FOR a AND b FOR THE EXPONENTIAL FUNCTION y = abx.
To graph the line:
Y= VARS 5:Statistics EQ 1:RegEQ GRAPH
YOU SHOULD GET A CURVE THROUGH THE SCATTER PLOT.
Appendix K
M & M Lab
Exponential Growth & Decay Name_________________________
Date _____________ Period_____
Don’t eat the M & M’s yet!
Materials Needed:
50-100 M&M’s 1 paper or plastic cup
1 graphing calculator 1 paper towel or paper plate
Growth
A. Gather the data.
1. Start with 4 M&M’s in the cup. This is recorded as trial #1.
2. Shake the cup and pour the M&M’s onto the paper towel. Count the number of M&M’s that have the M showing. Be careful with the yellow M&M’s; it is hard to see the M.
3. Add a new M&M for each one with an M showing. Record the total number of M&M’s in the table below.
4. Repeat steps b and c, recording the new total of M&M’s each time, until you have completed 7 trials.
|Trial # |Number of M&M’s |
|1 |4 |
|2 | |
|3 | |
|4 | |
|5 | |
|6 | |
|7 | |
B. Graph the data. Create a scatter plot of trials (x) and total number of M&M’s (y) using the axes above. Be sure to label the axes and use an appropriate scale.
C. Use the graphing calculator to find the equation that will model the data from your table.
1. Clear lists. (2nd, MEM, 4:ClrAllLists, ENTER)
2. Enter data. (STAT, 1:Edit, and enter data into L1 and L2)
3. Find the values for a and b of the exponential function y = abx. (STAT, CALC, 0:ExpReg, ENTER)
4. Graph the exponential function found in the previous step. (Y=, VARS, 5:Statistics, EQ, 1:RegEQ, ENTER, GRAPH)
5. Compare the graph to the scatter plot.
D. Questions:
1. Write the exponential equation for this data:______________________
2. Predict the number of M&M’s on trial #9:_______________________
3. Predict the number of trials needed to have 300 M&M’s:___________
4. Explain the meaning of a and b in the equation:___________________
______________________________________________________
5. Suppose you made the beginning number (4) trial #0. How would your graph have changed?________________________________________
______________________________________________________
6. How would the equation have changed?_________________________
______________________________________________________
☺Don’t eat the M&M’s Yet! ☺
Decay
A. Gather the Data
1. Count out your M&M’s and record this at trial #1. Place all of the M&M’s into the cup.
2. Shake the cup and pour the M&M’s out onto the paper towel. Remove the ones that have the M showing. You may dispose of these M&M’s however you choose!
3. Count the remaining M&M’s and record this as trial #2. Then place them back into the cup.
4. Repeat steps 2 and 3 until the number of M&M’s remaining is less than 5 but greater than 0.
|Trial # |Number of M&M’s |
|1 |4 |
|2 | |
|3 | |
|4 | |
|5 | |
|6 | |
|7 | |
|8 | |
|9 | |
|10 | |
B. Graph the data: Create a scatter plot of trials (x) and number of M&M’s remaining (y) using the axes above. Be sure to label the axes and use an appropriate scale.
C. Use the graphing calculator to find the equation that will model the data from your table. Follow the directions from the previous exercise that modeled exponential growth.
D. Questions:
1. Write the exponential equation for this data:___________________
2. Use your mathematical model to predict the number of M&M’s that would be remaining after trial # 8:_______________________
3. Use the equation to predict the number of M&M’s remaining after trial # –1 (two times before trial number 1):___________________
4. Use the equation to predict the trial when there would be 900 M&M’s:_______________________________________________
5. Explain the meaning of a and b in the equation:________________
______________________________________________________
6. According to the graph and/or the equation, will the number of M&M’s remaining every equal 0? Why or why not?____________
______________________________________________________
☺You may now eat the M&M’s!☺
Appendix L
[pic]
CBL Experiment
Exploring Exponential Equations
Name
Date
Class
[pic]
[pic]
Thinking About the Problem
Remember the last time you drank a hot beverage? You probably had to let it cool for several minutes so that you could drink it without burning your tongue. You may not have realized it, but your hot beverage cooled according to a principle called Newton's Law of Cooling. In this exploration, you will use a CBL System and a temperature probe to investigate this principle.
30 60 90 120 150
Time (in minutes)
Suppose a cup of hot water (180° F) is placed in a room whose temperature is 70°. How long do you think it will take for the water to cool to room temperature? How do you think the hot water cools? Will it cool at a constant rate as shown in the figure at upper left? Will it cool rapidly at first, then more slowly, as shown in the figure at middle left? Or will the hot water retain its heat for a while, and then cool rapidly, as shown in the lower figure?
t = l + Ckt
One person who considered these questions was the English scientist and mathematician Sir Isaac Newton (1642-1727). After performing several experiments and studying the results, Newton determined that the temperature of a cooling object can be modeled by the exponential equation
Newton's Law of Cooling
[pic]
30 60 90 120 150 Time (in minutes)
where T is the temperature of the cooling object at time t, L is the temperature of the surrounding medium, C is the difference between the object's temperature at time t = 0 and the temperature of the surrounding medium, and k is a constant associated with the cooling object.
30 60 90 120 150 Time (in minutes)
Exploring Exponential Equations 49
Suppose at time t = 0, the temperature of a cup of hot water is 180° F. If the surrounding air temperature is 70° F, do you need to know the value of k to determine C? Explain your reasoning. Suppose that 1 minute later (t = 1), the water's temperature is 170°. Explain how to find an exponential equation that models the temperature of the cooling object. Write the equation below.
[pic]
Equipment Needed
For this experiment, you need the following equipment.
Basic Experiment
• CBL System (power adapter AC-9201 optional)
• TI-82/83 graphing calculator with unit-to-unit link cable
• TI temperature probe (provided with the CBL)
• Coffee cup
• Hot water
• TI-GRAPHLINK
Experiment Set Up
Set up the experiment as described and shown in the figure below.
1. Use TI-GRAPH LINK to download programs cooling and cooldata into your TI-82/83.
2. Connect the CBL to the TI-82/83 using the unit-to-unit link cable and the I/O ports located at the bottom edge of each unit.
3. Connect the temperature probe to the Channel 1 port (CH1) located at the right of the top edge of the CBL.
4. Turn on the CBL.
[pic]
[pic]
Temperature Probe
CBL
50 Calculator-Based Laboratory
[pic]
Recording the Results
1. The data collected in this experiment is stored in lists LI and L2. The times, in seconds, are stored in LI and the temperatures, in degrees Fahrenheit, are stored in L2. The graph you obtained in the experiment represents the temperatures with respect to time. If available, use TI-GRAPH LINK to print the graph. Otherwise, sketch it in the space provided below. Label the printout or sketch to reflect your data.
2. Use the graph to verbally describe how the thermometer cooled. Which of the graphs on page 49 best represents the data obtained in this experiment?
3. The room temperature and the initial water temperature are in list L4. The room temperature is L4(l) and the initial water temperature is L4(2). Record each temperature below.
Room temperature:
Initial water temperature:
52 Calculator-Based Laboratory
Note: A collection of temperatures is also stored in L3. These temperatures represent the temperature (also in degrees Fahrenheit) of the cooling thermometer minus the room temperature. You must use the data in L3 when performing an exponential regression analysis.
[pic]
Analyzing the Results
1. Use the following TI-82/83 keystrokes to perform an exponential regression analysis on the data in lists LI and L3.
| STAT | |">] | ALPHA | |~A~| | 2nd | | LI | [7j | 2nd | |U] |~ENTER |
See Appendix for TI-83 keystrokes.
Write the resulting equation below.
See Appendix for TI-83 keystrokes.
2. Graph the exponential equation obtained in Exercise 1. To do so, first clear any equations in the Y= editor. Then place the cursor on Yl and enter the following keystrokes to "paste" the equation to Yl and graph it.
If available, use TI-GRAPH LINK to print the display. Otherwise, sketch it in the space provided. How does the graph of exponential equation compare to the scatter plot?
____________________________________________________________________________________________________________________________________________________________________________________________________________________________
3. According to Newton's Law of Cooling, the temperature of a cooling object can be modeled by the exponential equation
T = L + Ckl
where T, L, C, k, and t are defined on page 49. Using the equation obtained in Exercise 1, identify C and k. Then use the room temperature, L, to write the exponential equation that models the temperature of the cooling thermometer. Write the model below.
C = T =
k =
Exploring Exponential Equations 53
[pic]
4. Do the values of L, C, and k seem reasonable? Why must k be less than 1? Explain your reasoning.
5. Graph the equation obtained in Exercise 3 and the scatter plot on the same viewing screen. Does the exponential equation appear to be a good fit? (You may want to change the window dimensions to better compare the graph and the scatter plot.)
6. Use the model to determine the thermometer's temperature 2 minutes after its removal from the hot water. According to your model, how long will it take for the thermometer to cool to room temperature?
7. According to Newton's Law of Cooling, the thermometer's temperature will never actually reach room temperature? Why?
8. Does the initial water temperature have any effect on the value of k? Investigate by repeating the experiment several times using the same water. (The water will cool between experiments.) Record any results and conclusions below. On what do you think the value of k depends.
CBL Explorations in Algebra. (1996) Meridian Creative Group. Erie, PA. pp 49-54.
Appendix M
ACTIVITY 7
Probing Intensity
A friend of the author's, who shall remain nameless to protect what's left of his dignity, had (as a child) a novel technique for preparing bright-colored, flavored drink mixes. This fellow, who grew up to become a respected scientist in the eyes of some, would pour the package of drink mix into a large, clear glass pitcher and then add water until the mixture was the "correct color." Then, like many ten-year-olds, he added sugar to taste. Usually, the only thing correct about the final mixture was the color. He depended upon the color of the liquid to indicate that he had used the proper amount of water. Sometimes this is a good method and sometimes it isn't.
When you measure the intensity of the color of a liquid by passing light through it, you are performing colorimetry. The intensity of the color of a liquid solution depends upon the concentration of the solution. If you are studying a solute dissolved in a solution, color intensity can provide important information about the nature of the solute.
Science Note
The colorimeter is a simple device that measures the amount of light that passes through a volume of liquid. To use a colorimeter, first pour liquid into a special flat-sided test tube called a cuvette, and place the cuvette in the colorimeter. Then direct colored light at the cuvette. (You will use a color of light that is complimentary to the liquid's color. In this case your liquid is green, so you will shine a red light through it.) The solution will absorb some of the light. The remaining light passes through the cuvette to the other side, where it activates a photocell. The photocell emits an electrical signal in proportion to the light. The colorimeter interprets the signal as an absorbance value—measuring the amount of light being absorbed by the liquid—and also as a percent transmittance value— measuring the amount of light passing through the liquid.
[pic]
In this activity you will use a colorimeter to measure the intensity of color of a solution consisting of food coloring and water. As you add food coloring to the colored water solution, you will measure the quantity of light that passes through the liquid. This is called percent transmittance (%T). The relationship between the concentration of a dissolved substance and the percent transmittance of light is a logarithmic function. You will evaluate the data from your testing to determine the logarithmic function that best describes your experiment.
Prediction
1. What is the independent variable in this experiment? ____________
2. What is the dependent variable in this experiment?
3. Predict the graph of your data. Sketch your prediction in the box on the right. Label the axes to indicate the independent and dependent variables.
Materials
• CBL System
Graph of your predicted results
• TI-82 or TI-83 graphing calculator containing the program MATHSCI and its seven subroutines
• Vernier colorimeter and accompanying plastic cuvettes
• adapter cable to connect the colorimeter with the CBL System
• graduated cylinder, 250-mL or 500-mL capacity
ACTIVITY 7 PROBING INTENSITY
• beaker or flask, 400-mL or 500-mL capacity, or 20-oz. plastic soda bottle
• green food coloring
• stirrer, plastic or glass
• medicine dropper or plastic Beral pipette
• tissues or low-lint towels
Procedure
Step 1 Use a graduated cylinder to measure 250 mL of tap water and transfer the water to a small beaker or flask. Add 2 drops of green food coloring to the water and stir the solution thoroughly.
[pic]
Step 2 Connect the colorimeter to the gray adapter cable and connect the cable to channel 1 of a CBL. Connect the CBL to your graphing calculator with a black link cable. Press the cable into each device firmly.
Step 3 Turn on the CBL and the graphing calculator. Press PRGM and select MATHSCI. Press until you see the MAIN MENU.
SELECT SENSOR
1: TEMPERAT URE 2: LIGHT
3: VOLTAGE
4: MOTION
5: PRESSURE
6: MICROPHONE
7: MORE SENSORS
Step 4 Set up the CBL for the colorimeter. To calibrate the colorimeter:
a. First prepare a blank by filling a cuvette 3/4 full with tap water.
To properly use a cuvette:
• Clean and dry the cuvette on the outside with a tissue.
• Hold the cuvette only by the ribbed sides.
• Make sure the liquid in the cuvette is free of bubbles.
• Place the cuvette in the colorimeter with a ribbed side facing you.
ACTIVITY 7 PROBING INTENSITY
b. Place the blank cuvette in the cuvette slot of the colorimeter and close the lid. Turn the wavelength knob of the colorimeter to the 0%T position. When the voltage reading shown on the CBL screen stabilizes, press [trigger) on the CBL and enter 0 in your calculator.
c. Turn the wavelength knob of the colorimeter to the Red LED position (635 nm). When the displayed voltage reading stabilizes,
press [trigger and enter 100 in your calculator. Note: Do not
touch the wavelength knob of the colorimeter for the remainder of the experiment.
d. Press | enter) to return to the MAIN MENU.
Select 2: COLLECT DATA, then select 2: TRIGGER/PROMPT. Follow the screen instructions until the calculator and CBL are ready to collect data.
Step 5
DATA COLLECTION
1:TIME GRAPH
2:TRIGGER/PRQMPT 3: MONITOR INPUT 4:RETURN
Step 6
Remove the blank cuvette from the colorimeter and empty it. Use a medicine dropper or a plastic Beral pipette to transfer a sample of the green water to the cuvette. (Fill the cuvette to about 3/4 full.) Place the cuvette in the colorimeter and close the lid. When the percent transmittance reading (a value
between 1 and 100) stabilizes, press trigger] . Enter 2 as the value, which refers to the number of drops of food coloring in
the solution. Press ENTER and follow the screen instructions to
Step 7
continue the experiment.
Remove the cuvette from the colorimeter and pour the green water back into the container.
Step 8 Add 1 drop of green food coloring to the container of green water.
Stir the solution thoroughly with a glass stirring rod. If you are using a soda bottle as the container, place the cap on the bottle and invert the bottle 3 or 4 times to mix the solution.
ACTIVITY 7 PROBING INTENSITY
Step 9 Transfer a few milliliters of green water to the cuvette and place the cuvette in the colorimeter. Take a second reading, similar to Step 6, then pour the sample back into the beaker. Remember that the value you enter in the calculator is the total number of drops of food coloring that you have added to your solution.
Step 10 Repeat Steps 8 and 9 until you have added a total of 6 drops of food coloring to the solution and you have taken 5 readings. When you have finished collecting data, select 2: STOP AND GRAPH on the calculator to examine your results. Press | enter] to return to the MAIN MENU. Follow the screen instructions to exit the program.
Step 11 Clean up your work area. Thoroughly rinse the cuvette, beaker, and graduated cylinder. Place these items on a paper towel to air dry. Dispose of your green water as directed by your teacher.
Analyzing the Results
1. On your calculator, display the graph of percent transmittance versus number of drops of food coloring added. This means your y-values should come from list 3 and your x-values should come from list 1. Sketch the graph of your results in the graphing box on the right.
Graph of your actual results
2. Describe any differences between the graph of your results and the graph of your prediction. Explain the differences.
3. Find the logarithmic function that best fits your data.
a. When you have a function that you think models the data well, record it here. ____________________________
ACTIVITY 7 PROBING INTENSITY
b. Explain how you found the function.
4. Display the graph of absorbance versus number of drops of food coloring added. You will need to adjust your STAT PLOTS menu to get this graph. Absorbance values are recorded in list 2 of your calculator. Find a function that best models this graph and record the function here.
5. Find a function to describe the relationship of absorbance versus percent transmittance. Set up your STAT PLOTS menu so that you can view a graph with list 3 as x-values and list 2 (absorbance values) as y-values.
6. Suppose you have tested a sample of green water and discovered that its percent transmittance of red light is 37.5%. Use your best-fit functions to answer the following questions.
a. What is the sample's absorbance? Explain your answer. ________
b. If the original volume of the sample was 500 mL, how much food coloring was mixed with the water? Explain how you found your answer.
7. (Research) Plan an experiment that tests a solution of a different color, considering the three colors of light available in the colorimeter.
TEACHER NOTES 7
Probing Intensity
Activity Overview
In this experiment students measure the amount of light that passes through samples of colored liquids of varying concentrations. Students use a device called a colorimeter that directs a beam of light through a cuvette. (A cuvette is a special type of test tube with flat sides.) The colorimeter will actually record three lists of data in the calculator. List 1 will show the number of drops of food coloring in the liquid, list 2 will show values for the absorbance of light by the liquid, and list 3 will show the percent transmittance of light. The equations students discover to model the relationships are both logarithmic and linear functions, depending on which lists they compare. In Analyzing the Results, students can do Problems 4-6 outside of class, if they have access to a calculator and the data.
Sensor Needed
Vernier colorimeter. In this case you will need the gray adapter cable to connect this to the CBL.
Program Needed
MATHSCI
Other Materials Needed
• green food coloring
• medicine droppers or plastic Beral pipettes for transferring the green liquid
• tissues or low-lint towels to dry off the cuvettes during testing
Math Curriculum Match
The relationship between percent transmittance of light and absorbance of light is best modeled with the logarithmic function A = 2 log(% T], where A = absorbance and % T= percent transmittance. When students plot absorbance versus concentration, the best-fit function should be a linear function. When they plot transmittance versus concentration, the best-fit function is also logarithmic.
Approximate Time Line for Lesson (in Minutes)
|SETUP |PREDICTION |PROCEDURE |CURVE FITTING |DISCUSSION |CLEANUP |TOTAL |
|15-20 |5 |10 |15 |10-15 |3 |60-70 |
TEACHER NOTES / ACTIVITY 7 PROBING INTENSITY
Percent transmittance versus number of drops of food coloring
Answers and Comments on Student Materials
Prediction
1. The independent variable in this initial experiment is drops of green food coloring added. (This is proportional to concentration.)
2. The dependent variable in this initial experiment is percent transmittance of light.
3. Answers will vary. The graph should show percent transmittance decreasing as the number of drops increases. The Introduction directs the students to seek a logarithmic function, so the graph should follow a decreasing log function. (This would be something of the form y = ~a log x + b, with a > 0.)
Procedure Hints
• If graduated cylinders are not available, use plastic 20-oz. carbonated beverage bottles to hold the green water. Instruct your students to fill the bottles to the bottom of the label; this will be about 250 mL of water.
• Plastic Beral pipettes work best for transferring the green water to the colorimeter cuvettes. A filled Beral pipette bulb is the proper amount of liquid to fill a cuvette (about 3 mL).
Analyzing the Results
1. Answers will vary, based on the type of food coloring used and its concentration. The first graph in 2 shows the percent transmittance versus number of drops of the sample results. The graph also includes the best-fit curve described below in 3a.
2. Answers will vary. Students may simply observe differences in slope between their predictions and their test results.
[pic]
[pic]
Absorbance versus drops of food coloring
[pic]
Absorbance versus percent transmittance
3a. Answers will vary. A function that fits the sample data is y= 129.8 - 49.31nx. This is graphed with the sample data in the first graph above.
3b. Answers will vary. This function was found using guess-and-check. An intercept value of 100 was used first and a logarithmic function y = -a logx+ 100 was graphed. Then the a-value and y-intercept were adjusted to best fit the data. A negative coefficient was chosen for the log term because the data show a decreasing graph.
4. Answers will vary. The plot of absorbance versus number of drops should be linear, as shown above in the middle graph. A function that matches the sample data is y = 0.092x - 0.161. The function for the sample data is displayed with the data in the graph.
5. Percent transmittance and absorbance are related by a logarithmic function. The third graph above shows this relationship, along with the sample data and a best-fit
TEACHER NOTES / ACTIVITY 7 PROBING INTENSITY
r
function for the data. The best-fit function is y = 2 - 0.434 In x. This is equivalent to y = 2 - logx. All students will get this same best-fit function, regardless of their original data. This is because the colorimeter uses this function to calculate the absorbance values from the percent transmittance values.
6a. Using the function y = 2 - 0.434 In x that models absorbance against percent transmittance, the absorbance is 2 - 0.4341n(37.5) or 0.427 when the percent transmittance is 37.5.
6b. Answers will vary. To solve this, use the function that compares percent transmittance with drops of food coloring added. In the sample data this function is y = 129.8 - 49.31nx from 3a above. Because the percent transmittance is 37.5, this number is substituted for y, producing the equation 37.5 = 129.8 - 49.3 In x. Solving for x, you get lnx= (37.5 - 129.8)/(-49.3), or x = e1.872, or x = 6.505. This means the number of drops added is 6.5. These equations model the experiment for 250 mL of green water, so for 500 mL of green water, 13 drops of food coloring must have been added.
7. Three interesting extensions can be set up using powdered drink mixes as the source of colored liquid.
• A lime (green) drink mix can be prepared, with or without adding sugar, according to the label directions. The colorimetry test can be done by making several samples that are dilutions of the original mixture. Start with 100 mL of the green drink, and dilute it by removing 25 mL and adding 25 mL of water. Continue dilutions in this manner for 5-7 tests.
• Prepare a solution of green liquid that contains a certain number of drops of food coloring and give it to your students as an unknown to test. Prepare a large sample, at least 1 liter, so that the students must use their test results to extrapolate an answer.
• A different color liquid can be used, requiring students to decide which of the three available wavelengths in the colorimeter works best for the new liquid.
Randall, Jack. (1998) Sensor Sensibility: Algebra Explorations with a CBL, a TI-82 or TI-83, and Sensors. Key Curriculum Press. Berkeley, CA. pp.55-63.
Appendix N
Modeling Exponential and Power Functions Examples
Example 1: The table give the number of cell-phone subscribers (in millions) from 1988 to 1997 where x is the number of years since 1987.
|x |y |
|1 |1.6 |
|2 |2.7 |
|3 |4.4 |
|4 |6.4 |
|5 |8.9 |
|6 |13.1 |
|7 |19.3 |
|8 |28.2 |
|9 |38.2 |
|10 |48.7 |
|ln x |ln y |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
|x |ln y |
|1 | |
|2 | |
|3 | |
|4 | |
|5 | |
|6 | |
|7 | |
|8 | |
|9 | |
|10 | |
• If (x, ln y) is more linear, then the best model is an exponential function.
• If (ln x, ln y) is more linear, then the best model is a power function.
Example 2: The table give the mean distance x from the sun (in astronomical units) and the period y (in Earth years) of the six planets closest to the sun.
|x |y |
|0.387 |0.241 |
|0.723 |0.615 |
|1.000 |1.000 |
|1.524 |1.881 |
|5.203 |11.862 |
|9.539 |29.458 |
Appendix O
Modeling Exponential and Power Functions Worksheet
Name_____________________________
Date_______________Period_________
Write an exponential function in the form [pic]whose graph passes through:
1. (1, 4) (2, 12)
2. (6, 8) (7, 32)
3. (3, 13.5) (5, 30.375)
Write a power function of the form [pic]whose graph passes through:
4. (2, 1) (6, 5)
5. (2, 10) (8, 25)
6. (2.9, 9.4) (7.3, 12.8)
7. You have just created your own Web site. You are keeping track of the number of hits (visits) to the site. The table shows the number of hits (y) in each of the first 10 months (x).
|X |Y |
|1 |22 |
|2 |39 |
|3 |70 |
|4 |126 |
|5 |227 |
|6 |408 |
|7 |735 |
|8 |1322 |
|9 |2380 |
|10 |4285 |
a.) Graph the scatter plot (x, y).
b.) Graph (x, ln y).
c.) Graph (ln x, ln y).
d.) Which model is more appropriate: exponential or power?
e.) Find the equation of the model.
f.) According to your model, how many hits do you expect in the 12th month?
g.) How many hits do you expect in the 34th month? What is wrong with that number?
8. The table shows the atomic number x and the melting point y (in degrees Celsius) for the alkali metals.
a.) Graph the scatter plot (x, y).
|Metal |X |Y |
|Lithium |3 |180.5 |
|Sodium |11 |97.8 |
|Potassium |19 |63.7 |
|Rubidium |37 |38.9 |
|Cesium |55 |28.5 |
b.) Graph (x, ln y).
c.) Graph (ln x, ln y).
d.) Which model is more appropriate: exponential or power?
e.) Find the equation for the model.
f.) Using your model, predict the melting point of Francium, whose atomic number is 87.
9. The table shows the cumulative number s of different stamps in the United States from 1889 to 1989 where t represents the number of years since 1889.
a.) Graph the scatter plot (x, y).
|t |s |
| 0 |218 |
|10 |293 |
|20 |374 |
|30 |541 |
|40 |681 |
|50 |858 |
|60 |986 |
|70 |1138 |
|80 |1138 |
|90 |1794 |
|100 |2438 |
b.) Graph (x, ln y).
c.) Graph (ln x, ln y).
d.) Which model is more appropriate: exponential or power?
e.) Find the equation for the model.
f.) Using your model, estimate the number of stamps in the United States in the year 2010.
g.) In what year would you predict the cumulative number of stamps in the United States to exceed 3000?
10. The table below shows the femur circumference C (in millimeters) and the weight W (in kilograms) of several animals.
|Animal |C (mm) |W (kg) |
|Meadow mouse |5.5 |0.047 |
|Guinea pig |15 |0.385 |
|Otter |28 |9.68 |
|Cheetah |68.7 |38 |
|Warthog |72 |90.5 |
|Nyala |97 |134.5 |
|Grizzly bear |106.5 |256 |
|Kudu |135 |301 |
|Giraffe |173 |710 |
a.) Graph the scatter plot (C, W).
b.) Graph (C, ln W).
c.) Graph (ln C, ln W).
d.) Which model is more appropriate: exponential or power?
e.) Find the equation for the model.
f.) The table below shows the femur circumference C (in millimeters) of four animals. Use your model to estimate the weight of each animal.
|Animal |C (mm) |W (kg) |
|Raccoon |28 | |
|Cougar |60.25 | |
|Bison |167.5 | |
|Hippopotamus |208 | |
Appendix P
Project Data
In order to design this module, several aspects needed to be tested to see if they were likely to succeed. We chose to focus on the collection and culturing of native soil bacterial populations and analyzing their growth using our constructed spectrophotometer. We chose to test three bacterial strains isolated: a red and white strain found in the Pullman area, and a yellow halophilic strain provided by Dr. Peyton (Fig. 1). The adjusted values calculated using the spectrophotometer are seen for the homemade media (Table 1) and the research media (Table 2).
Fig. 1. Streaked Isolates from Pullman and Soap Lake.
[pic]
Fig. 1 Isolated bacterial strains. A.) Halophilic bacterial media containing 12.5% NaCl and pH 9.0. B.) LB (Appendix D) bacterial media using standard protocol. Both plates were streaked with each bacterial isolate to compare their ability to grow on different media. In A. only the halophilic strain was able to grow. Conversely, in B. the native bacteria were able to grow and the halophilic strain was nonviable.
Table 1. Growth Curves Using Media 1 (See Appendix D)
|Day |0 |1 |2 |3 |6 |7 |8 |9 |10 |13 |
|red |0.56 |1.13 |0.93 |0.94 |0.91 |0.96 |0.97 |0.99 |0.97 |1.00 |
|white |0.59 |0.89 |0.93 |0.92 |0.93 |1.03 |1.03 |1.14 |1.14 |1.09 |
|yellow |0.54 |0.79 |0.84 |0.80 |0.86 |0.89 |0.89 |0.93 |0.91 |1.32 |
|control w/ salt |0.59 |0.75 |0.86 |0.84 |0.88 |0.86 |0.88 |0.92 |0.90 |0.93 |
|control w/out salt |0.72 |0.81 |0.92 |0.88 |0.99 |0.95 |0.97 |0.95 |1.00 |1.02 |
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The data indicates the homemade media is not optimal for culturing these strains of bacteria, as the growth was very limited over a large stretch of days (Fig. 2). However, when these same bacteria are cultured in a standard laboratory media (LB) the growth rates increase significantly. These growth rates more closely match the expected growth curves and are better suited for the mathematical application desired (Fig. 3). It is our recommendation that such media be used to obtain more effective teaching results.
Table 2. Growth Curves Using Luria Broth
|Day |1 |2 |5 |
|red 2 |0.937 |0.964 |4.77 |
|white 2 |1.006 |1.057 |7.77 |
|yellow 2 |1.476 |1.47 |1.85 |
|control w/ salt 2 |0.923 |0.87 |0.91 |
|control w/out salt 2 |0.96 |0.945 |0.93 |
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Room
temperature
Room
temperature
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y = ________________________
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A
B
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