From Dragonflywoman's blog

Click on this image or this link to Dragonflywoman’s blog to learn how to preserve insects in hand sanitizer….what a cool way to prepare insect specimens for the classroom.

http://dragonflywoman.wordpress.com/2011/02/21/hand-sanitizer-preservation/

BTW,  you’ll find a lot of great insect resources on her web site.  I think you’ll be impressed.

Cook Off!

All right, I confess.  I love cooking shows.  I can’t resist them.  As I enjoy cooking myself, I find it inspiring to watch well trained and creative food gurus work their magic.  How exactly do they hold the knife?  In their estimation, how much is a “handful”?  What pots, pans, and kitchen gadgets do they use?  When the directions say, “simmer until reduced”, what does it look like exactly?

When I was at the gym yesterday, I saw that Rachael Ray (the host of a regularly scheduled television cooking show, for those of you who don’t know) was featuring the World’s Biggest Cooking Demo on her show, I couldn’t help but get sucked in.  Sure it was a corny tactic, but I have to say, it was pretty darned clever.  The producer’s plan was for Rachael to prepare a chicken dish outdoors, live, in front of her entire New York City studio audience, with each audience member positioned at a mobile cooking station, following along with her. Hundreds of little cooking stations were set up, equipped with a hotplate, pans, implements, and all the ingredients required for the dish.  It was quite the scene – all of those audience-member-chefs, lined up, following along with Rachael.

But that’s not all – hundreds more cooks kept up with the proceedings in live Los Angeles studio audience and even more followed along with the demo in their homes, watching it on television, and Skyping in their questions.  Occasionally, Rachael would take a break from the demo (while the sauce was simmering) to take a Skype call from a viewer in Cleveland or Tampa.  The caller was projected on the big screen, beamed in from his or her kitchen, working away at the same recipe, asking for a clarification or offering a suggestion.

I thought the whole thing was brilliant.  Who doesn’t like to feel a part of something larger than themselves?  So, why not leverage that and turn it into an event?  While watching the proceedings, another thing came clear to me –  there was infinite variation and adaptation at work.  As the camera scanned the 100′s of cooks putting the chicken dish together, you could see the color and consistency variation in the sauce; some toasted their corn muffins, some didn’t; and the Skype callers had all sorts of ideas for varying the recipe, improvising on the procedure, and making it their own.  Subliminally, we all got the message that there was no one right way to do – the recipe was a guideline and experimentation off the basic plan was endless.

By now I’ll bet you can tell where I’m going with this….how about a National Lab-Off?  Imagine thousands of high school students all over the U.S. doing a photosynthesis lab together on one promoted day of the academic year.  One master teacher leading the event, providing a game plan from which everyone could improvise, experiment, and collaborate.  Live video feed piped into classrooms all over the country. A producer to manage the video feeds and Skype calls with questions.  A post-event blogging session to pool data, interpret results, and discuss conclusions?  What a way to generate enthusiasm for investigation while at the same time encouraging the use of participatory media tools!  Man, if Rachael Ray can do it with honey mustard chicken, surely it could be done with a biology lab?

Wickliff, J. L., and R. M. Chasson. 1964. Measurement of photosynthesis in plant tissues using bicarbonate solutions. BioScience 14, no. 3: 32–33.

In this article I saw this graph of a photosynthesis light response curve that got me to thinking:

Last year, the UKanTeach program where I teach acquired a couple of PAR (photosynthetically active radiation) meters to measure photon flux. PAR meters are typically on the expensive side but this model from Apogee runs about $300. I hadn’t taken time to try them out and decided that now was the time.

Yesterday, I went out the north side of Haworth Hall and picked an ivy (Hedera helix) leaf that was growing in deep shade under a shrub.

I picked a shade adapted leaf figuring that a leaf adapted to shade would likely reach photosaturation earlier than a sun adapted leaf. I wasn’t sure whether or not my light source was bright enough to induce photosaturation.

My light source is a clamp shop light with an 8 inch reflector and an 100 watt equivalent compact fluorescent bulb. Actually I found that if I put my meter within a couple of inches of the bulb I can get a flux reading equivalent to a summer’s day. I was sure my light was bright enough for the leaf I had picked.

I modified the technique that I presented here by placing the infiltrated disks in shallow petri dishes instead of plastic cups. I also modified the data collection procedure. Instead of counting disks floating at the end of each minute, I actually attempted to time each disk–a bit of a challenge that I wasn’t quite up to the first time. I should have used a video camera or at least used a computer timer program capable of timing 10 or more “laps” or intervals.

It is real easy to record the first movements of the disks with this technique.

In low light conditions, I started by carefully cutting about 80 disks from one leaf. I then infiltrated ten disks at a time with a dilute bicarbonate solution with a vacuum created with a 10 ml syringe. I placed the 10 sunken disks in separate petri dishes with a total of 30 mls of bicarbonate solution. The dishes with the disks were then placed under a box lid to exclude any light. I then tested 6 of the sets of 10 disks under different light intensities. The data from the highest light intensity are not included because I neglected to use a water heat sink filter to keep the infiltration solution temperature constant. The higher temperatures on this replication affected the outcome. It was only when the light was very close to the petri dish that this was a problem but I need to account for this next time.

Here’s the results:

Note that I’ve plotted plus or minus two estimated Standard Errors for each mean. I was impressed. This is a classic response curve and the parameters of this curve are consistent with data reported in the literature for shade grown English Ivy. I’m more convinced than ever that the floating leaf disk assay is a very valuable tool for a biology teaching laboratory. With this technique students can start their exploration of photosynthesis but the same technique is powerful enough to explore more sophisticated concepts.

Miniposter--Jai Hoyer

Background and Rationale:

Almost 20 years ago, I was fortunate to be invited to my first Bioquest Workshop at Beloit College. Maura Flannery covered the Bioquest experience in several her columns in the American Biology Teacher. These workshops challenge and inspire you as you work with a number of like-minded biology educators working on the edge of new developments. What really caught me off guard was the intensity of the learning experience. Before the end of the first full day, each working group had to produce a scientific poster presentation. This was my first, personal experience with building a poster so I’m glad that I don’t really have a record of it. I talked to John Jungck about the poster requirements—he told me that the students in his labs prepare a poster for each laboratory–rather than a lab-write up and they have to defend/present them in poster sessions. I immediately saw that a poster would help me evaluate my student’s lab experience while provide a bit of authenticity to my students doing science. That fall I had my students do a poster session that was displayed in the science hall. It was a big success with one exception. For my high school class, the experience was a bit too intense and too time consuming. It turned out that we could only work in one big poster session that year. One of the little bits of clarity of thought that comes from teaching for decades instead of years is the realization that students need to practice, practice, practice—doing anything just once is not enough. I thought about abandoning the poster session since it was too time consuming. However, I witness great learning by all levels of students with this tool. I didn’t want to abandon it. With this thought rolling around in my mind, I was primed as I visited one of my wife, Carol’s, teacher workshops. She’s a science teacher, too. In this workshop she was presenting an idea to help elementary teachers develop science fair project—a mini-science fair poster. This idea involved the used of a trifolded piece of 11″ x 17″ paper. The teachers were inputting their “required” science fair heading with post-it notes. Revision was a breeze. The teachers learned the importance of brevity with completion. They added graphs and images by gluing their graph to a small post-it. It was all so tidy, so elegant, so inviting, I probably stared a little long, struck dumb by the simplicity of the mini-poster. Once I came to my senses I realized that the mini-poster was my answer–a way to incorporate authentic peer review, formative assessment in my science classes. My high school classes could be like John’s college classes.

Making Mini-posters:

Putting the miniposter together

Over the years, mini-posters have evolved into the following. We take two colored (for aesthetics file folders, trim off the tabs and glue them so that one panel from each overlap—leaving a trifold, mini-poster framework. Each student gets one of these. For these posters we go ahead and permanently glue on headings that include prompts to remind the students what should be included in each section. Later, they can design their own posters from scratch. The image at the top of the page and the ones following will give you an idea. By using post-it notes the posters can easily be revised and we also reuse the poster template several times over the year. Don’t feel that you have to follow this design–feel free to innovate.

Implementing Mini-posters:

Defending the miniposter

For the first mini-poster experience, I give my students as much as a class period to work up a poster after completing an original research investigation. (We do quite a few of these early in the school year with others periodically throughout the rest of the year). Sometimes poster work is by groups and sometimes by individuals. Once the posters are ready, the class has a mini-poster session. The class is divided up in half or in groups. Half the class (or a fraction) then stays with their posters to defend and explain them while the other half play the part of the critical audience. To guide the critic, I provide each “evaluator” with a one page rubric and require them to score the poster after a short presentation. I restrict the “presentation” to about 5 minutes and make sure that there is an audience for every poster. We then rotate around the room through a couple of rounds before switching places. The poster presenters become the critical audience and the evaluators become presenters. We then repeat the process. By the end of the hour every poster has been peer-reviewed and scored with a rubric–formative assessment at its best. The atmosphere is really jumping with the students generally enjoying presenting their original work to their peers. The feedback is impressive. At this point I step in and point out that I will be evaluating their posters for a grade (summative assessment) but they have until tomorrow (or next week) to revise their posters based on peer review—oh, and I’ll use the same rubric. The process works very well for me and my students and my guess is that it will for yours as well. You’ll naturally have to tweak it a bit—please do. If you find mini-posters work for you, come back here and leave a comment.

The images are from our UKanTeach Research Methods course first assignment—a weekend research investigation.  Thanks to the Research Methods course for the images.

Here’s a file that illustrates what a miniposter might look like constructed in MS Word.

Links to websites for advice on making scientific posters:

http://www.swarthmore.edu/NatSci/cpurrin1/posteradvice.htm

http://www.ncsu.edu/project/posters/NewSite/index.html

http://people.eku.edu/ritchisong/posterpres.html

http://www.tc.umn.edu/~schne006/tutorials/poster_design/

http://www.the-aps.org/careers/careers1/GradProf/gposter.htm

On numerous occasions I have argued that trying to model H-W equilibrium in classroom with activities such as the AP Biology H-W lab, the M & M’s labs (http://www.accessexcellence.org/AE/AEPC/WWC/1994/mmlab.php) or with beans suffer from too small of sample size (population) or the models are simply too tedious for the students to explore.  Computer spreadsheets provide a unique environment that allow students to build and test their own models on how a population’s gene pool can change.  The testing, in particular, provides for a powerful learning experience for teacher and student.

Most spreadsheets have a “Random” function that can generate random numbers to model stochastic events.  Like flipping a coin or drawing a card at random in the AP Biology Lab 8 H-W lab the “Random” function serves as the basis for our spreadsheet model. Unlike our physical models/simulations (like the M and M’s lab) the computer can generate thousands of samples in a very short time.  The benefits to learning are worth the challenge of trying to learn how to build the spreadsheet.

In this post, I’ll present the essential parts of an EXCEL spreadsheet (other spreadsheets will work as well) that can be used to explore some of the first principles of the effects of population size on genetic drift.  In addition, this post is long-winded because I’m attempting to provide the strategies and questions I would use to encourage my students to develop their own computer-based models.  This is not presented as the definitive spreadsheet model or approach but rather a rather simplistic model to be constructed and modified by your students.  The idea is that if the students can find their way through building this model it can serve as a foundation as they extend the model to explore more of the parameters that affect H-W.

I’ve tried to break down the more complex model into a series of manageable steps.  I’ll cover the extensions to this model and others in future posts.  BTW, it takes longer to read this post than it takes to make this relatively simple spreadsheet. I suggest that you bring up EXCEL or some other spreadsheet in a different window and try to create this worksheet as you follow follow along.  Be thinking what questions you will ask your students so they can develop their own version of this spreadsheet.  Once you’ve mastered this and can create or modify it at will, then try it out with your students–they can handle this level of difficulty. They just don’t know it, yet–that’s your challenge as their instructor. And when they do succeed, with your guidance, they will have an effective tool to explore the basic principles of H-W equilibrium–one they have created themselves.

Step by step instructions follow, below the fold:

There are a number of steps to model building.  Otto and Day (http://www.amazon.com/Biologists-Mathematical-Modeling-Ecology-Evolution/dp/0691123446), in their book on mathematical modelling suggest the following steps:

1. Formulate the question.
2. Determine the basic ingredients.
3. Qualitatively describe the biological system.
4. Quantitatively describe the biological system.
5. Analyze the equations.
6. Checks and balances.
7. Relate the results back to the question.

Let’s get started working with the first four steps of model building.

Formulate the question:

Even before the computers enter the picture I would ask my students a simple, review question specific to Mendellian inheritance patterns.  Something like:

“Hypothetically speaking, you and your spouse recently decided to have your DNA analyzed by one of the new companies providing genetic testing.  Unknown to either of you since neither side of your family has had a known case of cystic fibrosis, you find out from the test that you are both carriers for CF.  What is the chance that your child will be born with CF?  What is the chance your child will be a carrier of the CF allele?”

As they work this out I’d ask a number of other questions that, step by step, move to this target–>”We’ve been studying inheritance patterns in families, how are inheritance patterns for the entire population different?”  One question that might help get us to this goal would be something like:  “How do recessive alleles stay in a population?”  “Don’t they gradually disappear?”  Obviously, I’m playing to a commonly held misconception but I want this on the table as the students investigate the question—”How do inheritance patterns or gene frequencies change in a population?”

Determine the basic ingredients.

Next I present some questions that might  help the students simplify the model.  I’m shooting for a single generation, model with discrete time steps, with an infinite gene pool, gametes picked from the gene pool at random, single locus, sexual reproduction, two allele, simple dominance, limited population size, no mutation, no selection, and no migration.

“What’s gene system should we explore or model?  (ans:  single locus, two alleles)  Diploid vs Haploid?  Sexually reproducing?  What’s the inheritance pattern?  (ans: simple dominance) How will we represent this?  (ans:  A and B alleles instead of A and a) What do you expect to change?  How could we model the change?  Change implies…? (ans: time as a variable–model in discrete steps to avoid differential equations)  How should we study alleles in a population–what kind of term/variable? ” (ans:  gene or allele frequencies–an operational definition).

At this point I’d ask a series of questions that would lead to the conclusion that computers would make a good tool for simulating gene frequencies in a population.  Finally, the discussion would finish with a consensus from the students to explore the use of spreadsheets to model the population.  (This could be relatively easy if you have used spreadsheets in the past.)

Qualitatively describe the biological system.

This is a critical area to cover with students as they develop their models.  Here, you will want to emphasize simplifications, assumptions and the limitations of the model.  Again, using series of questions such as:

“Imagine a hypothetical organism, can you describe its life cycle?  Can you describe how it reproduces?  Can you diagram the stages of this life cycle?”

To make this initial exploration into an HW model there’s some important assumptions that need to be made:  All the gametes go into one, infinite pool and all have a chance of taking part in fertilization or formation of a zygote.  For now, all zygotes live to juveniles, all juveniles live to adults and none enter or leave the population and there’s no mutation.  For the first model attempt use questions to come to a class consensus that only one generation will be explored.  Here’s what a diagram might look like this.

I’m not sure if this is the time to cover processes like selection, migration or mutation but I think not.  In my mind, at this point we just want to create a single generation model–where the gametes are selected at random for each zygote.

Quantitatively describe the biological system.

Build the Basic Spreadsheet together as a class:

Here’s one possible script:

Where should we start?  Let’s start with the frequency of two alleles in the population.  (usually I wouldn’t answer this question but sometimes it’s just best to just get going)

I’ve created a blue zone from A2 to D3—How do you do that?   Go ahead and create this blue zone.  (Highlight the cells, right click and select format cells) The blue zone on this spreadsheet represents the gene pool–you might want to label this and you can if you wish in row 1.  (At this point I’d tell the class that they don’t have to recreate the same spreadsheet—but it’s an easier way to begin.)

Note that the value for p is entered in cell “D2″ and the value for q is calculated by a formula in cell “D3″.  Make sure you enter a formula to calculate the value for q.  (you may need to offer some instruction, here.) Don’t forget to label these cells.

According to our model, the gene pool is assumed to be infinite and the selection for gametes for the next generation is assumed to be random. To accomplish this in the spreadsheet we call on the RANDOM function. Let’s see how it works.  If you were to pick an empty cell on the spreadsheet and enter the following function: =RAND() What do you get?  Hit the F9 key (manual recalculation) several times and tell me what you get. (a random number returned that was somewhere between 0 and 1.)  Our entire model is going to be based on this RANDOM function.  Unfortunately I’m not sure how random it really is but for our purposes it will work.

In cell “E5″ let’s generate a random number, compare it to the value of p, and then place either an “A” gamete or an “B” gamete in the cell.  We’ll need two functions to do this:  The RANDOM function and the IF function.  The smart thing would be to have you look this up but I think we’ll just put this into the spreadsheet.

Note that the function that is entered in cell “E5″ is: =IF(RAND()<=D$2,”A”,”B”)

Be sure to include the “$” in front of the “2″ in the cell address “D2″ ,  it will save time later when building on to this spreadsheet.

Which basically says, if a random number between 0 and 1 is less than or equal to the value of p then put an “A” gamete in this cell or if it is not less than or equal to the value of p put an “B” gamete in this cell. “IF” functions and “RAND” functions are very powerful tools when you try to build models for biology. Now create the same formula in cell “F5″–don’t just copy it sideways or if you do make sure that it is formatted exactly like “E5″.  When you have this completed then press the “F9″ key on your windows keyboard to force a recalculation of your spreadsheet. If you have entered the functions correctly in the two cells you should see changing values in the two cells.  (This is part of the testing and retesting that you have to do while model building that tends to reinforce the general principles in the learner.)

Finally, copy these two formulas down for about 16 rows that will represent 16 offspring for this generation.

We’ll put the zygotes in cell “G5″ . The zygote is a combination of the two randomly selected gametes. In spreadsheet vernacular you want to concantenate the values in the two cells: In cell “G5″ enter the function: =CONCANTENATE(E5,F5) and then copy this formula down as far down as you have gametes.

The next columns on the sheet, “H”, “I” and “J” are used for bookkeeping–keeping track of the numbers of each zygote’s genotype. They are rather complex functions that use IF functions to help us count the different genotypes of the zygotes.  Here goes:

The function in cell “H5″ is: =IF(G5=”AA”,1,0)

which basically states: if the value in cell “G5″ is “AA” then put a 1 in this cell, if not then put a 0.

Enter a very similar function in cell “J5″:  =IF(G5=”BB”),1,0)

Can you interpret this formula–what does it say in English?  (If the value in cell “G5″ is”BB” then put a 1 in this cell, if not then put a 0.

Now let’s tackle the nested “IF” function. This is needed to test for either “AB” or “BA”

In cell “I5″ enter the nested function: =IF(G5=”AB”,1,(IF(G5=”BA”,1,0))

This example requires an extra set of parentheses–something that is necessary to nest functions. This function basically says: if the value in cell “”G5″ is exactly equal to “AB” then put a 1; if not then if the value in cell “G5″ is exactly “BA” then put a one; if it is neither then put a 0 in this cell….

Copy these three formulas down for all the rows you have produced gametes.

Enter the labels for the columns you’ve been working on, “gametes” in cell “E4″, the label “zygote” in cell “G5″, the label “AA” in cell “H4, the label “AB” in cell “I4″, and the label “BB” in cell “J4″

There you have it. Copy the cells “C5″ through “H5″ down for as many zygotes as you’d like in the first generation. Sum up the values in the “F”, “G”, and “H” columns to summarize the genotype frequencies in the next generation, make a histogram and you end up with something like this:

With a model like this you can vary the number of offspring by inserting new rows and copying the formulas or by deleting rows to investigate the effect the size of the population has on the gene frequencies in the next generation. Those of you using the AP Biology Lab manual can use this spreadsheet to answer the questions in the Lab section 8B, Case I  and it should be pretty easy to modify it to answer others. That’s your challenge.  We’ll explore expanding this model in future posts.

In case you had trouble interpreting these instructions I’m also embedding a Zoho collaborative spreadsheet that you can click on to explore this model and how the cell formulas are entered:

http://sheet.zoho.com/public/ksbioteacher1/untitled-4

BW

Earlier posts in this series:

Sparrow Lab

Exponential Growth

Logistic Growth

Once my students have successfully modeled logistic growth, we sometimes wrap things up with discussions on the power and limitations of models followed with a sequee into human population growth models and predictions.  However, as often as not, if the class has begun to embrace modeling I’ll take them on a momentary side trip.  To begin this exploration, I ask them to identify the parameters in the equation that have the most impact as the parameter changes.  I ask that they systematically try out changes to N, to K and to r.  Generally, the students conclude that changes in N and K are pretty straight forward and predictable but that changes to r change the overall shape of the growth curve more dramatically.  Interestingly, since the inititial values for r that the class have used to this point have been less than “1″, I usually have to persuade them to try out values greater than “1″.  I ask them to record in their notebook, the shape of the growth curve for various values of r between 1 and 4.  Pretty soon I hear some “ohhhh’s” and some “wow’s” coming from different students as they discover the same type of patterns that Robert May (R.M. May (1976). “Simple mathematical models with very complicated dynamics“. Nature 261: 459) did back in the early seventies when he explored the effects of different r values on the logistic.  You can find more info at:

http://en.wikipedia.org/wiki/Logistic_map

There are some difference between their models and accepted models, but soon the students find that there is a point where there are two stable points that the “population” oscillates between–then four and then chaos…..

From Wikipedia

Now, I’ve got a problem—how far do we explore?  Generally, I leave this particular topic at this point with a few quick words on complexity and how small differences in intitial conditions can have profound effects on processes through time.  Mostly, my goal here is to introduce students to an entire other field that they might find fascinating and be able to link to their math, to their biology and to their physics.  To that end I challenge them to explore this topic on their own time–and many do, each year, bringing back all kinds of links, fractals, and applications of complexity theory in biology.

Earlier posts in this series:

Sparrow Lab

Exponential Growth

At the end of the exponential growth post I mentioned that mathematical models can be additive–perhaps I should have said modular.  The exponential equation developed in the earlier sheet now serves as the core for more sophisticated models.

At this point with my students I enter a conversation that explores what they see in the real world.  Do populations continue to grow exponentially?  Why not?  What factors might limit population size?  Eventually, using guiding questions we follow a path that leads to a new concept:  carrying capacity.   At this point, with student input, I sketch a graph on the board that has the x-axis labeled time and the y-axis labeled population size.  I then draw a horizontal line across the top of the graph that I label carrying capacity.  I ask the student to do the same on a piece of paper and then challenge them to sketch a line that represents a population that grows exponentially at first but as the population size approaches the carrying capacity the population growth slows and the population size levels off.   Eventually, the class agrees that a likely scenario would be an S-shaped line, with an increasing slope early on, with a transition zone where the slope changes to a decreasing slope and an eventual leveling.

With the target in mind, I bring the class back to their earlier spreadsheet model of exponential growth that had two terms:  N and r.   I ask a number of question such as:  “Which of the two terms change as the exponential equation is recalculated”  “Which term is constant?”  “If we wanted to modify the exponential growth curve into the S-shaped curve what has to happen to r?” (no longer constant)   At this point I introduce a new variable to the work:  “K” which represents carrying capacity.  (Naturally, there is further discussion about carrying capacity in the real world and in the model.)

Now, for the hard part—having the students come up with the logistic expression themselves.  First I remind the students about the algebraic form of the exponential equation that they represented their earlier spreadsheet:

Nt = N_(t-1) + r*N_(t-1)

The discussion has already focused on the “r” term which is in the second expression.  I ask the questions such:  “What part of the graph is population growth maximal?  minimal?”  “How can we change ‘r’ to maximize growth? minimize growth?”  “Now if the spreadsheet has a constant value of ‘r’ how might we change that value during the calculations?”    At this point I will introduce the idea of adding another expression to the equation–the logistic.  “Is there some mathematical expression that we could add to this equation that maximizes ‘r’ early but minimizes ‘r’ in later generations?”  “Can you think of an expression that includes just the N variable and the K variable that can be multiplied times ‘r’ to fill the needs of the model?”  or  “Can you think of an expression that is approximately equal to “1″ when N (the population size) is small but approximately equal to “0″ when N approaches K in size?”  At this point I let the students “discover” this expression themselves.  I ask them to try out the expressions they think will work in their spreadsheet.  To evaluate their proposed expression put it in the spreadsheet and use the graph produced to evaluate whether the expression works as planned.

The first time I tried this, the students took most of an hour and went through quite a bit of frustration.  I’m not really sure why I thought they could “empirically” determine this expression or what I thought they’d get out of it but I realized part of the value of the exercise when all of a sudden, one of the girls jumped up and yelled “Yeaaaah, I’ve got it”.  I decided to not have her share her strategy with the others—but instead prompted them to keep trying.  Eventually the entire class came around to the logistic expression:  (K-N)/K    Definitely a powerful experience.  The students learn that they can solve seemingly impossible problems with hard work but they also learn how to think about mathematical models in of biology.  It’s fairly easy to discuss  now, the limitations and the power of the model.  BTW, that first student is now a professional biologist.

I hope that you try to create this spreadsheet yourself before you ask students to do so.  Here is an example of how the spreadsheet model might be formulated.

Link to the spreadsheet in case the embed feature is not working.

Now for Part Two:

After the students build their own spreadsheet models of the hypothetical sparrow population, as a class we discuss the parameters that taken together determine population growth or decline.  I guide the discussion with questions until the students are able to articulate the four factors that determine population growth:  birth rate, death rate and migration (emmigration and immigration).  I am careful to make sure the discussion includes reviewing a working definition of a population and that the factors identified are rates and therefore have a time element to consider.  At this point we revisit the sparrow spreadsheet model and identify how these four parameters are taken into account in the actual cells of the spreadsheet.  Students quickly identify that there is no migration terms and that the birth rate is taken care of when each pair of sparrows produce 10 offspring (column D).  They have a more difficult time with the death rate.  There is no explicit cell with a death rate parameter but only the offspring in column D move to the next year–death is taken care of by omission.

The discussion then moves to asking the students to consolidate these four factors into one term–a per cent of increase or “r“.  We also establish a variable for the population size at any particular time interval: “N_t“.  The students are now challenged to represent the exponential population growth in a single equation with the variables “r” and “N_t“.

Eventually the class arrives at the following:

It’s now up to them to create the spreadsheet using the formula.  Interpreting normal algebraic notation into spreadsheet notation is a bit of a challenge but they usually figure it out.  By iterating the formula (using the results from one time interval as the basis for the next) the students can explore and create models that without computers would require a familiarity with calculus.  Once the students have created their simple model I have them expand it to 300 generations or time intervals and graph the results.  When trying this with your students make sure that you don’t get too explicit with your help—students have to work at building this model but it is doable for most.  You should try it as well before checking on the spreadsheet embedded here:

If the spreadsheet is not loading you can find it at: Exponential Model

There is one spreadsheet technique that you need to be aware of to make this model–the difference between relative reference and fixed reference in a cell’s formula. Since the reference to “r” always points to the same cell, it should not change. The default in a spreadsheet formula reference is “relative”, which changes. You can make a cell reference fixed by adding a dollar sign in front of both components of a cell’s address. For example referring to cell: B1 is a relative reference but referring to $B$1 is a fixed reference.

Normally when I explore this topic in my class, we can pretty easily get through the sparrow population model and the exponential model in one hour. Modeling is an additive process and this is only the start. Note that the procedure thus far has only added a bit of complexity at each step–with only rudimentary math operations. The next step will be to explore the logistic model. I try and reserve at least a day for it along with a homework assignment. I’ll cover the logistic in the next post.

BW

BW

These partnerships are not onlybeneficial for the individuals involved, but are wonderful opportunities for much needed public relations opportunties.  Eventhough I got a couple of bee stings out of the outing, the student’s comments after seeing drones, queen cells, and larvae make it all worth while.  I just took this picture yesterday.  Exploring the production of Defensin by Bees is a great partnership between these two people.  The bee keeper is very interesting in organic farming practice, and the student is looking at the bee’s natural responses to infection.  A match made in heaven.