# The Most Visually Appealing “50”

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.

Our 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.

# Stumbling Into an Interesting Example (Inverse Functions)

I was working on preparing problems for precalculus class on inverse functions. I started to type the following run-of-the-mill problem type:

If f(x) = blah and g(x) = bleh, show f(x) and g(x) are inverse functions.

And the work usually goes something like this:

f(g(x)) = f(bleh) = algebra kung fu happens here = x
g(f(x)) = g(blah) = some more algebra kung fu = x
Then conclude f(x) and g(x) are inverses.

Without thinking about it, I typed f(x) = -x - 6. Then I wrote the statement on my pad of paper, started working... exchanged the y and x, solved for y.... and got the exact same function.

y = -x - 6. Hmmm.

A part of me wondered if I had made a careless error. Double checked. Nope. No error. I wondered if the graph of the function was its own reflection across the y = x line.

Sure enough... Since y = -x - 6 is perpendicular to the line y=x, it will be its own reflection across the y=x line and consequently its own inverse. In fact, this made me think of an interesting question to pose to my students...

"Can you define a class of linear functions that are all their own inverses?"

In hindsight, perhaps I should be more mindful when constructing tasks for my students. But then again, this would be a great discussion to have with my students. Why does the function end up being its own inverse? Can we think of non-linear functions that are their own inverse? Can we define these classes of functions carefully?

# Midwest Noyce Conference: “Using Statistics in Math Classes”

This post contains resources for the talk "Using Statistics in Mathematics Classes" given by Jason Vitosh (Falls City High School, Falls City, NE) and myself at the Midwest Regional Noyce Conference on Thursday, October 2 from 2:15 pm - 3:00 pm.

Click on the link below to access the presentation file containing resources, images, and links.

10-2-14 Stats is not Math

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# Awesome Bowl Question: Segments in a Cube

We held our first bowl team session in MTPS today: an friendly, informal competition dividing the class into two groups of roughly equal 'ability.' The bowl competition we attend at the University of Nebraska-Lincoln in November consists of games of 15 questions with 30 seconds to answer each question.

Here is a question we spent some time on in class today.

Here is a screenshot from my Promethean board of the work we did as a group with students:

And here is a screenshot from the Geogebra sheet students constructed in front of the class at my computer terminal to demonstrate what is going on in this problem. If you would prefer to download the Geogebra sheet for your own tinkering, you can find it here.

No one got this question right in the allotted 30 seconds, so we spent some time developing the various parts of the expression. We color coded the edges on the cube (segments AB, AD, and AG) orange. We color coded the diagonals on faces of the cube (segments AC, AH, and AJ) green. We color coded the segment inside the cube (segment AI) black.

The kids had a lot of fun with building the Geogebra sheet and then trying to reconcile between sqrt(2) and the crudely rounded segment length 1.41. Or the sqrt(3) and 1.732.

The pink stuff at the top of Promethean board screen shot is our efforts to generalize this question. I tell the kids we are trying to "hack all possible problems." We use the term invariant and that we are looking to write a question that covers all possible question types. For example, what would happen to the question if s = 2? Or if s=3? Or if s=n? Is it possible to answer the question before the moderator finishes reading it? If so, at what point can we be confident we can buzz in and answer correctly?

# Differentiation in Math Class: Make Students Write

My elective math class has all grade levels represented within it. While not ideal, this is a feature of our scheduling system. So I have freshmen in Algebra I all the way up to seniors that have already completed Calculus AB. This poses a huge classroom differentiation challenge each Monday, Wednesday and Friday we hold class. Here is an instructional strategy I use that gets students writing about the mathematics they do in class.

Students worked on one of four things in the computer lab last Friday.

1. Construct 2013 Probe I Problem 7 diagram (2D)

2. Construct 2013 Probe I Problem 23 diagram (2D)

3. Construct 2013 Probe I Problem 11 diagram (3D)

4. Work on Alcumus problems independently

After we spent approximately 55 minutes in the lab, we returned to my classroom for a writing activity.

Here is the writing prompt I put on the board:
(Writing exercise on a separate sheet of paper to turn in to me)
Think about what you learned about the diagram or diagram(s) you built in Geogebra. Write a letter to the you of October 12. How did building diagrams in Geogebra help you understand the problem better?

I put eight minutes on the clock and informed students they would need to continuously write for the eight minute timeframe. Here are some samples of student writing from the activity. For convenience, I have inserted another copy of the problem. Immediately below each problem appears student writing associated with the problem.

The Students revealed their thinking about these math problems throughout their writing. While some students chose to concentrate on the construction process in Geogebra, others also revealed some of the mathematical structure they encountered while making the diagrams.

Writing samples from students that worked on Alcumus:

These writing samples revealed to me the depth of student thinking going on in the classroom. If I could have a superpower, I would be a mind reader. Then I wouldn't have to guess at what my students are thinking. Having the students write for an extended period of time gives me insight into how they are seeing the mathematics and gives me ideas on how I can help further their understanding and guide them as they struggle.

I collected these writings immediately after students completed them. I ran the pages through my ScanSnap scanner and converted them to a PDF for me to review later. I told students we would get these writings back out in a month's time, emphasizing the need for specificity on what they were working on and what they learned that day.

Going forward, these writings help me be more efficient with respect to differentiating classroom instruction. We don't need to be working on the exact same thing at the exact same time at the exact same pace for the students to engage in meaningful problem solving.

# Piecewise Functions: Geogebra vs Desmos

I was trying to write an item for an assessment where I would give a student a graph of a piecewise function and ask them questions about the domain, range, and to evaluate the output value for a specific input value - for example, find f(-3). The purpose of this post isn't really to pit Geogebra and Desmos against one another; rather, I want to make note of some of the things I was thinking about as I tried to make a piecewise function graph in each program. (Disclaimer: I am not an expert at this. I have much more experience with Geogebra than with Desmos. I want to see what the differences are between the programs to figure out when to use each of these powerful tools to enhance my instruction in mathematics. I am sharing my thinking about this task.)

Here is the graph I made using Desmos:

As I typed the function syntax into Desmos, I thought the editor was a little more user friendly than Geogebra. I typed <= and the editor automatically generated the less than or equal to signs for the restrictions on x. When I wasn't sure what to type, I browsed the examples of projects submitted by Desmos users found on the Desmos homepage. Ideally, I want to capture this graph and place it on an assessment. A photocopier may not pick up on the sections of the function on the graph given the lack of density (being able to make the segments and curves thicker). The way to make these curves denser was not immediately obvious to me. The circle centered at (1, 1/3) with radius 1/10 is my effort to place an open circle on the graph. To clean up the image, one thing I could try is modifying the restriction on x [for example, writing 1.2<=x<=4 instead of 1<=x<=4] so the user does not see the part of the function jutting into the circle shown at the left.

Here is the same graph I made using Geogebra:

After making the graph in Desmos, I assumed I could use similar syntax to make the graph in Geogebra. Using similar syntax, I had a problem with the restrictions in Geogebra. Each function has a default y-value of 0 for values of x outside the restriction. Pictures are, after all, worth a thousand words... here are the three functions shown individually. Take a look at the x-axis.

Here is the exact syntax I typed into the Geogebra input bar for each of the above pictures.
(1 / 4 (x - 1)² + 1) (-3 ≤ x ≤ -1)
(x - 2) (-1  <  x  <  1)
1 / 3 x (1 ≤ x ≤ 4)
I incorrectly assumed the syntax would be similar to that of Desmos. I knew from experience I could clean this issue up by using Condition to Show Object in the Object Properties menu if I had to, but I couldn't remember exactly how. I went to Youtube and found a video on graphing piecewise functions in Geogebra:

Below is an image of the corrected Geogebra graph using the appropriate If[ ] commands to define the rules f(x), g(x), and h(x).
Here is the corrected syntax I typed into the Input Bar to define f(x), g(x), and h(x):
If[-3 ≤ x ≤ -1,1 / 4 (x - 1)² + 1]
If[-1  <  x  <  1,x - 2]
If[1 ≤ x ≤ 4,1 / 3 x]
This approach eliminated the x-axis issues from the improper syntax I used at first. These graphs show some of the thinking I do day-to-day as a mathematics teacher trying to construct examples to display in class and problems to use in assessments. If somebody reading this has any advice that could help me become more effective with using Desmos or Geogebra for this purpose, please email me at saaberg@sbps.net or find me on Twitter (@ShelbyAaberg). **Update! See below for additional support on Desmos use. Thanks to Eric Berger (@teachwithcode) and Desmos.com (@Desmos).

Here is the additional resource from @Desmos.

# Nested Cylinders and Water Problem (UNL Math Day 2013 Probe I)

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.

Nested Cylinders Problem

Here's the Geogebra Sheet (built in the Geogebra 5.0 beta with 3-D graphing) if you'd like to take a look.

# Sorting Lists in Geogebra

Here is a prime example why Twitter is a great collaboration resource for math teachers. Last night I was killing time while waiting for a haircut. Reading through some tweets, I noticed a chat going on with the hashtag #ggbchat. With some luck, I caught the very end of the session and posted a question about something that's been bugging me about Geogebra.

I use Geogebra to analyze student summative assessment data in my classes. I like to sort the data to guide me when deciding which students I should group together for class activities. Sorting the data inside a Geogebra spreadsheet would eliminate an extra step for me (specifically, entering the data into Excel and sorting prior to copying & pasting data into Geogebra). I would think the software should allow a user to select a list, right-click, and then be given the option to sort the column of data. Here's a solution to the problem I faced, compliments of Geogebra guru John Golden (@mathhombre).

John also followed up with an idea to create a "Sort" button using a script.

Here is a screencapture of John's suggestion for making a Sort button.

# Pharmacy, Coumadin Dosage & Counting Theory for Students

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
Let
A = 5 mg dose of a drug (an arbitrary concentration)
B = 2.5 mg dose of a drug

Let
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.

Where
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.