Engaging students: Combinations

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 Heidee Nicoll. Her topic, from probability: combinations.

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How could you as a teacher create an activity or project that involves your topic?

As a teacher, I would give my students an activity where, with a partner, they would be in charge of creating an ice cream shop.  Each ice cream shop has large cones, which can hold two scoops of ice cream, and six different flavors of ice cream.  Each shop would be required to make a list of all the different cone options available.  (Note: cones with two scoops of the same flavor are not allowed.)  The groups would calculate the total number of combinations, and try to find any patterns in their work.  I would ask them how to calculate the number of options for 7 flavors of ice cream, and then ask them to find a general rule or pattern for calculating the total for n flavors, and have them try their formula a few times to see if it gives them the correct answer.  As a bonus, I would also ask them how many flavors of ice cream they would need to be able to advertise at least 100 different cone combinations.

 

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Historia Mathematica, a scientific journal, has an article called “The roots of combinatorics,” which describes records of ancient civilizations’ work in combinations and permutations.  I would share with my students the first part of this description of the medical treatise of Susruta, without reading the last sentence that gives the answers:

“It seems that, from a very early time, the Hindus became accustomed to considering questions involving permutations and combinations. A typical example occurs in the medical treatise of Susruta, which may be as old as the 6th century B.C., although it is difficult to date with any certainty. In Chapter LX111 of an English translation [Bishnagratna 19631] we find a discussion of the various kinds of taste which can be made by combining six basic qualities: sweet, acid, saline, pungent, bitter, and astringent. There is a systematic list of combinations: six taken separately, fifteen in twos, twenty in threes, fifteen in fours, six in fives, and one taken all together” (Biggs 114).

I would ask them to estimate the number of combinations of any size group within those “six basic qualities” without doing any actual calculations.  Once they had all made their estimates, as a class we would do the calculations and comment on the accuracy of our earlier estimates.

 

 

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How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Sonic commercials boast that their fast food restaurant offers more than 168,000 drink combinations.  This commercial shows a man trying to calculate the total number of options after buying a drink:

I would show my students the commercial, as well as images of Sonic menus and advertisements for their drinks, such as the following:

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sonic2

sonic3

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The Wall Street Journal also has an article about the accuracy of the company’s claim to 168,000 drink options, found at http://blogs.wsj.com/numbers/counting-the-drink-combos-at-a-sonic-drive-in-230/.    The author talks about the number of base soft drinks and additional flavorings available, and says that according to the math, Sonic’s number should be well over 168,000 and closer to 700,000.  He describes the claim of a publicist who works for Sonic that 168,000 was the number of options available for no more than 6 add-ins, which the company deemed a reasonable number.  The article also notes the difference between reasonable combinations and literally all combinations, which could spur a good discussion in the classroom about context and its importance in real world problems.

 

References

 

Biggs, N.l. “The Roots of Combinatorics.” Historia Mathematica 6.2 (1979): 109-36. Web. 08 Sept. 2016.

 

Carl Bialik. “Counting the Drink Combos at a Sonic Drive-In.” The Wall Street Journal. N.p., 27 Nov. 2007. Web. 08 Sept. 2016.

 

http://www.youtube.com/channel/UC9fSZEMOuJjptiXVsYf8SqA. “TV Commercial Spot – Sonic Drive In Sonic Splash Sodas – Calculator Phone – This Is How You Sonic.” YouTube. YouTube, 29 Oct. 2014. Web. 08 Sept. 2016.

Combinatorics and Jason’s Deli (Part 2)

Jason’s Deli is one of my family’s favorite places for an inexpensive meal. Recently, I saw the following placard at our table advertising their salad bar:

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The small print says “Math performed by actual rocket scientist”; let’s see how the rocket scientist actually did this calculation.

The advertisement says that there are 50+ possible ingredients; however, to actually get a single number of combinations, let’s say there are exactly 50 ingredients. Lettuce will serve as the base, and so the 5 ingredients that go on top of the lettuce will need to be chosen from the other 49 ingredients.

Also, order is not important for this problem… for example, it doesn’t matter if the tomatoes go on first or last if tomatoes are selected for the salad.

Therefore, the number of possible ingredients is

\displaystyle {49 \choose 5},

or the number in the 5th column of the 49th row of Pascal’s triangle. Rather than actually finding the 49th row of Pascal’s triangle by direct addition, it’s simpler to use factorials:

\displaystyle {49 \choose 5} = \displaystyle \frac{49!}{5! \times 44!} = \displaystyle \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44!}{5 \times 4 \times 3 \times 2 \times 1 \times 44!}

= \displaystyle \frac{49 \times 48 \times 47 \times 46 \times 45}{5 \times 4 \times 3 \times 2 \times 1}

= 49 \times 12 \times 47 \times 23 \times 3

= 1,906,884.

Under the assumption that there are exactly 50 ingredients, the rocket scientist actually got this right.

Combinatorics and Jason’s Deli (Part 1)

Jason’s Deli is one of my family’s favorite places for an inexpensive meal. Recently, I saw the following placard at our table advertising their salad bar:

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I share this in the hopes that this might be reasonably engaging for students learning about different methods of counting.

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. Here’s my series on the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, 2b, and 2c: The 1089 trick.

Part 3a, 3b, and 3c: A geometric magic trick (see also here).

Part 4a, 4b, 4c, and 4d: A trick using binary numbers.

Part 5a, 5b, 5c, 5d: Predicting a digit that’s been erased from a number.

Part 6: Finale.

Part 7: The Fitch-Cheney 5-card trick.

Part 8a, 8b, 8c: A trick using Pascal’s triangle.

 

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:

 

Collaborative Mathematics: Challenge 12

I’m a few months late with this, but my colleague Jason Ermer at Collaborative Mathematics has published Challenge 12 on his website: http://www.collaborativemathematics.org/

Hands on SET

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight “Hands-on SET®,” by Hannah Gordon, Rebecca Gordon, and Elizabeth McMahon. Here’s the abstract:

SET® is a fun, fast-paced game that contains a surprising amount of mathematics. We will look in particular at hands-on activities in combinatorics and probability, finite geometry, and linear algebra for students at various levels. We also include a fun extension to the game that illustrates some of the power of thinking mathematically about the game.

The full article can be found here: http://dx.doi.org/10.1080/10511970.2013.764368

Full reference: Hannah Gordon, Rebecca Gordon & Elizabeth McMahon (2013) Hands-on SET®, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 23:7, 646-658, DOI: 10.1080/10511970.2013.764368

Introduction to the Catalan numbers

From the YouTube description:

Alissa S. Crans, Associate Professor of Mathematics at Loyola Marymount University, introduces viewers to the Catalan numbers, which take on a variety of different guises as they provide the solution to numerous problems throughout mathematics.

More on the Catalan numbers can be found at MathWorld and at Wikipedia and at http://www-math.mit.edu/~rstan/ec/. This video is accessible to the general public, including gifted elementary school students.

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:

Password strength and combinatorics

xkcdpassword_strength

Source: http://www.xkcd.com/936/