Categories
Algebra

Breaking Apart Sums and Differences of Perfect Cubes

The first few days of math class…an awkward time for both students and teacher.  The kids haven’t picked up on my mannerisms yet, aren’t sure why I fly around the room like a maniac, and worse, they aren’t laughing at the jokes.  I tend to use the first few days of any class seeing how far I can put my foot on the gas…what do my students understand?  Where are there gaps?  Who will I need to sit on during year?  Without exception, I dedicate at least part of the 1st day with students at boards, shouting out review problems.  The problems are strategically chosen to allow for initial success, dust off some cobwebs, provide for discussion when we hit some road blocks, and most importantly let students know that it will be perfectly acceptable to struggle in my class….as long as you keep trying.

I’m trying a differenmt apporach with an Algebra 2 course I am teaching this semester, and hoping to build some discovery and communication moments right in the first few days.  As their first day assignment, students fill out a Google form with information about themselves: hobbies, goals, clubs, etc.  As part of the form, I am adding this task: “Tell me everything you know about this graph:”

cubic

For day 2, I’m hoping the responses will provide a review of vocabulary (intercepts, roots, solutions, even rotational symmetry perhaps?), and some table-setting for what’s to come.  In our district, I can expect that Algebra 2 students will have a solid background in linear functions and basic polynomial operations, mostly limited to quadratics.  Cubics for the most past have not been explored yet.  And while polynomial multiplication and factoring are not new, rarely do students see polynomial division before algebra 2, so I will bring this into the discussion as a new idea.  NOTE: our district uses the Everyday Math program in the early grades, which stresses partial quotients.  Wondering how this will play when I attempt polynomial division…update may be coming!

DEVELOPING FORMULAS FOR SUMS AND DIFFERENCES OF PERFECT CUBES

Starting algebra 2, students should be able to “read and recognize” the following polynomial patterns: difference of squares and perfect-square trinomials.  But beyond this, I want students to be able to relate factored form to graphs, which often seems to be marginalized in the drive to practice process.  So, one of my first lesson openers will be a short and sweet challenge.  Does the following polynomial factor?

In their teams (my students always sit in groups), I will provide some time for students to consider this problem, and observe their trials.  I expect that will have a few groups attempt (x-2)^3, which will end badly, but hopefully lead to more trials.

CubicSo, how do we cross the bridge to the formulas for differences, and sums, of perfect cubes.  Time to start looking at some graphs, in particular the functional form of the given expression:

What do we notice with this graph?  And what characteristics will be helpful with the factoring problem at hand?  Here is where I hope students drive the discussion:

  • This graph has an x-intercept of 2.
  • This means that x-2 is a factor.
  • There are no other obvious intercepts, but we can employ long division here.

So, x^3-8 DOES factor.  Do other cubics factor?  How?

With their teams, students will now be given a few more cubics to factor:

 

What patterns do we notice?  Can we develop a general rule for factoring difference of cubes, and even sums of cubes?

Guiding the discussion towards a generalization, without students feeling forced-fed, is part of the art of teaching.  Hoping these first day discussions tie together lots of previous knowledge with a discovery moment.  I am not sure how it will go, but I hope to set the table that nothing is given for free.  Show me what you know!

Categories
Technology

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…..it’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.

Categories
Algebra High School

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:

Board

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.

Enigma

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:

LXFAVPBNAQMHIZJPBMMRCSWOI

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.

Flowchart

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!