Here is a problem I have been working on this afternoon. I worked on building a model in Geogebra to help students understand the situation.
2013 Probe I Problem #3
A cylindrical can with radius of 6 inches is filled with 2 inches of water. When a smaller cylindrical can 4 inches tall is placed inside of the first with its bottom lying on the bottom of the first, the water in the first can rises a further inch to 3 inches. What is the radius of the smaller can?
A. 3 B. sqrt(12) C. sqrt(15) D. 4 E. sqrt(18)
Here is a screenshot of the Geogebra model I constructed.
Here's a PDF of my typed-up solution.
Here's the Geogebra Sheet (built in the Geogebra 5.0 beta with 3-D graphing) if you'd like to take a look.
My wife is a pharmacist at a hospital. She and her co-workers often provide me with some really interesting math problems. Clinical pharmacists often need mathematics to effectively work with complex patient cases. Modeling drug interactions becomes tricky as the number of prescriptions for a particular patient increases. As the number of prescriptions increases, the amount of time a pharmacist must spend doing drug interaction research also increases. Time is money, so herein lies the problem.
One of my wife’s co-workers asked her to ask me, “What is the number of possible permutations of seven days?” After looking at a handwritten note and talking to the pharmacist directly, here’s a more thorough description of the problem.
Coumadin is a drug used to treat issues associated with blood clots.
Risks come with any anti-clotting drug. If the patient has a car accident, for example, bleeding risk increases dramatically and can have dire consequences. The benefits associated with Coumadin usage must be weighed with the bleeding risk. Computing the correct treatment scheme – a schedule and selection of doses – is important to the patient’s safety. Doctors and pharmacists determine dosage based on a target value: INR target value of 2.5, target range of 2.0 – 3.0 (a confidence interval!!), and if the INR goes above 4.0, there is no greater therapeutic benefit to the dose and patient bleeding risk increases beyond any benefit.
Dosages for this particular drug vary dependent on many factors. However, for the sake of this problem, the pharmacist in question wishes to investigate how much time it will take to write an Excel spreadsheet to determine the different possible treatment schemes. We will assume the simple case: either a patient takes a dose (pill) on a particular day, or a patient does not.
Back to the original question: why does the pharmacist ask for possible “permutations” of the days of the week? Because these would correspond to patient dosage schedules. For example, if the patient takes a dose three days a week, they might take that dose on Monday – Tuesday – Wednesday and not take a pill the rest of the days of the week. Or they might take a dose Tuesday – Thursday – Saturday. All the possibilities would not correspond to permutations, despite the wording of the original question. What we really need to consider are combinations. Permutation means order matters, so we would treat Tuesday – Thursday – Saturday and Thursday – Tuesday – Saturday as different events, when in reality they would be the same treatment schedule in a given week.
If the patient takes a dosage of Coumadin three days a week, then the possible number of treatment schedules would be 7 choose 3.
There are 35 different possible ways to choose 3 days out of the 7 days in a week.
How would we count the number of possible treatment schedules assuming a patient either takes a dose or does not take a dose each day? We need to consider all the ways to select different groupings of 7 days. Enter Pascal’s Triangle & binomial coefficients.
The bottom row above corresponds to each case. 7 choose 0 would correspond to a patient not taking any Coumadin. There’s only one way for that to happen: the patient takes no doses. This would be the trivial case. We aren’t concerned with the patients not taking any doses. The blue number, 128, is the sum of the values in the row with 1, 7, 21, 35, 35, 21, 7, and 1.
Written a little more formally, we have
This value, 127, is 128 – 1, which is , where the 2 corresponds to the number of daily outcomes, either dose or no dose. The 7 corresponds to the number of days in week, and the subtracted 1 corresponds to the trivial case where of the 7 days, the patient takes doses on 0 days.
(Tangential side note: This value, 127, is the difference of a power of 2 and 1. I am immediately reminded of the original Legend of Zelda game on the Nintendo Entertainment System. The maximum number of rupees – currency – a player could obtain is 255, which is 256 – 1, and seems to be related to binary storage limitations of the game.)
This was the answer the pharmacist needed to communicate the number of different outcomes his Excel spreadsheet would need to consider. Some of the treatment schemes might be impractical, so instead of considering 127 different possibilities, he will argue they should boil the cases down to 10 or 12 common dosage schemes.
But for me, the math doesn’t stop there.
What if a person is taking a drug with a more elaborate dosing scheme? I wanted to put a structure on the next case up in complexity, the case where a patient might take one of two different doses on any given day. Here is the structure I used when reasoning through this case initially.
Case 2: Two different dosages on any given day
A = 5 mg dose of a drug (an arbitrary concentration)
B = 2.5 mg dose of a drug
1 = Monday
2 = Tuesday
3 = Wednesday
4 = Thursday
5 = Friday
6 = Saturday
7 = Sunday
A patient might miss a dose. Let C = no dose. Or, they might be instructed to take nothing on a particular day. Either way, the patient takes no dose. The table below describes all the cases for a week.
To count all the possible outcomes efficiently, we have 3 independent choices each day. At least, we will treat each treatment choice as independent although this may or may not be practically true. The total number of possible treatment schedules, then, would be
But we would also throw out the trivial case (no treatments on any day) by subtracting 1. So our total number of possible treatment schedules would be 2,186.
I then wondered if I could write a function to count the number of treatment schedules for any possible number of different dosages.
d = the number of different dosages a patient may take
t = number of days in the timeframe of reference
7 above for days in a week; we could change this to 30 for days in a given month
The number of cases grows quickly as the number of different dosages increases.
This problem is a great practical example of mathematics used in an authentic example. I can extend this problem to exponential decay and dosing with antibiotics. Kids need to know there's a reason a person needs to take an entire course of antibiotics, even if they are feeling better midway through the treatment course.
I will pose this question to my students when we start our unit on counting theory this coming school year. I need to spend more time this summer finding authentic applications locally.
Here is information from my NATM Pre-Professional presentation.
Clssrm Mngmnt 2013-14 NATMPre_Prof
If you would like any additional information, please let me know. Also, here is a link to a previous post I made specifically for the NATM annual conference in Kearney on 9/30.
NATM 2013 (the Nebraska Association of Teachers of Mathematics annual conference) had some great presentations and provided great resources for math teachers at all levels.
One of the best presentations I saw was given by Kathy Niedbalski on Sunday night before many of the participants showed up. The Youtube video below gives the content of Kathy's talk.
I could not resist the urge to give this activity to my MTPS students immediately. The students played games independently for 15-20 minutes, then insisted on trying to 'hack' the game (student speak for finding the optimal strategy). This activity was a GREAT way to introduce graph theory to the students.
Exhibit A: Students posing with their thoughts on the board.
The greatest value my class found in this game was the introduction to breaking a complex problem into smaller solvable problems. The computational thinking strategies the students demonstrated actually seemed to be straight out of a computer programming class. I am excited to see what comes out of our typed summaries in the coming days.
Below is a list of ten resources I think will benefit Nebraska elementary, middle school, and high school math teachers. If you know of any resources, please feel free to leave them in the comments section on this post. I made this list with Nebraska Association of Teachers of Mathematics (NATM) Kearney Annual Conference participants in mind.
IPEVO offers great low-cost technology solutions for educators. Pictured at left is the Point 2 View Document Camera, which offers a low-budget portable document camera for teachers. This is the only product I own from IPEVO, but I have heard great things about the other doc cams and products on the site.
Make tasks in your class time dependent with Online Stopwatch. This free site has different timers and tones from which to choose. If I want a writing or discussion task to be five minutes, I set the count down timer and focus instead on circulating the room and providing feedback and guidance to students.
A desktop scanner allows the teacher to quickly take exemplary work a student has created in class and convert it to a PDF on the spot. The model I use is a Fujitsu ScanSnap S1300i. Many entities that recognize exceptional teaching, such as the National Board for Professional Teaching Standards, require applicants to include student work examples. I have used my iPad in the past to take photos of student work to convert to PDF, but sometimes the ambient lighting can lead to a fuzzy or distorted image. The ScanSnap lets me scan documents directly to my Google Drive and converts the scan automatically to a searchable PDF using text recognition software. This is one of the single greatest personal purchases I have made as a teacher.
Digital file management is critical to teacher success. Keeping organized files allows a teacher to later reflect on lesson plans and activities to maximize efficiency. Dropbox is a reliable file management system that lets educators ditch the thumb drive. Our math department has a Dropbox account we use to post our common lesson plans and assessments. Our teachers can access these files from anywhere. Teachers that prefer to craft materials in the comfort of their own home can do so using Dropbox. Changes to files are made automatically for all users on a shared folder to see.
The following passage comes directly from NCTM's website.
Wolfram Alpha provides students free access to Mathematica, a powerful software package. Wolfram Alpha gives instantaneous access to a Computer Algebra System (CAS). In the spirit of UNL Math Day, above is an image of the prime factorization of 2013, something bowl participants may wish to memorize. Wolfram Alpha also offers some great mathematical oddities like the Nicolas Cage Curve.
While Nebraska has not adopted the Common Core State Standards for Mathematics [yet!], the CCSSM does provide a beautiful outline for a complete picture of K-12 mathematics. The CCSSM shows a progression of skills that build upon each other. Bill McCallum at the University of Arizona is one of the visionaries of the Common Core math movement. I had the pleasure of hearing him speak at a conference at UNL in 2011. Although ironic a conference on the Common Core was held in a state that does not use the Common Core, Dr. McCallum shared a prolific resource for teachers called the Illustrative Mathematics Project.
I haven't had much time to play around with Desmos yet, but there's a reason why Twitter and the MTBoS both rave about this free product. As a teacher productivity tool, Desmos has great potential. Another added bonus is that Desmos works really well on an iPad right now.
Few things in the development of math instruction historically can stack up against Hohenwarter's masterpiece. The Geogebra Tube community allows teachers to download free materials prepared by other teachers and make the materials their own. It does not matter whether you teach kindergarten or high school calculus, there is something on this site you can use to improve your math instruction immediately at no cost. A couple of Geogebra Kung Fu Masters I know in Nebraska are Jerel Welker (LPS), Dan Schaben (Arapahoe), and Matthew James (LPS).
Many teachers may challenge me on being bold enough to say there's something better out there than Geogebra, but I feel strongly enough to say the Art of Problem Solving website is one of the best resources available to math teachers. Students often express frustration in school because math problems are boring. Much of the thinking is done for the students, particularly in textbook problems. Dan Meyer has a great talk on the thinking we unintentionally do for students.
Two wonderful differentiation resources are available to teachers on the Art of Problem Solving website: Alcumus and For the Win.
Alcumus tracks student progress through curriculum strands. For the frustrated student, the interface offers side quests and tracks experience points like a video game. For example, a student might earn 500 XP by clicking "I Give Up" 250 times in an hour. Students can track their own progress and work through problems at their own pace. The interface gives feedback and shows detailed solutions for every problem.
For the Win! allows teachers to set up localized games among students in a computer lab. The problems come from robust sources like MATHCOUNTS and the American Mathematics Competition. For the Win! lets students make mistakes in a safe environment.
The Art of Problem Solving site has a Youtube channel with videos instructing students on how to do challenging problems from various competitive exams. If you are interested in starting a math club at your high school, this Youtube channel is an invaluable resource for AMC preparation.
Please feel free to share additional resources with me. I appreciate your time and thank you on behalf of your students for taking a look at this blog post. If you would like to email me directly at school, you can find a link to my contact information here.
The "Home Depot Garden Club Photo" investigation described in an earlier post has been dubbed the "Berry Bush Problem" by my students, likely in an effort to conserve syllables. Another reason for the renaming is due to the mislabeling of a plant at the Home Depot: what we thought was a blackberry bush was actually a blueberry bush. The activity associated with the photograph will be the basis for my presentation on 9/30 in Kearney at the Nebraska Association of Teachers of Mathematics (NATM) Annual Meeting. The description in the conference listing reads:
CCSSM Practice Standards 2 and 4: Using Images to Maximize Student Engagement: This session will model how to use an image as a hook and build a lesson which engages students in the listed practice standards. Participants will formulate questions and conduct investigations based on these questions. The featured image is one the presenter took while shopping at Home Depot. Geogebra will be used in this session. Topics may include, but are not limited to, volume, area, linear functions, temperature, and modeling. Participants would benefit from bringing a laptop or tablet device with wireless capability.
For reference, here are the two Standards for Mathematical Practice featured:
CCSS.Math.Practice.MP2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
CCSS.Math.Practice.MP4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
I learned a great deal from working alongside my students and acting as a facilitator. Here is a subset of the outcomes I saw in my classroom:
- Transfer of mathematical authority from teacher to student.
- Ownership and intrinsic motivation in the revision process.
- Students developing healthy strategies for working through point of frustration.
- Procedure viewed as process rather than the culmination of the mathematics.
- Rich discussion regarding potential sources of error, particularly roundoff error in Geogebra.
- Collaboration and mentoring between experienced and inexperienced math students.
- Understanding we make sacrifices in modeling that may be revisited later.
Initial student video:
Andrew, Matt, Nik, Ryan
At the NATM talk, I will show examples of Geogebra sheets and PDFs written by other students in class and provide insight on these students' mathematical backgrounds. I will also offer information on how I assess such a project in a way that encourages revision.
Revised student video:
Andrew, Matt, Nik, Ryan