Engaging students: Permutations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Sarah McCall. Her topic, from probability: permutations.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now?

In high school math, word problems are essentially unavoidable. They can be a pain, but they do help students to be able to see applications of what they are learning as well as good problem solving skills. So, if we must make use of word problems, we might as well make them as engaging/fun as possible. Some examples of ones that I found and would use in my classroom:

  1. Permutation Peter went to the grocery store yesterday and met a super cute girl. He was able to get her phone number (written on the back of his receipt), but today when he went to call her he couldn’t find it anywhere! He knows that it consisted of 7 digits between 0 and 9. Help Permutation Peter by figuring out how many combinations of phone numbers there are.
  2. Every McDonald’s Big Mac consists of 10 layers: 2 patties, 3 buns, lettuce, cheese, onions, special sauce, and pickles. How many different ways are there to arrange a Big Mac?

 

 

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How has this topic appeared in pop culture?

Many students are easily confused when they first learn the difference between permutations and combinations, because for most permutations is an unfamiliar concept. One way to show students that they have actually seen permutations before in everyday life is with a Rubik’s cube. To use this in class, I would have students pass around a Rubik’s cube, while I explained that each of the possible arrangements of the Rubik’s cube is a permutation. I would also present to them (and explain) the equation that allows you to find the total number of possibilities (linked below) which yields approximately 43 quintillion permutations. This means it would be virtually impossible for someone to solve it just by randomly turning the faces. Who says you won’t use math in the real world!

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How can technology be used to effectively engage students with this topic?
In a day and age where a majority of our population is absorbed in technology, I believe that one of the most effective ways to reach high school students is to encourage the constructive use of technology in the classroom instead of fighting it. Khan academy is one of the best resources out there for confusing mathematics topics, because it engages students in a format that is familiar to them (YouTube); not to mention it may be effective for students’ learning to hear a different voice explaining topics other than their normal teacher. In my classroom, I would have my students use their phones, laptops, or tablets to work through khan academy’s permutation videos, examples, and practice problems (link listed below).

References

https://www.quora.com/How-are-permutations-applied-in-real-life

https://prezi.com/q3aaem0k2xie/permutations-in-the-real-world

https://ruwix.com/the-rubiks-cube/mathematics-of-the-rubiks-cube-permutation-group

https://www.khanacademy.org/math/precalculus/prob-comb/combinatorics-precalc/v/permutation-formula

Fun With Permutations and Asimov’s Three Laws of Robotics

I’m not a big fan of science fiction, but I know enough to know that Isaac Asimov was one of the great science fiction novelists of the 20th century. The following was written by him in the October 1980 issue of The Magazine of Fantasy and Science Fiction and was reprinted in his book Counting the Eons, which was published in 1983. (I’m now holding the battered and torn pages of my copy of this book; I devoured Asimov’s musings on mathematics and science when I was young.)

Robotics has become a sufficiently well development technology to warrant articles and books on its history and I have watched this in amazement, and in some disbelief, because I invented it.

No, not the technology, the word.

In October 1941, I wrote a robot story entitled “Runaround,” first published in the March 1942 issue of Astounding Science Fiction, in which I recited, for the first time, my Three Laws of Robotics. Here they are:

  1. A robot must not injure a human being or, through inaction, allow a human being to come to harm.
  2. A robot must obey the orders give it by human beings except where those orders would conflict with the First Law.
  3. A robot must protect its own existence, except where such protection would conflict with the First or Second Laws.

Clearly, the order in which the Three Laws of Robotics matters. Shuffling the order leads to 3! = 6 possible permutations, and xkcd recently had some fun about what the consequences would be of those permutations.

Source: http://www.xkcd.com/1613/

My Mathematical Magic Show: Part 7

This mathematical trick, which may well be the best mathematical magic trick ever devised, was not part of my Pi Day magic show. However, it should have been. Here’s a description of the trick, modified from the description at http://mathoverflow.net/questions/20667/generalization-of-finch-cheneys-5-card-trick:

The magician walks out of the room. A volunteer from the crowd chooses any five cards at random from a deck, and hands them to your assistant so that nobody else can see them. The assistant glances at them briefly and hands one card back, which the volunteer then places face down on the table to one side. The assistant quickly place the remaining four cards face up on the table, in a row from left to right. After all of this is completed, the magician re-enters the room, inspects the faces of the four cards, and promptly names the hidden fifth card.

In turns out that the trick is a clever application of permutations (there are 3! = 6 possible ways of ordering 3 objects) and the pigeon-hole principle (if each object belongs to one of four categories and there are five objects, then at least two objects must belong to the same category). These principles from discrete mathematics (specifically, combinatorics) make possible the Fitch-Cheney 5-Card Trick.

Unlike the other tricks in this series, the Fitch-Cheney 5-Card Trick requires a well-trained assistant (or a smartphone app that plays the role of the assistant).

A great description of how this trick works can be found at Math With Bad Drawings. For a deeper look at some of the mathematics behind this trick, I give the following references:

 

Fun with combinatorics

I found the following videos through UpWorthy: http://www.upworthy.com/see-this-teachers-amazing-response-to-the-question-but-when-are-we-gonna-have-to-use-this. Hats off to this wonderful middle school math teacher for engaging his students in some surprisingly rich problems.

Part 1 (be sure to read the comments in the original YouTube video to see why the answer isn’t 2^{10} \cdot 10!):

Part 2:

Permutations and Combinations

Courtesy of Math with Bad Drawings is a nice derivation of the formulas for permutations and combinations:

http://mathwithbaddrawings.com/2014/02/17/permu-combi-what-now-or-the-mathematics-of-my-wedding-playlist/