Yesterday was Pi Day, and as a high school science teacher, this is a big deal. Not as big of a deal as it is for math teachers, but until the practical but not-yet-widely-accepted Tau catches on, it’s the best mathematical constant we can celebrate during the school year.
My wife Carol and I about to devour some “banarama” and “shady wake” pie at Muddy’s Grindhouse here in midtown Memphis! I also made sure to share some Pi puns with my students during the day and remind them what Pi means (it’s the ratio of circumference divided by diameter of ANY CIRCLE EVERRRR).
But that’s not why I’m writing this post. This post will use my second-favorite mathematical constant to bolster your economic future. If you readily look for applications of geometry in your everyday life, then you may already know what I’m about to say. If not, then this may be the most important statement you are reading at this very moment. Here it is:
You should always buy the biggest pizza.
For this post, I’ll be using prices from this menu from the Memphis Pizza Cafe on Broad Avenue. Here are the prices for the 10″, 13″, and 15″ “Hey Meat!” pizza:
At first glance, it seems like each inch of pizza diameter costs you about a dollar. It’s just under $10 for the 10″. But the deal seems to get worse as you add more inches. The 15″ costs $16.35, which is more than $1 per inch! It’s a WORSE deal than the smaller pizza!
Pizza prices are misleading though. Most pizza locations (unless they use the ambiguous sizes of small/medium/large) list pizza sizes by the diameter of the pizza. But of course, you aren’t eating a linear pizza, you are eating a circular pizza. The AREA, not the diameter, is what you care about. And it turns out that each additional inch of diameter nets you an increasing amount of area. For example, adding an inch to the diameter of the 10″ pizza increases the total area by a little, but adding an inch to the diameter of the 15″ pizza increases the total area by a lot!
The equation for area is:
A = π r², where r is the radius (i.e., half the diameter)
Here is my calculation for area of the 10″ diameter pizza:
A = π r²; r = 5″
A = (3.1415)(5″)²
A = 78.5 in²
The 78.5 in² (10″ diameter) pizza costs $9.90. Divide the cost by the area, and you get $0.126/in², or 12.6¢/in². Not bad, only 12.6 cents for every square inch of pizza!
I did the same calculation for the other two sizes, getting the following results:
10″ pizza: 12.6¢/in²
13″ pizza: 9.91¢/in²
15″ pizza: 9.25¢/in²
These numbers would be like the “unit price” you find at grocery stores, telling you the price per unit (such as ounce or gram) that you can compare to other brands/sizes. Look at that! The 15″ pizza costs you only 70% the price per square inch of the 10″ pizza!
So, unless you hate leftover pizza, it’s always the best deal to get the biggest pizza possible! Share that pizza with some other folks and bask in the glory of savings! Thank goodness for π and for geometry!
A friend of mine shared this photo. It stirred up in me a lot of thoughts about disability (I have a complete L2 spinal cord injury, meaning I’m paralyzed from the waist down). Everything I wrote here applies to a non-disabled person thinking about a disabled person, or a disabled person thinking about a more-disabled person. If you have questions or comments on this topic, please feel free to share.
Can she* really “kick [someone’s] ass”? I don’t think I would ever talk like that unless I really felt that I had the upper hand in a potential fight and felt the need to try to intimidate another person.
I also don’t resonate very well with unrealistic claims. There is a need for balance of optimism and reality. If the girl in the picture has studied martial arts or learned to shoot a gun or some other similar thing, then okay, maybe her statement is empowering and true. But if not, she probably can’t do much ass kicking, and her physical disability undoubtedly decreases her ability to win a fight.
I would get comments in college all the time from people about upper body strength. They would say something about how my arms must be really strong because I push a wheelchair all the time. Those were actually, in a counter-intuitive way, back-handed compliments. It diminished the hard work I had put in to weight training, cardio training, and two-and-a-half-hour basketball practices five days a week.
I don’t want to be special or inspirational because of my disability. We can think very highly of disabled folks because of our low expectations. It’s soft bigotry. It would be similar to a person being very impressed by a black person getting an A on a test but not being surprised by a white person getting the same grade. That lower expectation placed upon the black person is similar to the low expectations placed upon disabled people. If someone thinks it’s simply astounding that I can open a door, drive a car, or push up a ramp**, then that person (even with good intentions) is being the opposite of empowering. Instead of sending the intended message (you are awesome and do lots of cool stuff!), it sends a very demeaning message (I thought you sucked, but actually you are surprisingly okay!).
I don’t mean to be harsh. I appreciate compliments. But I also think it’s important to spread this message. Here’s what you. Instead:
1. Look for actual positive attributes of the person. Ask this question: If the person were not disabled, would I still think this attribute is noteworthy? Then compliment that.
2. Don’t feel sorry for anyone. Don’t say, “You’re amazing…if I were in your situation, I’d probably give up.” That’s pretty insulting to the people you’re speaking to; you have no place to claim that the other person’s life is that terrible. That also underestimates your own perseverance.
3. Don’t belittle the experiences of disabled people by portraying them as heroes or saints just because they have a disability. See them as real people. Viewing someone as a hero can be just as demeaning as viewing someone as inferior.
*After writing this post, someone informed me that the girl in the picture is batgirl. So maybe she can kick some ass…
**Another friend asked me about folks that are recovering from an injury in rehab. That is a very different situation. Most of my post applies to people who are no longer in the adjustment phase. For a person in rehab, opening a door or driving a car can be a huge accomplishment. But, years after learning how to open a door while in a wheelchair or drive with hand controls, those tasks are common-place and no longer great accomplishments.
I downloaded an animation app for mac called Hype. My first thought was to check out Adobe, but $50/month is a bit steep for me (or $20/month with the teacher discount). Hype is just a $50 one-time purchase.
Well, the short of this is that I want to do simple animations/videos/simulations/etc for my classes.
I kept this first one simple because, well, I’m brand new to this and wanted to make an easy first project to show off. I’ll have my students make a motion map and position and velocity graphs for it as a Do Now (warmup) this week.
Here are manipulatives I made and bought (i.e., tic tacs) to help students memorize the steps of action potentials! I have resources and a more detailed description of the lesson cycle posted below.
Developing the Action Potential Model
We go through the action potential steps together first. We have a section on cell membranes at the beginning of the year, so they have a good understand of the basics of facilitated diffusion and active transport. A lot of the “direct instruction” here is me telling them what channel opens in each section, then they figure out what the ions will do. I’ll post a photo of the “notes” we build in this process once I figure out how to convert a Doceri file into an image file…
Practicing the Steps
Next, we use the manipulatives to practice the steps of the action potential. The goal here is for the students to familiarize themselves with the process and memorize the steps. It’s more or less repetition, but that’s not allllways a bad thing.
The case study works well because students have to analyze what happens when a toxin throws off one of the action potential steps. For example, preventing voltage-gated sodium channels from opening will prevent depolarization, therefor no action potentials can happen.
The case also requires students to connect this all back to various functional types of neurons (sensory, inter-, and motor) and neuron properties (effects of myelin, etc.). That all really gets back to the idea of interleaving as well!
For the buggy paradigm lab, I wanted the graphs to all look different so we could compare and contrast. To get some negative positions, I made a common origin out of tape and staggered the tables. I also made at least one group run their buggy the opposite direction so we could figure out the meaning of a negative slope on the position-time graph.
After collecting data, students graphed it on engineering paper and then whiteboards the results. Lots of aha moments during the board meeting!
“Anatomy and Physiology is just a lot of rote memorization, isn’t it?”
I have been asked that a handful of times, and my gut instinct is a resounding “no.” Then, I admit that there is a fair amount of memorization in my A&P class. That’s just part of the package.
But memorization is not the heart of the class nor the point of the class. We do case studies to figure out what disease is ailing a patient. We analyze flow charts to determine causes of abnormal hormone levels. We learn how to read and speak like physiologists, breaking down long words like “electroencephalogram” and “tachycardia” into their roots to determine their meaning. We diagram homeostatic feedback loops to determine how certain pathologies disrupt normal processes in the body.
It’s not about memorization. Is history about memorizing dates of events? Are math and physics about memorizing equations? Is music about memorizing scales?
Last year, in an attempt to make the class seem less like an exercise in fact recollection, I set out to determine what the big ideas in physiology are. Homeostasis was an obvious one. That’s what our organs do: keep our body’s internal conditions relatively constant so that we don’t die. The interdependence of organ systems was also important. Even though we study the body one system at a time, they all depend on each other to function. Finally, in my class there is a big emphasis on using our knowledge of the organ systems’ functions to better understand diseases that occur when those functions are disrupted. Hence, my three big ideas in A&P:
Of course, there are other big ideas in physiology as well: “structure determines function” is a common one. But I think the three I chose will bring a lot of cohesion to the subject.
How will I use these big ideas in class?
I put the three posters up on my wall (using my 11″ by 17″ color printer). I plan on referencing these throughout each unit.
I also plan on doing several activities that help students summarize what we are learning. An example of this would be making a mind map showing how each of the systems are dependent on each other. I’ll have more info on this in later posts when we get to those activities.
Finally, at the end of the year, my students write an essay using one of the big ideas as a thesis statement, backing it up with examples we have learned throughout the year.
If you are a fellow A&P teacher, feel free to use and edit the posters. Links below.
EDIT: I now have my own adaptive immunity video! For my classes, I now use my own video instead of the Bozeman video.
The human immune system is extremely intricate and complex.
In a high school A&P class, it’s an extremely important but somewhat abstract process. Like every process in physiology, we have to simplify it with a model, which we can add to as we learn more (for example, my students who go on to biology or other pre-med majors will learn a lot more details about innate and adaptive immunity than we do in my A&P class).
In past years, this has been a section where I do almost all of the talking. Because of the complexity, I felt like I had to explain it all super clearly for students to understand. Of course, most students still did not understand because they didn’t have to construct their own meaning and do their own sense-making.
Here is a basic summary of the process:
Innate Immunity includes the first lines of defense, such as the physical barrier or skin, the chemical barrier of antibacterial substances, and cells that move to infection sites to kill pathogens and activate adaptive immunity (described in steps 2-5).
Adaptive immunity starts when an antigen-presenting cell (APC), usually either a dendritic cell or macrophage, eats and processes a pathogen and shows this antigen to a helper T-cell.
The helper T-cells then take this information and activate B-cells and other T-cells.
Activated B-cells then produce specific antibodies that disable and mark pathogens by binding to their antigens.
Activated T-cells (called cytotoxic or killer T-cells) destroy infected human cells, which helps stop the spread of the pathogen.
As you can see, if you’re not familiar with the process already, it’s pretty abstract and confusing at first.
This year, I did it very different. The rest of this post will explain the lesson cycle and what I learned from it. In all, this lesson takes 2 to 2.5 class periods of 50 minutes each.
Direct instruction using a video to flip the classroom
I started off by assigning Paul Andersen’s Immune System video as homework. Students were required to take notes on the video. This can be a bit daunting, so I showed them this slide to give them a guide for important terms they will need to know. (That slide also tells students to look up the difference between antigens, pathogens, and antibodies. In a rare critique of Paul’s excellent videos, this particular video does not differentiate between an antigen and a pathogen, which is confusing for students at first.) In short, I’m using a Bozeman science video to flip the classroom.
Small group work in class
In class the next day (our first official day with this system), I had students in groups of two sketch out a diagram to explain the processes of innate and adaptive immunity. About half the groups more or less copied the diagram from the video onto their whiteboard, but it still made them think about the process. Other groups really explained the process in their own semi-unique designs, including a house and a castle metaphor (which was a bit confusing, but still creative and helpful for the groups making them). Here are a few examples below:
Example 1: A particularly thorough board with good explanations.
Example 2: Another good board, but simpler. This required me to ask the group a lot more probing questions, such as, “Do the T helper cells become B cells? Or do they just communicate? What’s going on there?” If they couldn’t answer right away, I would leave them with a few good questions (unanswered) and come back later. When I got back to them, they usually had figured it out and were eager to share what they had learned.
Example 3: A house metaphor. It was a bit of a stretch, but overall it worked for this group. Paul, in his video, uses a castle metaphor, which inspired this house metaphor. There were some good connections made, though I think the metaphor breaks down once you get into the different types of cells.
As they worked, I asked them questions about what they had written. This revealed a lot of gaps in their understanding, and they had to discuss or go back and read more information about the process. It went really well.
At one point, a group of students could not figure out the difference between antigens and pathogens, but another group that had figured out the difference took pride in going around and sharing their work with other groups. It was beautiful. Here is part of that group’s board that clearly shows the difference:
Here is another picture of a member of that group explaining the difference to a different group.
Whole group board meeting:
After students had made their boards, we gathered in a circle to present and discuss what they had drawn. This is something I’m still learning how to best facilitate. It can be a bit awkward at first. If you want to know more about how board meeting work, here is a great post by Kelly O’Shea, who uses whiteboards in her physics classes. I try to model how to ask good questions. It gets really quiet for a bit, as students wait for me to lead the group. Then….I leave (i.e., move out of the circle to another part of the room). This was hard to do…hard to give up some control.
But something magic happened when I left. They took over and started asking great questions and sharing explanations. My absence was the best thing for helping them start to own their own learning. Of course, I still did interject and ask a question here and there to give them some specific things to discuss. One example of this was asking them what the difference between humoral and cell-mediated immunity. I asked, then left the circle again.
Here is a picture of one of my classes during their board meeting:
Now that students have a pretty good grasp of how innate and adaptive immunity works, I finish this lesson cycle with a case study that I wrote about a man who doesn’t know that he has HIV. The case helps reinforce how immunity works and gives students an organism-level example of the effects of a compromised immune system on the cellular level. If you want to use the case study I wrote in your class (and edit if needed), please do! I would be honored.
Students work on the case study individually mostly, but they are allowed to discuss answers or work in a small group if they wish. If they don’t get finished with the case, I’ll have them finish the rest as homework. The next day, we’ll discuss the case study briefly, I will give a short quiz, and then we’ll move on to the lymphatic organs.
If you are reading this, you probably have noticed something…
This blog is pretty empty.
I only have a couple posts at this point. It’s the end of the school year, and we are wrapping things up here at school. Seniors have only two more days, then they are gone. Graduation is Monday. All other grades have one and half regular weeks, then finals week. Then, summer!
It’s a weird time to start a teaching blog.
My goal is to do at least two posts per week about things going on in my classes. I’m waiting until the beginning of next year to really get that started. Starting from the beginning, I hope to chronicle each of the units that I go through in my classes, so that other A&P and Physics teachers can see what’s going on and get ideas. I have learned a ton from other teachers who do the same thing.
I also plan to post (nearly) all of the resources and materials I use in my classes.
Finding a great resource online that you can use in class usually feels like winning the lottery, and I want to share all of the lottery tickets I have collected and created. So much of what I do as a teacher has come from other great teachers, and I want to give back!
What if I’m not a teacher?
That’s okay! The blog might not be as interesting to you, though. But if you want to, I’d still love for you to stick around and see some of the things going on in my classes! Also, feel free to share this with other teachers you know!
So in the meantime, have a great summer, and I hope to see you in a few months for a new school year!
Today, we wrapped up the digestive system and introduced the urinary system. It was a shortened schedule day, so we didn’t get too far. We only have about a week to spend on this unit because it’s the end of the school year.
The exciting thing about this system’s introduction is that I didn’t have to do too much…
We discussed the functions of the urinary system first. They already knew the primary functions from when we talked about blood pressure regulation back in the cardiovascular system and from just knowing what urine is. I had to add the regulation of pH to our function list because we haven’t discussed pH much at all…if only we had more time in the year. We still won’t get into how pH is regulated unfortunately.
I gave them today’s notes sheet, and in groups they were able to identify almost all of the organs without my help. I didn’t have to tell them what anything was. We had seen most of the organs in diagrams of other systems at different points. They had seen the location of the bladder, urethra, and ureters (sagittal view) back during the reproductive system. The aorta and vena cava were obvious from studying the cardiovascular system. The adrenals were from the endocrine system. And the kidneys are easy to label.
Obviously, the system gets much more complex than just labeling the system’s organs, the adjacent organs, and naming the general functions of the system. We’ll discuss the parts of the bladder, structure and function of nephrons, and urine production the rest of the week.
But today it was nice that they knew so much coming into the first day of the system! Not too revolutionary, but I’m always glad to cut out some unneeded lecturing/direct instruction.
In Anatomy & Physiology, we are about to start section 10.4 Capillary and Fluid Exchange. This is a complicated section dealing that helps to identify the location of water in the body (60-40-20 rule), describe the structure of capillaries, and explain the factors (blood pressure and osmosis) that affect edema (i.e., swelling).
We needed a refresher on osmosis, which we talked about way back in our first unit (an amalgam of topics ranging from directional terms to homeostasis to body systems). So as a review, I had students in groups whiteboard a diagram and description of osmosis.
They included an intentional mistake in their presentations (a la Kelly Oshea’s Mistake Game), which helped produce a more interesting conversation about the process and helps to normalize error. There were a handful of unintentional mistakes/differences/misconceptions as well, which is always welcome!
One of the mistakes in the whiteboard shown above was that the water molecules (solvent) are shown only moving toward the area of higher solute concentration. In reality, water molecules move both ways (so they needed an up- and down-arrow), but the net movement is toward the higher solute concentration. With whiteboarding and the mistake game, we got to talk about this and clear up these kinds of misunderstandings!
On to capillary fluid exchange and edema tomorrow!