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):

Staplers

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.

Bob

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.