Monthly Archives: November 2013

Student-Created Polynomial Digital Notebooks

Over half-way done my first time teaching Algebra 2 in over 8 years (under block scheduling), and it’s amazing how much technology has changed many of my former approaches.  Nearing a chapter on polynomial functions, I was somewhat dreading the experience.  This is a pretty dry chapter…synthetic division, rational root theorem, complex conjugate roots…there’s a lot of rules and regulations here, and not much room for engagement.  Here’s what I settled upon to avoid dry lectures, promote student ownership, and encourage a serious review of resources:


On day 1 of the unit, I gave students their assignment for the chapter, which you are welcome to download: Polynomial Notebooks.  The link describes the assignment:

This chapter contains many landmark theorems and ideas for analyzing polynomial functions.  Your job is to create a digital notebook of the ideas below, using a Google Document to house your information, and eventually share with the class.  You may use online resources, or examples you create, to serve as examples for each idea.  The goal of this document is to serve as a resource which demonstrates an understanding of the ideas, and which helps you study their meaning and usage.

To be honest, this did not start off as I expected, as my students were simply not accustomed to not being fed material.  The idea of seeking outside resources, being able to weigh their merits, and summarize them for use was foreign.  But our second day in the computer lab was fruitful, as students had great questions about the Intermediate Value Theorem and Descartes’ Rule of Signs; what was really fasicnating were the student attempts to think about the ideas in language which made sense to them: “Does this mean that…?”, “Does this example work here…”.

Not all class time was spent doing research.  We started class with sythetic division problems done on boards around the room.  Also, I developed 3 “food for thought” questions each day during the unit, which helped drive discussions.

  • Name a polynomial with roots  2i and -3 (multiplicity 2).
  • How many possible negative real zeroes does the following polynomial have?
  • Show that polynomial given must have a real zero between 2 and 3.

Enjoy all of these Food for Thought questions: food_for_thought_daily_qs2

It’s also interesting how different groups have chosen to format their notebooks.  While some are going through my list in a linear fashion, others have developed connections and examples to help make sense of the rules.  One ambitious young man has developed an outline of the topics, rearranging the ideas, providing examples, and organizing links.

Tomorrow we will have a group quiz on the material, and I feel like this experience has helped students personalize their needs, and think about their gaps.  I also feel more confident in the aiblity of the class to think about these  polynomial rules as a whole, rathern than as a set of disconnected ideas.  Hoping for a good day!




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.

Counting Principles and “The Price is Right”

I have a confession to make…’s really quite embarassing…

I’m a Price is Right nerd.

{sigh} wow, feels so good to get that off my chest.

priceSince I was a little kid, I loved watching the Price is Right.  I know all the games, many of the prices, and of course can name the “back in the day” models without batting an eye (Dian, Holly, Janice).  I even made the pilgramage to Television City a few years back to see Bob Barker in his last years of hosting.

Now, as a stat teacher, I have used a number of Price is Right games in the classroom as probability lessons.  I’ve given a number of talks using Plinko as the centerpiece of a lesson.  Almost all of Price’s games have some probability element.  Here are a few games you can discuss on your classes, starting with basic ideas, and moving up to more complex counting principles.


DoublePricesThis is the most simple probability game on the show.  The contestant is shown a prize, and two possible prices for the prize.  If the contestant guesses blindly, they then have a 50% of choosing the correct price, and winning the prize.  In all of these games, the pricing aspect is a “clue” to the player, which hopefully increases their chance of winning.  But in all of these examples, we will look at the games as random chance experiments.


This game is only slightly more  difficult than Double Prices.  In it, the contestant is shown three prizes, each with a price tag, one of which is an incorrect price.  If the contestant identifies the incorrect price, they win all 3 prizes.  Given random guessing, a contestant has a 1/3 chance of winning.


Here’s where we start looking at some more interesting counting methods.  In this game, a contestant can win a large prize and a smaller prize by correctly giving the price to the smaller prize.

SafeThe smaller prize has 3 distinct digits in its price, which the contestant is given.  They must place the digits in the correct order to find the price.  With 3 digits to place in order, we have 3! = 6 possible prices.  BUT, in this game, the price always ends in zero (they don’t tell you this, but it’s always true), which means this is essentially a 50-50 game.  For example, if the 3 given digits are 0, 9 and 5 – then there are only two possible prices, $950 or $590.


A contestant has a chance to win a prize with 4-digits in its price.  A “dummy” price is given, like $5447, and the contestant must determine if each digit in the actual price of the prize is higher or lower than the digit in the dummy price.

BonkersTo make their guesses, the contestant places markers either above or below each digit in the dummy price.  If they are correct with all 4 digits, they win the prize.  If they are wrong, they can go back and make changes.  A total of 30 seconds is given to make as many guesses as they can, running back and forth between the game board and the guessing buzzer.

Each digit has 2 outcomes, higher or lower.  Since there are 4 digits, there are 2 x 2 x 2 x 2 = 16 different outcomes.  With only 30 seconds to make guesses, this game often comes down to how well the contestant uses their time to maximize the number of guesses out of the 16.


In this game, a contestant can win 3 different prizes: one with 2 -digits in its price, one with 3-digits, and a car with a 5-digit price.

The contestant is first shown 3 digits, 2 of which make up the price of the first prize.  The goal is to use as few “chances” as possible to get the correct price, which allows the player to move on the the next prize.  With 3 digits to choose from, there are in theory 3 x 2 = 6 possibilities.  But like Safecrackers above, the price will always end in zero; so there are only 2 real choices.

For example: if the given digits are 0, 3 and 5, then the only real possibilities are $30 or $50.  Often, the frustration in this game is associated with contestants who don’t know that all the prices always end in zero…which causes me to yell at my television.

ChancesMoving onto the next prize, 4 digits are given, 3 of which make up the next price.  In theory, there would be 4 x 3 x 2 = 24 choices here.  But again, given that the price will always end in zero, there are only 3 x 2 = 6 viable choices.

The goal in this game is to economize your Chances, so that you have a good number left to play for the car.  5 digits are given, all of which must be used in the car price.  In theory, this gives 5 x 4 x 3 x 2 = 120 choices.  But there are 2 ideas at play here: the price will end in zero AND the price will always begin wth 1 or 2.  Depending on the assortment of digits given, this reduces the number of possible prices a player needs to assess.


DisksThis is one of the most difficult games on the show to win, and is often played for a luxury car.  The game is played with 8 wooden disks, which are placed in a bag and shaken.  5 of the disks have number on them; digits in the price of the car.  The other 3 disks have red “strikes” on them.  A disk is drawn, and if a number is drawn, the player must tell which position in the car’s price the number represents.  If they are correct, the disk is removed from circulation.  And if the player is able to complete the price of the car before drawing all of the strikes, they win the car.

This game is a bit tricky to analyze, because often numbers are drawn repeatedly, as the contestant tries to narrow down where digits go in the price.  For an in-class analysis, let’s assume that the player knows the price, and is trying to just draw the digits.

If you are assuming perfect play, then we could simply list all of the possible ways to arrange the numbers and strikes.  Let N indicate a number and X indicate a strike.  So you could have:

NNNXXNXN (Loss, since 3 X’s occur before all N’s are drawn)
NNXNNXXN (Loss)…..

The number of ways to play the game is then 8C3, which is 56.  Now, this may seem like a small number, but I am treating the numbers in the bag as similar objects, since we are assuming the contestant places them correctly.

BoardNow, I could go through and count the number of these 56 that produce wins, but I think it might be simpler than that.  The game really comes down to the last item on the list.  If the last item is an X, then you will have won the game.  If the last item is an N, then you have lost.  The chance that the last item is an X 3/8, so the probability of winning the game, assuming perfect play, is 3/8.

So, this game only has  37.5% chance of victory IF the contestant plays perfectly.  Add in that often the contestant often must struggle to position the digits, and you see why the game is so difficult to win.


LineUpThis game is played for a car, with 3 smaller prizes as well.  The 3 smaller prizes have prices with 3 -digits, 2 digits, and 3-digits.  The prices of these smaller prizes are used to fill in the middle 3-digits in the price of the car, as shown on the game board here.  This give 3 x 2 x 3 = 18 possible outcomes.  The nice part about this game is that the contestant is given a second chance, and is told how many digits they have correct after the first attempt.  If the player needs their second attempt, it would be interesting to analyze how many choices of the 18 remain, given that they have 0, 1 or 2 of the digits correct.

There are plenty of other games on the show which also have basic counting prinicple ideas worth exploring.  Some quick hits:

  • Balance Game – how many different ways can 2 bags be chosen from the given 3.
  • Dice Game – how likely is it to roll correct digits?  When should I choose higher or lower?
  • Golden Road – how difficult is it to advance to and win the big price at the end of the Golden Road?
  • Let Em Roll – how many different ways can the 5 special dice be rolled?
  • Make Your Move – how many different possibilities exist for moving tre sliders?
  • Race Game – how many ways can the price tags be placed?
  • Take Two – how many ways are there to choose 2 prizes from the given 4?

And some in-class ideas:

  • Let students choose a game to analyze.  Create a poster and share with the class.
  • Start each day of your probability unit with “a Game a Day”.  Start with the easy games, and move it to the more complex ones.
  • Have a contest where students design there own pricing games.

Thanks to my friends at for the fun pictures.

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!