PBL: Problem Based Learning

A few years ago, one of our local elementary schools had a 50th anniversary celebration. The principal of the school contacted me regarding an anniversary photo the school wished to take with students and staff. The principal asked our Math Theory & Problem Solving class to come up with the "most aesthetically" pleasing dimensions for the photo. Our class was tasked with using mathematical methods to systematically design the dimensions of both digits.

Here's a quote from the newspaper article about the photo before it was taken:

Browning, who has taught at the school since 1976, starting as a music instructor and then principal in 1998, said Friday would begin with the weekly assembly in the gym.

“Each class will perform a song from each decade, starting with the 1960s,” she said. Browning added the day-long celebration for students would also include special drawings for prizes, carnival games and the group photograph – if it’s not raining. The students and staff will form a “5-0” and have their picture taken by Downey’s Photography from the Scottsbluff Fire Department’s aerial ladder truck.

The principal emailed some information about how many students and staff would be involved in the photo, the approximate height of the aerial ladder, and that the photo would be taken in a field adjacent to the elementary school.

MonaLisaGoldenRatioOur class used mathematics to figure out the size of the viewing field of the camera (based on conservative estimates regarding the camera lens viewing angle). The students did some research and found information on the Golden Ratio, a number that appears over and over again in artwork.

The students wrote instructions for the staff members to utilize when organizing the photograph. We discussed the challenge of keeping that many students organized and engaged for a sufficient period of time in which the photographer could take the photo.

Below is a screenshot of the photo taken to celebrate the 50th anniversary of Westmoor Elementary.

(photo by Downey Photography)



After the photo was taken, I asked my students, "If we compute the ratio of the width of each number to the height of each number, how close is the actual value to our recommendation of the golden ratio (approximately 1.618)?"

Below are screenshots from the Geogebra worksheet examining how close the dimensions are to the target value of 1.618.


Screenshot with the initial question. Students can use measurement tools to judge whether or not the 5 and 0 in the photo meet the desired dimensions.


Clicking on the checkbox in the Geogebra sheet (Show / Hide Measurements and Ratios) reveals the details in evaluating how close the ratio of height to width for each digit is to the Golden Ratio.


Here are the details to the Geogebra sheet. If you would like to download the Geogebra sheet and mess with the values to see what happens, you can find the Geogebra sheet here.

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.


One thing I hope my students learn in school is how a person can unlock their creative potential and pursue their passions into adulthood. To this end, our Math Theory & Problem Solving (MTPS) class took a field trip to do some data collection. I would like to give special thanks to Daryl Payne for allowing our MTPS class to enjoy data collection (racing cars). Daryl's creativity and passion for racing inspired the students to re-imagine what is possible in the world outside school. The video below shows the electric car race track where students raced.

We spent our lunch period having a pizza party prior to racing cars as a reward for the work students have been doing in class. My original motivation in this trip was collecting data and trying to determine how a person could use statistics to potentially detect cheating through exceptional lap times. However, there are also many other mathematical and statistical ideas we can explore with what we learned on this trip.

Here are some things that students wondered about and could lead to mathematical investigations:

  • How much longer is the outside lane than the inside lane?
  • How does the electronic timing system work?
  • How much voltage/current is being supplied to each car?
  • What would a person have to know about electronics and circuitry in order to build such a track?
  • What is the difference between cars with magnets and cars without? (Cars with magnets can maintain higher speeds around the turns, for example)
  • What amount of voltage causes a car to fly off the track?
  • How should we determine the best racer? The fastest lap time? The best median lap time?
  • Is there a difference between the performance of the blue guest car and the silver guest car? If so, how could we detect this difference numerically?
  • How does the "KILL POWER" switch work?


Below, Mr. Payne gives the students some guidelines to follow while they practice racing on the race track.

This short video shows the beginning of a head-to-head race between students.


I am interested to see the types of mathematical investigations that spring up from our field trip. We will take our race data and use it to determine the best racers.


Here is a link to the original post on the broken pipe in a classroom problem.

This is the email I received from a teacher at a different school:

I don't have all the details and may not be able to get them... a pipe in my room burst on Monday and caused water damage in the hallway end and a total of 6 classrooms in full or in part... I have been told varying amounts as to how high the water was in my room and mere estimation by our custodial staff on how much water they themselves eliminated... the plumber said that the little copper pipe was spewing 12 gal per min...

Some of the outcomes of this activity:

  • Students gained a stronger understanding of Excel. Many students chose to write formulas and use the fill handle to brute force the time it would take to fill the various bathtubs.
  • Students deduced the connection between casework, writing out values, finding a trend, and generalizing through trying to write formulas for the volume of each bathtub as time passes.
  • Students discovered finding the "center" of a semicircle is more challenging than they anticipated. (see the half-cylinder tub on the original worksheet) Several used Geogebra to find the center through geometric construction.
  • One of the students made some progress on understanding 3-dimensional graphing with Geogebra 5.0 Beta.

Here are some examples from student work on Excel:

This student (below) chose to write the functions for fill and drain as linear functions in Geogebra:

Here is some work by a student trying to find the center of a semicircle:

A student modeled a theoretical room using Geogebra 5.0. (Apologies in advance for the sound quality.)

The students submitted written reports summarizing their thinking in this activity. I wanted to see a progression in their thinking from specific cases - like the percentage of a tub filled after 10 minutes - to generalized cases.


A teacher from another school across the state emailed me last week. She described a terrible facilities issue she has dealt with for days.

I don't have all the details and may not be able to get them... a pipe in my room burst on Monday and caused water damage in the hallway end and a total of 6 classrooms in full or in part... I have been told varying amounts as to how high the water was in my room and mere estimation by our custodial staff on how much water they themselves eliminated... the plumber said that the little copper pipe was spewing 12 gal per min...

We had been working on a different project during MTPS (Math Theory Problem Solving) class that I didn't want students working on without me being present... I had a sub coming in last Friday and decided this would be a great opportunity for students to do some modeling work in the computer lab.

Take a moment to consider how elaborate this situation is. Water accumulates within a room. Many places exist where the water could escape. Some places are obvious, like underneath the door. Other places are not as obvious, such as through electrical outlets or through the cracks in the drywall where the tile meets the wall. There are many potential sources for error. Modeling simple cases is easy, flow in is positive, flow out negative. But this is definitely a problem from the world outside school (I've never been a fan of the term 'real world'... that would imply high school isn't real... and I remember sitting through some interminable classes with a very real feeling of when will this class ever end...)

I worked to write a worksheet that would help students identify some of the potential complications in modeling how water would accumulate in various rooms. I thought about using theoretical "bathtubs" to simplify the computations and help the kids understand some of the complexities they would encounter in modeling a real room.


The students have spent one 46 minute period and one 90 minute period in the computer lab working on modeling the situation. The source worksheet appears below.

The Broken Pipe Problem-1

Students will have another 90 minute period in the computer lab Wednesday, then 30 minutes on Friday before each student gives a 3 minute presentation on Friday about their lab work and findings. I will share some of the students' work on this dilemma later.

P. S. On the worksheet, I use Google images to retrieve pictures of each of the bathtubs. The copy did not turn out as nicely as I had hoped, so I traced over the images and scanned the resulting worksheet. I told the students the half-cylinder tub should be oriented in a way where the curved side is tangent to the floor. (The 3D image on the worksheet makes it look like the tub is tilted, but I did not intend for the half cylinder tub to be tilted). The values for the time in minutes were arbitrary. All of my students have chosen to set up tables of values in Excel so they can address some of the interesting questions (like how many minutes will it take to fill the rectangular prism bathtub?)


XmasTreeI received an email from a teacher at a school far, far away (names have been changed to protect the innocent!!). She has an interesting logic puzzle related to a gift exchange situation within her own family. This problem has really resonated with the kids. They have tried making physical representations and even attacked the problem using Microsoft Excel.

The original email is below.


Here is the situation. My husband’s family draws names for Christmas and everyone gets one other person a gift. You cannot have yourself and you cannot have your spouse. There are four couples: Bobby and Becky, Tim and Tiffany, Dave and Darla, Molly and Michael. My mother-in-law (Becky) emailed last night and told us that Bobby forgot who he has. He thought he had Dave, so she emailed me (Tiffany) and my husband, Molly and Michael to see if we had him. I (Tiffany) have Dave. So now we don’t know how to figure out who Bobby has without everyone just telling who they have. I would like to avoid this. What I know is: I have Dave and Tim has Michael. Becky has Molly. So, Bobby could have me, Tim, or Darla.

What do you think – possible or not possible?

The question is whether it is possible, given the information provided, to determine who Bobby has as his assigned gift recipient without revealing the identities of all gift recipients. Try it out and see what you can come up with!!


In my opinion, this is one of the greatest lessons I have done in the math classroom. Not only did this activity scream STEM, the kids learned first hand the challenges a small business owner can face and that mathematics is useful in many different disciplines.

Back story: I was at a birthday party for my friend Erik's girlfriend. We got to talking about the new automotive shop he is building. The shop uses a waste oil heater. As they change oil in vehicles, they dump the waste oil into a container. The container has a pump that feeds the waste oil into a long metal tube that burns the oil. The exhaust from the metal tube is piped out of the shop. The air outside the metal tube is heated and blown into the shop area.

Because the shop is close to a municipal water source, the 215 gallon waste oil tank must be inside an enclosure that can contain an equal volume of oil in the event the tank were to rupture and its contents completely empty out. Erik needed to figure out the optimal dimensions of a cinder block enclosure for the desired tank location.

I asked if our MTPS class could serve as consultants and determine the dimensions. I went to the shop building, took photos, and posted them on my school website. Below are the photos I took and showed the students during a Monday class.


Exhibit A: Dimensions of the tank

Tank Scale Location

Exhibit B: Images to give sense of scale and desired tank location

6 Water Port, Needs 12 inch Clearance

Exhibit C: The water valve, which needs minimum 12 inches clearance

The students began work on modeling the cinder block enclosure needed to contain the oil in the event the tank is compromised. I had many issues in the Monday class trying to answer questions about the constraints. I invited Erik to attend my Tuesday class and describe the tank and the cinder block enclosure requirements.

Erik Nemnich from Nemnich Automotive describes the oil tank storage problem to students during the Tuesday MTPS class. Erik points out in the event of a tank rupture at the bottom of the tank, we can still include the volume of the tank when determining the dimensions of the cinder block oil containment.

The students worked for an approximate total of 100 minutes in class. They had to email their solutions by 8:00 am Wednesday morning. Here are some samples of student solutions.

Oil Tank Solution Samples

Students had to deal with many challenges. Many students wondered how the cinder block would be staggered to provide greater structural integrity. If we simply stack blocks one on top of the other, the resulting structure would be unstable. Since the blocks are 8" x 8" x 16", we can use an interlacing pattern at the corners of the structure to create this desired staggering without cutting block. Designs needed to avoid cutting the cinder block for obvious reasons.

Students had to perform conversions between cubic feet and gallons. Students modeled the position of the structure with Geogebra. Constraints on the placement wall included the location of the water valve, the location of the garage door sensor, the location of a nearby parts washer, and the amount of space the structure would protrude into the shop floor and workspace.

Below is an image of the finished oil tank storage cinder block enclosure.


 Exhibit D: The finished oil storage tank containment area

Special thanks go to Erik Nemnich at Nemnich Automotive for the opportunity.

Please read these two posts first to familiarize yourself with the problem.
Mailbox Problem
Mailbox Problem - The Sequel

Our class acquired a copy of the current postal route the mail carrier follows. The total driving distance in the development is 4.4 miles. (The tabular values actually sum to 4.7 miles, but there is some round-off error due to distances being rounded to the nearest tenth of a mile). An image of the current postal route appears below.













Our proposed route, which includes the administrator's desired mailbox location, saves the post office mail carrier approximately one mile per day. A PDF copy of our collaborative class summary report appears below. The administrator can use the information to approach the postal service about the mailbox location.

Mailbox Problem Write Up (Online)

Below is a Google Earth video created by the students displaying the optimal route they propose.




Disclaimer: I am not a computer programmer nor do I teach a class on programming. I do have two programming units within what I teach. One unit is on programming the TI-84 graphing calculator. The other is on using ALICE to teach simple programming constructs. Much of this is independent reading, independent learning, and a lot of trial and error.

Much of the technology learning teachers do arises from need. The need to do something drives the technology teachers choose to incorporate or not incorporate into their practice. The particular need about which I am writing came from my experience as a middle school football coach. Scottsbluff lies on the western end of the Nebraska Panhandle. Playoff sites are a big concern to coaches and teachers alike. Having an away game leads to lost instructional time, higher transportation cost, and big draws on the pool of available substitute teachers. Coaches desire to know the short list of potential opponents given the time demand film exchange presents. For many years, the coaching staff would try and break the playoff seeding up into smaller problems and finding certain minimums. For example, if A beats B, then we might play C, D, or E. After seeing their methodology, I knew I could be more efficient using technology, specifically Microsoft Excel. How could I resist a really complex, interconnected system of math problems?

Let me begin by explaining the playoff system within the classification our school falls. Nebraska has the following 11-man football classifications based on school enrollment numbers: A is the largest, followed by B, then C1, C2, D1, and D2. If you want to see the full blown points system, click here. The short version: the playoff seeding system is based on two things. First, the team's winning percentage puts them into one of three tiers. In a nine game season, teams with 6, 7, 8, or 9 wins are considered a Tier 1 team. Teams with 3, 4, or 5 wins are considered a Tier 2 team. Teams with 2, 1, or no wins are considered a Tier 3 team. Second, the opponent's tier at the end of the season determines the quality points ("Power Points") assigned each team for wins and losses.

Win against a Tier 1: 50 points
Win against a Tier 2: 45 points
Win against a Tier 3: 40 points

Lose against a Tier 1: 38 points
Lose against a Tier 2: 33 points
Lose against a Tier 3: 28 points


At the end of the season, the team with the highest "Power Point" average across the nine game season is seeded #1. Of 32 Class B teams, 16 will make the eight game first round of the playoffs. The higher seed will host a playoff game in the first round. This is a big deal in a state that is nearly 500 miles across from east to west, with 400 of those miles west of Lincoln, NE. Single elimination games are played until the final two teams play the championship game in Memorial Stadium at the University of Nebraska - Lincoln. You can probably imagine, with the fan based Nebraska enjoys, how high a privilege playing in Memorial Stadium is to high school football players across Nebraska.

An additional twist lies in the districts across Nebraska. There are 8 four team districts, labeled B1, B2, ..., B8. The district champions of these eight districts will automatically qualify for the playoffs. Roughly speaking, a team could start the season 0-6, win all of its last three games, and qualify for the playoffs. This adds an additional layer of complexity to the problem. Just because a team makes the top 16 Power Point seeds does not guarantee the team makes the playoffs.










Where will our kids be in the first round this year?

You can view the complete Class B football schedule from the Nebraska School Activities Association (NSAA) here. If you want to toy with the data yourself, you can find the comma-delimited .csv file at this link.


The data is already entered into Excel, ripe for the picking. This is the time of year I am called on by our coaching staff to provide an accurate projection for who will play who in certain scenarios. On the final night of the regular season, I can expect to receive text messages from coaches and friends around the state, asking for the playoff seeds and sites. In both the 2011 and 2012 football seasons, I had the projections nearly two hours prior to any major newspaper, the Omaha World Herald and Lincoln Journal Star included.

I will share the methods I have used to make these predictions below. I set out in 2011 to make a spreadsheet the coaches could use to forecast the first round seeds and sites. First, the coaches would want a simple table of the week 9 games to enter wins and losses. Next, the sheet would have to be completely dynamic, since every team affects all other teams on its schedule. Last, it would need to compute the average power points and rank the teams.

Since my programming background is limited, I approached this problem through trial and error. I knew I could write "if-then" statements to compute Power Points for teams on the bubble going into Week 9. Specifically, teams that are on the bubble in Week 9 are those that enter Week 9 with either 5 wins or 3 wins.

5-3 then wins = 6-3 = Tier 1
5-3 then loses = 5-4 = Tier 2
3-5 then wins = 4-5 = Tier 2
3-5 then loses = 3-6 = Tier 3

These bubble teams wreak havoc on the Power Points of every team on their schedule. I wrote "If-Then" statements to model the impact of these bubble teams on their opponents.


Here is an example of such a statement for a Week 9 Bubble Team:
=IF(X3="W", 50-5*(R15-1), 38-5*(R15-1))
As you can imagine, writing these statements for every team, tabulating the win/loss percentage, making the predictive table, and making the seeds is incredibly time consuming. Below you can find a copy of the Excel sheet I made last year to model the playoffs.

Final B Playoffs Example (from 2012 season)

This year, I am trying to work smarter, not harder. I am working with a student to learn a little about Visual Basic and to use Macros to set up an Excel spreadsheet I can use year after year. In theory, I should be able to download the NSAA source spreadsheet, copy and paste the win-loss data into my spreadsheet, and have the spreadsheet do all the heavy lifting by a combination of explicit formulas and string operations in Visual Basic. I will post the sheet in the coming days once it is complete.




Click here to see the original "Mailbox Placement" post. You will want to see the details before reading below.

My students have spent some time trying to model the mailbox problem. Let's examine the similarities and differences across the various models students built with Geogebra.


A photo collage of students in action. I used Fotor to make this.

Similar actions among students
- Tying the source image to three points to preserve scale
- Using vertices to represent mailboxes
- Using edges, segments, and curves to represent driving routes
- Using arrow heads on segments to indicate directed paths
- Using color to mark different paths and different directions
- Breaking the entire graph into smaller sub-graphs in an effort to solve the smaller sub-graphs

Different actions among students


- Measuring distances on the graph and applying scale factors to determine true driving distance (this information will help the student find an optimal path with respect to distance)












Using check boxes to give the user the ability to remove layers of the graph (for example, the background photo) to help the user keep information organized














-Offering the user the ability to hide the mailboxes (I thought this was a great feature because it allows the user to focus on the neighborhood layout without the clutter)






- Solving smaller sub-graphs and then taking the union of these solutions to solve the larger problem



- Seeking alternative maps that offer more or fewer details







The administrator came to class Thursday to monitor work for a short time and offer additional background information to assist students with modeling. Our class has learned the actual mail route may have been assigned arbitrarily - that is, without the aid of software. The administrator is working to determine the mail route the delivery person takes. If we are able to acquire this information, we could work to find a path which contains the administrator's preferred mailbox placement that also shortens the total driving distance - a win for everyone.

Our class will spend two 90 minute class periods this week finishing up modeling work and coming to a class consensus on the optimal path. I will post the findings as they become available.