#MathEd – Math for My Nephew

A while ago I was asked to provide some resources I use to teach math for my nephew (well, nephew-in-law). I decided to gather them together into a blog post and thus get a two-fer.

Associations

Mathematical Association of America (http://www.maa.org/) – Their stated mission is “to advance the mathematical sciences, especially at the collegiate level.” Membership is open to students and teachers (K-12 and college), starting at $35/year (student with proof of status) and going up to $249/year (Member Plus). I’m a member and for me the real value of membership is access to a wide range of publications plus discounts on books, both e-book and printed. (Disclaimer: I currently write book reviews for the MAA web site but I am not compensated.) You can follow them on Twitter at @maanow and on Facebook as maanews.

National Council of Teachers of Mathematics (http://www.nctm.org/) – Like the MAA, the NCTM offers memberships to students, teachers (primary through college) and to organizations. They also offer the option of an ‘e-membership’ at each level for a slight discount. Membership annual dues range from $44 (Student and Emeritus) to $144 (Full Individual Membership). Membership gives you access to a host of instructional materials, NCTM’s ‘e-standards’ and NCTM’s E-Seminars, 60 minute on-demand video presentations on a variety of math education topics. You can follow them on Twitter at @NCTM or on Facebook as NCTM Illuminations.

Blogs

Math with Bad Drawings – This is one of my favorite sites. Ben Orlin provides an entertaining and educational view of math from a teachers perspective. The title comes from each post being illustrated by a series of stick drawings on a whiteboard. I wrote a review of his site here. I’ve used some of his posts as jumping-off points for discussions in my math class. You can follow Ben on Twitter as @benorlin. He’s also on Facebook but doesn’t appear to be too active.

MathBabe – Cathy O’Neil is a former Wall Street quantitative analyst who left for the wilds of higher education. She mainly blogs about Big Data and math education in higher ed. I wrote a review of her site here. A smart, interesting, fun site. You can also follow Cathy on Twitter.

Finding Ways to Nguyen Students Over – Fawn Nguyen is a California math teacher who has one of the best math blogs I’ve seen for primary and secondary educators. She clearly works hard to present math to her students in innovative and entertaining ways and she shares these techniques on her blog. (I wrote a review of her site here.) However, the real hidden gems of her site for educators are the affiliated sites like Visual Patterns, Math Munch and Would You Rather. These sites are a treasure trove of fun math problems and exercises you can share with your students or work on your own.

The NRICH Project – Not a blog per se, but The NRICH Project was started by the University of Cambridge, according to their “About” page:

“to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.”

(More here.) The content is divided into material for teachers and students. Each category is further divided into primary and secondary education. The idea is to present tasks that target multiple learning styles. These are known as “rich tasks” and you can get more information about them from this article.  Teaching materials are printable and downloadable for ease of use. You can also follow the NRICH project on Twitter  and on Facebook. You can sign up on their mailing list to get updates. They also provide a guide for parents and caregivers.

Project Euler – Also not a blog but a cool site for more advanced math fans. (Yes, there are some of us out there.) It’s a collection of almost 500 math problems, some of which may require basic programming skills. Each problem builds on insights gained solving previous problems so I advise doing them in order. I’ve posted a few of the problems on my site. Working on the problems presents interesting insights and is definitely mind-expanding.

Videos

NumberPhile – This is a nice resource of short, entertaining videos about specific math concepts. I do a short segment at the beginning of my classes called “Math Minute” and I’ve used several of these as inspiration for material. You can also follow them on Twitter and Facebook.

ComputerPhile – Not strictly related to math, but computer science is based on math. Similar to NumberPhile, ComputerPhile provides short videos about topics like undecidability, cryptography and how computers use math to do animation. You can follow them on Twitter or on Facebook.

I’m always looking for good math resources, both for my classroom and this blog. If you have any interesting leads, post them in comments!

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#WeHateMath One Man’s Meat is Another Man’s Poisson

Recently I got to thinking about my grad school course on system performance. The instructor had a day job at Bell Labs where he did the math that he tried to teach to us. It was all about things like queuing theory, think time and all sorts of statistical and stochastic analysis of information systems. (It was pretty overwhelming at the time but I’d like to take the class again.)

Anyway…one of the concepts that I took away from the class was about Poisson distributions. It got me wondering about how I would explain this concept to my College Mathematics students.

Let me explain.

College Mathematics at my school is a 100-level class for non-technical students. Most of my class are in programs like Medical Assisting, Graphic Design and Business Administration. This is the course description:

This course develops problem-solving and decision-making strategies using mathematical tools from arithmetic, algebra, geometry, and statistics. Topics include consumer mathematics, key concepts in statistics and probability, sets of numbers, and geometry. Upon successful completion of this course, students will be able to apply mathematical tools and methods to solve real-world problems.

Basically, it’s a functional numeracy class. Don’t get me wrong — it’s fun to teach and I enjoy the challenge of presenting math concepts in an interesting and accessible way. One of the ways I do this is start each class session with a ‘Math Minute’ where I run through a quick math concept to give my students a little something to ponder.

The Poisson distribution was developed by Siméon Denis Poisson in 1837. According to StatTrek:

A Poisson distribution is the probability distribution that results from a Poisson experiment.

Okay, a Poisson something-or-other comes from a Poisson something-else. Could you vague that up for me?:

A Poisson experiment is a statistical experiment that has the following properties:

  • The experiment results in outcomes that can be classified as successes or failures.
  • The average number of successes (μ) that occurs in a specified region is known.
  • The probability that a success will occur is proportional to the size of the region.
  • The probability that a success will occur in an extremely small region is virtually zero.

Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.

That’s a little better. Let’s look at the classic example of a Poisson distribution: bus arrivals.

On your route, a bus comes by, on average, every ten minutes. On this particular day, you arrive at the bus stop and there is no bus to be seen. How long, on average, do you think you’ll have to wait for the next bus?

At this point, your audience is expected to say “Why, five minutes, of course.” The standard answer is actually ten minutes. The problem with this example is that it doesn’t give the right picture so the actual answer doesn’t fit our mathematical intuition. That’s why at this point the explainer hauls out the equation:

Via StatTrek:

Poisson Formula: Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e) (μx) / x!

where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

And everyone’s eyes glaze over.

I’m not complaining but I’m looking for a way to explain this to a non-technical crowd. So the bus stop example can be much more effective if we just tweak a few parameters. A Poisson distribution is generated out of experimental data so let’s adjust our mental picture.

Instead of being the person waiting for the bus, let’s instead picture ourselves sitting on a park bench across the street from the bus stop. In one hand we have a stopwatch and the other a spreadsheet. We can start our experiment any time but let’s say that we begin when a bus arrives. As we observe busses and passengers, we record two things:

  1. The time the bus arrives
  2. The time that each person arrives at the bus stop.

We need to gather enough data for a useful answer, so we’re going to be here for a while. It could be an hour, two hours, twelve, whatever. At some point we get bored and go home to crunch our data. Lo and behold, we discover that the bus interarrival time does in fact average ten minutes but the average wait time is greater than five minutes.

Why is that? Well, from our perspective as an outside observer, this phenomenon makes a lot more sense.  Let’s look at our bus arrivals over a timeline:

bus 1  –  5 mins. – bus 2  – 15 mins.- bus 3 – 10 mins.bus 420 mins.bus 5 and so on…

As an individual waiting at the bus stop we didn’t have this perspective. But when we look at it from outside, we have a better picture of what’s going on. More passengers arrive during the longer intervals than during the short ones. As a result, the wait times for these passengers are going to skew our average wait time, giving us our previously unintuitive result of greater than five minutes.

Uncertainty is part of our lives, so it makes sense that it would show up in math as well. Probability (and statistics) don’t eliminate uncertainty but rathergive us a useful way to acknowledge  and honor uncertainty.

 

#WeHateMath The Sniper’s Tale

I’d like to tell you a story about a man named Rob Furlong. Mr. Furlong is a Canadian, formerly of Her Majesty’s 3rd Battallion (Princess Patricia’s Canadian Light Infantry) Furlong’s job with the Canadian military was one that most of us (okay, mostly us guys) would think was kind of cool.

Rob Furlong was a sniper.

Well, not just a sniper. It turns out that Furlong was THE sniper. In 2002, he set a record for the longest confirmed combat kill, at 2,657 yards (2,430 meters). Coincidentally (or not), the previous record holder, Master Corporal Aron Perry, was on Furlong’s team. (Must be something in the poutine.) By the way, Furlong’s record was broken in 2009 by British sniper Craig Harrison. Perhaps afternoon tea trumps poutine.

What does this have to do with math? Well, if we take our education about snipers from popular culture, Furlong and his companions are some kind of Jedi Master/Zen Monk types who rely on mystical instincts to hold a stick very steadily for a long time.

Not so much.

Consider gravity. Over 2,657 yards, Furlong knew his bullet was going to drop almost 300 feet so he had to aim high. Wind resistance? Check, adjust for that. By the way, what’s the temperature out? Turns out that gunpowder burns at different rates at different temperatures. Speaking of air, how high above sea level are we? I only ask because the bullet travels farther in the thinner air at higher altitudes. Also, what direction are you facing? If your target is to the east, you’ll have to aim a bit lower, due to the Coriolis Effect.


In other words, Rob Furlong is a highly trained mathematician who can kill you from over one and a half miles away.

#WeHateMath Book Review – Good Math by Mark C. Chu-Carroll

Talking (or writing) about math in an interesting, engaging way is hard. It’s also something that is very important. Math is more important now than ever before but we haven’t changed the way we teach it in decades.

That’s why it’s a pleasure to find a book like Good Math by Mark C. Chu-Carroll. As a self-described ‘geeky kid’, Chu-Carroll took an interest in the work that his father, a physicist, would bring home from work. This started Chu-Carroll’s life-long love of math and inspired him to create the blog Good Math/Bad Math. Good Math (the book) is his attempt to bring his love of math to a wider audience.

One of the qualities that makes this book a delight is that you don’t need much more than high school algebra to follow along. (Naturally, the more math background you have, the more you’ll get out of it.) In addition, you don’t need to read the entire book in sequence. It’s designed for you to dip in at any point, find something that catches your eye and then move on. Some sections refer to other parts of the book but you can follow them or not as you wish. In addition, some sections include computer code if you want to extend yourself and experiment a little bit.

The writing is friendly and accessible, with plenty of diagrams and examples. The book is divided into six parts: Numbers, Funny Numbers, Writing Numbers, Logic, Sets and Mechanical Math. Chu-Carroll presents a broad look at the foundations of modern mathematics. I recommend this book to anyone who likes a good casual read that makes you smarter.

 

I’d also like to take a moment to give a shout-out to The Pragmatic Bookshelf. I’ve loved their stuff since reading The Pragmatic Programmer and they’ve only gotten better with their expansion into e-books. As you might expect, most of their publications are aimed at programmers but they have content aimed at beginners as well as books to help you ‘take care of your body and expand your mind’.


(Chu-Carroll, M. C. (2013). Good math: A geek’s guide to the beauty of numbers, logic, and computation. Dallas, TX: Pragmatic Programmers.)

#WeHateMath Project Euler 9 – Pythagorean Triplets

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

For example, 32 + 42 = 9 + 16 = 25 = 52.

There exists exactly one Pythagorean triplet for which a + b + c = 1000.

Find the product abc.

Let’s get the problem part out of the way first. It turns out to be pretty easy to generate Pythagorean Triplets (sometimes called Pythagorean triples):

Pick two positive integers, m and n, ,where m < n. Generate your three values like so:

Of course, doing this by hand is annoying and this is just the sort of thing for which spreadsheets were made. I set up the formulas and then starting fiddling with values of m and n and it just took a few tries to range in on the answer:

m

5

n

20

a

375

b

200

c

425

Sum(a,b,c)

1,000

Product(a,b,c)

31,875,000

a^2

140,625

b^2

40,000

a^2 plus b^2

180,625

c^2

180,625

Readers know by now that the point of the Project Euler posts is not to simply get the answer to the problem but to figure out the point of the problem. In this case, that’s not obvious. Though Pythagorean Triples are a surprisingly hot topic of discussion among mathematicians if my casual research is anything to go by. I even found a paper documenting a new way to generate triples that suggests an interesting application of the math to cryptology. It’s especially surprising when you consider that these triples have been around for over 2000 years.

The classic Pythagorean triples are (3, 4, 5) and (5, 12, 13). You can generate new triples by using multiples of each element. In our answer above, (375, 200, 425) is a multiple of (3, 4, 5). There are others, however. For example (4, 3, 5) is also a valid triplet but can’t be generated by the traditional methods.

Pythagorean Triples are connected to both Fibonacci numbers and primes, which we’ve discussed previously.. The two classic triples each have Fibonacci values as two of the three values. In addition, up to two of the legs can be prime numbers but because at least one of the legs must be even, there is no triple where all three elements are primes.

This is a great example of pure mathematics that may (or may not) someday find some practical application. I view mathematics as a way to perform experiments on the universe, but unlike traditional laboratory science, my lab bench is conveniently located between my ears. So the next time someone tells you that studying math is impractical, remind them:


#WeHateMath Project Euler 8 – Math Makes Everything Better

Find the greatest product of five consecutive digits in the 1000-digit number.

73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450

At first glance this appears to be arithmetic which, as we all know, is boring and mechanical and completely beneath us as humans. Well, actually, at first glance the part of our brains that does arithmetic (you know, the boring part) sees this awful number and just wants to shut down. Since I knew it had to be more than just arithmetic, I tucked it in the back of my mind to ferment a bit before tackling it again.

After a bit of thought, I can see the trick here. The problem statement refers to a 1000-digit number but we’re supposed to deal with individual sets of digits. So we can use math to make this more interesting (since math makes everything better) and simply regard this as a string of 1000 digits.

How does this help? First of all, any string of digits with a zero in it we can disregard. In fact, we can discard any ten consecutive digits that have a zero in the middle since any five digit sequence will multiply out to zero. So we look for digits with this pattern:

XXXX0XXXX

Let’s take another look at our number, clearing out digits within four positions of any zero:

731671765XXXX0XXXX19225119674426574742355349194934
969XXXX0XXXXXXX0XXXX23957XXXX0XXXXXX0XXXX478851843
858XXXX0XXXX12949495XXXX0XXXX95833XXXX0XX0XXXX
XXXX0XXXX4715852XXXX0X0XXXXXXXX0XXXX9522XXXX0XXXX7
668966XXXX0XXXX44523161731XXXX0X0XXXX1121722383113
622298934XXXX0X0XXXX336276614XXXX0XXXX486645238XXX
X0XXXX0XXXXXX0XXXXXXXX0XXXXX0XXXXX0XXXXXXX0XXX0XXXX
0XXXX27121883998XXXX0XXXX274XXXX0XXXXXX0XXX0XXXX6
6572XXXX00X0XXXX78XXXX0XXXXXXX0XXXX2XXXX0XXXX52243
52XXXX0XXXXXX0XXX0XXX0XXXX586XXXX0XXXX415722155397
5369781797784XXXX0XXXX51XXXX0XXXX69321978468622482
8397224137XXXX0XX0XXXX0XXXX0XXXX968652414XXXX00XXXX21XXXX0XXXX0XXXX
XX000XXXX2XXXX0XXXX41227588666881
164271714799244429XXXX0XXXX65674813919123162824586
17866458359124566529476545682848912883XXXX0XXX00XXXX
XXX0XXXXX0XXXX6321XXXX0XXX0XXXXXXX0XXXX6XXXX0X0X
0XXXXX0XXX0XXXX554443629XXXX0XXXX79927244XXXX0XXXX
XXXX0XXXXXX0XXXX9133875XXXX00XXX0XXXX99XXXX0XXXX0X
0XXXX116XXXX0XX0X0XXXXX00XXXXX983XXXX000XXXX5729725
716362695618XXXX0XXXX52XXXX00XXXXXXXX0XX0XXXXXXXX0

Okay, I’ve eliminated all strings of digits whose product will be zero so our new string of digits becomes:

73167176519225119674426574742355349194934969239574788518438581294949595833471852952276689664452316173111217223831136222989343362766144866452382712188399827466572785224352586415722155397536978179778451693219784686224828397224137968652142124122758866688116427171479924442965674813919123162824586178664583591245665294765456828489128836321655444362979927244913387599116983572972571636269561852

(I could have written a script to do this for me but pattern-matching is part of my evolution so I can do it myself without much trouble.)

This is a much more manageable number (still 389 digits long, though) but if we’re ultimately going to be doing arithmetic (and it looks like we’ll have to eventually) we have to think about having a machine do the work and that involves explaining the task to a machine in a way that it can understand. But this brings us to a question:

Do we really have to do any arithmetic at all?

A bit of thought tells us that we may be able to avoid arithmetic after all.  Here’s the logic:

Start scanning the string of digits and split off each set of five. For example, if our string was:

73167176

We would pull out the substrings:

73167

31671

16717

67176

Now, remember, I don’t really want to any arithmetic if I can help it so I do a bit of math thinking and rearrange the numbers in sorted order:

Original           Sorted

73167                77631

31671                76311

16717                77611

67176               77661

I can multiply numbers in any order I want so rearranging the digits won’t change the outcome. However, by sorting numerically, the number with the largest magnitude will automatically give me the largest product. 77661 is the largest number in my sequence here but let’s test my hypothesis just to check my thinking:

Original                     Sorted               Product

73167                          77631                882

31671                          76311                 126

16717                          77611                294

67176                         77661                 1764

Sorting has the added benefit of letting us eliminate strings that have the same digits (and thus the same product). So I only need to find the largest five digit number (after sorting) and then get the product of those five digits. I have now knocked my job down to doing a single multiplication and I bet I can get my computer to do even that little job for me if I’m feeling particularly lazy.

So I just whipped up a quick script to dump out 386 separate five digit strings of digits and sort each set of strings numerically. Then I copied and pasted the output into a spreadsheet and did a quick numerical sort to find the largest numerical quantity. Here is my result:

99972

It’s then a simple matter of using my hand-dandy calculator and determining the product, which is 10,206.

So we took a problem that looked like it would require almost 1000 arithmetic operations and knocked it down to a single calculation with just a bit of thought about the conditions of the problem.

If anyone is interested, I’ve posted my script here.

See, math makes everything better!

The Math of Shopping

It’s easy to identify people who can’t count to ten. They’re in front of you in the supermarket express lane.

M. Grundler

Let’s set the scene, shall we? You keep a list of household items that you need and about once a week or so you head out to the supermarket to stock up. Also, since you’re probably not independently wealthy, you probably have a budget for how much you want to spend on these items so let’s say you don’t want to go over $100.

Now you’re at your local supermarket and your goal is to get the items on the list without exceeding your grocery budget for the week.

In other words, you’re doing math.

There’s a field of mathematics known as operations research which, simply put, is the math of decision-making. Specifically, what you’re doing is solving what is known as the Knapsack Problem:

“A tourist wants to make a good trip at the weekend with his friends. They will go to the mountains to see the wonders of nature, so he needs to pack well for the trip. He has a good knapsack for carrying things, but knows that he can carry a maximum of only 4kg in it and it will have to last the whole day. He creates a list of what he wants to bring for the trip but the total weight of all items is too much. He then decides to add columns to his initial list detailing their weights and a numerical value representing how important the item is for the trip.

The tourist can choose to take any combination of items from the list, but only one of each item is available. He may not cut or diminish the items, so he can only take whole units of any item.

Which items does the tourist carry in his knapsack so that their total weight does not exceed 400 dag [4 kg], and their total value is maximised?”

The use of the metric system aside, this is exactly what we do whenever we go grocery shopping. Operations research is the mathematics of minimizing the stuff that you don’t want (don’t spend more than $100) and maximizing the things you do want (getting as many items on your shopping list as possible).

But consider this: Operations research has only been around since the second World War but humans have been (metaphorically) grocery-shopping for centuries. So math is not this Other that is separate from our daily lives but instead comes directly from who we are and what we do as humans.