# Tag Archives: counting

## Encourage Generalization and Communication with these Math Challenges

A comment from a recent post of mine on differentiation asked what I do with students who complete tasks early.  In every course I have ever taught (usually Algebra 2, Prob/Stat, Algebra) I have used weekly problem-solving challenges, no matter what the level of student.  Often, my intent in these problems was to develop written communication skills in mathematics, and have students begin to reflect upon their own writing style.  Have students complete the challenge, critique their writing and provide a path for improvement, and have students turn in their best works as part of a portfolio at the end of the semester.

In this post, I focus on tasks involving counting, number theory, and algebra.  The problems here are ones I have assigned, graded, revised, and enjoyed over the past 15 years.  I’ll have some more tasks in a later post.  Click the title to download the PDF document.

PATHS:  How many ways are there from start to finish?  I love this problem, because there are multiple ways to approach it.  Combinations give the result, but there is also a Pascal’s Triangle approach, or as a permutation with identical items.  And Polya’s strategy of starting small and working your way up is key to this one.

SOME ZEROES:  I have always enjoyed giving this problem, as you you can have rich conversations about simple number facts and the commutative property.  And the student explanations will range from the ridiculous to the intriguing.  When I started giving this problem 15 years ago, some students would use Excel to try to simply compute the answer, which often “broke” Excel and gave a wrong answer.  I have given up on trying to follow the technology, and have given a similar problem as a follow-up on a quiz.

AVERAGE SPEED:  One of my favorites because of its simple premise, and a result that is counter-intuitive.  Also, can be easily differentiated.  For some students, choosing distances and testing serves as a good starting point, while students with advanced algebraic skills can dive right into the abstract.

LAST DIGIT:  A premise simple enough for grade 6, yet complex enough to challenge older students if you ask for a general formula.  It’s also easy to adapt this problem and use it as an opener for class.

INVERSES:  In this challenge, students must find a matrix which is its own inverse, of which there are many, many possibilities.  How will your students ensure that their matrix is unique?

Feel free to contact me for solutions, tips, or more ideas.

## 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.  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: