Monthly Archives: August 2013

Twitter Resources

Resources from today’s Twitter PD session at Hatboro-Horsham HS.

Hatboro-Horsham “Coaches’ Corner” from April 2013.

Cybrary Man’s Twitter Resource site:  this is the encyclopedia of all things Twitter.  From articles to read, classroom resources, hashtags to follow, and chats to join…’s all here!

60 Inspiring Examples of Twitter in the Classroom: from the 21st Century Fluency Project

50 Ways to Use Twitter in the Classroom: from TeachHub

22 Effective Ways: moving up Bloom’s Taxonomy using Twitter

“The Power of Twitter Chats” video – teachers and administrators share their experiences with Twitter chats.

Talking Inverses and the Enigma Machine

Here is a challenge which has appeared on my classroom board, in various forms, over the past 10 years:


Can you decode the message?  In 10 years, I have given out zero gift cards….so good luck.  More info on this challenge below.

A trip today to the Franklin Institute science museum in philadelphia reminded me some of cryptography nuggets you can use in math class; in particular, discussion starters for inverses, and code-breaking using matrices.  One of the first artifacts we encountered in the exhibit was the Enigma machine shown below, which I fawned over like a teenage girl at a One Direction concert.


The Engima machine is a coding machine, used primarily during World War II, to both code and decode messages.  Messages were typed using a standard keyboard.  The electric signals from the keyboard passed through a system of rotors and plugs, and lit up a letter, which was recorded.  There were a number of variations of the machine over the war years, and the Allied forces employed many mathematicians, many working through Blechtley Park in London, to intrcept and de-code messages.

Consider this intercepted message:


How would you even start to decode this message?  Does a one-to-one correspondance seem reasonable?  How else can letters be coded?

You can try your hand with some coding using this Enigma Simulator, which shows the coding rotors, inputs and outputs.  But here’s the neat thing about the Enigma machine: the machine is used to both code AND decode messages, using similar procedures, which are outlind here.


So, now you have everything you need to decode my message it seems.  You have a message, and a device.  Oh, but those pesky rotors.  If they aren’t set correctly, then the machine is of little help.  Working through this issue was the task of many of the mathematicians during WWII.  And I want you to be successful!  Set those pesky rotors to R-J-L (my initials), and start typing!  You can also copy and paste the message, but it is far more fun to watch the rotors do their work as you type.

Embedded in all of this crypography history are some neat math discussions:

  • After looking at some messages and their coded outputs, is there a ONE-TO-ONE correspondance here?  For example, does the letter E in a coded message always map to the same decoded output letter?
  • Are there any patterns we can use to help decode the message? Any predictable behavior?
  • A message is coded using a rotor setting.  Then this coded message is typed, using the same rotor settings, and we get back the original message.  The Enigma machine is its own INVERSE!  How exciting is that!  How many ideas or devices do we know of which are their own inverse?

Here are some sites with additional information relating to the Engima:

Exploring the Enigma, from +Plus Magazine.  Good student reading, with guiding questions.

This Numberphile Video has a demonstrations of the gears and plugboard of the Enigma, and some explanation of combinations.

In my next post, we’ll look at Hill’s Cipher, a cryptography application of matrices, and think about my Best Buy challenge!

A New Start for The Blog?

So, the name of this blog is “MathCoachBlog”.  I picked the blog name about a year-and-a-half ago, as I was working in my district’s curriculum office, and hoped to use this forum to share ideas, resources and experiences.  I treasure the opportunities I have had to share with others, the kind feedback people have given me about many of the activities, and the many friends I have made through this blog and twitter.

But here’s the thing: I’m not an academic coach anymore.  After 2 years working with some great people in my office, I have chosen to go back to the classroom.  This was purely my decision, and I am looking forward to implementing many of the ideas and resources I have been encountered in the last 2 years.  I’m thinking of it as a mid-career re-set, and in some ways I am more energized to teach classes than I ever was before.

But the blog….keep the name?  Change the name?  Keep it?  Change it?  We’ll get back to that….

This week, I had to set-up a new classroom for the first time in a long time.  I was in the same room for 13 years before, and had to pack a lot of stuff when I moved into an office.  So, time to dust off the cobwebs, think about what’s important, and do some moving-in.  Here are some elements I like to have in my classroom.  What are some neat things you like to have to create a positive classroom culture?

I like to tell a lot of stories in my class, think about anecdtoes from previous years, and keep in contact with as many former students as I can.  For AP Statistics, the “Wall of 5’s” has always been a topic of conversation, and a goal for many students who strive to eventually be “immortalized”.  It’s a nice hook the AP classes.  Thanks to my colleague Joel, who kept the wall alive the past 2 years, and has now made a duplicate wall for his classroom.

Wall of Fives

Along the same lines, I like to have lots of pictures from previous years around.  Many of these are from our annual Stats Fair, and are great conversation starters.

Stats Fair

The Wall of Badges: more chances to talk about my experiences as a Siemens STEM Fellow, an AP reader, and conference junkie.


Cool art.  Escher works always generate buzz.  Now with 100% more Legoes!


T-shirts from math contests our math club has attended provide just the right amount of geek-pride.


And finally, the oragami art a graduate made for me is the best gift ever, and gets it own spot in the room.  Special appearance by John McClain – a “Secret Santa” gift.


So, about the name of the blog…  Since I announced my return to the classroom, I have had lots of conversations with colleagues in my department, and know I am blessed to work with many fantastic people.  Sometimes we don’t agree on things, and that’s healthy.  And I’m thrilled to be able to implement so many of the great new things I have learned, and continue to share them out to you.  So, I’m no longer a coach in my district, but I think that, in many ways, I way wind up being a more effective coach to my friends online through the sharing of classroom ideas.  So, MATHCOACHBLOG LIVES ON!

Also, it’s sort of a pain to change the name,….so there ‘s that.

Globe-Trotting With

The Common Core places increased imporance on statistics in middle school, beyond the tasks of creating simple data displays often encountered in middle-school texts.  The new standards require that students be able to describe distributions, compare samples to populations, and design simulations:

  • CCSS.Math.Content.6.SP.B.5c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered
  • CCSS.Math.Content.7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

It’s all great table-setting for AP Statistics down the road, and working with authentic, interesting data.  In this activity, students use an online resource to perform a simulation in order to find the proportion of the earth’s surface covered with land (as opposed to water).  This is not a new activity, as a number of teachers suggest using an inflatable globe and some classroom tossing to reach estimates.  But I think the method below uses some web tools in a novel way, and encourages some authentic geography discussions.


Before diving into any simulation, I like to gather ideas from my students, to see if they have any initial estimates or backround about the problem.  Using sites like Padlet or TodaysMeet are great for encouraging and archiving discussions and participation, or you can go old-school and just record initial estimates on the board.  Asking an initial questions like: “Do you know the percentage of Earth which is covered by water” to start the discussion.  The start a discussion of sample size: “Would it be better to sample 20 points on earth, or 40 points, or 100 points?  What factors would be part of your decision?”


In this simulation, students will sample a random point on the earth’s surface, record whether the point is covered by land or water, and repeat for a given sample size.  For this, we will use the site, which generates random events, mostly things like numbers and dice, and their Random Coordinate Generator, which chooses a random location on earth and displays it using Google Earth.

Google Earth

The site works quickly, and the water/land data can often be determined without issue.  It is also easy to zoom in and out to take care of those “close calls”.  Quite a more accurate measure than the thumb-check data from the inflatable globes.  Very few issues arose in my trials with this method, but the biggest snag is Antarctica, which is land, but often appears light blue on Google Earth.  Also, a few rare occasions produced a data point above the North Pole on the map, which I discarded.  For your class, have each student generate a sample of size 10, and look at the proportion of land hits.  Below, I did 10 trials for 4 different sample sizes:


The next steps depend on the sophisitcation and grade-level of your class.  But in general, we want to know which sample size provides the best estimates.  How do you know?  Have students write explanations which defend a particular sample size.

Multiple sources (Circle graph from ChartsBin,NOAA Information) verify that about 29% of the Earth’s surface is land.  Do our trials verify this?  How often were our trials within 10% of this 29% mark?

Within 10%

As more data is collected, free site like StatKey can be used to generate appropriate graphs and statistics.



I see this as an improvement of an existing AP Stats exploration, and opening activity for Confidence Intervals for proportions, and extension into the behaviors of CI’s.  For those playing along at home, here are the calculations for 2 standard deviations (Margin of Error) for my given samples:


And the corresponding intervals, showing how often my sample proportions were captured within each interval:


Model Classroom Resources for Siemens STEM 2013

Online Calculator: Desmos

Balloon Activity:

Bob’s Eyeball:

Resources for the Conic Sections Activity:

Blog Post


Global Math Department Presentation Slides


The Daily Desmos: teacher-written graphing challenges

Geogebra: dynamic construction software.  Download, or run in java.

Geogebratube: teacher-created demonstrations


Visual Patterns – Challenge your students to find and summarize patterns.

Graphing Stories – video starters, from linear to exponential

101qs – picture and video openers for promoting inquiry.  Contribute your own!

Mathtwitterblogosphere – network of teachers dedicated to sharing resources and classroom experiences

How are We Connected? Meaningful Adjacencies.

It’s Siemens STEM Academy Week at Discovery HQ in Silver Spring, MD, and I am enthused, excited, engaged, tired, and looking forward to more.  This is my third year participating in this program, and the experiences just get better and better.  Fifty educators from around the country are here to experience sessions related to STEM education, and make new connections.  Check out #stemin13 on Twitter to follow the action through Friday, and look back on previous sessions.  Also, experience last year’s STEM Academy through my blog posts from last year:

Digital Storytelling with Hall Davidson

Let’s Play Plinko, my STEM 2012 Presentation

Flipping the Classroom with Lodge McCammon

imageThis year, the Fellows heard from Discovery Channel personality Danny Forster, host of “Build it Bigger”, who shared his experiences traveling the world examining interesting engineering and architetural feats, and the thrilling (and sometimes gasp-inducing) views of those experiences.

Part of Danny’s talk focused on the 9/11 Memorial in New York City; in particular, the fascinating method the memorial designers chose for arranging the names of those memorialized around the base.

What methods could designers have used to arrange the names?  An alphabetic approach seems reasonable, accessible, and un-complicated.  But how does this method honor friendships?  Co-workers?  Fiancees?

The designers decided on an approach which has since been named “meaningful adjacencies”.  Victims’ families were contacted and asked to name up to 5 victims with whom their loved one shared a relationship.  This data was then used to create connections, with the goal that names listed on the memorial base would be connected to – adjaent to – as many meaningful relationships as possible.  With almost 3,000 names to consider, this became a large optimization problem, and a fascinating one to discuss with your classes.


Here are some reading and resources to get you started:

Jer Thorp explains his methodology for creating the algorithm used to analyze the relationship data.  Great visuals for networking and a video example of the algorith in action.

PBS interview with Jer Thorp and Jake Barton

Scientific American summary of the Meaningful Adjacency method

9/11 Memorial Names Guide

“Rise of Freedom Flashback” – 9 minute video from

“Names Come to Life” from The Rising by Discovery Channel

How can you introduce this complex problem to your classes?  Have students think about different ways we could arrange the class.  Certainly, an alphabetical method is reasonable, but how else could we arrange students?  By height?  By desire to sit in certain seats?  Are there students who we would want, or need, to arrange near each other?  Are there others we should keep separated?

Try this with your class:

Have each students write their name on an index card.  Then, under their name, have each student list their 5 favorite movies (or TV shows, or books…whatever).  Collect the cards, and spread them across a floor, or using magnets, to a magnetic board.  How can we arrange students to ensure that student with similar interests appear close to each other.  Consider this hypothetical class example, where favorite movies of students are considered.

STEP 1 – We found a number of students who liked the Smurfs, and others liked Monsters University.  They have been placed near each other:


STEP 2 – Digging deeper, we found that some of the Monsters U lovers also liked the Smurfs, and vice-versa.  These students have been placed strategically to ensure connections.


STEP 3 – Some students also enjoy Despicable Me, and are placed next to each other.


STEP 4 – One of these students also enjoys Monsters University, so connections are made.


STEP 5 – But a problem arises when we look at Donna, who seems distant from Bob, Aiden and Joe.


Here’s where would we revise our model, and experiment with new ideas.  Can we ever have a model which has perfect adjacencies?  How can we maximize these adjacencies.  Have teams of students consider, develop, and defend their models and vote on the best.  Extend this into an introduction to network models and matrix representations, or just use it as a class-building activity on your first day.