Month: November 2012

Categories
Algebra

Look Through the Eyes of Your Students

Here in Pennsylvania, many high schools are gearing up for the upcoming Keystone Exam in Algebra 1.  In this first year of Keystones, the Algebra 1 exam is being given not only to students as they complete Algebra 1, but also to 10th and 11th graders who have passed the course in the past.  The state has provided a number of sample items, which we have been using in math classes to help our students prepare.

A discussion of one of these released items not only revealed a common algebra misconception, but also generated thoughts of how teachers may see problems differently that their students.  Here is the question:

Problem

So, how could teachers and students view this problem differently?

HOW I SEE THIS PROBLEM:

Teacher Problem

My teacher eyes tell me immediately that this is a factoring problem.  I’m not sure how it is going to factor yet, but I am pretty confident that the answer will be C or D.  Choices A and B are not really even up for debate here.

My radar perked up when a colleague shared that a handful of students in one of her classes chose B.  B???  How the heck did they get B????

HOW SOME STUDENTS SEE THIS PROBLEM

Problem Student

My old nemesis….cancelling across addition and subtraction signs, how nice to see you again!

So, while I immediately see the problem as two expressions which will separately require factoring, I need to remember that students don’t always view problems the same way.  Being able to identify, discuss, and personalize these common errors are all part of the wonderful art of teaching.

And while illegal cancelling will be a struggle for students long after I retire, I often use the exercise below to generate discussion.

HOW TO TURN $100 INTO $199 (LEGALLY?)

Cancelling

So, either I have made a math error here, or I have a great method for generating some side income here (and why am I sharing it with you, anyway???).

Categories
Middle School

The Take-Away Game

A recent visit to a 6th-grade classroom gave me a chance to introduce a simple game I have used in the past as an-going challenge.  Even after a few pop-ins to this 6th grade class, I am still undefeated, and don’t plan on giving up my championship belt anytime soon!

THE TAKE-AWAY GAME – Rules

On a board, or piece of paper, draw 23 X’s.  Players will alternate turns, and on each turn a player must erase 1, 2 or 3 X’s.  The winner is the player who erases the last X.

It’s an easy game to understand.  An example is given here:

With a class, I will give students a chance to use dry-erase boards and play against each other.  Then, as students begin to understand the game, they are allowed to challenge me.  This usually ends badly (for them), as I know the tricks to the game.  I start by asking the player if they would like to go first, or allow me to go first.  Since kids are usually nice, they will allow me to go first, and this sets them up for certain doom.  Also, I will use my best poker skills to agonize over my moves, though I know exactly where I want to go with my moves.

Eventually, students will gather around to suggest moves.  Their first realization is that if I get the board down to 4 Xs, I will win.  This will then extend to 8 remaining.  With some classes, I have placed a fist behind my back, and done a thumbs-up to signal those watching when I know I have the game won.  Shoot me an e-mail if you need thorough instructions on how to win.

As students master the game, we can ask some extension questions:

  • Does the number of X’s we draw change the game?  What if we use 25, 35, or 50 X’s?
  • What if we could erase 4, 5, 6 or n number of X’s?  How would the strategy change?

For now, play the game with your students, and I look forward to retaining my Inter-Galactic Take-Away Game BeltChampionship Belt!

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!