Categories
Statistics

Shuffle Up and Deal, and Deal, and Deal….

Take a look at the video below, where Stephen Fry, host of the British panel show QI, alleges to do something never done before:

EDIT:  Seems as though the YouTube folk removed the video clip.  Try this link instead, and let’s hope it lasts:

QI Card Shuffling Clip

What a great “hook” for a probability or counting principles unit. Some thoughts about how to use this in your class.

1.  The result given in the video can be expressed as

If we were to shuffle the cards once every second, with each arrangement occurring once, how long would it take for use to go through every possible arrangement?  A neat example of something “big”, which is accessible and easy to discuss.

2.  The online poker site PokerStars is celebrating it’s 10th anniversary, and is offering a prize to the players who participate in their 100 billionth hand (assumed to occur around the 10th anniversary).  At this rate, how long should it take PokerStars to go through all possible arrangements?

3.  As an extension, challenge your class to find the number of possible arrangements of a deck of Pinochle cards. Cards The main differences with a Pincohle deck are that there are only 48 cards, and each card (like the 9 of diamonds) appears twice in the deck.  This problem introduce the idea of permutations with duplicate items.  In this case, we start with 48!, but then must divide out the double-count which occur with the repeat items.  We divide by two for each instance of a repeat item, and the number of permutations is given by:

4.  Let’s evaluate Mr. Fry’s conjecture:

Were you to imagine if every star in our galaxy had a trillion planets, each with a trillion people living on them, and each of these people had a trillion packs of cards, and somehow they managed to shuffle them all a thousand times a second and they had been doing that since the Big Bang, they would just now begin to repeat shuffles

To summarize, we are looking at this many shuffles per second:

Dividing by the number of possible shuffles yields:

The number of seconds in each year is given by:

Which implies we would have to shuffle for this many years:


Great exercises in laws of exponents for your students.  Share your thoughts and ideas about this fascinating video!

Categories
Algebra

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.

Categories
Uncategorized

One-Variable Equations – Let’s Look at Visual Methods

This past week, I worked with 2 Algebra 1A classes, which were working through solving one-variable equations.  By the time of my visit, students had worked through solving equations which had a variable on both sides, some which required students to combine like terms, and some where the distributive property was required.

The students in the class had been exposed to the traditional series of worksheets, drills, and various games to keep the class motivated.  Homework the previous night consisted of problems like the one below:

My challenge to the class was to work with me to think about these one-variable equations differently.  Rather than thinking of the equation as a whole, we have two expressions, one on the left and one on the right.  While by this point in algebra 1 the students do not know how to graph a linear equation, this lesson provides a good opportunity to open that door.  By this point, students do understand that:

  • Expressions can be evaluated, given a value for x, and this provides an output (y).
  • Function tables provide ordered pairs which can be plotted
  • While we often are asked to “plug in” a specific value for x, we can plug any value into most expressions and obtain an output.

So, let’s use this background to our advantage and bring consider a graphical method for solving one-variable equations.  The class I worked with used laptops, and worked through some examples using the Desmos Calculator.  

Calculator

Students generally understood that we had strong interest in that point of intersection.  But what does the x represent?  What is the y?  Which one represents the solution we seek?  And what a great review of ordered pairs, more work with functions, and a preview of linear functions.  Students worked through function table with me on this worksheet, and used the grapher to work through problems on their own.

class 1

Class 2

Working with this class produced some rich discussions, particularly when the class was faced with a problem which had no solution, or a problem which had a “messy” solution.  Discussing the parallel lines in teams strengthened the students understanding of “no solution” problems, and problems with messy fractional solutions became possible.

The activity also provided a nice segue to inequalities.  By problem 6 on the worksheet given above, a number of students noted that they could look at when one line was “above” the other.  And some nice foreshadowing of linear systems has been built into early units.

Should students be proficient in traditional algorithms for solving equations?  Of course!  But looking at algebraic ideas using multiple representations often lets students personalize their learning by assessing methods and making connections between tables, equations and graphs.