Category Archives: Middle School

Estimation and Anchoring

A recent post by my Stats-teacher friend Anthony, “Wisdom of the Crowd“, reminded me of an estimation activity I have used many times in my 9th grade Stats class.  The activity is based on a chapter from John Allen Paulos’ book A Mathematician Reads the Newspaper.

You’ll need two groups of students; 2 different classes will do.  Each student uses an index card or a scrap of paper to write responses to 2 survey questions. I warn the students beforehand that the questions may seem strange: just do your best to answer as best you can.

  • Question 1: Do you believe the population of Argentina is MORE or LESS than 10 million people?
  • Question 2: Estimate the population of Argentina.

Allow a few moments between the questions for the inevitable blank stares and mumbling.  Then collect the responses.

For the second group, you will ask the same two questions, except that the first question will replace 10 million with 50 million.  After you have data from both groups, write it on the board or print it and hand it out. It’s time to analyze and compare. Challenge students to communicate thoughts about center and spread. Also, which group’s data do they feel does a better job of estimating question 2?  It’s a neat activity, and while you will receive some strange responses as estimates, and students will generally guess higher on question 2 if they have been anchored to the 50 million number.  Some guidelines for this activity are avilable.  Have fun!

According to Google, the actual population of Argentina is around 41 million.



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!




The Common Core and Simulation Models

The Common Core Standards provide an exciting opportunity for statistics education, with inference concepts starting informally in middle school and sampling distributions with inference moving into the high school mainstream. Under the “Probability and Statistics” strand, we find the following:

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

But many of our high school teacher colleagues will need supports to help their students think statistically.  AP Stats teachers, often the loneliest folks in their departments, will now need to share their expertise with non-stats inclined colleagues.  Below are snapshots from a lesson on sampling and margin of error I used with my 9th grade classes, with many ideas adapted from my AP Stats teaching experiences.


Student in this class had already been exposed to a probability unit, which included conditional probabilities, binomial settings and the normal distribution.  After a discussion of sampling techniques, I wanted to conceptualize margin of error before diving into formulas.  Here’s what I did:

  • Class is broken in two large groups
  • Each side of the room was given the task of flippling virtual coins, using a graphing calculator
  • For one side of the room, 25 coins were flipped.  For the other side, 100 coins were flipped.
  • “Coins” were sorted and counted, and the proportion of heads communicated to team captain, who recorded the result on a group graph.

The results for the two groups are shown here.Coins

The class discussion the moves to a comparison of the graphs.  How are they similar?  How are they different?  Clearly, both graphs center about the true proportion (.5).  The shapes also seem similar.  But the vaiability seems to be different.  How likely is it to have 60% of coins show heads, if 25 coins are flipped?  Does this likelihood change with 100 coins?  We’re moving towards sampling distributions and inference topics, without all the scary-sounding language.


In the next part of the lesson, I wanted students to explore public-opinion polls, choose a topic of interest to them, and think about the associated language and structure. The site is ideal for this.  The site collects and summarizes poll results from many sources, categorized by topic.

Polling Report

Student pairs were asked to find one poll of interest to them, record the source and the number of people surveyed, and record it on the back board.  We then looked for common threads:

  • Sources: we found many repeats in the sources chosen (Rasmussen, Gallup, Washington Post, FoxNews) and discussed reliable sources of information.
  • Sample sizes: most surveys had about 1,000 participants, with one poll using 2,000.

Can 1,000 people possible allow us to represent a population?  How reliable can we expect a poll to be if just 1,000 people are surveyed?  Time to move on to the last stage of the lesson – simulation.


The great free site StatKey allows us to look at sampling distributions easily and discuss our observations.  In my class, pairs had a netbook for exploring the site.  We started with the poll about college football above, where 47% of those surveyed registered “support”.  If 47% is the true proportion, and we survey 1000 people, how close to 47% should we expect to see in our survey?  Is 46% plausible?  How about 44%, or even as low as 40%?  We can set the population parameter, and the sample size, then it’s time to draw some samples:


Most of the possible samples center about 47%, but here’s the follow-up for class:

A large majority of the possible samples seem to be within ___ % of 47%.

Certainly, we see that ALL of the surveys are within 10%, but can we narrow that window?  After some dicsussion, the class agreed that 3% was a reasonable window for capturing a large portion of the surveys.  How many of the samples here are within 3% of the “true” proportion of 47%?  StatKey can take of that work for us:


As the unit progressed, StatKey was always shining up on the board to verify our margin of error calculations, and provide a link to the sampling distribution ideas.


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

The Puzzle Solutions You’ve Been Waiting For….

This post will present solutions to two puzzles I have presented here on the blog, and some ideas for extending the learning with your students.


This is a game I proposed a while back on the blog, and it took my friend Anthony from Twitter’s gentle nudging to ask for a solution.  Here’s a re-cap of the game:

  • 23 marks are placed on a board
  • On each turn, a player must remove 1, 2 or 3 marks
  • The player who clears the board wins

I often challenge classes with this game, letting kids play me and try to figure out the secrets.  The beauty here is that my students are often polite souls, and will let me go first…which leads them to doom.  It will often take a few days before a student can conquer me.

Here’s a video example:

And here is the message I sent back to Anthony, who requested the secrets to the game.

Here’s the secret to the take-away game – if I can make it so that the number of dots remaining is a multiple of 4, then I will win the game – guaranteed.  If they take away 3, I remove 1.  When they remove 2, I remove 2.  If they remove 1, I remove 3.  The game is all about groups of 4.  

With 23 dots to start the game, I will offer the student a choice; who plays first.
If they let me play first (which they often do because they want to be polite), then I erase 3 dots (leaving 20) and wait for their eventual demise.  If they play first, then I need them to slip up and make it so that I can control the multiples of 4.  If they erase 2 on their first move, then I remove 1, leaving 20.  
It often takes a few days of playings at the end of class for kids to develop the strategy.  They will first realize that being left with 4 is sure doom.  Then 8….then eventually they get it.
Have fun
And here are some ways to tweak the game, and see if students can develop strategies:
  • Change the number of starting marks
  • Change the number of marks allowed to be removed
  • Play with 3 players (actually, I’ve never tried this…)
I posted about the Hot Seat game recently.  Time to reveal the secrets to this interesting pattern game –

This table shows the winning chair in the game, based on the number of chairs which start the game.  Note that if the number of chairs is a power of 2, the winning chair will equal the number of chairs.  From any multiple of 2, the winning chair goes up by 2, until the next power of 2 is reached.

How would your students express this pattern?  What vocabulary would they need to use in order to communicate the pattern?  After a class develops the data table, challenge students to write a concise rule which will identify the winning chair.  Then, have students trade their explanations and critique them.

Could an algebra student develop a formula which outputs the winning chair, if the number of chairs is given?  

Take a look at the winning chairs, plotted vs the number of chairs.  Can we write a function which follows the pattern?  Share your function ideas in the comments, and enjoy the challenge.


My “Fake World” Task

Dan Meyer’s recent post on “fake world” math tasks has me thinking about many of the openers and games I have used in my classroom.  I have written about The Take-Away Game before, and I still use it often…until the kids learn how to beat me and the strategy is revealed.  This next one is not so much a game, but more of a task, similar in some ways to the Locker Problem.


In this task, chairs are placed in a circle.  Chairs will be removed from the circle using the following rules:

  • Chair #1 is removed first.
  • The next remaining chair is skipped, and the next chair removed.
  • This continues, with chairs skipped and removed until only one chair remains.
  • Once a chair is removed, it is “out” of the circle
  • Whoever is sitting in the last remaining chair “wins”

Here’s a brief Doceri video which shows some game playings:

Like the “Take-Away Game”, I can’t recall where I first encountered this problem.  They have both been sitting in my files for over a dozen years.  If anyone can name a source, I’d be happy to award some credit.

Why I enjoy this problem:

  1. It’s not intimidating.  We have a chance to draw, get out blocks, magnets….whatever we want to use to model the problem.  Great for working in teams.
  2. I can let the problem marinate.  On one day, I may ask the class “Where should I sit if there are 8 chairs?”, and come back the next day with “How about 24 chairs?”  If it seems like discussion is flowing, I can put my foot on the gas.
  3. I can use this problem with all levels of students.  If we need to create a data table and look for a pattern as a class, that can happen.  If my honors kids want to fly with it, that can also happen.
  4. The answer is not obvious, but a clear pattern eventually emerges if you model enough circles.  And there will be some nice vocabulary opportunities as the payoff.

There are a number of ways to express the solution.  Later this week, I will post the “answer”.  Until then, have fun moving around the furniture.

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.