Tag Archives: functions

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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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.

Class Opener – Day 65 – Kohl’s Coupons

After a weekend away from composite functions, today’s opener was designed to bring functions back into discussion, disguised as an innocent-looking shopping problem:

It’s the day of the big department store sale, and you have two coupons you have clipped from the newspaper.  One coupon takes $10 off the price of any item, while the other takes 30% off the price.  In what order should these discounts be taken for you to realize the maximum savings?

After a few minutes of table talk, just about all groups agreed that taking the 30% off first would seem an optimal strategy.  But when asked to provide justification, groups took much different paths.

Some felt choosing a dollar value would provide adequete justification:

trial trial2

How many values are needed to convince ourselves that this strategy is optimal? Is it possible that one strategy is best for some prices, while the other is best for others?

Another group shared the “I know I am right…just because” method

explain

Not very elegant…nor very convincing. But a ray of sunshine appears from the other side of the room, as a group considers defining functions to represent the discounts….but stops just short of pursuing them as a proof.

functions

The eventual “proof” done via composite functions shows that not only is one method superior – it will always be superior by 3 dollars. Add in a domain restriction that our starting value must be at least 10 dollars, and we have successfully reviewed all of our scary function vocabulary.

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.

Doing the Translation Dance

Last month, I wrote about my talk on Encouraging Perseverance in Math Class, given at the Fall, 2012 ATMOPAV conference.  But earlier that same day, I had the opportunity to hear Scott Steketee‘s thoughts on functions: “Function Dances: Using Transformations to Make Variables Vary and Functions Behave”.

Steketee

I have found that the approach many teachers take to functions is one of notation only.  That by simply introducing the f(x) and g(x) symbols, and “covering” domain and range, algebraic functions will be understood.  Scott’s presentation provided ideas for introducing the concept of  function, without all of the scary symbols, through dynamic Sketchpad files.  The group worked through a number of progressively intricate functional relationships on iPads.  In this first example, students can grab points and look for relationships.  Some points will not move when dragged, as they are “dependent” upon other points’ movements.    Also, the dependencies vary, from simple linear relationships, to a few which require dilations or reflections.

iPad1

Later, we were introduced to the Sketchpad “Translations Dances”.  As one point (below, the point on the green outline) travels about its “domain”, we are challenged to trace the “range” of the translated point p.  These start off innocently enough, but become more diabolical as the translations begin to include reflections and rotations.

iPad2

These were addictive and appropriate uses for the iPad, and I was able to easily load the files into iPad’s Sketch Explorer through my DropBox account.

The second half of Scott’s talk was more kinesthetic, social, and potentially embarrassing, as the group split into partnerships to choreograph dances based on transformations.  My partner acted as the independent variable, and I (the dependent variable) followed her actions, using lines in the floor to act as  axis of reflection.  This would be a fun way to expose kids to functional ideas, but I made sure that no photographic evidence of my dancing ability exists!

What I appreciated most about Scott’s sketches and dances is that they allow teachers to develop functional ideas without having to wade through all of the complex language.  Through play and exploration, students can summarize their observations, and begin to characterize the relationships.  As students begin to understand the relationships between variables, we then can discuss the need to have special notation to express them.  Finally, dilations and reflections, which are often over-looked in our curriculum, become the stars of the show through fun (and addicting) Sketchpad games.  My screen grabs here certainly don’t do Scott’s files justice, so download them, play around, and enjoy the dances!

How to “Break” Your Graphing Calculator

A conversation with a colleague on function operations reminded me of one of my favorite “Oh Wow!” moments from math class.

You’ll need a TI-83 or 84.  This is a case where the Nspires are too nice for our use.

Enter functions as shown below.  We are interested in the rational function which comes from dividing the two previous functions:

Screen1

Since the denominator is a factor of the numerator, we can simplify the expression, resulting in a linear function when we graph.  But what about x=-3?  What happens there?  Let’s explore a bit:

Screen 2

Zoom in even more….doesn’t seem all that interesting…..

Screen4

Pretty boring.  But tell your students to keep zooming.  And then….

….wait for it……

…whoa!!!!!!….

Screen 5

Zoom in even more….

Screen 6

This is a nice reminder that, while we may simplify a function, we are still looking at a quotient and need to consider the parent functions used in the division.  And the “noise” we get here is a result of the algorithms TI calculators utilize to plot the points.  Actually, if somebody has a better explanation for the noise, I’d love to hear it.  Some “new” calculators will now show the hole in this function, like my new man-crush, the Desmos calculator:

Hole

But the noise is more fun.

Composite Functions and E.S.P.

An opener I have used in Algebra 1 encourages discussion of composite functions, but more importantly, allows me to show of my amazing ESP powers!

To start, you’ll need a wizard and/or magic hat.  A cape is also acceptable.  If all else fails, the cardboard crown from Burger King will suffice.

Wizard

Have students choose any number, and write it down, out of view of you…the wizard!  Encourage students to choose a number which is easy to work with.  Now, step-by-step, have students perform the following operations, keeping track of their answers as they go along, and always hiding their paper from you.

  1. Take your number, and double it.
  2. Take your result, and add 4 to it.
  3. Take your result, and divide it by 2.
  4. Take your result, and add 4 to it.
  5. Take your result, and double it.

Now, have students share with you their result.  Using your incredible predictive powers, you will be able divine their original number.  Utilize temple massaging and mysterious gestures to maximize the effect…..

If a student says “46”, their original number is 17.

If a student says “84”, their original number is 36.

If a student says “20”, their original number is 4.

I usually do this activity twice with a class.  The first time around, I suggest the students use a simple number.  In the second round, I’ll allow them to try to stump me, which usually leads to some awful mental math experiences for me.

So, what’s happening here?  In this activity, students are asked to do three different things to their results, which can be described by functions:

  • let f(x) = 2x
  • let g(x) = x + 4
  • let h(x) = x / 2

The “secret” to this ESP puzzle is revealed in this 2-minute video, used making the great FREE iPad app, Doceri:

So, as students reveal the result, your job is to subtract 12 from it, then divide by 2.  This will always yield the original number, unless the student mis-applied any steps along the way (this happens!).  Also, in the second go-round, I will always have a student whose answer is 50.98, or something worse.  Have your mental thinking cap on!

This leads to a great discussion of the “inside-out” nature of composite functions, and the ability to recycle functions.  Challenge your students to come up with their own function puzzles, and amaze the world with their math ESP!