Categories
Algebra Technology

Those Funky, Funky Exponential Functions!

A neat math discussion came from an unexpected place today when a teacher in my department sought me out with the TI Nspire of one of her students in hand. The student was in a pre-calc class where exponential functions were being examined, and attempted the graph the following:

What should we expect to see?  How does this graph behave?  Here is what my new Nspire app gave me, which matches what the student calculator showed.

Nspire1

There seems to be a little funkiness around the origin which confused the Nspire, but the bigger issue is that the meat and potatoes of the graph is just wrong.  This about these values of this function and you’ll see why:

This function be-bops around in a quite interesting manner, and the TI-84 shows the graph nicely, as individual dots.  After going through some usual diagnostics in my head, and the list of dumb things kids sometimes do to calculators which cause them to act funny, the problem seems to be with Nspire. But this got me thinking about this strange function, and it’s behavior.  What happens if x = 1/2?  2/3?  3/2?  What’s the domain of this function?  And how do some of my other online math tool friends handle this one?

Wolfram|Alpha is our first contestant.  Show me your stuff:

Wolfram

How cool!  And what a neat discussion of complex numbers, and an interesting overlap between real and complex parts.  Wondering if anyone has insight into the domain and range though.  Is it that this function has no domain?  Or is it that  the domain is simply too difficult to express nicely?

Next up is my old friend Desmos.  I know you won’t let me down.  First, entering the function, Desmos does nothing (trust me, no screencap…nothing happens).  But, activate the table and you can plot some points.  I also added a few of my own at the end of the table:

Desmos

A good effort, but wish there was some indication of the graph’s behavior without the table.

Overall, this is a tricky little function with a lot to talk about.  Put it on the board for your classes and let them think about:

  • What rational values cause the function to be undefined over the reals?
  • What rational values cause the function to have negative value?  Positive value?

Then, the plot THICKENS!  Later in the day, I was showing of my Nspire app and the goofy function to some math friends at a meeting, when an English teacher collegue joined the fray.  After giving us the obligatory “what a bunch of geeks” staredown, she grabbed my iPad and gave a few finger swipes at the graphs, changing the window values and……

Nspire2

Holy crud!  How cool, yet….pretty much not useful at all!  During the day, I also sent a note out to TI about what I had found, and a response was given later in the day.  Thanks for getting back to me so quickly TI folk!

Categories
Algebra

This Week’s Required Reading for Algebra Teachers!

Mid-April, that time of year where teachers and students start to see the finish-line of the school year.  Everyone feels the burdens…state testing, class distractions, covering all the “material”….teachers have a lot on their plate.  But it’s also a great time to reflect upon the past year, work in teams to consider best-practice, and plan changes for next year.  Two intriguing blog posts by Grant Wiggins this week should be required reading for all secondary math teachers.

First, Grant Wiggins rants against courses we call algebra 1.  What could be wrong with Algebra 1?  We all took it, we all agree kids “need” it, and isn’t a proven gate-keeper to college success?

Algebra, as we teach it, is a death march through endless disconnected technical tools and tips, out of context. It would be like signing up for carpentry and spending an entire year being taught all the tools that have ever existed in a toolbox, and being quizzed on their names – but without ever experiencing what you can craft with such tools or how to decide which tools to use when in the face of a design problem.

Amen, brother.  In algebra, we move from the unit of linear functions, to the unit on systems of equations, to the unit on exponents, then the unit on polynomials. At the end of each unit, we duitfully give the unit test, get some number score back, then move on to the next unit.  We have trained students to think this way:  that algebra means mastering one skill, then the next.  How often do we provide rich tasks which allow students to reflect upon their cumulative skills set?  I appreciate the work of many math folk out there to change the nature of Algebra 1 from a rigid sequence of skills to a course which encourages application and reflection, driven by interesting, authentic problems.  Some examples of outstanding math educators working to promote inquiry in math class are listed at the end of this post.

worksheets

For many special education students, chunking is a device used to “help” students in algebra.  By continued pounding of square pegs into round holes, using worksheets of similar problems (i.e. solving a one-variable equation, with variables on both sides), students can achieve temporary, recordable “success”.  The students most in need of seeing auhentic problems are often those least likely to move past the chunking, and into authenticity.  Fortunately, to help sort out the madness, Grant Wiggins provided a second great article of required reading for math teachers this week:

Grant Wiggins on turning math classes into bits of disconnected microstandards.

What’s so harmful about taking a broad subject like Algebra and breaking it into pieces?  What is the consequence?

Take a complex whole, divide into the simplest and most reductionist bits, string them together and call it a curriculum. Though well-intentioned, it leads to fractured, boring, and useless learning of superficial bits.

Hallelujah!  Make sure you check out Grant’s driver-ed analogy for the full effect.  More ammunition for us to develop math courses rich with interesting, relevant tasks, where algebra is the tool, not the star of the show.

Fortunately, there are many educators out there working to develop tasks which develop algebraic thinking, and encourage the use of algebra as the tool, rather than the exercise.  Keep them in your toolbox for future planning.

Dan Meyer: the king of perplexity.  If you aren’t visiting Dan’s blog at least semi-regularly, then start now.  And check out his spreadsheet of tasks for the math classroom.  In the same theme, visit Timon Piccini, and his many on-point 3-act tasks.

Sam Shah:  Sam leans more towards the pre-calc, calc end of the math spectrum, but I apprecaite Sam’s constant self-reflection and great ideas for engaging kids in math discussions.

Kate Nowack:  sometimes task-oriented, sometimes ranting on policy, but always interesting.

NCTM’s reasoning and sense-making task library has a number of problems around which algebraic ideas can be wrapped.

Categories
Statistics

Expected Value and Analyzing Decisions, part 1

As an A.P. Statstics reader, I am excited to see the increased emphasis on statistics and probability in the Common Core standards.  Ever better, the standards specifically ask our students to be able to reach conclusions based on data:

CCSS.Math.Content.HSS-MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

CCSS.Math.Content.HSS-MD.B.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

This is a great start, and requires that we move beyond just flipping coins and drawing beads from bags.  We need to get our students writing about what they see, and provide strategies for developing clear, statistical thinking.

In this first post, we’ll look at a famous game show, and examine possible decisions.  Next, the game of Monopoly will provide a more complex argument in expected value.  Finally,  in a third post, we’ll look at past Advanced Placement Statistics items and adapt them for use in non-AP clasrooms.

The game show “Let’s Make a Deal” provides a simple example in decision-making based on values and probability.  At the end of the show, contestants who won prizes during the show are asked if they would like to give up their prizes for a chance at a bigger prize by choosing one of three doors:

Should a player give up their prizes?  How much does the “big deal” need to be worth in order for a player to be tempted to give up what they won?  How about the other two doors…do they matter?  Show the first video to your class, and let them debate whether they would go for it, and take a class vote.

In the next stage, we can show what all 3 doors have behind them, and begin to consider the game as a whole:

Does this new information change any decisions?  How willing are you to risk your prizes if you know there are some other nice prizes to be won, which may or may not have the same value of what you already have?  The slides below let you walk through a discussion with your class.  Use dry erase boards or even Google Forms to have students share their ideas.

There are a number of approaches which can be taken here.  Hopefully your students develop one of these on their own, or have creative, new ideas.

  • In this example, 1/3 of the doors hold the Big Deal, but another door has a prize of essentially equal value, the “medium” prize.  It may be difficult for us to predict the value of the middle prize, but it seems plusible that 2/3 of the doors will have value at least equal to what we are giving up.
  • In the worst-case scnenario, the non “Big Deal” doors have minimal value to us, essentially zero.  So, we have 1/3 chance of selecting the Big Deal, and 2/3 chance of winning nothing.  We can compute the expected value:

  • In the video above, we have some insight into the three doors on the show, and there always seems to be a “middle” prize and a “small” prize.  We can compute the expected value based on this information:

In any case, it seems reasonable for us to consider giving up the $7,000, but next we can think about the limits of our decisions.  What is the most you would give up to go for the Big Deal?  Can students create a general rule for making the decision?

And finally, a few parting shots for discssion:

  • It’s easy to think of the prizes as a cash value only, but does the prize you are giving up matter?  What if you always wanted to go to Paris?  Does that add to the value?  Or are there prizes you would never give up?
  • Expected value gives a nice summary of what we should expect to see in the long run, after many, many trials.  But on Let’s Make a Deal, you only have one shot…the chance of a lifetime.  Does this change your approach to the decision-making?

Would enjoy hearing your class experiences in sharing the Let’s Make a Deal examples.  Stay tuned for the next post on Monopoly.