Monthly Archives: September 2013

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.

10. IPEVO - Low-cost tech solutions for educators

IPEVO_Doc_CamIPEVO 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.



9. Online Stopwatch

Online_StopwatchMake 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.

8. A Desktop Scanner

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 modelDesktop_Scanner 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.

7. Dropbox

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 sysDropboxtem 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.


6. NCTM Core Math Tools

The following passage comes directly from NCTM's website.



5. Wolfram Alpha Computational Knowledge Engine








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.


4. Common Core State Standards for Mathematics

While Nebraska has not adopted the Common Core State Standards for Mathematics [yet!], the CCSSM does provide a CommonCorebeautiful 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.


3. Desmos

DesmosI 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.


2. Geogebra

GeogebraFew 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).


1. Art of Problem Solving Website

Art_of_Problem_Solving_SiteMany 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.



Students really enjoy learning about scatter plots, correlation, and the least squares regression line. They find it fascinating that as ice cream sales in urban areas increase, so do homicide rates, and the lurking variable in the background - ambient temperature and its effect on behavior! - is the culprit. The content gives students the opportunity to develop a robust BS detector (calm down, BS stands for Bad Statistics).

One data set that challenges the student's notions on regression and the LSRL is Anscombe's Quartet. We are using the activity in AP Stats class today.

APS Anscombes Quartet Activity

Anscombe_PDF_ImageBelow is a sample student response to the worksheet. The paragraphs students wrote at the bottom of the page revealed many things about the students' current understanding of LSRL concepts.

Captured_Student_Work_Example_Anscombes_QuartetExhibit A: Sample student response to the task.

I wanted to show students that because the coefficient of correlation depends on several non-resistant measures, including the standard deviation of both x and y as well as the mean of both x and y, a single data point can have a devastating impact on the LSRL. Specifically, one data point can completely reverse the direction of a linear model. The images below support this claim. I showed the students on a dynamic Geogebra worksheet posted by another teacher.

LSRL_Initial_ImageExhibit B: Initial LSRL. The source site can be found here.

Drag a point or two to an extreme of the screen and suddenly the direction of the relationship changes.

LSRL_Image_2Exhibit C: Moving points and checking the effect on the sum of the areas of the squares whose side lengths are the residuals for each point.

Coupling the Anscombe's Quartet worksheet with the LSRL activity in Geogebra convinced the students the coefficient of correlation, r, and the coefficient of determination, r^2, do not necessarily provide a complete summary of bivariate data. Furthermore, both measures do not always tell us definitively whether a linear model is appropriate for a bivariate data set.

In my collection of problem resources, I have a book full of MENSA problems. One particular puzzle caught my eye a couple years ago.



Exhibit A: An innocent looking trapezoid, just waiting to be cut up.

Forgive the distortion caused by the scanned page. The intent of the question author is to offer what appears to be, for all practical purposes, an isosceles trapezoid. Showing this problem to students seldom leads students to uncover what the author of the book claims is the solution.








This problem struck a chord with me. I wondered whether it would be possible to find other solutions. I wanted my students to investigate the possibilities and, if there were no other ways to do this, to rule out cases with mathematical reasoning. Several of my students came up with what appeared to be a solution, but it seemed like an accident of distorting the sketch a student made. I challenged the students to rigorously show whether or not the solution works in all cases, or if there were certain dimensional restrictions or conditions under which the students' solution worked. One of my students went on to model this problem in Geogebra.


 Exhibit C: A student models a solution in Geogebra. Is this solution a special case?

We should be careful about the language we use in class as math teachers. Had I told my students to find "THE" answer, they may not have uncovered the possibilities of the colored trapezoid above. My students went on to write inequality statements under which the second solution exists or does not exist. When I posed the problem to my students, I asked for "AN ANSWER," a far cry different from "THE" answer.

Amazing how definite a difference an indefinite article can make.

Today's post relates to what I did in one of my classes today. Effective teachers maximize class time. Administrators sometimes refer to this notion as "bell-to-bell teaching." Many teachers misinterpret this term to mean delivering instruction all the way to the bell. Direct instruction all the way up to the bell misses out on a great instructional opportunity: closure.

Since human beings tend to remember beginnings and endings - sometimes referred to as primacy for the former and recency for the latter - providing strong closure activities increases the probability students retain content long term. Michelle Haiken has a great blog post on potential closure activities here. My post will focus on one way I try to conserve class time.

I struggled in my first few years of teaching with how to hand back papers in an efficient manner. I avoid seating my students alphabetically. I dreaded the first day of school each year since my last name starts with two A's. I really don't like sitting at the front of a class. I prefer to sit not necessarily in the back of a classroom but on the boundary of a classroom. I tend to seat my students randomly early in the semester since I know many teachers choose to let the alphabet rule the day. A randomized seating arrangement does have an unintended consequence: handing back papers can torch two to three minutes of class time unnecessarily. Here is an example of how I approach the class period after an exam and distribute papers to conserve class time.

SaveClassTimeByLayingOutPapersExhibit A: AP Stats exams laid out on tables. On top of each exam is a printed grade summary for the student.

I use a desktop scanner to scan my answer key into a searchable PDF. Next, I enter the data into Geogebra to construct a boxplot and dotplot of exam scores, along with statistics on the exam. This allows students to compare their own performance with the entire class anonymously.

Exam_Summary_ExampleExhibit B: An example display of exam results from one of my classes. Protecting the identities of the students that performed poorly is really important to keeping those students receptive to feedback.

After analyzing the summary with students, I will go through the most frequently missed problems with students on the scanned answer key PDF. I identify the problems we spend time on through item analysis. If I am teaching multiple sections of a particular class, I will compare the boxplots and summary statistics across the sections and reflect on my teaching practice if I note any marked differences in median or standard deviation. I will post more information later on what I like to do with analyzing exam data and modifying my practice according to results.


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Our school district started a new alternative school this year. I went to the alternative school during our professional days. In retrospect, my visit was serendipitous to say the least.

I walked into an English teacher's classroom. After pausing momentarily, she asked, "Where are you going?" As I stepped out of the room, I replied, "I left my phone in the car. I need to get my camera." I returned and took a photograph of the classroom layout.Classroom_Seating_ProblemExhibit A: English classroom. Note the students in the back left corner of the background. These two students will bump into each other as they pull their chairs away from the table.

The English teacher asked me, "Is there a mathematical way I could figure out the best way to arrange my tables?" I asked her how many students does the arrangement need to seat. She said 19 students. I posed the problem to my MTPS students with only the photograph as a reference. Students began modeling the problem in several ways: drawing on graph paper, cutting out paper trapezoids and taping them to notebook paper, applying masking tape on the floor of the classroom to represent the tables, even using manipulatives.

Taylors_diagramExhibit B: A physical model of a possible table configuration, affectionately referred to as "Taylor's Diagram"

Before going to the lab to model with Geogebra, I did receive some clarification on the table and classroom dimensions. I included them in a follow-up file for students.

Seating_Problem_with_DimensionsExhibit C: The table photograph with dimensions listed.

The English teacher also added there is a tenth table available should we need to use it. I will publish some of the students' modeling work later. For now, I am interested in potential solutions readers will pose.

This summer I took an excellent class at UNL about using Geogebra. The final project required us to construct something useful for our classroom. I initially set out to make a free throw simulator that would capture results to a spreadsheet for presenting simulations in AP Statistics. As I worked on this Geogebra sheet, I became fascinated with making an animation for both a made free throw (easy) and a missed free throw (not as easy).


Exhibit A: Screenshot of the free throw. I am six feet tall. The scale is in inches.

Free_Throw_ParabolasExhibit B: Screenshot of the quadratic functions governing the missed free throw shot. The ball traveling from right to left is actually a reflection of a hidden ball.

There are several copies of the basketball that appear at various times. I learned a great deal through trial and error on this project.

Here is the Geogebra sheet for you to download and use. If you make improvements to the Geogebra file, please let me know.

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In 2010, I had the privilege of attending a USATF Level 2 Coaching School. One of the interesting presentations was by a field event coach using Dartfish. The video featured a world class rotational shot putter. He lost a competition despite tying with another competitor for his best throw. The tiebreaker goes to the next best throw. This shot putter fouled the other five of his six throws at the competition. Using Dartfish, the coach superimposed the video of the athlete's best throw with his worst throw - the throw in which the athlete fell out of the ring.

The video reminded me of playing Mario Kart on the Super Nintendo. After setting a personal record on a particular track, a player would race against his or her own 'ghost,' a recording of his or her best race. This throws video looked the same, where the best throw was the ghost. The software synchronized the two throws perfectly for diagnostic comparison.

A person could easily get lost in the details, wondering how it was possible for this elite athlete to fall out of the ring. The instructor encouraged us to pay close attention as he played the clip in slow motion reverse. The catastrophic error, the fall out of the ring, could be traced back to the thrower's first step with the left foot. The first step on the worst throw was roughly three inches different than the first step on the best throw. The instructor called this a cascading error. He stated many coaches spend too much time coaching later errors that are the result of the initial error. The trick, he said, is to diagnose and correct the initial cascading error.

This idea stuck with me. In my experience coaching football, basketball, and track & field, I have found good teaching and good coaching have a lot in common. Conversations with inexperienced and pre-service teachers often include classroom management strategies. Teacher preparation programs often point to classroom management as a struggle for new teachers.

These ideas bring me back to the notion of a cascading error. If your spider sense is tingling, and you sense a child is about to misbehave, try to diagnose the cascading error before the behavior blows up. Diagnosing a cascading behavior means identifying the root cause of the misbehavior and working to correct the child before misbehavior really gets started. Below is a wonderful exercise for both new and experienced teachers alike. Edward Sabornie at NC State has a video aimed at identifying good and bad behaviors by both students and teacher. Think about cascading errors while diagnosing the behaviors by both teacher and students. What would you do to prevent student misbehavior in this room? Click on the image below to access the video.








What does an effective classroom manager do differently than an ineffective classroom manager? The teacher skilled at classroom management:

  • Leverages student relationships to correct behavior and optimize conditions for learning
  • Communicates classroom rules and procedures effectively and efficiently
  • Creates and executes engaging lessons that reach all students
  • Maximizes and conserves time in class for learning
  • Lets administrators handle discipline but not before conference one-on-one with the student and placing a phone call home to the parent or guardian

My approach to classroom management is an amalgamation of ideas from many different sources. These are my views on effective classroom management.

  1. If your lessons are interesting and engaging, and the kids are thinking about the content, the probability of having a behavioral episode in your classroom approaches zero. Administrators conducting observations for evaluation purposes often record passive compliant behaviors as engagement. I have seen many students in my classroom and while observing other teachers feigning attention in a lesson. Passive compliance does not imply engagement. Even active compliance does not necessarily imply engagement in the lesson. If students are actively wrestling with the content, many issues resolve themselves. If you know a lesson may have a long lecture period (by long I mean more than 5 minutes), embed opportunities to socialize within the lesson (e.g. "Turn to your table partner and explain which of the three sampling techniques we should use in this experimental design to minimize bias; I will call on two people after the minute timer is up.")
  2. Establishing consistent procedures and high expectations translates to student success and teacher efficacy. I attended a presentation a few years ago by Harry and Rosemary Wong. They have a wonderful book titled The First Days of School. Harry and Rosemary Wong assert students coming to class on the first day have the following questions that a teacher must address to start the teacher-student relationship strong.
  • Am I in the right room?
  • Where do I sit?
  • What are the rules in this room?
  • What will I be doing this semester?
  • How will I be graded?
  • Who is my teacher as a person?
  • Will my teacher treat me fairly?

Another big idea I took from this book and presentation is the difference between classroom discipline and classroom procedures. Classroom discipline concerns how students behave. Discipline has penalties and rewards. I have yet to meet a teacher that felt truly triumphant after a power struggle with a student. Classroom procedures, in contrast, concern how things are done and have no penalties nor rewards. Procedures put the focus on what the students should do.

3. Assume everyone is on the same basic level. If students are to learn how to behave properly inside and outside school, we must maximize on the time students spend in our classrooms. It is gambling at best to assume students 'already know how to behave' or 'students should know how to behave.' Assume nothing and carefully communicate your expectations of your students. Students are bound to forget your procedures. Keep calm and gently remind students, particularly in the first three weeks of class, how you would like them to request to leave the room. Students tend to have several classes, particularly at the high school level. The student is not trying to offend you; he or she simply has a lot to remember. Model the behavior you want to see, and be prepared to teach students how to behave. As the old adage goes, "an ounce of prevention is worth a pound of cure."

 4. A phone call home is the single most powerful tool at a teacher's disposal. Use it early and whenever possible. I can remember feeling anxious and nervous calling home, particularly in my first three years of teaching. My view on the matter is definitely more complete now that I have a daughter of my own. If my child were misbehaving or struggling in school, I would want to know about it. I would want the teacher to call me. Calling home in week one, no matter how awkward the interaction, makes a tough conversation about academic performance in week six go more smoothly. Telling a parent or guardian you would like help connecting with his or her child gives you an ally at home, increasing the likelihood the student will correct his or her behavior in class.

5. Use positive reinforcement when students perform a procedure correctly. As one of my assistant principals told me this summer, "If we wanted to catch students doing something wrong, we could probably do that every single minute of the day." Here is an example of positive reinforcement In my classroom, I post the materials students need to have ready at the bell. As I walk in the room after greeting students in the hallway, I will quickly survey the room to catch students ready for class. "I see Ben and Alexis are ready for class. I know that because both of them have their graphing calculator and pencil on their table. Oh look! They even have their textbooks open to page 98. Man, that helps me out a bunch. Thanks for being ready." Kids won't see it as cheesy if you don't sell it as being cheesy. Make the comment and move the focus to the objectives for the day.

6. Any person visiting your classroom will know within 30 seconds whether or not you are effectively managing your students. Lew Romagnano, in his book Wrestling with Change: The Dilemmas of Teaching Real Mathematics, describes his experience as a researcher working with a practicing teacher. In the gap between theory and practice, Romagnano asserts while teachers do have problems to solve, there will also be unsolvable dilemmas in the classroom teachers must manage. Managing these dilemmas effectively will maximize the probability students will learn in your classroom.

7. Cold calling is highly effective in keeping students on task. Seasoned law enforcement agents and war veterans will say time almost seemed to slow down during firefights. As a teacher builds more teaching experience, the cognitive load during teaching does decrease slightly over time. I can validate this claim empirically; having taught how to solve a system of equations by substitution for several years, the cognitive load has gone down considerably from the first time I taught the lesson. Instead of being comfortable while teaching and going through the motions, I choose to focus on how many times I have called on each student. I am reading the body language of every single student in my room and looking for off-task facial expressions and body language. I will call on students I know do not have the answer. I will call on students I see change posture from sitting upright to slouching. I will call on students whose line of sight has drifted as soon as I identify it. I never let a student get away with saying, "I don't know." If a student knows they can be called on at any time, the student will likely choose to stay on task to avoid embarrassment in front of his or her peers.

8. Proximity is a powerful tool. More than management by wandering around, move with purpose about your room. Teachers that sit at the computer for the entire period lose out on valuable opportunities to assess students as they are working. Roaming the rows purposefully lets you spot both potential cascading behaviors and content struggles. Actively collect evidence as you roam the room on which students are getting it and which students are not.

Please share your effective classroom management strategies in the comments section.

One particular problem type that appears periodically on the UNL Math Day PROBE I involves tying an animal to a building with a leash. The student then must make determinations regarding the area in which the animal can roam. Below are the problems that have appeared on the PROBE I since 1990.

1994 PROBE I #18
A square shack 30 feet by 30 feet is in the middle of a large field. A goat is tethered to one corner of the shack by a chain 60 feet long. She cannot get into, onto, or under the shack, but can graze anywhere else she can reach on her chain. What is the area of the portion of the field she can graze?


2001 PROBE I #2
A dog is tethered to the corner of the outside of a ten foot by ten foot building by a leash which is 14 feet long. How much area (outside of the building) can the dog roam?





2002 PROBE I #25
A dog is tied to the corner of a ten-foot by 20-foot shed on a rope having length 60 feet. Assume the dog starts out as pictured below and winds his way around the shed counterclockwise as far as he can go. What is the total area (in square feet) swept out by the rope?





With my students, I approach the three problems in similar ways. I want students to successfully obtain the correct solution. Next, I want them to write the area in which the animal can roam as a function of other features of the diagram. What matters here? The length of the leash? The dimensions of the building? The shape of the building? The direction of the rotation? Writing a function gives the student an opportunity to work towards solving for all possible cases and unearthing the features of the setting we can treat as invariant.

Geogebra provides a great medium for modeling this problem. A fun exercise involves asking what happens to the area in which the animal can roam as the length of the leash exceeds the perimeter of the figure. Modeling the overlap becomes a challenge quickly. Jerel Welker, a math coach in Lincoln Public Schools in Lincoln, NE, has used the most recent of the three listed problems as a wonderful professional development opportunity for middle school and high school math teachers.

What other features of these problems can we augment to challenge our students?

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