Tag: inverse

Categories
Algebra

Inverse Function Partner Share

We’re working through functions in my college-prep pre-calculus class; meaning a more rigorous treatment of domain, range, and composition  ideas than what students experienced in earlier courses. As I was about to start inverses last week, I sought an activity which would provide some discovery, some personalization, and less of me rambling on.

These are the times when searching the MTBoS (math-twitter-blog o’sphere) leads to some exciting leads, and the search for inverse functions ideas didn’t disappoint – leading me to Sam Shah’s blog, and an awesome discussion of inverse functions which I turned into a sharing activity. A great list of blogs and MTBoS folks appears on this Weebly site.

To start, I wrote a function on the board, and asked students to think about the sequence of steps needed to evaluate the function:

The class was easily able to generate, and agree upon a list of steps:

  1. Square the input
  2. Multiply by three
  3. Add 1

From here, I asked the class to divide into teams of 2. Each partnership was then given two functions on printed slips (shown below) to examine: list the steps of the function, and provide 3 ordered pairs which satisfy the function.

THE FUNCTIONS:

inverses

Notice that the functions are arranged so that A and B in each set are inverses.  Partners were given two different functions, but never an inverse pair. So a team could get 2A and 4B, but not 3A and 3B.

My plan was to complete this entire activity in one class period, BUT weather took hold. They day we started we had a two-hour delay, and the next two days were lost due to snow, then a weekend. SO, the best-laid intentions of activity, sharing and resolution became activity…..then 5 days later.

As we started the next class day, I asked students to review their given functions (and re-familiarize themselves), then seek out the teams who had the other half of the function pair and share information. So a team which had 2B sought out 2A, and so on.

After the sharing, a classwide discussion of the pairs was then seamless. Students clearly saw the relationships beteen the inverse pairs and the idea of “undoing” steps, and we could now apply formal definitions and procedures with an enhanced understanding. Also, by sharing ordered pairs, students saw the domain-range relationship between functions and their inverses, and this made graphing tasks much easier. I’m definitely doing this again!

Finally, notice that pair 2A / 2B features a quadratic / square root. While we didn’t dive right in at the time, this set the trap for a discussion of one-to-one fucntions and the horizontal line test the next day.

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Categories
Algebra Class Openers

Class Opener – Day 67 – Verifying Inverse Functions

We’re finishing up our unit on function operations.  Yesterday we developed the definition of inverse functions (using only linear functions) and graphed to demonstrate the symmetry. Time to see what we have learned:

functions

Many students’ instincts let them down on the first pair, believing them to be inverses. But after my prodding that they somehow verify the inverse relationship, we soon verified that f(g(x)) did not produce the result we desired.   The second example was then complete easily.

But what about that third problem?  They sure seem like inverses.  One student offered his proof for the pair:

They are inverses just because I know.

Sometimes ideas in math are just that obvious, and maybe we don’t need to prove them specifically.

On the board, we “proved” that both f(g(x)) and g(f(x)) both seem to simplify as x.  And a few numerical examples help show this:

  • f(g(5)) = 5
  • g(f(10)) = 10
  • g(f( -6 )) = 6……. ruh roh……

Students in my class have not been exposed to a formal definition of the square root function, and this led to a nice discussion of absolute value, and the need to restrict the domain in order to consider inverses. Planting seeds for algebra 2, which many of my students will take next semester, is always a bonus.

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!