Monthly Archives: February 2013

Hooks for Inverse Functions

While browsing through Dan Meyer’s most recent post on great classroom action, I found a link to a picture which put a smile on my face, from the blog Your Poisonous Cousin (cool name):


A high school colleague of mine uses the Dr. Seuss story “The Sneetches” to hook students into a conversation about inverse functions.  This AP Calculus Blog has a nice summary of inverses and pictures of our Sneetch friends.

Enjoy the Sneetches with this video, discuss the star-applying machine, and the charlatan who sells its eventual inverse.

An Open Letter to My TI Friends

The good folks at Texas Instruments, at long last, have released an app for their popular Nspire product.  For geeky math folks like me, this was met with “I want to play.  I want to play.  I want to play!!!!”.  That is, until I visited the app store and found our that the app costs $29.99.  {insert sad face}

NspireSo…download, or don’t download?  I have been a sucker for all things TI for some time now, and the TI folks were kind enough to host me for their Fast Track program a few years back, where I received training on the Navigator system.  I’ve done many training sessions at my school for staff on graphing calculators, spoke at the T^3 conference one year, and wrote a grant last year for a class lab of Nspire CX’s.  When it has come to TI products in my school district…I’m all in!

HeartsBut despite out long relationship together, Texas Instruments, I’m thinking it may be time for us to break up.  It’s not you…it’s me.  See, I don’t see a future in this relationship, and I don’t know who you are anymore.  Remember last year when I bought all those glossy, snazzy Nspire-CX’s?  That was fun, and we have done some great lessons together.  But now I see you making the TI-84 color with new bells and whistles, and I can’t help but feel a little twinge of jealousy.  I don’t know what product I’m supposed to tell my students to buy anymore.  Some days you are Nspire, some days you are 84, and now this new app which a student could never possibly use on an SAT or AP exam….I just don’t know.

And your Nspire software?  I told all of my friends about how great you were, and bought a whole bunch of you in my grant last year.  But let’s face it, you take up way too much memory in my computer, and run way too slow at times.  And while the tns files are cool, and your new app plays them, I get tired of waiting for you sometimes.  Oh, and that free software offer on your website?  The one where I get free software if I buy the app?  I can’t help but feel a little hurt that you forgot about us who have purchased your software {sigh}…

Desmos PiSee, the thing is…I’m seeing someone else.  Her name is Desmos, and she is really cool.  I’ve told all my teacher friends about her, and they agree that she is really fast and reliable.  And while she doesn’t have all of your features, she is working on it.  We’re growing a nice relationship together.  She even makes me Pi when I need it.  And she is free!  (Note: OK, maybe this isn’t the best line for a break-up letter….but the Desmos calculator is free…check it out!)

I’m looking forward to seeing you at the T^3 conference next month, and I hope we can talk about our relationship.  But I don’t know if I see a future between us.

I hope we can still be friends.


Encourage Generalization and Communication with these Math Challenges

A comment from a recent post of mine on differentiation asked what I do with students who complete tasks early.  In every course I have ever taught (usually Algebra 2, Prob/Stat, Algebra) I have used weekly problem-solving challenges, no matter what the level of student.  Often, my intent in these problems was to develop written communication skills in mathematics, and have students begin to reflect upon their own writing style.  Have students complete the challenge, critique their writing and provide a path for improvement, and have students turn in their best works as part of a portfolio at the end of the semester.

In this post, I focus on tasks involving counting, number theory, and algebra.  The problems here are ones I have assigned, graded, revised, and enjoyed over the past 15 years.  I’ll have some more tasks in a later post.  Click the title to download the PDF document.

PATHS:  How many ways are there from start to finish?  I love this problem, because there are multiple ways to approach it.  Combinations give the result, but there is also a Pascal’s Triangle approach, or as a permutation with identical items.  And Polya’s strategy of starting small and working your way up is key to this one.

SOME ZEROES:  I have always enjoyed giving this problem, as you you can have rich conversations about simple number facts and the commutative property.  And the student explanations will range from the ridiculous to the intriguing.  When I started giving this problem 15 years ago, some students would use Excel to try to simply compute the answer, which often “broke” Excel and gave a wrong answer.  I have given up on trying to follow the technology, and have given a similar problem as a follow-up on a quiz.

AVERAGE SPEED:  One of my favorites because of its simple premise, and a result that is counter-intuitive.  Also, can be easily differentiated.  For some students, choosing distances and testing serves as a good starting point, while students with advanced algebraic skills can dive right into the abstract.

LAST DIGIT:  A premise simple enough for grade 6, yet complex enough to challenge older students if you ask for a general formula.  It’s also easy to adapt this problem and use it as an opener for class.

INVERSES:  In this challenge, students must find a matrix which is its own inverse, of which there are many, many possibilities.  How will your students ensure that their matrix is unique?

Feel free to contact me for solutions, tips, or more ideas.

All about ATMOPAV

One of the many hats I wear is that of second vice-president of ATMOPAV, the Association of Teachers of Mathematics of Philadelphia and Vicinity.  The organization hosts conferences in the fall and spring, provides awards and grants for classroom teachers, and publishes a newsletter 3 times a year.  Find out more about our organization on our website.

This past fall, we were all excited when the ATMOPAV newsletter was recognized by NCTM as best chapter newsletter, and our fantastic editor Lynn Hughes will accept recognition at the NCTM national conference in April.  And now, for the first time, our newsletter is available for pdf download.  The Winter, 2013 edition features:

  • “Graph Paper Racing” , “Electing the National Donut” and a review of Dragon Box, by Lynn Hughes
  • Pedagogical Ideas, Chapter 1 by Beth Benzing
  • “Fun and Engaging Bar Graphs” by Rich Murray
  • Technology Corner – thoughts on the TI-84 OS by Margaret Deckman
  • And, a guide for getting started on Twitter by yours truly!

Visit the ATMOPAV website, and click the Newsletter tab to download and enjoy our award-winning publication.

A New Really, Really Big Prime Was Found! Woo Hoo!

On January 25, the largest prime to be discovered to date was “found”.  I use the word found in quotes, because the special number found was determined as part of the Great Internet Mersenne Prime Search (GIMPS).  Mersenne primes are prime numbers of the form:

where P is itself a prime number.  The  numbers 3, 7 and 31 are Mersenne Primes, corresponding to P’s of 2, 3 and 5.  In the GIMPS program, anyone with a computer can help search for new Mersenne primes by installing a program on their computer, which runs quietly in the background.  The recently found prime is the 48th Mersenne Prime to be found, and has over 17 million digits.  It’s not often that math is news, so this is a great opportunity to use something “ripped from the headlines”.

This is big, exciting news in the math community!  And the finding was heralded by an article on, which included the following passage:

Prime numbers, which are divisible only by themselves and one, have little mathematical importance. Yet the oddities have long fascinated amateur and professional mathematicians.

First, thanks to Fox News for pouring cold water on this big discovery.  Prime numbers are hardly oddities, and play a big role in much of the math we all do.  And, prime numbers, while often taught as a trivial notion, play a vital role in our national, world, and personal security.  In fact, the whole idea of RSA cryptography, that which keeps our data secure in thsi data-driven world, relies on prime numbers.  A quick search turned up some useful site which explain the role of the prime number.  Enjoy them, share the news with your students, and celebrate the discovery!

Prime Numbers and Cryptography – Clay Math Institute

RSA explained using paper and pencil

RSA numbers – from Wolfram Mathworld

RSA cryptosystem – prime glossary

Absolute Value Inequalities and the Human Number Line

In most Algebra 1 courses, the topic of Absolute Value inequalities comes at the end of a longer unit on inequalities.  We shade our number lines, attend to our open or closed circles, and start to hit the wall a bit with the routine.  So, as we begin to think about introducing absolute values, let’s get our students up and moving.  Here’s how:


Print out or scribble out cards with the integers -12 to +12, or use my handy integer card set.  This will give 25 cards, and you can adjust the cards based on the size of your class.  Give each student, except 1, an integer card.  The student who does not get a card will act as the observer during the activity, and will verify the class’ actions.

In a hallway, or outside on a nice day, have students sit in order from lowest to highest.  The students are making a human number line.  It is important that their card be clearly visible at all times.   The class observer should verify that all students are seated in order, with somewhat equal space between them.


With students seated, the teacher holds up an inequality.  Any student holding an integer which is a solution to the inequality will stand, thus making a human solution set.  The job of the observer is to verify the correctness of his class-mates solution.  My inequality cards file starts with two warm-up problems, to make sure the instructions are understood, before we start to head into the absolute value inequalities.


One of my colleagues used this activity with his class recently, having students step forward if they were a solution.  He also added a twist I hadn’t thought of: having students hold their hands over their head to make an “open circle”, if they were a boundary number.


As the class builds the solution sets for the absolute value inequalities, have the observer describe the graph.  What do greater-than problems “look” like?  How about less-than problems?  What sorts of problems tend to veer off (to infinity) in both directions?  What sorts of problems are bounded?  Here are some other teaching tips and ideas for this activity:

  • Have students trade cards, or totally re-mix after 2 problems.  If you don’t, the students with “end” cards can simply follow the crowd.
  • This is a great time to find a class leader to be observer, or uncover a hidden talent of a shy student.  Keep those cards visible.  Give them the responsibility to keep things orderly.
  • If you have room, take pictures of the human number lines, and use them later as a review, or to keep around the class to build the team spirit.
  • Using this activity a lot, or with many classes?  I always thought it would be neat to have integer shirts for this, and to use through the year.



After we have developed some ideas about absolute value inequalities and their solution sets, it’s time to start formalizing our thoughts.  If you need more hands-on practice, click on the graph link below to try a Desmos demonstration with sliders.

Compare the absolute value function (in blue), to the constant function (in green).  The comparison (in red) allows us to look at make greater/less-comparisons.

Let me hear about your Human Number Line experiences!