Tag Archives: math

When was the last time an argument changed your mind?

I mean, really.

Imagine you are watching a debate for the presidential candidates and you go into the debate with certain opinions on how things should be run – say: tax code. Then, after sitting and watching the candidates outline their rationale (right?! I know I’m reaching here) you think to yourself, ‘huh. Well, that guy just changed my mind.’

Does this ever really happen?

Fairly certain this guy has never been persuaded to a different opinion

Fairly certain this guy has never been persuaded to a different opinion

Can people who believe in a ‘flat tax’ be persuaded that a ‘progressive tax’ structure is more fair and more worthy of their support? (I threw in the ‘and’ there because you can be shown the rationale for something and agree with it without changing your position)

Can proponents of a ‘pathway to citizenship’ be convinced that it’s simply too impractical to actually be enacted?

Can pro-lifers be converted to pro-choice by the right argument?

(as a side note, I wrote the above statements in a completely arbitrary manner, because I recognize that people also seek out ‘echo chambers’ for their own ways of thinking, which may be a part of the problem as a whole. Anyway, I don’t mean to deter a reader because they see words like ‘pro-choice’ or ‘pro-life’.)

Kepler could have applied himself better...

Kepler could have applied himself better…

Sometimes I question whether the Greeks were just wasting their time spending all that energy thinking about rhetoric. They didn’t persuade the Romans to stay out of their lands and to not steal their whole pantheon of gods. Maybe if they spent a little more time practicing their phalanx formations and a little less worrying about whether there was really a place filled with Perfect Forms (I’m looking at you, Plato) that we vaguely remember from before the time we were born, they might have effected a more sturdy border guard.


I changed my mind today about something. (I’m still working on changing it about some other things that would make my life easier, but I’m off to a good start) I got an email pointing me to the following post by Brett Berry on Medium this morning. 5 x 3 = 5 + 5 + 5 Was Marked Wrong
My first reaction was to be upset with the teacher who gave this kid points off for correct answers. I opened the article in order to satisfy my own desire for hearing an echo chamber of my thoughts only to find that the author took a different stance.

I kept reading because I was determined to write a comment to express my ire – but, you know, wanted to make sure that I could point out the best examples of the author’s flawed thinking first. I first saw that he was making a reasonable argument, but felt like it was still wrong. Then I saw how his examples supported his way of thinking and was starting to lament that he was making it more difficult for me to undercut him. Finally, he added that, depending on the order that things were taught, the answer could be considered correct under some circumstances, but that it was better to teach the meaning of the maths stepwise in order to law the proper framework for future lessons.

I give up. You win, Math Guy.

Not only did you change my mind on this issue, but you also laid the framework for me to re-examine my whole approach to Common Core.


Posted by on October 31, 2015 in Uncategorized


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Screen Shot 2015-08-11 at 12.31.32 PMThanks to Parker Malloy for her article pointing me to this video of a brilliant sketch by Key and Peele in which teachers are treated like professional athletes.

“… His father living paycheck-to-paycheck as a humble pro football player — the kid was a natural mathlete.”

and my favorite part, the BMW endorsed by AP English Literature Teacher, Ruby Ruhf. “Meet the new teacher’s pet.”

I know. It’s ridiculous. But does it show how ridiculous it would be to treat teachers (not to mention other ‘blue collar heroes’) like superstars? Or does it show how ridiculous it is to treat athletes like superstars? Maybe an athlete is a hero that kids can look up to as a role model. But are athletes the only ones who should be recognized in this way?

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Posted by on August 11, 2015 in Uncategorized


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It crawls… It creeps… It eats you alive!

Compounded Interest

How money or value or perhaps even The Blob grows. In the simplest of all settings, interest can be as straight forward as, I’ll loan you $10 for a hamburger today, and you pay me back $11 on Tuesday. In this case, Wimpy is taking out a loan of $10 at 10% simple interest.

That’s great for a cartoon, but it’s not the case in most real-life examples of interest. In real life, a loan you take today might accrue interest every year, month, day, or even continuously. Although the last case is one of the most interesting ones as it deals with a special number ‘e’, I just want to address the more intuitive cases.


Clean Shaven Man

Let’s say Wimpy is keen on that hamburger today, but he won’t have any cash until Tuesday again. This time, his usual rubes are all cash strapped as well, so poor Wimpy has to go to a formal lender. This lender is OK with the loan, but as insurance against Wimpy’s ability to pay back the loan on time, he insists on compounding interest every week.

“Let’s be clear about this Wimpy,” his loan officer says, as he walks him through the conditions. “You can have your $10 today at 10% interest. The loan is due on Tuesday, and that will come to a total of $11. If you can pay it off then, great. But if you need more time, we will be compounding the interest – that means that you will essentially be getting a new loan of $11, at the same 10% rate.”

“OK,” says wimpy and leaves his mark on the loan document.


This is exactly the right way to think about it.

  1. initial loan is made: $10 at 10%, due in one week.
  2. If the loan continues, another 10% is charged on the new total.
  3. Week after week, this goes on until Wimpy can pay up or he’s referred to collections and they repossess his barbershop.


Mathematically, this takes the Principal (loan amount ) and multiplies it by 10% every week.

  1. week 0: $10
  2. week 1:$10 + 10% = Principal x 1.1                                                          -> $11.00
  3. week 2: ($10 + 10%) x 110% = (Principal x 1.1)2                                              -> $12.10
  4. week 3: (($10 + 10%) x 110%) x 110% = (Principal x 1.1)3       -> $13.31


This can be generalized by the formula:

Amount owed at time      t = P (1 + R)t

Where P = principal

R = rate (expressed as a decimal)

T = the number of times interest is compounded

(whether its days, years, months, whatever)

(10)(1+.1)3               -> $13.31


This goes for any compounded growth.


oooo – Air Conditioning!

The Blob arrived in Downingtown, PA in 1958. At first it was just something riding into town on a meteorite. But soon after, an old man touched it and got it stuck to himself. Steve McQueen comes to the rescue and gets the old fellow into town to see a doctor. Meanwhile, it becomes evident that the blob is not letting go, and is hurting terribly. Dr. Hallen decides to amputate, but before he can, the blob grows large enough to eat the old man, then a nurse, and then the doctor.

From then on, the thing just keeps growing. Let’s say it grows at a rate of about 50% an hour and use the same formula…


  1. hour 0: 100g
  2. hour 1:100g + 50% = Principal x 1.5                                             -> 150g
  3. hour 2: (100 + 50%) x 150% = (Principal x 1.5)2                                               -> 225g
  4. hour 3: ((100 + 50%) x 150%) x 150% = (Principal x 1.5)3        -> 338g
  5. hour 24:         ——-à                                                                       ->1,683,411g


You can really see how this thing gets huge fast (or at least massive, we never talked about the density of this thing).

Graphically, the blob’s growth looks like this:


One troubling thing is that this could also represent the balance on a credit card that isn’t attended to.

“It crawls… It creeps… It eats you alive!”

-Tagline, The Blob 1958


For a good explanation of interest, compound interest, and ‘e’ – check out Khan Academy’s lectures on this or this site that does a great job illustrating the difference between several types of interest.



Posted by on June 13, 2014 in Uncategorized


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Undead Mathematics

ImageMultiple Zombies, the second installment of the undead mathematics series is now available at the iTunes Book store. Get this book and the first installment, In Parts:Fractional Zombies, free for a limited time.

Both books are written for an audience of about 2-3rd grade. Multiple Zombies is specifically for those learning multiplication of numbers 0-10. As such, it introduces the concept of multiplication as addition of sets and includes practice problems tied to the evolving story. Each book requires an iOS or Mac device running iBooks to read and follows the story of two friends as they battle their way through a town suffering from a nasty case of zombies.



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Posted by on March 9, 2014 in Uncategorized


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ImageIn population genetics there are two equations that allow us to estimate the frequency of alleles within a population and also to estimate the number of homozygotes vs heterozygotes for a recessive trait. These equations are known today as the Hardy-Weinberg equations because they were simultaneous proposed by two independent scientists. Like many equations, they assume a model that is not exactly reflective of the real world, however they do lend us an understanding of the rules of the system.

The two equations are:

q + p = 1

q2 +2pq +q2 = 1

It’s that easy. In each of these equations p stands for the frequency of one allele in a population and q stands for the frequency of the other allele. Assuming there are only two alleles, they must add up to 100%, represented by the decimal number 1 here.

In order to use these equations, certain conditions must be adhered to.

  1. No gene flow (immigration / emigration)
  2. No sexual selection
  3. No survival selection
  4. No mutations
  5. No genetic drift

 The last one is the one that has been interesting me lately.

 What is genetic drift? What it describes are statistical anomalies, like a run of ‘Red’ on the Roulette Wheel or an unexpectedly long string of ‘Heads’ when tossing a coin.

 What happens during genetic drift is that one allele becomes favored just because of such a statistical swing. But unlike roulette or coin tosses, when an allele loses out for a number of generations, it stands a diminishing chance of being seen again. The statistical anomaly becomes ‘hard-coded’ and self-reinforcing, such that eventually alleles disappear.

The key is that small samples allow genetic drift to happen more often, while larger populations tend to not see this occur. Using out coin toss example, if you toss a coin ten times, it is not especially surprising when you get 8 ‘heads’ and 2 ‘tails’. Whereas, in a toss of 1000 coins, getting 800 ‘heads’ is nearly inconceivable.

I encountered this while coding a genetics simulation program (note: my simulation uses a Wright-Fisher model that has distinct, non-overlapping generations).  I wrote the program and started testing it by allowing random breeding to occur over 100 generations or so. I started using only 100 animals in my simulation, but regularly saw one allele outcompete all others, meaning that the population had lost diversity.

Below is an example with 100 organisms with four alleles for the gene breeding randomly for 200 generations.


I was sure it was a problem with my algorithm. Then I started increasing the number of animals and the ‘problem’ went away.

Here’s a second experiment at the other end of the spectrum using 50,000 animals also with four alleles breeding for 200 generations. I’ve forced Excel to graph this out on the same axis.


All this, just to demonstrate to myself that the prohibition against genetic drift is actually another way of saying, “This only works with large populations.” 

What interested me is how to know whether your population is large enough to ‘resist’ genetic drift. And, how quickly will genetic drift drive alleles to fixation / loss?

“The expected number of generations for fixation to occur is proportional to the population size, such that fixation is predicted to occur much more rapidly in smaller populations.”

Not surprisingly, there is an equation designed to predict the time (# of generations) before an allele is lost by drift.

The expected time for the neutral allele to be lost through genetic drift can be calculated as


where T is the number of generations, Ne is the effective population size, and p is the initial frequency for the given allele.

(this section is informed greatly by the work of Otto and Whitlock at the University of Columbia, Vancouver. ) 

Sometimes having a computer simulation comes in handy to help get a better look at how these rules apply given different populations. I’d like to get this simulation built into a simple app for either desktop or mobile device to make public, but I have been having a lot of difficulty making the leap from a program running in the console to something worth sharing.

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Posted by on August 8, 2013 in Uncategorized


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Limited Time Promotion – 100% off

Until Sunday, my iBooks, The Thirteenth Labor of Heracles, In Parts and The Curse of Sisyphus are free in the iTunes Store. Click the titles for more information.


Can I hez Brainz?!!!


Posted by on July 19, 2013 in Uncategorized


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New Coding Challenge – The Quincunx!

ImageThe Quincunx – A triangular pegboard that will create a nice normal distribution as balls are dropped from the top and bounce down randomly over the triangular array of pegs.

Society’s greatest achievement, The Price Is Right, demonstrates the use of a plinko board in this video with the most excited player ever.

The coding challenge is to design a quincunx that demonstrates each of the following four points… No animation is required, simply (1)show the board as an array of X’s with a (2) user-determined number of rows (1-20) and the (3)resulting bins filling with integers as a (4)user-determined number of balls (1-100,000) is dropped. This time, I’m awarding prizes to the cleanest, most clearly documented entries in each language represented on Codecademy (Ruby, JS, Python).

As always, the prizes are bragging rights, presentation of your code on my blog with full attribution to you and a promo copy of any of my eBooks on iPad for you to share with the youngster in your life (or keep yourself). Each of my books presents educational material  in the form of a story (Heracles and the Gas Laws, Sisyphus and the Laws of Motion, Zombies and Fractions).

Happy Coding!

(submit your entries as links in the comments below)

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Posted by on July 10, 2013 in Uncategorized


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Coding Challenge

Recently, there have been a couple new revelations about number theory published in Science Within the article was a pair of theories about prime numbers that I had never heard before, one of which was:

Goldbach’s conjecture, [which] makes two assertions: that every even number greater than 2 is the sum of two primes, and that every odd number greater than 5 is the sum of three primes.

I thought it would be fun to start with the first part of this problem and write a program to accept user input in the form of an even integer > 2  and then look for the two primes whose sum is equal to the user provided:

Goldbach_partitions_of_the_even_integers_from_4_to_28_300pxprime1 + prime2 = user input

where prime1 and prime2 may be any prime number (even the same number twice)

I could easily see this escaping the processing power of my machine if the numbers get high, but I think it shouldn’t be too hard to at least write a code that could look for them and demonstrate whether this worked with known input.

Are you up for a quick challenge?

zombie locker

Learn Fractions with Zombies

If so, submit your documented answer here as a comment. Feel free to use any language you would like (I just did it in C++, but I’m eager to see better answers than my own). My favorite submissions will win a free copy of  my iBook, In Parts, Tales of Fractional Zombies, which you can enjoy yourself or regift to a youngster in your life who wants a fun way to learn the concept of fractions.

You can use these links as resources to help check your work:

prime numbers       prime checker

If you are new to coding and are looking for a coding environment to work in, check out this posting for help setting up a C++ coding environment using Xcode (on your mac)


Posted by on June 19, 2013 in Codecademy, Coding


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Release of ‘The Curse of Sisyphus’


The Curse of Sisyphus

The Curse of Sisyphus has been released and is available on the iTunes iBookstore. To celebrate the release, this, and its companion volume, The Thirteenth Labor of Heracles are both free until Sunday.

Zeus is not one to be trifled with. And Sisyphus has been a thorn in his side, defying him at every turn, yet escaping every punishment with uncanny cunning. But this time, the mortal has gone too far and Zeus has a special punishment befitting Sisyphus’ persistence.

The Curse of Sisyphus is the tale unlike others you may have heard about him before. Here you can find out exactly how Sisyphus defied Zeus yet again – and learn about the physics of motion, gravitation and orbit at the same time.

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Posted by on June 13, 2013 in Uncategorized


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The Curse of Sisyphus -Coming Soon to iTunes

ImageThe Curse of Sisyphus, from DownHouse Software is coming soon to iTunes.

Sisyphus’ is a tale of cleverness and cunning in which the malevolent King Sisyphus offends  and then repeatedly infuriates Zeus  until the King of the Gods is forced to personally curse Sisyphus to a punishment befitting his crimes: To slave beneath a stone, pushing it each day to the peak of a great mountain in the underworld and then have it come crashing down upon him, leaving him to repeat the task again … and again … and again, throughout eternity.

But Zeus has tried to hold Sisyphus captive before only to find that the clever human is not so easily trapped.

In this volume, Sisyphus taps into the vengeance of scorned brother, the wisdom of an oracle and the might of a demigod as he masters the rules governing gravitation and motion to escape his punishment.

Look for it in the iTunes book store this May.


Also, take a look at In Parts: A Tale of Fractional Zombies, free in the iTunes Bookstore now until Saturday!

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Posted by on April 23, 2013 in Uncategorized


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