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.
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.
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.
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.
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.
Editor’s Note: So far this semester, the most popular single post on the BioBlog is this September 2013 peer-review piece from our blogfather, Brad Williamson. Also this is a reposting of a reposting. Blogception! Enjoy this, and if you use mini-posters in your classes, share your experience with us in the comments!
This is a reposting of a post that first appeared on the NABT BioBlog:
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.
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 miniposter-headers 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.
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 Sample-miniposter might look like constructed in MS Word.
Links to websites for advice on making scientific posters:
Welcome to the KABT blog segment, “In My Classroom”. This is a segment that will post about every two weeks from a different member. In 250 words or less, share one thing that you are currently doing in your classroom. That’s it.
The idea is that we all do cool stuff in our rooms and to some people there have been cool things so long that it feels like they are old news. However, there are new teachers that may be hearing things for the first time and veterans that benefit from reminders. So let’s share things, new and old alike. When you’re tagged you have two weeks to post the next entry. Your established staple of a lab or idea might be just what someone needs. So be brief, be timely and share it out! Here we go:
Investigating Energy Flow with ZOMBIES!
It’s the zombie apocalypse! You have a safe fenced-in area that is impenetrable to the zombies. But, you also cannot leave the fenced in area. If you had time to prepare this land, what would you plant? What livestock would you have? (Note: Students have the option of doing a Mars Biodome if they do not want to do the zombie apocalypse.)
Student groups are all given the same 11 x 17 inch grid paper. Each square equals 100 square feet. Each student needs a housing structure(s) that equal 20×25 squares.
Sustain as many humans as possible using the land space given. The group who can sustain the highest number of people wins. The criteria for sustainability is 2,000 calories per day, per adult (730,000 calories per year). (Note: No stockpiling allowed).
Students need to find the total number of producer calories from all their crops. (Find the calories / square foot for each food, and then multiple by the number of total square feet.)
Then, students need to calculate how many of those producer calories are actually available for human consumption. To do so, students must figuring out how many of those producer calories their livestock will consume per year.
Next, students need to find the total number of anaimal calories produced. They calculate how many calories of meat (or eggs/dairy) each animal produces. (To simplify, one could assume the entire weight of the animal is meat.) Students do this for each type of livestock and add it together to find the total number of livestock calories produced. (If you have any secondary consumers, they will take a whole other set of calculations!)
Next, students find out how many calories their land produced for human consumption. They take the number of plant calories available for humans and add it to the total number of animal calories produced. Then, they divide that by 730,000 (the total number of calories needed per human per year) to see how many humans they can support.
Getting the Numbers
To make it easier, you could provide a list of several crop and livestock options with their calorie information. But, for me, one of the best parts of this project was having it open ended for the students. I have my students find the information on their own, but they have to back it up with a credible source. This gets pretty competitive, so the students really hold each other accountable.
Here are some important questions that we discussed after completing this project:
Why do we lose calories when we feed them to livestock?
What is the “best” crop? (calories vs. nutrients)
Should we be putting plant calories into livestock?
What are the pros and cons of having livestock?
What would be the “best” livestock? (For example, for many reasons crickets are much more energy efficient than cows.)
What does this make believe scenario have to do with the real world?
Tips and Suggestions
I suggest you have a running list of “rules” that you as a group decide upon throughout the project. For instance, someone will probably ask if it’s okay to do a rooftop garden. Whatever you decide, you should keep documentation of the “rules” your class makes. The students get pretty competitive and this is helpful.
To simplify our model, we assumed a lot. 1) People only need calories to survive, not certain nutrients. 2) We have sufficient water, fertilizer, and everything else needed to grow the crops. 3) We can store crops up to one year, and there is no limit to the type of crops that can be planted due to climate, etc. 4) Animals can only eat the part of the plant that humans eat. 5) All animals reproduce each year. 6) We eat the entire weight of the animal in meat. And more. But, these assumptions lead to fantastic discussions! I have students write about them for part of the end paper. They are also great opportunities for extensions.
Even with all of the assumptions and simplifications, the students were really able to “get it” in terms of energy transfer and the 10% rule.
If you’d like a more detailed description or have any questions, please e-mail me. firstname.lastname@example.org
I know KELLY KLUTHE has some cool stuff to share! Tag, you’re it!