Category Archives: High School

When Student Choice is a Struggle

Like most of the East Coast, schools here still have quite a ways to go before enjoying summer. I see my students for one more full week before final exam review begins and finals are given; a time which becomes more crazy as I travel to Kansas City for the AP Stats reading (or…Stats Christmas in June!)

It’s a starange time of year for AP Stats.  The College Board exam was given on May 9, and students took a final exam in my class before then, so we have been done with new material for some time now.  With a full 3 weeks (or more) between the exam and the end of the school year, it’s a time to take my foot off the gas from day-to-day material, but I still need to see my kids engaged in statistics.  Our culminating event, Stats Fair, provides a chance to highlight our program and keep the statistical ball rolling.  There’s really only one requirement for Stat Fair: design a project of your choosing which serves as evidence of your statistical learning. At the Fair, students show off their work to invited guests and fellow students (you can see pictures from previous fairs on my school website).  Teams must also provide printed documentation of their project to me.  It’s a great opportunity to be creative, study something you are passionate about, and explore something new.  There’s just one little problem…

Most student project ideas suck

Yep.  After a year of learning about experimental design, the role of randomness, and all sorts of nifty confidence intervals, many of my 17 year-old students will revert back to their 6th grade dopplegangers; proposing scientific studies of their peers’ favorite colors or chocolate chip cookie preference or how much honors’ kids backpacks weigh. Sigh….

Maybe I’m just jaded.  I warn the students early-on that it is likely I will reject their first 5 stats fair ideas.  It’s not that I am intentionally trying to be mean, rather I want my students to pick something memorable, something they could speak passionately about in front of others.  Working with students to develop their concepts could be the most frustrating part of my academic year.  Why is it so difficult for students to develop a “good” concept?

  • Despite a year full of examples and articles, it’s still a tough leap to the “real world” of teenagers.
  • Developing a good concept takes deep thought, revision, patience and reflection; not always teenage qualities.
  • The best concepts often contain a high dose of creativity – not something we are always accustomed to in math class.
  • It’s the end of the year, and the beach awaits

But all is not lost!  Today’s class started with a rousing success: a student, who had earlier proposed a study of NBA player ages (which was going nowhere), finally moved towards one of his passions – music. Using an app on his iphone, he tested the ability of peers to detect high and low pitches in mHz.  This led him today to some independent study online of the human ear, and reflection on the data he had gathered.

Another group is using their passion for fashion to see just how “skinny” jeans are these days, comparing waist sizes from different stores.  Some interesting data coming from this.  Another group is testing the “locally grown produce” claim of supermarkets…neat stuff!  And I’m looking forward to the random study of our school’s wireless device access – just how slow is it?  It’s the interesting projects which keep me coming back, and make this class memorable – like the team a few years back who entered and won the American Stats Association poster competition with their Bacterial Soap review.

Stats Fair is next Friday.  Look forward to sharing pictures and reflections!


Your Official Guide to Math Classroom Decorations

null_zps9fabad2bThe most recent challenge by the MTBOS (Math-Twitter-Blog-oSphere) is to share what’s on your classroom walls.  (Follow the action on twitter, #MTBoS30)

This post will go beyond my own classroom, and take you on a tour of many classrooms of my colleagues.  Here I present to you the Official Guide to Math Classroom Decoration.

To rank these items, I will be using the “Justin Scale”, an internationally-accepted scale of math beauty.  It is based on the works of Justin Aion, who is an expert on classroom decoration.  Seriously, you should be following Justin’s Blog for his daily classroom obsessions.

Here’s how the “Justin Scale” works

  • 1 Justin = an insult to scotch tape
  • 2 Justins = better than having a blank wall; marginally stimulating mathematically
  • 3 Justins = setting the tone for an engaging math experience
  • 4 Justins = cool beans!

You can see it’s pretty scientific.  Now, on to the decor!


null_zpsc7a23222In the history of math posters, has any student ever looked at one of these and thought “hey, so THAT’S how you add fractions”…seriously?  Sure, these posters are well-intentioned, but they are boring as heck and suck any imagination out of math class.  Also, I have to cover them up anytime the SAT comes around.





I like to have items around my room which tell a story. Maybe they are stories of past students or experiences; other times they remind me of math nuggets I pull out once a semester. These shirts are from a number of Muhlenberg College Math Contests from the past few years, each with a neat math concept from the year of the contest.  On the left, the 28th year celebrated 28, a perfect number. 27 is a cubic formula, and the 31st features the Towers of Hanoi.   Full disclosure, I designed the 16th shirt as an undergrad.




null_zpsbff2ac29Go to any math conference and you’ll find gaggles of math teachers walking around the vendor area with swag bags, free stuff the many companies have for you. TI posters are one of the most popular items, and you’ll find many math classrooms sporting these artifacts of math boredom.  “It was free, therefore I must place it on my wall”

These posters fill lots of space and give your room the right dose of geekiness.  And a reminder of the vast machine TI is.  Have any english teachers ever placed a large photo of a typewriter on their wall?  Nope.


1justin_zps94600af5 (1)



A338E759-516A-4B97-8AAE-EB7C225B9AB1_zpsxravcheaSo many cool infographics to choose from, so little toner. Love posting these guys all over my room; love it even more when I find kids checking them out just before the bell.  But they are a pain to print, and they age badly.











Usually purchased by rookie teachers, you will find these posters at your local teacher supply store.  Hunting season for these posters is short, running from early August to mid-September, so get yours while they last.  “Is that a cat hanging from a tree”….why yes, yes it is….


1justin_zps94600af5 (1).JPG


You don’t need to try hard to find neat stuff for your classroom.  A colleague of mine, who often teaches geometry, has pictures of neat things above his board.  Here’s your challenge: find your favorite items from, print them, and post them all over the place. The conversations start themselves.





Anytime you can post, share and provide inspiration through student work, it’s bonus time.  Here, I share pictures from many past years of our AP Statistics Fair.  These often lead to stories of projects past, and where many of these students are now in their colleges and careers.  As we get later in the year, student work will take over many of the empty spaces on the walls. Also, I have a John McClane action figure on this board….and you can’t blaspeheme Nakatomi Plaza….never forget!




Chi-Squared Tests: Rock, Paper, Scissors.

At the beginning of the school year, I shared a post about a fun Rock, Paper, Scissors applet on the New York Times website.  Back then, my class used the applet to collect data for 2-way tables, and considered appropriate methods for displaying the data set.

Fast-forward 6 months: my AP Statistics class is knee-deep into hypothesis testing, and we’re now up to Chi-Squared tests.  These are some of my favorite tests, as the data is often richer than what we find in tests for means or proportions.  Here’s how we used the Rock-Paper-Scissors applet to produce data:

  • Teams played the game in veteran mode.
  • In round 1, teams were given 3 minutes to play the game normally, which I’ve labeled the “guts” method.
  • In round 2, teams were given 3 minutes to play the game randomly, using “randint” on their calculator to generate a digit from 1 to 3, which corresponds to a move.

We then considered an appropriate test for assessing the data.  This comes on the heels of Chi-Squared Goodness of Fit tests.  But here we have two samples, and we want to determine if the proportions are similar in both samples: this was our first test for homogeneity, and it was easy to move through the mechanics of the test.


Doug Page also shared a worksheet he has developed for using the Rock, Paper, Scissors applet.  I do not have contact info for Doug, but I hope he provides some details on his success in the comments.

This activity will now become a yearly staple in my AP Stats arsenal!  Enjoy.

Some of My Students Failed Today! Woo Hoo!

A new semester has just begin here at my high school, and one of my classes is a co-taught course we call Prob/Stat.  The Prob/Stat course is one we offer to our 9th graders, as a follow up for Algebra 1.  It includes concepts in probability and statistics, along with algebraic concepts like systems, polynomial operations, and matrices.  The students in this academic class will take the Pennsylvania Keystone Exam in May, a graduation requirement, so this course is quite important for them.

My math department colleague and I, along with both co-teachers, agreed that we did not want this to “feel” like math courses they had taken up until now.  We wanted our students to become more reflective in their approach, think about their strengths and weaknesses, and devlelop their own learning paths.  We have embraced Standards-Based Grading and a policy of re-dos and retakes to help meet our ideals for this course.

On the first day of the class, I wanted to set the tone that communication and discussion would be valued in my classroom.  I asked the students to arrange their desks in a circle, which brought many questioning looks and rolling eyes.  But once we established our circle (actually, it was more like an oblong), I passed around small slips to paper to every student.  I asked the group to list any factors which had caused them to not perform well in their past math classes.  Many students were willing to share their stories: “I don’t do homework”, “Teacher X didn’t like me.”, “I don’t like to ask for help.”…the list was rich.  Placing a trash can in the center of the floor, I instructed students to ball up their slips of paper, and toss them into the bucket…they are in the past! Chum BucketI stole this idea from my time at the Siemens STEM Academy, where we started the week by catapulting our educations hold-backs into the chum bucket (it was Shark Week at Discovery Ed).  You can read more about the chum bucket activity on the Siemens STEM Institute blog.

Next, I asked the students to write something they could do, moving forward, to improve their math outlook.  What an awesome conversation!  One student shared her fear of reading problems in math, but a desire to work through it and seek help.  Many students confessed their need to complete assignments.  Others communciated the need to start self-advocating, asking more questions.


For many students in my class, this is their first experience with Standards-Based Grading.  Before the course began, I took all course concepts and arranged them into 4 anchors, mimicking the anchor language of the PA Keystone Algebra 1 content.  Each anchor contains 5-7 standards, written as “I can” statements.  The document also contains room for multiple attempts on the same standard.  As students complete notes or assignments, I instruct them to write the standard we are working on clearly at the top of the page.

In this course, we start off with the probability sections, so we actually led off with 4.5 “I can find the probability of a simple event”.  Probability is a topic which haunts students of all ages, sizes, and ability levels.  And while many students did just fine on their first quiz, a number of students struggled.  Under normal circumstances, this would cause deep sighs from me, and steamrolling on. But, to be honest: I HAVE NEVER FELT MORE ENERGIZED ABOUT STUDENTS STRUGGLING IN MY CLASS!

All students in the class have their own binder, which houses the Standards Tracker, and all assessments. During the next few class meetings, my co-teacher and I will develop groups for small group instruction to discuss mis-conceptions, and work towards the re-do on their 4.5 quiz.  At the same time, we have moved forward into 4.6, multi-stage events.  We are striving to set-aside time each Friday to be reflection and redo time, in order to establish regularity with these new grading concepts.  I find myself looking forward to students dicussing their needs, and working with them to do better next time.  It’s early in the semester, but already things feel different.

Check out some of my earlier blog posts on Redos, Retakes, and Standards-Based Grading:

Rick Wormeli – Redos and Retakes

Standards-Based Grading, twitter chat recap

Quality Assignments, #sbgchat

Matched Pairs with Hallway Bowling

The experimental design unit in AP Statistics is a fun one, with lots of opportunities to design activities, discuss possibilities and collect data.  For a few years, a “Hallway Bowling” activity I created has been one of my favorites for discussing matched-pairs experiments.  This year, I added a new wrinkle to this activity day, in order to economize class time.  As students entered the class, they drew a playing card, each having one of three suits which determined their group assignment for the day.  Each group had 7 or 8 students.  Groups then rotated through 3 stations, with 15 minutes on the clock, and with each activity designed to review a different aspect of the chapter.

In Station 1, students met with me in a small group, where we discussed experimental design, writing ideas and experiment trees on desks.  This was a departure from whole-class discussions, and more students had the chance to share their ideas on experiments dealing with clothes washing temperatures and drug trials.  Experimental design vocabulary like blocking and matched-pairs were clarified, and the small-group discussions were rich.  At the end of the day, the students shared how much they liked being able to share in a more intimate setting.

In Station 2, the group completed an actual AP item dealing with experimental design.  Papers were collected as a group, and I will randomly choose 2 paper from the group to grade.  Students knew this going into the activity, and this procedure holds all students accountable for the group grade/

In Station 3, the group went out of the room to play and collect data with “Hallway Bowling”.  15 minutes was enough time for students to practice, play, and collect data.

You can down loading the rules here:  Hallway Bowling

Here’s how Bowling works.

  • 2 markers are placed 5 meters apart (I had pre-taped blue X’s on the floor)
  • players stand behind one marker, and roll a golf ball as close to the other marker as possible.
  • During the data recording, players will roll 4 times; alternating hands and measuring the disatance to the marker.

BowlingAfter the activity, a whole-class discussion is held to talk about Hallway Bowling as an experiment.  What are we trying to prove?  How does our activity provide data for the experiment?  Where is the randomization?  What could be done to improve the design?  Here, we are looking to encourage “matched-pairs thinking”; where all subjects are exposed to both treatments (rolling with dominant and non-dominant hands), and we are interested in those differences.  We can also consider blocking here if we feel that males and females may be effected diffferently by the treatments.  We can also revisit the data later when we look at hypothesis testing procedures.

And about that data we collected?  My kids entered their data into a Google form.  There are some great comparisons to consider: right hand vs left hand, boys vs girls.  But how did the distances come out for dominant hands vs non-dominant hands?

Graph 1

Note the difference in medians here.  But can we directly compare individual player performances?  To do this, we can subtract dominant and non-dominant hand scores, and observe the differences:

Graph 2

If players are truly better with their cominant hands, we should see many negative differences here.  We see over 50% negative, but is there enough evidence to prove a mean difference for ALL players?  Time to start linking to inference.

So have fun with hallway bowling, and try some classroom stations!

A Day in The MTBoS Life

MTBoSThis week’s assignment in the 8-week Explore the MathTwitterBlogosphere project is to provide “A Day in The Life” of a math teacher.  It’s Monday morning, and here is my day….

It’s 6:30AM, and I’ve just arrived at my desk.  I’ve always been an early-riser; I don’t like feeling stressed in the morning so it’s always been custom for me to be in my classroom well before kids start rolling in at 7:20.  Weekend e-mails include a student seeking guidance on entering the PA Statistics Poster competition, an update on the Math Madness competition for our math club, and a few items from my local math teacher group, ATMOPAV.  Today in Algebra 2, we will be doing a test review for an exam on exponential and logarithmic functions.  Thanks to the great site Problem Attic, I was able to quickly assemble an assortment of review items for hanging around the room.

We have block scheduling here.  My day consists of 3 assigned periods, and a prep at the end of the day.  This semester, I teach all honors students and will have a more traditional schedule in the spring.

7:30, Homeroom – such a strange time of the day.  A group of students I see for only 5 minutes a day wander in, their attendance is verified, then they move on with their day.  Is there any “real life” analogy for this?  The DMV perhaps? Very difficult to get to know kids this way.  On Friday,Amish I had a conversation with a homeroom student who was excited to travel into the city to visit the Philly-famous Reading Terminal Market.  I encouraged her to visit the Amish people and their fine foods, but the student admitted an irrational fear of the Amish folks.  This led to an assignment from me: have a meaningful conversation with an Amish person, and report out to the homeroom.  I’m happy to share that Kianna talked to a few Amish people, found them “fun”, and is no longer scared of them.  A good start to the morning.

7:40, Learning Resource Center – my day begins with an assigned period in our LRC, where I help out any students seeking math assistance. Some come from study halls, others will visit if they have been absent a few days.  It’s a neat assignment, but can be stressful when a student comes seeking Pre-Calc help, and you realize that you haven’t thought about trig identities for a long time.

This morning’s deep discussion: how many spaces are you supposed to put after the period at the end of a typed sentence?  Young folks say 1; old-heads say 2.  Damn you Twitter and your shortening of everything.

Desks9:00, Algebra 2 – Today is test review day for exponential and log functions.  What I like to do on days like these is to post problems through the room, let students wander, have conversations, work through problems, and ask questions.  What I don’t want to have happen is to have students working quietly and isolated.  Many students have the same needs and misconceptions; I strive to create an environment where those questions bubble to the surface, and it is OK to need help.  At the end of class, we did a quick review of polynomial division as a table-setter for the next unit.  I love having students write on desks, as I can wander around the class and assess work.  It’s a great strategy for facilitating group discussion; just have a bottle of Formula 409 handy.

During this period, my AP Statistics colleague e-mailed me with an issue which will alter my plan for next period.  For years, I have used the “Against All Odds” video series for part of my Statistics class.  My favorite video provides a summary of the Physicians Health Study of 1981.  I love this video, as so much of the vocabulary we stress in experimental design is discussed in a real-life application.  The video is old, to be sure, but effective.  As of last week, the video ran fine on the site.  But today, all of these old videos from the series have been replaced with newer versions.  What to do when a resource you have used for so many years seems to have disappeared?  I have 30 minutes to figure this out…

10:30 AP Statistics – With a quick preview of a new experimental design video snuck in while my Algebra 2 students completed their review, I am set to go.  For homework, these students completed an actual AP exam item from last year which deals with survey design and bias.  This is a problem I graded last year in Kansas City in my role as AP reader, and I saw about 1,500 student responses.  It’s great to be able to discuss the grading procedure with students, and the exercise of working through the College Board rubrics and discussing them so intimately has improved my instructional practice.

ONE OF MY FAVORITE STRATEGIES: To go over the problem, I use my handy 24-sided die to choose a student at random.  Their problem is placed under the document camera and critiqued by the class.  This can be intimidating for the student, but I assure the class that everyone eventually will have their work assessed via camera during the year.   By this point in the year, I hope students have been through this enough times to see the positive value in peer evaluation.  I often start classes now by handing out index cards and asking a quick understanding question.  For example, a day after we had gone over the required elements when describing a scatterplot, the day’s opener asked students to describe a relationship and the use of r-squared.  Many examples went under the camera, and we had a snapshot of where we are as a class.

Doc Cam

The new video on experimental design is nice, but not as great as the older one.  The experimental design chapter is one of my favorites, with so many opportunities to think creatively.  Hoping to talk share our “old wives’ tales” project for this unit in a later post.  Looking forward to my “hallway bowling” activity for next time, which provides need for matched-pairs.

12:00 – Directed study – All students here have have a half-period directed-study built into their schedule.  Most teachers are assigned one to watch.  Fortunately, many of the kids in my directed study are in orchestra, so they choose to leave and go practice.  Many days I will have visitors seeking math help, but today is pretty quiet.

1:15 – After a lunch-time spent dissecting the Eagles victory with colleagues, I had my prep period at the end of the day.  Most of my time was spent writing and revision tomorrow’s Algebra 2 test.  Such a tough balance trying to develop an exam with enough rigor for honors students, yet be a fair measure of student growth.  The students should be in good shape.  Getting ready for the next chapter in stats, thinking out how my portfolio project for Algebra 2 will work, and a track and field discussion with a colleague round out the time.

Time passes…nice walk…dinner…Monday Night Football and….

9:00PM – #alg2chat – one of many weekly twitter chats I keep my eye on, this is a nice community of folks who share their successes, pains, ideas, and resources.  This week, the discussion bounced around from completing the square (where a twitter colleague looks forward to trying the box method), to synthetic division (and Dr. James Tanton’s railing against it), to a discussion of matrices and where they fit in a HS math sequence (answer – all over the place).  For me, this is the most powerful aspect of the MTBoS, having a network of enthusiastic educators looking to share ideas and make their lessons better.

Applications of Quadratic Functions

My class just completed its unit on quadratic functions, where we looked at all of the old favorites: completing-the-square, quadratic formula, -b/2a.  We also looked at “sideways” parabolas (those of the form x=…), and the formal definition of a parabola, including the focus and directrix.

But beyond the important algebraic processes to be mastered, I wanted students to appreciate the many vital applications of parabolas.  Towards the end of the unit, I posted the following task on Edmodo:

Research an application of parabolas, and explain how the properties of parabolas make them an effective shape for your chosen application, including specific vocabulary. Find a picture of your parabola in action, and use an application like Geogebra to find its equation. Turn in electronically as a single slide or page which could be posted.

Many of the questions students had about the assignment dealt with my intentionally non-specific instructions: Do I have to use the focus? What if I can’t find anything about the directrix? How do you use geogebra? How much do I need to write?  This was one assignment where I needed to “play dumb”; I wanted to students to think about what was essential, and craft explanations carefully.


On the day the assignment was due, I printed out all student papers.  I told students that the assignment would be worth 20 points, but I wanted their advice on how I should grade it.  What should I look for?  What evidence of understanding should I see?  After a rough start, where we argued the benefits of “creativity”, the class settled on a nice list of 4 ideas:

  • Proper use of vocab: while not all vocabulary words are required, those that are mentioned should be used properly.
  • Understanding of the application: pretty self-explanatory
  • Equation: is it correct? does it model the situation?
  • Structure / Flow: this was the compromise for creativity.  A good paper should have a logical structure which a reader should be able to follow.

The next step was to assign points to each category.  I told students that the assignment was worth 20 points.  With their group, I told students to come up with a method for allocating the 20 points.  After a minute or two, all groups had contributed their point-allocation ideas, which were recorded on the board:


I then took a pseudo-average from the group results, and a 7-7-3-3 point structure was agreed upon.


Armed with this rubric, I placed students in groups of 3, and used a version of Kate Nowak’s great speed-dating method to have students peer-assess their work.  With the clock set for 4 minutes, students shared their parabola discoveries, and discussed ideas for improving their paper.  After 4 minutes, the students (labeled A, B and C) moved to a new group.

  • A’s moved one group to their left
  • B’s moved one group to their right
  • C’s stayed at their current group

So, in 12 minutes, students had a chance to have their work evaluated by 6 peers, and see how their work stacked up to the class standard.  At the end of the “speed dates” I gave students 2 more days to revise their papers, based on their peer reflections, and this revised paper would be what was graded.

The class found some great applications of parabolas, and the chance to reflect and revise not only made the papers so much sharper, but also allowed students to share each other’s applications.  Some examples were:

Bridge Design:


Satellite dishes:


And solar cookers:


And bringing in the solar cooker application allowed me to share this video about a home-made solar cooker.  Cool stuff for kids to see!