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.


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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….


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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.