Data Analysis in a Natural Selection Simulation

+/-1 SEM bars added

I really like the HHMI Biointeractive activity “Battling Beetles”. I have used it, in some iteration (see below), for the last 6 years to model certain aspects of natural selection. There is an extension where you can explore genetic drift and Hardy-Weinberg equilibrium calculations, though I have never done that with my 9th graders. If you stop at that point, the lab is lacking a bit in quantitative analysis. Students calculate phenotypic frequencies, but there is so much more you can do.  I used the lab to introduce the idea of a null hypothesis and standard error to my students this year, and I may never go back!

 

We set up our lab notebooks with a title, purpose/objective statements, and a data table. I provided students with an initial hypothesis (the null hypothesis), and ask them to generate an alternate hypothesis to mine (alternative hypothesis). I didn’t initially use the terms ‘null’ and ‘alternative’ for the hypotheses because, honestly, it wouldn’t have an impact on their success, and those are vocabulary words we can visit after demonstrating the main focus of the lesson. When you’re 14, and you’re trying to remember information from 6 other classes, even simple jargon can bog things down.  I had students take a random sample of 10 “male beetles” of each shell color, we smashed them together according the HHMI procedure, and students reported the surviving frequencies to me.

Once I had the sample frequencies, I used a Google Sheet to find averages and standard error, and reported those to my students. Having earlier emphasized “good” science as falsifiable, tentative and fallible, we began to talk about “confidence” and “significance” in research. What really seemed to work was this analogy: if your parents give you a curfew of 10:30 and you get home at 10:31, were you home on time? It isn’t a perfect comparison, and it is definitely something I’ll regret when my daughter is a few years older, but that seemed to click for most students. 10:31 isn’t 10:30, but if we’re being honest with each other, there isn’t a real difference between the two. After all, most people would unconsciously round 10:31 down to 10:30 without thinking. We calculated the average frequency changed from 0.5 for blue M&M’s to 0.53, and orange conversely moved from 0.5 to 0.47. So I asked them again: Does blue have an advantage? Is our result significant?

Error bars represent 95% C.I. (+/- 0.044) for our data.

Short story, no; we failed to reject the null hypothesis. Unless you are using a 70% confidence interval, our result is not significantly different based on 36 samples. But it was neat to see the interval shrink during the day. After each class period, we added a few more samples, and the standard error measurement moved from 0.05 to 0.03 to 0.02. It was a really powerful way to emphasize the importance of sample size in scientific endeavors. 

Should the pattern (cross-cutting concept!) hold across 20 more samples, the intervals would no longer overlap, and we could start to see something interesting. So if anyone has a giant bag of M&M’s lying around and you want to contribute to our data set, copy this sheet, add your results, and share it back my way. Hope we can collaborate!

Email results, comments, questions to Drew Ising at aising@usd348.com or drewising@gmail.com

–Versions of Battling Beetles Lab I’ve Tried–

HHMI Original

My “Student Worksheet” Edit

Lab Instructions Google Doc

Lab Notebook Intro. from 2017-18

Lab Notebook Data from 2017-18

Assessing the Science and Engineering Practices

I have been thinking a lot about the message that I want to send to students about science and reflecting on my own understanding of what science is. In my short two years as a teacher a lot of kids have come into my room conditioned into memorizing words and concepts until a test. They see science classes as more challenging versions of the memorization-regurgitation cycle and often have insecurities about science. As a student it took me a really long time to realize that science isn’t about memorizing processes or vocabulary but about the feeling I get in my head when I don’t know something yet but know that there is something to be learned. It’s about the confusion that happens when you have data that doesn’t come out you expected it to and you don’t understand why, or the excitement when you can connect two ideas you didn’t realize were related to each other before. I only realized these things when  I had mentors in college who asked me questions that I couldn’t answer by regurgitating vocabulary words. They taught me how to learn rather than how to be taught, and I gained so much confidence. No matter how difficult the concept, I had gained some kind of magic comfort in my abilities to work through problems and struggle through sense-making because I had sort of re-focused my education on the act of learning versus the things I learned.

But how do I get 15 year-olds who have been trained from a young age to read their books, do their vocabulary words, and memorize what the teacher tells them to change their ways and actually do this science? How do I give them the the science magic that I found during my college years? Thankfully I am not the only educator who has asked these questions and the creators of NGSS built in science and engineering practices to the standards. I’ve always planned my lessons with the science and engineering practices in mind but I’ve never really told my students what the practices are or how you exactly do those things. So this year I’ve promised myself that I’m going to be more deliberate about this. I made colorful posters with the practices on them and hung them in my room, and have told my students and their parents multiple times that I value the practices. I don’t think that these practices are THE ANSWER to helping students understand real science but I think they are a good place to build from. 

I’m going to value these skills in my classroom and I added a grade book category just for them. My goal is to assess my students on one of the practices at least once a week and to be very explicit and clear with them what these skills look like.In an attempt to briefly outline mastery, proficient, and developing skills I put together a rubric that includes all 8 standards. I plan on using the rubric as a general guideline to grade various different projects or tasks, varying from exit slips or bell ringers to longer in-class activities. If I want to assess a certain practice more in-depth I will break it down into its own more detailed rubric, but for now this is what I’ve got. I’ve attached my first and second drafts of these rubrics in attempt to show how my thought process changed. I love google docs and have given all viewers of these documents the ability to add comments…please do so! I am more happy with iteration 2 but am not sure that everything is student friendly or actually what those skills look like. Big thanks to Camden Hanzlick-Burton and Michael Ralph and others on the KABT Facebook page who encouraged and pushed my thinking before I was quite ready to make a blog post.

TLDR: Science is awesome! How do I get students to stop memorizing and do science? I made some rubrics to assess science and engineering skills but think they could use some improvement: HELP!

DRAFT 1

DRAFT 2

Trying Something New With Grades

I have wanted to change the way I assess students for a while. I have made changes to how and when I grade assignments, the format of tests, and how understanding is communicated during and after lab activities. But in the end, I was still grading students the same way I always had, the same way I was in school, and the same way students have for quite a while. Kid accumulated points, some assignments were weighted more than others, and students who turned in most of their work on time (regardless of quality) tended to do well. This school year, I am not doing that. I will probably fail spectacularly. Luckily I have administrators who are supporting me, knowing I am trying to do what is best for our students. I am going to try this first with my AP Biology students, since I share the Biology 1 classes with two other teachers, and hope this leads to a wider transition.

I will share what I am doing, but I need your help. After reading through my plan, send me a message or leave a comment with your feedback. What looks good? What should I change? What have you tried and can share to improve my students’ experience? 

via GIPHY

I am basing my course assessment off a document shared by AP Biology/Calculus teacher Chi Klein. The College Board shares, as part of the curriculum framework, “Essential Knowledge” statements and has recommended “Learning Objectives” from them. Ms. Klein compiled and organized those learning objectives into a document that could be shared with her students. I will be sharing a GoogleDoc with my students in the first days of class which they will use over the course of the school year.

As is the case in most standards-based and “gradeless” classes I have seen, students will be responsible for justifying their level of mastery over the content. The “Learning Objectives” document I will share with them covers 149 content standards. Students will be able to earn up to four points for each standard based on their mastery of the content, meaning we’d have 596 possible points by the end of the school year. Here is what I’m thinking for my mastery levels (category title suggestions welcomed):

Level of Mastery

Example Activities

Knowledge

Notes, Guided Readings, Discussions

Comprehension

Class activities, Worksheets, POGILs, Article Annotations, Quizzes

Application

Experiments, Virtual Labs, Demonstrations, etc.

Synthesis

Summative Exams, Projects, etc.

I envision the initial knowledge mastery as being pretty straight-forward to demonstrate. For the successive levels, I have been torn as what threshold to use for mastery. If a student wants to use an assignment, lab, test question, etc., do I require them to have earned all possible points? I have been considering at least 90% on a given assignment/test item before a student can try to use it to justify mastery. As an example, if I have a free response item on our evolution test with 10 possible points, a student would need at least 9 points before they could use that in a grade conference. If a student only earned 6 points, they would have to revise their response and get new feedback on the item before trying to use it again during their next conference.

So students are still earning points, and the points they earn as a percentage of the overall points possible still determine their final grade. Not very earth shattering there. How they are being assessed, and what is being assessed is different than how I have ever done this before. There is a much greater burden of responsibility (and independence) placed on the student. My feedback is going to need to be both more flexible and more timely to allow students to complete any needed revisions. If not, I will be setting my students up for a very difficult experience.

The one final change is, at least for my AP Biology class, I am moving away from the traditional 90/80/70/60 scale for grades. The purpose of the AP class, to me, is to prepare students for post-secondary success and to show well on the AP Biology test.  So I want the rigor of the class to match the rigor of the expectations and examination. As anyone who has taken or taught AP Biology can attest, this won’t be difficult. I also want my scoring to reflect that of an AP test. If a student has an A in my class, I want them to have an expectation to earn a 5 on the test. If they have a C in my class, they might expect to earn a 3 (which in Kansas would now get them college credit; good change KSBOE/Regents!). Going back through all the data I could find on the correlation of raw exam scores to 5-point AP Scores, here is what I am going to roll with this year:I am going into this completely aware that revisions will happen when I get AP scores back in the summer. If I have a student who earned 499 points in class, but only got a 3 on the exam, I will need to reconsider either the point range for that grade, or how I let students demonstrate mastery. Again, I am very lucky to have administrators who are willing to let me take this chance, fully aware of I will likely make mistakes.

As for pacing, I am planning on emphasizing one Big Idea each quarter. We’ll start with Big Idea 1 (evolution), which will be more teacher-centered as my students (and I) learn how to function in this new system. As the school year progresses, I hope to transition to a more student-centered model with Big Idea 4 being largely personalized by each individual. Shouts to David Knuffke and Camden Burton for the inspiration here.

This will be my 11th year in the classroom, and 5th teaching AP Biology, and I am finally to a point where I am comfortable enough with my knowledge and abilities to make some changes. I hope this will be a better and more accurate way of assessing student knowledge and mastery, providing more meaning to the grade students earn in my class. But what do you think? What feedback can you give me? I’d love to hear from you in the comments, social media (@ItsIsing), or you can email me (drewising@gmail).

Here goes nothing…

via GIPHY

–Documents of Note and Muses–
Syllabus: Ising APBio2017
Student Learning Objectives: GoogleDoc
Camden’s BioBlog Post: http://www.kabt.org/2015/02/23/my-biology-objectives/
Kelly’s Gradeless Classroom: http://www.kabt.org/2015/06/26/the-great-gradeless-experiment-1/
Bob Kuhn’s 52-Week Gradeless Blog: https://medium.com/@mszczepanik/52-weeks-of-grade-less-week-1-the-journey-begins-da3e03739a7e
David Knuffke’s Published Thoughts on SBG: http://www.knuffke.com/search?q=standards

 

NOW ACCEPTING 2017 Fall Conference Session Proposals

Friends, Members, and Colleagues,

The Kansas Association of Biology Teachers would like to encourage you to submit a session proposal for our upcoming fall conference. We are being hosted by the Sternberg Museum (Fort Hays State University, Hays, KS) Saturday, September 9th (more information to follow soon). Whether you are a seasoned presenter or a first-timer, an individual or a group, we’d love to have everyone share something with us. Our strength is in the innovation and openness of our classrooms, and we can’t wait to see what amazing stuff is going on across our state.

Proposals will be accepted from 21 July-8 August. Presenters will be notified of proposal status no later than 11 August. 

2017 FALL CONFERENCE PROPOSALS

If you have any questions regarding the conference, proposals, Shark Week, etc, contact Drew (andrewising@gmail) or Sara (sarahettenbach@gmail).

 

In My Classroom: NESC Videos are helpful

I have a student-teacher this semester, and he asked to teach our evolution unit as his “portfolio” unit. He is, at this point, mostly being left on his own to plan, assess, and manage the classroom. Our students were all on board for the Geologic Time Scale and natural selection (and it’s accompanying demonstrations and labs).

However, as we started talking phylogenies and focusing on ancestry, a handful of students started asking why people thought we evolved from monkeys, and why monkeys weren’t evolving into humans. I knew as a more experienced teacher (who had made many mistakes already while teaching students), that this kind of questioning is preventable with some different organization of your unit. But I was interested in how he would confront this in his classroom because it would tell me a lot about his progress and readiness to handle his own classes. As a cooperating instructor, I was interested in how he would respond to this. As a fellow biology teacher, I could sympathize with how he was probably feeling; even if you do everything perfectly, address every misconception, incorporate the nature of science into every lesson, this type of question is always going to get asked by somebody. So what did he do? He impressed me.

I have used “tree-thinking” quizzes and other resources available from Understanding Evolution but have never used any of their video clips. My student teacher had some productive discussions about making conclusions from evidence, why scientific explanations have to be falsifiable, and what it means to have a “common ancestor”. He followed all of that up with this video:

I had never seen this before, but our students really responded well to it. It is definitely something that I will be using in the future!

More Understanding Evolution and National Evolutionary Synthesis Center videos can be found here.

And perhaps it is time to remove my padawan’s braid.

TBT: Protein Synthesis Models (In My Classroom)

EDITOR’S NOTE: THIS POST ORIGINALLY APPEARED IN FEBRUARY 2015 AS THE 3RD INSTALLMENT OF THE “IN MY CLASSROOM” SERIES. KABT MEMBER IN EXILE, CAMDEN BURTON, SHARED THIS ACTIVITY WHERE HE HAD HIS STUDENTS COMPARE AND CRITIQUE MODELS. ENJOY THIS KABT CLASSIC!

Thanks to a little idea from Brad I thought I would try something with my AP Biology students this week that I saw him try with his BIO 100 students at KU earlier.

We’re currently marching our way through the mind-bending terror that is protein synthesis. So we’ve gone over the whole process a bit but to make sure we were not getting lost in the details I gave them this:

Blank central dogma 1Blank central dogma 2

Two different models of the same process. Nothing earth-shatteringly innovative but how I framed it and worked with it was unique to me. I didn’t just say it was a worksheet to complete. I framed it as 2 different models of the same process. If they wanted to use the picture in their book that was ok because the diagram in their Campbell book also looked different. What I was surprised with was how much students struggle translating [pun] knowledge across models. Students struggled with labeling processes versus structures, labeling the same structure that was differently drawn in two models, and especially when one model added or removed details (like introns and exons).

The other cool part was that afterwards when students shared their answers on the board, they had lengthy discussion about what was “right”. For example, two students argued whether the 4th answer from the top was “pre-mRNA” or “mRNA” and explained why they thought that. After looking to me I shared that by their explanations both could be right. That’s what I think was cool, students argued different answers where with the proper explanations, either could be right. So because of that, I would avoid giving an “word bank”.

Also, at the very end I created a list on the board titled “limitations” and I had them share what was limiting about these diagrams. Some thoughts were “no nucleotides were shown entering RNA polymerase”, “no other cell components were shown”, “the ribosome on top only had room for one tRNA”, “no mRNA cap or tail were shown”, and many more.

I found this exercise useful because I struggle giving students modeling opportunities (especially non-physical ones) and this was a simple way for students to get practice comparing/contrasting models while also discussing the usefulness and limitations of them.

Alright, for the 4th installment I nominate el presidente himself, Noah Busch.

A 3D Gene Expression Lesson on Epigenetics

Disclaimer: As far as standards go, I really like the Next Generation Science Standards. Particularly important to me is the emphasis it places on learning not just the content (disciplinary core ideas), but how scientists work/think (science practices) and connections between ideas (cross-cutting concepts). Over the last 3-4 years, I have been giving my favorite activities and labs an NGSS facelift to modify them to better fit this framework. I am going to share with you a lesson that I feel address all 3 dimensions of the NGSS.

 

Is your lesson “3D”? Use the NGSS Lesson Screener tool to find out.  LINK

Many students really enjoy their genetics units, but one of the more difficult things to understand is gene expression. Several years ago, I would have presented my students with the “central dogma”, given some notes over transcription and translation, then worked through a few scaffolds to get them to understand how amino acid chains are produced. After reading Survival of the Sickest in 2008, I started to mention that epigenetics was a thing, though I didn’t have my students investigate it with any depth.

With the introduction of the Next Generation Science Standards, an emphasis has been placed on understanding the implications of the processes in the classic dogma without getting overly concerned about what specific enzymes might be doing at a given time. This has freed up more time to explore the regulation of gene expression, including epigenetics. There are a number of amazing resources out there (like this… and this… and this…), but here is how I cover gene regulation with my 9th grade biology students:

This format is something I have adapted (with few changes) from an NGSS training put on by Matt Krehbiel and Stephen Moulding, which I attended thanks to KSDE. I like this because it is flexible, provides students with the entire trajectory of the lesson from the beginning, and can double as a lesson plan. Can you guess the reasoning behind the color-coded words? That, too, is explicit, though it is in most cases more for my own benefit. RED words are commands for the students. It tells them how they should address the problem and how I will assess their work. The GREEN words relate to cross-cutting concepts (in this case, systems/system models and patterns), while the BLUE(ish) words are science practices.

Depending on how much time you have available, this could take 2 to 4 50-minute class periods (or 1-2 block periods if you’re lucky enough to roll with that schedule).  I like to use more time for this because I have designed discussion and collaboration into the process, but the “Gather Information” and (obviously) “Individual Performance” sections could be done by students on their own and wouldn’t require a classroom. Devoting a little extra class time will also allow for you to conduct ad hoc informal formative assessments (read over a kid’s shoulder and ask them questions) as you move around your room.

Part 1: Gathering Information

Have you listened to the RadioLab episode, “Inheritance”? If not, you should do that. I find that RL is a good way to indoctrinate your students into the world of science podcasts. And this episode is one of my favorites. 

I really like reading with my students, asking them questions that get them thinking deeper as they go, so I usually devote an entire class period to reading an article on epigenetics. I break my class into three groups with each group reading a different article, and students will (for the most part) self-select based on the length or difficulty of the reading.  I use readings pulled from Discover Magazine, Nature Education, Nat Geo’s Phenomena blogs. Students sit around large tables and talk and write and sketch as they read. There is structure and agency, direction and freedom, and I love those days. But if you’re in a hurry (in my opinion, one of the worst reasons to do something), I guess you could assign the reading as homework.

via GIPHY

Part 2: Thinking Deeper

To really understand something, you need to really dig into it. This section is meant to be collaborative. If I have some really outstanding students grouped together, I will encourage them to divide the work in this section between them, then teach their group members in a scaffold.  I wouldn’t normally do this with an extension/research-based activity because I want to make sure each student has a chance to interact with each aspect of the activity. If I can’t trust all the group members to produce the same quality of work, I won’t recommend the divide-and-conquer approach.

When dealing with my AP/College Biology students, I would word a question like #5 differently. With the general biology kids, I recognize most of them will not end up in a biology-centric career. They will, however, be citizens of the world, and voters. So I try to incorporate questions where they reflect on their emotional response to the content. I know it is popular to think of scientists as unfeeling, opinion-less automatons, but that is disingenuous. I live with a scientist, trust me. I use experiences like this to really emphasize the importance of evidence-based, empirical thinking and using data to drive decision-making.

Part 3: Individual Performance

How do you know if your students “get it”?  A lot of the time, when using a science notebook or interactive journal, it might be several days before you go back and read everything your students wrote (and maybe, sometimes, you still don’t read everything). What I like to do is tell students they will have 15 minutes to produce the best possible answer after I give them 5 minutes to discuss with their classmates how they will address the last couple of items for this assignment. Once the writing starts, I am walking the room, reading over shoulders, and looking for patterns. Are there any things that I think they should have gotten, but most people are missing? Are we particularly strong in certain areas? Are students adding models to their answers in support? This lets me know if I need to reteach something or if we can move on.

I also look for answers that are good, but might be missing one bit of information to take it over-the-top. It is a good rule of thumb to think that, if one student is making a mistake, there are other students making the same error. I will then (not so) randomly ask students to read exactly what they have written down. By using an answer that is mostly correct, it takes some of the stigma away from making a mistake. We can then have a discussion with the class to see if we can identify where the answer can be changed or added to, and praise the parts of the answer that were done well. Students with sub-par responses are encouraged to add to their answers, and we learn more together.

Conclusion 

If you are still with me, what do you think? What does this activity do well? Where can I get better? What are my students missing? If you would like to modify/use this activity, you can find a GoogleSlides version here. Send me an email (andrewising[at]gmail) or tweet (@ItsIsing) and let me know how it went!

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/

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.

Model Building and Building on Models

I make my students build and use models on a daily basis in my classrooms. I think that I have a better than average grasp on the Next Generation Science Standards, their practice and three-dimensional lesson planning. But I have apparently never thought to throw a bunch of vocabulary words at my students and give them the time to really struggle to connect them into a cohesive model with their groups.  And at the end of a session on Cognitive Models, presented by AP/IB Biology teachers Lee Ferguson and Ryan Reardon, that is exactly what we did.

nabtmodel

To start, the instructions were sparse: Create connections and uncover relationships between pancreatic β-cells and photosynthesis. My group was made up of six other AP Biology teachers from 4 states, none of us with any idea where to start. There was some discussion about the significance of the color of each card, which it ends up wasn’t important… there just wasn’t time to sort them before the session.  We eventually found the word “Metabolism”, which we all agreed was the one thing that all the cards shared. From there, we tried to make shorter stacks of cards that were related. For example, “Hyperglycemia”, “Blood sugar rises”, and “insulin”.

Once we had all the cards grouped, we tried to place them into a pseudo-concept map. In our classrooms, I would have probably done this on a big whiteboard so we could draw arrows and write connecting terms, but my group guess that the Sheridan didn’t want us writing on their table cloths. 🙂  As we went, we had to stop and rearrange our map several times and each time we edited the map, members of the group were justifying why some cards had to stay or move.  It was a really great conversation and I learned some things about feedback loops that I don’t think I had ever known.

At the end of the process, we were encouraged to go look at what the other tables had put together and reflect on our map. To my surprise, none of the other groups had anything resembling our model. Talking to some of the other groups, I don’t think that anyone had a model that I think failed to achieve the original objective. It was really a powerful reminder that students, no matter the amount of information they may possess, each approach a problem from a unique viewpoint. And when you have people put together information, even people that all know “the right answer”, there are many ways to arrive at that conclusion.

Needless to say, next week when we start preparing for our next test in my 9th grade Biology class, my students are getting a stack of 3×5 cards tossed on to their tables. I can’t wait to hear their conversations and see what they create!

This post is part of a series of posts from KABT members reflecting on some of the most important things they’ll bring back into their classrooms from the NABT 2016 Professional Development Conference.