Summary Post for Artificial Selection with Fastplants

Apparently, I never got around to providing a post that provides links to the series of posts I made back in 2013 that followed the AP Biology Artificial Selection Lab from start to finish with a number of hints along the way.  This post provides those links:

Growing Fastplants

Day Four

Day 7

Day 12

More of Day 12

Day 15

Day 17

Day 40

Second Generation

 

Plotting Error Bars in Google Sheets?…..on a scatter plot????

Robbyn Tuinstra, tri-athlete and AP Bio teacher extraordinaire recently had a question about putting error bars on scatter plot data in Google Sheets.  Several of us weighed in—a couple of us suggested it wasn’t possible, a couple of others pointed to a video where custom error bars were placed on a bar graph.  I mentioned that I had tried before to do this but gave up since I use other tools like Excel, Plotly and various stat programs.  Still this issue festered for a while and I finally had to try and attack it again.  I was partially successful.   I’ll describe what I have discovered but this also provides an opportunity to revisit suggested quantitative goals that the community might want to work towards.

First the type of experiment/data appropriate to this question.  Last year I produced a series of posts that featured a lengthy coverage of the types of data analysis and model application one might want to consider when doing a very simple lab–the yeast catalase floating disk lab.  You can find these posts on the Kansas Association of Biology Teachers Bioblog:  http://www.kabt.org/2017/02/06/summary-post-for-teaching-quantitative-skills/

I didn’t use google sheets in these posts but I will here.  Here is a data table of results that has already been transposed from disk rise time to rate of disk rise in floats per second.

This data table is typical of how we might record this types of data.  In the original postings I talked about how to plot this data and to do a curve fit.  Here’s one way to plot this data (in excel) using approx. 95% error bars (2 x SEM).

I think this is the type of data and plot that Robbyn was talking about.   The model for enzyme kinetics is known as the Michaelis-Menten equation and it can be used to fit the data.  I’m not sure we want to get into that in the AP Bio classroom but perhaps we do.  Nevertheless, I think we definitely should consider having students at least generate the graph.  The error bars are nice but I think when it comes to developing student argumentation from evidence that simply plotting all the data points along with the means is sufficient.  A plot that looks like this in Excel:


How do we do this in Google Sheets?

One of the first things to do to make this easier to plot is to change the data table into something like this:


Note that there is a column for the data points and a separate column for the means. This allows us to plot two dependent variable series on the graph.  We’ll use this strategy later.  Note that I have also added a 2 % substrate concentration and a 0% substrate concentration but I have left the rise time blank for these.  These x variables extend the range of of the x axis when we plot. 

Select these columns, choose Insert Graph and change to a scatter plot you end up with a plot that looks like this:

Here I’ve changed the size and color of the individual data points.


I won’t go into modifying your lablels, axis titles and titles.

Personally, I think this is more than adequate evidence to make the argument about the shape of this curve but I imagine in my classes we’d go for a non-linear curve fit (to help them justify the upper end math classes they are taking)

But perhaps, like Robbyn you want to include error bars instead of the data points for each substrate concentration.  This really doesn’t seem to be possible with simple menu options in Google Sheets.  (obviously, if you want to get into programming, it would be possible).  I did however find this work around.

First let’s change the data table again. Lets add a new column that has a calculated 2 x standard error of the means.  And another new column that includes values for [mean + (2 x SEM)] and [mean – (2 x SEM).]  Now the table looks like this:


Highlight the entire table, insert a chart BUT here is the thing.  If you highlight the data and let Google sheets determine the graph type it will pick Line Graph.  Let it this time.  That is key to what we need to draw the error bars.  You get something like this:

We have too many variables plotted.  We don’t need the individual data points now so we’ll get rid of those.  We will also turn off the plotting of the SEM (but not the plus or minus SEM).  Finally, select, use column A for labels (assuming you’ve put your substrate concentrations in column A.


Once that is done, we should be down to something that looks like this.  One variable plotted is the means and along with a line that connect plus 2 x SEM to minus 2 x SEM….

There you have it—a work around that works because by default Google sheets treats the blank cells in the plus or minus columns as null data–not zeros.  

p.s.

You can turn off that feature and the graph will look like this:


Obviously not what we want.  

Summary Post for Teaching Quantitative Skills

Part 1: Teaching Quantitative Skills using the Floating Disk Catalase Lab: Intro
Part 2- Teaching Quantitative Skills in a Lab Context: Getting Started in the Classroom
Part 3- Establishing an Experimental Procedure to Guide the Home Investigation
Part 4- Teaching Quantitative Skills: Data Analysis
Part 5- Curve Fitting AKA Model Fitting–the End Goal
Part 6- The Final Installment: Extending and Evaluating Quantitative Skills.
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These are links to the posts on Teaching Quantitative Skills with the Floating Disk Enzyme Lab

  1. http://www.kabt.org/2016/11/29/teaching-quantitative-skills-using-the-floating-disk-catalase-lab-intro/
  2. http://www.kabt.org/2016/12/01/teaching-quantitative-skills-in-a-lab-context-getting-started-in-the-classroom/
  3. http://www.kabt.org/2016/12/04/establishing-an-experimental-procedure-to-guide-the-home-investigation/
  4. http://www.kabt.org/2016/12/09/data-analysis/
  5. http://www.kabt.org/2016/12/18/curve-fitting-aka-model-fitting-the-end-goal/
  6. http://www.kabt.org/2017/01/06/the-final-installment-extending-and-evaluating-quantitative-skills/

The Final Installment: Extending and Evaluating Quantitative Skills.

A note:  You might want to scroll down, directly to Applying the NetLogo model to avoid my long winded setup and context)   

Getting Stuck in a Rut:

I grew up about 1 mile from the Santa Fe Trail which cuts diagonally across Kansas on its way from Independence, Mo. to Santa Fe, New Mexico.  And I have lived most of my adult life close to the trail.  Not everyone is familiar with this historical trail so here’s a quote from the Santa Fe Trail Association’s website that might put things into context:   “In 1821, the Santa Fe Trail became America’s first great international commercial highway, and for nearly sixty years thereafter was one of the nation’s great routes of adventure and western expansion. ”  For folks growing up on the plains, the trails are kind of a big deal.  For instance, along U.S. highway 400/50 in western Kansas you can pull over, park and walk in the ruts of the trail that still exist.  Here’s a Google Earth screen shot of the ruts trending to the SW. I have put a white polygon around the ruts.  Amazing, isn’t it?

 



More than 150 years have not erased these ruts.  How many wagons, people and livestock must have walked in these ruts, all with the same goal.  “Stuck in a rut” takes on additional meaning when you realize where the phrase comes from.  As you can see from this image as each of the ruts became “impassable” for the wagons they would start a new path parallel to it–still heading in the same direction with a focused goal.  Obviously, this highway opened up surrounding areas to Europeans but only if they got out of the ruts.  And just as obviously, this trail helped to set things in motion that eventually led to tragedy for the Native Americans.  That is another discussion.    But why bring up ruts on the Santa Fe trail as I finish out a series of posts about leveraging the yeast catalase floating disk lab to introduce and reinforce a plethora of quantitative skills to biology students?

Well, the short answer is that I think we, the teacher community, are particularly at risk of getting “stuck in a rut.”  Like the folks on the Santa Fe trail we are often looking for direct, point to point solutions for many of the challenges that surface in a classroom of students who all have different skills and backgrounds.  Take for example, “The Scientific Method”.  Here, was a simplification designed originally by Paul Brandwein to make science as a verb more accessible to both teachers and students.  Of course it was a simplification and of course, if Paul were still here, he’d be appalled at how one-dimensional this model has become.  We do that in science education–we make ruts—deep ruts.  Another example, that strikes close to home is the former AP Biology Lab manual–a series of labs that became known as the “Dirty Dozen” that folks felt they had to follow to the letter while almost always neglecting or ignoring the suggestions at the end of each laboratory for further, deeper investigations–another deep rut.

As many of you know, I’ve spent the last 9 years helping to prepare math and science teachers in the UKanTeach program.  In this program we introduce the students to the 5E lesson plan model to help them prepare effective (and efficient) lessons that are steeped in inquiry.  The design works fairly well and really serves as a great scaffold to build an effective lesson or series of lessons around.  Those of you familiar with the model may recognized that one could deconstruct these series of posts down into the 5E’s.  Engage, Explore, Explain, Extend, and Evaluate.  But, to avoid our basic nature of creating a rut to fall into, I’ve purposely left out any explicit designation, notation or label consistent with the 5E’s.  Part of that is because, I think you can see that some folk’s Extension activity might be another’s Evaluation activity.  It depends on the context in my opinion.  Perhaps more importantly, is that I don’t want to suggest that teaching quantitative skills is a linear process.  Following linear paths, creates ruts.  Instead, I hope I presented a multitude of paths or at least suggested that this lab is a very rich resource that opens all sorts of options you might consider.  It is up to you, the trail boss to decide how you plan to guide your class over the quantitative skills landscape and hopefully, you’ll find it rewarding to the point of taking quantitative skill instruction beyond what I’ve suggested here.

With that said, I am going to present material here that might fit more appropriately in the Extend or Evaluate phase of a 5-E lesson.  I see a couple of paths forward.  One takes the quantitative and content level skills learned in this exploration and applies them in an another laboratory investigation and the other takes those same skills but applies them in model-based environment.  Doubtlessly there are many other paths forward for you to discover but let’s focus on these for now.

A model environment that probes deeper into thinking about enzyme reaction kinetics:

But first some more history/ reminiscing.

In the 1980’s when personal computers first arrived on the educational scene one of the first applications were programs that provided simulations of biological phenomena.  I even wrote one that students could use to generate inheritance data with simulated fruit fly crosses.   I was pretty proud of it to the point that I actually marketed it for awhile.  Students had to choose their parent flies with unknown genotypes from primitive graphic images that provided phenotype information.  Once a cross was chosen, then the program would randomly according to the inheritance pattern generate about 48 fly images that represented the phenotypes possible.  The student had to infer genotypes from phenotypes.  However, when I wrote this program I created an option where the student could pick and choose the inheritance pattern to investigate.  So the program  only simulated data to confirm a given inheritance pattern. The data was realistic since it used a random function to generate gametes but it could have promoted more inquiry and scientific thinking.  I found this out when I cam across the Genetics Construction Kit (GCK) a piece of software written by John Calley and John Jungck.  This program lacked the graphics that mine had but it promoted inquiry much, much better.  Students didn’t start by choosing an inheritance patter.  Instead, they received a sample “vial” of flies with a number of different traits expressed.  They had to choose different flies, different traits and such and then design crosses, form hypotheses, look for data to support those hypotheses and go to work.  It was a revelation.  Even better to my way of thinking it “modeled” almost every type of inheritance you could study in those days.  Even more better—the program didn’t tell the student if they were right or wrong.  The student (and the teacher) had to look at the various crossing data to determine if the data supported their hypothesis.  This was an excellent educational tool to teach genetics.  If you and your students could meet the challenge I guarantee that the learning was deep.  (Collins, Angelo, and James H. Stewart. “The knowledge structure of Mendelian genetics.” The American Biology Teacher 51.3 (1989): 143-149.)   If you don’t have access to JSTOR you can find another paper by Angelo Collins on the GCK here:  Collins, Angelo. “Problem-Solving Rules for Genetics.” (1986).  I promoted the GCK heavily throughout the late 80’s and 90’s.  I still think it is an inspired piece of software.  The problem was that little bit about not providing answers.  Few of my teaching colleagues were comfortable with that.  I’d round up computers or a computer lab for a presentation or professional development.  Everything would be going smoothly.  There would be lots of ooh’s and aw’s as we worked through the introductory level and then everything would so south when the teachers would find out that even they couldn’t find out what the “real” answer was.  Over and over, I’d ask; “Who tells a research scientist when they are right?” but it wasn’t enough.  Teachers, then were not as comfortable without having the “real” answer in their back pocket.  I think that has changed now at least to some degree.

The software world has changed as well.  GCK went on to inspire Bioquest.  From their website:  “BioQUEST software modules and five other modules associated with existing computer software, all based on a unified underlying educational philosophy. This philosophy became known, in short, as BioQUEST’s 3P’s of investigative biology – problem posing, problem solving, and persuasion.”  Another early “best practice” educational application of computer technology was the software LOGO from Seymour Papert’s lab.  LOGO was agent based programming specifically targeted to students early in their educational trajectory.  My own children learned to program their turtles.  As the web developed and other software environments developed LOGO was adapted into NETLOGO.

Netlogo (Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.) is a modeling environment that is greatly underutilized in the educational community.  Netlogo is an agent based programming language.  There is a desktop version and there is a web browser version.  Agent based models can provide very interesting simulations models of real world. Agent based programming assigns various properties to individual “agents” along with a set of rules for how this agent interacts with other agents or with the “environment”.  No doubt you and your students will gain the most learning if you could explore coding and building your own model but unless we do this multiple times during the year, the time requirement to build these skills is usually prohibitive.  You don’t have to be a coder, though to utilize these tools.  Luckily the Netlogo community has already put together a model on Enzyme Kinetics that can extend your student’s understanding of enzymes.  (Stieff, M. and Wilensky, U. (2001). NetLogo Enzyme Kinetics model. http://ccl.northwestern.edu/netlogo/models/EnzymeKinetics. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.)

But it is not always clear how to take advantage of NetLogo simulations as a teaching resource.  Typically students tend load up one of the simulations, click on a few parameters, get some kind of graph that they don’t understand and then shut down—the program and their brain.  Models environments like this require systematic exploration much like a laboratory investigation.  And, just as you try to engage your student in lab based inquiry where they are thinking for themselves and not following a bunch of cookbook like instructions, you need to find ways to encourage your students to engage deeply with the model.  Parameters need to be changed in response to questions—not just to see what happens.  Seeing what happens with random changes can lead to insights but generally it is better to have a plan–to explore the model systematically.  This type if involvement or engagement by the student requires a bit of student motivation.  There are many sources of student motivation and Mihály Csíkszentmihályi‘s FLOW theory applied to education is a good place to start.  In FLOW theory the idea is to seek a good balance between the challenge before you and skills you bring to the table.  One of the important lessons I’ve learned over the years is that using model simulations requires every bit of set-up and preparation as a “normal” wet lab does.  And, more to the point for this example is that working with the actual physical/biological phenomenon ahead of time helps to create enough first-hand knowledge, beginning skills and such that working with models becomes more accessible to students.  Students aren’t used to working with software like this and it takes a bit of preparation to get them to explore the model in a productive way.  In this example of the floating disk catalase lab the students by this time will have explored models to develop a beginning conceptual understanding of enzyme reactions, designed and carried out experiments, collected and analyzed data, and perhaps have fit their data to mathematical expressions.  Hopefully, they are developing deep understandings of enzyme action that now need to be tested, reflected upon, and revised.  While continued exploration in the laboratory can provide that environment of reflection and revision the time and space limitations of a typical classroom likely prohibits a robust exploration.  This is where a simulation like the NetLogo Enzyme Kinetics can play a vital role in student learning.  Here the student or student teams can explore and manipulate all sorts of variables in a relatively short period of time.

Applying the NetLogo Model:

The NetLogo model is built on the same conceptual model for enzyme kinetics that we have explored before:


By Thomas Shafee (Own work) [CC BY 4.0 (http://creativecommons.org/licenses/by/4.0)], via Wikimedia Commons
 
Instead of generating an expression with Vmax and Km, though the agent based model assigns properties to the agents based on three “constants”.

Constant 1:  The rate of formation of the ES complex.

Constant 2:  The rate of dissociation of the ES complex back into E and S

Constant 3:  The rate of catalysis of the ES complex into E and P

You explore the model by changing these constants or changing the substrate concentration.  Changing the constants, changes the properties of the enzyme.

http://ccl.northwestern.edu/netlogo/models/EnzymeKinetics

I’ve put together a short video of that introduces how one might work with this model to create data similar to the data from the original wet lab.  You can find it here:

https://youtu.be/hPwqVlHvzMA

Here’s a M-M curve that I generated by changing the values of the constants and then seeing how those properties determined Enzyme rates/velocities at differing substrate concentrations.

In this curve I collected, 8 samples for each substrate concentration.

Here’s the data, generated by the model.  Looks a lot like the wet-lab data, doesn’t it?

And here is a curve fit to the Michaelis-Menten equation.  Note that the data from the NetLogo model has to be fitted to the idealized curve.

Note that the data from the NetLogo model has to be fitted to the idealized curve.

The thing is that I could go back into the Netlogo model and explore questions like, what happens if I lower the constant that describes the rate of Enzyme-Substrate formation relative to the constant that describes the dissociation of that complex?  Several questions come to mind.

Of course you don’t have to explore Netlogo as an extension or evaluation activity. You could have your students explore this spreadsheet from the Bioquest Esteem project:

Michaelis-Menten Enzyme Kinetics

Or if you are really ambitious you could have your students develop their own spreadsheet model like the one described in this paper from Bruist in the Journal of ChemEd.

Bruist, M.F. (1998). Use of a Spreadsheet To Simulate Enzyme Kinetics. Journal of Chemical Education, 75(3), 372.  http://biochemlab.org/wp-content/uploads/2011/06/Spreadsheets-in-enzyme-kinetics-Bruist.pdf

Or you could have your students explore the AP Biology Community’s own Jon Darkow Stella-based model for lactase enzyme activity:  https://sites.google.com/site/biologydarkow/lactase-enzyme-simulation  This is an HTML5 version created in the dynamic modeling system known as Stella.  

Practice, Practice, Practice (Curve fitting in the Photosynthesis Floating Leaf Disk lab)

To master any skill takes lots of practice–something we don’t provide enough of in academic classes.  We do in the performance and fine art classes but not so much in academics.  The excuse as to why not usually gets back to the extreme time limitation we face in the biology classroom.  Still with the right selection of lab topics skill practice is not only possible but highly productive.  For instance in this case, it turns out that the procedure, the data created, the data analysis, the curve fitting (to the same mathematical model) are all skill that can be applied to the Floating Leaf Disk lab, if the students explore how the intensity of light affects the rate of photosynthesis.

In 2015, Camden Burton and I presented some sample data sets from the Floating Leaf Disk lab at NABT.  Later I shared those in a series of posts on the AP Biology Community forum where a lively discussion on data analysis ensued.  If you are a member of the forum you can find the discussion here. 

One of the more problematic data sets we shared was data from a photoresponse curve experiment that explore how light intensity affects the rate of photosynthesis.  Here’s a photo of how light intensity was varied by varying the height of the stack of petri dishes.

 

 

Here’s the raw data for this lab using the lap timer on a smart phone:

 

The first step working with this data is to convert the lap times into cumulative times along with generating the descriptive stats.

 


Because the how fast a disk rises with this technique is inversely proportional to the actual rate of photosynthesis we need to convert this time into a rate by taking the inverse or the reciprocal.  And since this turns out to be a small decimal number with the units of float/sec, I’ve modified it by multiplying by 1000 seconds to get a rate unit of float per 1000 seconds.  The descriptive stats are converted/transformed in the same way.  This process of data transformation is not emphasized enough at the high school level in my opinion.

 


Graphing the means in this data table along with plus or minus 2 SEM error bars creates a graph something like this:

Which in my mind is a curve waiting to be fitted.  If you google something like “Photosynthesis Irradiance Curve” you’ll find a number of resources applicable to this experiment and guess what?  You’ll find that folks have been using the Michaelis-Menten equation to model the curve fitting.

I’ll let you explore the resources but here is the fit based on the Michaelis-Menten equation.  There is a modification to the Michaelis-Menten expression that we have to do for this particular lab.  Since this procedure actually is measuring the accumulation of oxygen as a product and some of the oxygen is being consumed at the same time for cellular respiration, we are actually measuring the net rate of photosynthesis.  To account for the oxygen consumed in respiration we need to add an additional term to the Michaelis-Menten equation.

I’ve changed the variables but the form of the equation is the same.  In the curve fitting that I have done, I have manually changed the value of R and let the solver vary Pmax and KI.

The fit for this set of data is not as good as we got for the catalase lab but it is not bad.

Interestingly, you can get a “good” fit to an exponential function as well–maybe even a better fit.  But, that is part of model fitting.  There is many biological reasons to consider that Michelis-Menten provides a model for photosynthesis but I can’t think of one for an exponential fit.  There are many ways to continue to modify the Michaelis Menten application to Photosynthesis Irradiance curves and you can find several with a bit of google searching.

Here’s one fit I managed in excel using the same techniques that we used earlier.

Here is a Desmos version you or your students can play with.

 

 https://www.desmos.com/calculator/dsfe5xfwyi


I think it is time to wrap this series of long winded posts up.  I hope, if you’ve read this far, that you have found some ideas to try in your class and I hope that despite the deep dive that an idea of how an increased emphasis on quantitative skills can also lead to an increase understanding of the content–at least it does for me.  Finally, I hope you and your students have a good time exploring data analysis—it really does feel good when the data works out like you think it should. 😉

 

Curve Fitting AKA Model Fitting–the End Goal

Curve Fitting AKA Model Fitting:
When I started this series of posts my goal was to see if I could generate precise data with a proven classroom lab.  The data precision that is possible with the yeast catalase lab provides a unique opportunity where data analysis skills can be productively explored, practiced and understood.  My contention was that this is the ideal lab to focus not just on content, not just on experimental design, but also to introduce relatively sophisticated data analysis.  To be up front about it, I had only a hint of how rich this lab is for doing just that.  Partly , this is because in my years of teaching high school biology I covered most of the enzyme content in class activities and with 3D visualizations, focusing on the shape of enzymes but neglecting enzyme kinetics.  That would be different if I were teaching today—I’d focus more on the quantitative aspects.  Why?  Well, it isn’t just to introduce the skills but it has more to do with how quantitative methods help to build a deeper understanding of the phenomena you are trying to study.  My claim is that your students will develop a deeper understanding of enzymes and how enzymes work in the grand scheme of things if they follow learning paths that are guided and supported by quantitative data.  This post is an example.
The last post focused on plotting the data points as rates, along with some indication of the variability in each measurement in a plot like this.
As I said before, I would certainly be happy if most of my students got to this point as long as they understood how this graph helps them to describe enzyme reactions and interpret others work.
But a graph like this begs to have a line of best fit–a curve that perhaps plots the relationship implied by our data points.
Something like this.

One of the early lessons on model building in my current Research Methods course involves taking data we have generated with a manipulative model (radioactive decay) to generate a predictive model.  The students plot their data points and then try to find the mathematical expression that will describe the process best.  Almost always, my students ask EXCEL to generate a line of best fit based on the data.  Sometimes they pick linear plots, sometimes exponential, sometimes log plots and sometime power plots.  These are all options in EXCEL to try and fit the data to some mathematical expression.  It should be obvious that the process of exponential decay is not best predicted with multiple types of expressions.  There should be one type of expression that most closely fits the actual physical phenomenon–a way of capturing what is actually going on.  Just picking a “treandline” based on how well it visually fits the current data without considering the actual phenomenon is a very common error or misconception.  You see, to pick or develop the best expression requires a deep understanding of the process being described.  In my half-life exercise, I have the students go back and consider the fundamental things or core principles that are going on.  Much like the process described by Jungck, Gaff and Weisstein:

“By linking mathematical manipulative models in a four-step process—1) use of physical manipulatives, 2) interactive exploration of computer simulations, 3) derivation of mathematical relationships from core principles, and 4) analysis of real data sets…”
Jungck, John R., Holly Gaff, and Anton E. Weisstein. “Mathematical manipulative models: In defense of “Beanbag Biology”.” CBE-Life Sciences Education 9.3 (2010): 201-211.
The point is that we are really fitting curves or finding a curve of best fit–we are really trying to see how well our model will fit the real data.  And that is why fitting this model takes this lab to an entirely new level.   But how are you going to build this mathematical model?
Remember that we started with models that were more conceptual or manipulative.  And we introduced a symbolic model as well that captured the core principles of enzyme action:

By Thomas Shafee (Own work) [CC BY 4.0 (http://creativecommons.org/licenses/by/4.0)], via Wikimedia Commons
Now how do we derive a mathematical expression from this?  I’m not suggesting that you should necessarily unless you feel comfortable doing so but I’ll bet there are kids in your class that can given a bit of guidance.  You may not feel comfortable providing the guidance.  But in this day of “just ask Google” you can provide that guidance in the form of a video discussion from the Khan Academy designed to help students prepare for the MCAT.  Don’t let that scare you off.  Here are two links that take the symbolic model and derive a mathematical expression–not just any expression—the Michaelis-Menten equation for enzyme kinetics. You or your students will no doubt need to view these more than once but the math is not that deep—not if your students are exploring calculus or advanced algebra.  It is really more about making assumptions and how those assumptions simplify things so that with regular algebra you can generate the Michaelis-Menten equation.
You can also find a worked out derivation here:  https://www.ncbi.nlm.nih.gov/books/NBK22430/  in this text excerpt from Biochemistry, 5th ed. Berg JM, Tymoczko JL, Stryer L.
New York: W H Freeman; 2002.
Of course, you don’t even have to go through the derivation you could just provide the equation.

The important thing is that students understand where this equation comes from—it doesn’t come out of thin air and it is based on the same core principles they uncovered or experienced if they did the toothpickase manipulation–it is just quantified now.  So how do I use this equation to actually see how well my data “fits”?  If it were a linear expression that would be easy in Excel or any spreadsheet package but what about non-linear trend lines?  I can tell you that this expression is not part of the trend line package you’ll find in spreadsheets.
I’ve got to admit, I spent too many years thinking that generating best-fit curves from non-linear expressions like the M-M equation was beyond the abilities of me or my students.  But again “Ask Google” comes to the rescue.  If you google “using solver for non-linear curve fitting regression” you’ll end up with lots of videos and even some specific to the Michaelis-Menten equation.  It turns out EXCEL (and I understand Google Sheets) has an add-on called Solver that helps you find the best fit line.  But what does that mean?  Well it means that you need to manipulate the parameters in the M-M equation to generate a line until it mostly fits your data–to see if the model is an accurate description of what you measured.  What parameters are these?
Look at the equation:
V0 equals the rate of the reaction at differing substrate concentrations–the vertical axis in the plots above.
Vmax equals the point at which all of the enzyme is complexed with the substrate–the maximum rate of the reaction with this particular enzyme at this particular enzyme concentration (that is enzyme concentration not substrate)

Km equals the concentration of the substrate where the rate of reaction is 1/2 of Vmax

[S]  equals the substrate concentration, in this case the H2O2
Two of these parameters are variables—one is our experimental or explanatory variable, the concentration of H2O2 and the other is our response variable, the rate of the reaction. Some folks prefer independent and dependent variable. This is what we graph on our axis.
The other two parameters are constants and the help to define the curve. More importantly, these are constants for this particular enzyme at this particular enzyme concentration for this particular reaction. These constants will be for different enzymes, different concentrations or reactions with inhibitors, competitors, etc. In other words it is these constants that help us to define our enzyme properties and provide a quantitative way to compare enzymes and enzyme reactions. You can google up tables of these values on the web. from: Biochemistry, 5th ed. Berg JM, Tymoczko JL, Stryer L.
So calculating these constants is a big deal and one that is not typically a goal in introductory biology but if you’ve come this far then why not?
This is where generating that line that best-fits the data based on the Michaelis-Menten equation comes in.
You can do this manually with some help from Solver in Excel.  (Google Sheets also is supposed to have a solver available but I haven’t tried it.
I have put together a short video on how to do this in Excel based on the data I generated for this lab.

I’ve also taken advantage of a web based math application DESMOS which is kind of a graphing calculator on the web.  While I can create sliders to manipulate the constants in the equation, Km and Vmax  to make a dynamic spreadsheet model it is a lot easier in DESMOS and DESMOS lets me share or embed the interactive equation. Scroll down in the left hand column to get to the sliders that change the constants.

You can also just go to Desmos and play with it there

I had to use A and B and x1 in my equation as symbols.

It is not that difficult to use DESMOS and with my example your students who are familiar with it will be able to make their own model with their own data within DESMOS.  Move the sliders around—they represent the values for   Km and Vmax  in the equation.  Notice how they change the shape of the graph.  This really brings home the point of how these constants can be used to quantitatively describe the properties of an enzyme and helps to make sense of the tables one finds about enzyme activity.  Also, notice the residuals that are plotted in green along the “x-axis”.  These residuals are how we fit the curve.  Each green dot is the result of taking the difference between the a point on theoretical line with particular constants and variable values and the actual data point.  That difference is squared.  A fit that puts the green dots close to zero is a very good fit.  (BTW, this is the same thing we do in EXCEL with the Solver tool.)  Watch as you try to minimize the total residuals as you move the sliders.  The other thing that you get with DESMOS is that if you zoom out you’ll find that this expression is actually a hyperbolic tangent…and not an exponential.  How is that important?

Well, think back to the beginning of this post when I talked about how my students often just choose their mathematical model on what line seems to fit the data the best–not on an equation developed from first principles like the Michaelis-Menten.

Looking at a plot of the data in this experiment before the curve fitting one might have proposed that an exponential equation might have produced the best fit.  In fact, I tried that out just for kicks.
This is what I got.

Here’s a close-up:

Thinking about the actual experiment and the properties of enzymes there are two things really wrong with this fit although you’ll notice that the “line” seems to go through the data points better than the fit to the Michaelis-Menten equation.  1.  Notice that the model line doesn’t go through zero.   Hmmmm.  Wouldn’t a solution with no Hydrogen peroxide not react with the yeast?  That should be tested by the students as a control as part of the experimental design but I can tell you that the disk will not rise in plain water so the plot line needs to go through the origin.  I can force that which I have in this fit:

But the second issue with this fit is still there.  That is the point where the plot has reached it’s maximum rate.  If I had generated data at a 3% substrate concentration I can promise you the rate would have been higher than 0.21 where this plot levels off.  While the exponential model looks like a good fit on first inspection it doesn’t hold up to closer inspection.  Most importantly the fit is mostly coincidental and not base on an equation developed from first principles.  By fitting the data to the mathematical model your students complete the modeling cycle described on page T34 in the AP Biology Investigative Labs Manual, in the Bean Biology paper cited above, and on page 85 in the AP Biology Quantitative Skills Guide.
Give model fitting a try—perhaps a little bit a time and not all at once.  Consider trying it out for yourself with data your students have generated or consider it as a way of differentiating you instruction.  I’ll wrap this up with a model fitted with data from Bob Kuhn’s class that they generated just this month.  He posted the data on the AP Biology forum and I created the fit.

The key thing here is that his enzyme concentration (yeast concentration) was quite a bit diluted compared to the data that I’ve been sharing.  Note how that has changed the Michaelis-Menten curve and note how knowing the Km and Vmax provides a quantitative way to actually compare these results.   (Both constants for this graph are different than for mine)
Hopefully, this sparks some questions for you and your students and opens up new paths for exploring enzymes in the classroom.  I’ll wrap this up next week with how one might assess student learning with one more modeling example.