## Back in Session

I’m trying a few new things this year in math.  I will try to summarize how each week goes throughout the year and highlight successes and failures.

This week I really tried to introduce the honors freshmen to “real” math.  That is, some basic proofs, how generalize things in math and expose them to some difficult questions that are easy to approach, but difficult or surprising in their answers. But all of it was done in a way that is accessible to the students and with high levels of participation.

I really like when a student shares a thought and I repeat their thinking outloud and ask if the others understand, not necessarily agree, but understand.  This seems to really get them thinking and communicating.

Some of the questions we explored were, Is 0.999… less than one or equal to one, why/how we know the square root of two is irrational (we actually did the proof in class, carefully), is zero odd or even, why can’t we divide by zero?

There is also a challenge question posted, in two parts.  Part 1:  Given that a and x are natural numbers, and a is less than or equal to x, and is greater than 1, could the following number be prime:  (x! + a)?

Part 2:  If = 99, how many values of  would be composite.

The purpose of all of it was to challenge their thinking, hopefully incite some curiosity and promote deeper understanding.

One thing I really wanted them to understand is that rational numbers could be expressed as a ratio of integers. The old way I would’ve quizzed them on this knowledge would be to say, Write the following numbers as fractions.  The better question is, Express the following as a ratio of integers, as that addresses the definitions of rational and irrational numbers.

The number zero was a little difficult for some, but since we discussed that division is best thought of as a question, the denominator times what is the numerator?, that went well for most.  For example, instead of reading 8/2 as “eight divided by two,” it is better to ask, “two times what is eight?”

This is most effective when showing why you cannot divide by zero.

I was pleasantly surprised by the show of knowledge and understanding when I asked them, on their quiz, to express the square root of two as a ratio of integers.  Many said that the square root of two could not be expressed as a ratio of integers, that’s why it is irrational.

The depth of that understanding is of little consequence really, but it is a big victory that they have taken something that they always memorized in the past, and now truly grasp.

How well this translates to them owning their learning will remain to be seen, but I think we’re off to a good start.

## Textbooks Should Be Resources, Not Curricula

Textbooks are Resources
NOT
Curricula

Teach like a good doctor practices medicine.  Be prepared for a wide variety of issues, but always seek for the root of the problem so that your remedies may be most effective.

One of the biggest hurdles in education, especially in High School, is getting those that wish to help to understand the nature of the problems faced in education today.  The education industry has been down the path that has arrived at this stop for over a century.  Those guiding the bus have landed us in the wrong part of town!

And those guiding the bus wish to help get us out of this part of town, but they seem to misunderstand what got us here in the first place.  It is the direction of their help that has landed us here.  The majority of support and research and resource we find within education is just more of the same old stuff, because it has gotten off of the same bus.  It’s not much good.

Consider the new wave of high school mathematics text books being published.  There is a series called Big Ideas, which is highly ironic.  The title of the book suggests it teaches the big ideas, the concepts.  The authors claim that is the aim of the book.  Each lesson and topic has a BIG IDEAS bullet and icon and yet, every single one of them is a procedure.

An example of this could involve graphing linear equations.  Students can learn all about graphing linear equations, can even be highly proficient with this, without ever knowing what a graph is.

The Big Ideas book, which does not stand out as particularly horrible compared to its competition, will say something like:

Big Idea:  To graph y = mx + b, plot the b term on the y-axis and then count the slope, m.

That’s not even an idea, much less a big one.  That’s a protocol.  If the problem looks like this, then do that.  Ever have students see a problem like (3x + 5) – (2x – 5) and what they do is distribute, use FOIL?  That’s because they’re accustomed to such protocol.  The problem looks like (3x + 5)(2x – 5) and since they just learned how to multiply, that’s what should be done here.

The best American textbook we have found so far is by Pearson.  It contains a few decent thinking strategies and some good “bell work” type activities.  But it contains far more things like this:

Write addition, subtraction, multiplication or division, to describe the operation displayed below.

1. 2 + 5 = 7       3 · 6 = 18 …

I kid you not!  That is a high school level math book, supposedly.

I teach a non-American curriculum designed and tested through Cambridge University.  Students that come through that program witness all of the things educators say they want to see happening with students.  They are confident problem solvers who can think on their feet, make use of information, they latch onto more efficient methods than what they previously understood, and they have high levels of retention of materials (recall).

The books Cambridge University approves to be used for their curriculum are entirely different in nature than our books here in the US.  They’re thin, don’t have silly eye-catching icons and are intended to be used as a resource.  The books are not designed to go “cover to cover,” but instead can be used in any order desired.  Math is connected in many different ways.  Depending on how it’s unfolded one topic we consider “higher” can be taught first and then our supporting, foundational topics can be introduced later.

The point is, the curriculum is not the book.  The books for Cambridge are written to support the curriculum.  The book is a resource.  Like resources, sometimes the more and the greater the variety, the better.  That’s why some formats like YouTube are so powerful for students.  There is a variety of techniques, styles, approaches and flavors all addressing the same thing on YouTube.  Students can navigate their way through, seeking understanding, which of course leads to more questions, which when chased leads to more understanding, and the cycle starts over again.

Textbooks, no matter how fantastic, do not do this.  I argue that no entity, be it a company, group or individual can offer such a service.  (Yes, YouTube and others host means of expression and information sharing, but their intent is not to address one topic or genre.)

But textbook companies have marketed themselves as providing such services.  There is a greater need for such a thing in Elementary Schools where teachers teach all subjects and topics.  Having a resource that a teacher can use, cover to cover, that structures their day, weeks, and year, balancing all of the various tasks they must cover, is a huge boon, especially for inexperienced teachers.

In High School, the game is entirely different.  Each subject is taught by an expert in that field (in an ideal world anyway).  Such a tool is not only not needed, it is impossible to use.

If students are learning a topic in English, say some writing component, and they lack some foundational skill, a textbook will not identify and then address that.  They try with remedial support materials, but they’re of the nature in the picture I shared earlier.  And remediation of such a fashion, in isolation, not embedded in new content, is completely ineffective.

The majority of what happens in a good high school lesson is unscripted.  Much of it will be anticipated and the teacher will be prepared for these things, but until the specific issues themselves crop up, the remedy is unknown.

It’s not unlike what a family doctor experiences.  The doctor can anticipate what is going to happen in an interview with a patient, but until they see it unfold, they don’t really know what the best remedy is.  But whatever the issue they either have a remedy or know where to turn.  Unfortunately, doctors don’t always do a good job identifying problems, and like bad teaching, they have remedies in mind before understanding the issue at hand.  The remedies, of course, fail, causing frustration while allowing those problems to fester.

Teaching high school math is similar to this.  A good teacher will have some beliefs about why students are confused, or what they don’t know and do know, but they explore a little bit to verify their beliefs.

It is my firm belief, though this statement is pirated for Sir Arthur Conan Doyle, that a teacher that can be replaced by a computer should be.

The educational industry has been moving to compartmentalize and create modular components of education itself.  That way, if one part of the system fails, that modular can be pulled out, and a replacement piece can be inserted.

But education is organic and to teach is to perform an act of charity.  A textbook, no matter how well written, can do perform organically and cannot itself be charitable.  A book, however, can be used as a resource, especially if the intentions of the authors is to create a resource.

The education industry does not want books to be a resource.  Many good teachers have left, positions are filled with unqualified or inexperienced people.  Those that are experienced and qualified are overwhelmed with the call to help support others.

If only we could fix all of that with a book that anybody could open and use, then our students would have a fighting chance.

## Why Does the Order of Operations Work?

Why does the order of operations help us arrive at the correct calculation?  How does it work, why is it PEMDAS?  Why not addition first, then multiplication then groups, or something else?

I took it upon myself to get to the bottom of this question because I realized that I am so familiar with manipulating Algebra and mathematical expressions that I know how to navigate the pitfalls.  That instills a sense of conceptual knowledge, but that was a false sense.  I did not understand why we followed PEMDAS, until now.

By fully understanding why we follow the order of operations we can gain insight into the way math is written and have better abilities to understand and explain ourselves to others.  So, if you’re a teacher, this is powerful because you’ll have a foundation that can be shared with others without them having to trip in all of the holes.  If you’re a student, this is a great piece of information because it will empower you to see mathematics more clearly.

Let’s start with the basics, addition and subtraction.  First off, subtraction is addition of negative integers.  We are taught “take-away,” but that’s not the whole story.  Addition and subtraction are the same operation.  We do them from left to right as a matter of convention, because we read from left to right.

But what is addition?  In order to unpack why the order of operations works we must understand this most basic question.  Well, addition, is repeated counting, nothing more.  Suppose you have 3 vials of zombie vaccine and someone donates another 6 to your cause.  Instead of laying them out and counting from the beginning, we can combine six and three to get the count of nine.

And what is 9?  Nine is | | | | | | | | |.

What about multiplication?  That’s just skip counting.  For example, say you now have four baskets, each with 7 vials of this zombie vaccine.  Four groups of seven is twenty-eight.  We could lay them all out and count them, we could lay them all out and add them, or we could multiply.

Now suppose someone gave you another 6 vials.  To calculate how many vials you had, you could not add the six to one of the baskets, and then multiply because not all of the baskets would have the same amount.  When we are multiplying we are skip counting by the same amount, however many times is appropriate.

For example:

4 baskets of 7 vials each

4 × 7 = 28

or

7 + 7 + 7 + 7 = 28

or

[ | | | | | | | ]   [ | | | | | | | ]   [ | | | | | | | ]   [ | | | | | | | ]

Consider the 4 × 7 method of calculation.  We are repeatedly counting by 7.  If we considered the scenario where we received an additional six vials, it would look like this:

4 × 7 + 6

If you add first, you end up with 4 baskets of 13 vials each, which is not the case.  We have four baskets of 7 vials each, plus six more vials.

Addition compacts the counting.  Multiplication compacts the addition of same sized groups of things.  If you add before you multiply, you are changing the size of the groups, or the number of the groups, when in fact, addition really only changes single pieces that belong in those groups, but not the groups themselves.

Division is multiplication by the reciprocal.  In one respect it can be thought of as skip subtracting, going backwards, but that doesn’t change how it fits with respect to the order of operations.  It is the same level of compacting counting as multiplication.

Exponents are repeated multiplication, of the same thing!

Consider:

3 + 6

4 × 7 = 7 + 7 + 7 + 7

74 = 7 × 7 × 7 × 7

This is one layer of further complexity.  Look at 7 × 7.  That is seven trucks each with seven boxes.  The next × 7 is like seven baskets per box.  The last × 7 is seven vials in each basket.

The 4 × 7 is just four baskets of seven vials.

The 3 + 9 is like having 3 vials and someone giving you six more.

So, let’s look at:

9 + 4 × 74

Remember that the 74 is seven trucks of seven boxes of seven baskets, each with seven vials!  Multiplication is skip counting and we are skip counting repeatedly by 7, four times.

Now, the 4 × 74 means that we have four groups of seven trucks, each with seven boxes, each with seven baskets containing seven vials of zombie vaccine each.

The 9 in the front means we have nine more vials of zombie vaccine.  To add the nine to the four first, would increase the number of trucks of vaccine we have!

Now parentheses don’t have a mathematical reason to go first, not any more than why we do math from left to right.  It’s convention.  We all agree that we group things with the highest priority with parentheses, so we do them first.

(9 + 4) × 74

The above expression means we have 9 and then four more groups of seven trucks with seven boxes with …  and so on.

Exponents are compacted multiplication, but the multiplication is of the same number.  The multiplication is compacting the addition.  The addition is compacting the counting.  Each layer kind of nests or layers groups of like sized things, allowing us to skip over repeated calculations that are the same.

We write mathematics with these operations because it is clean and clear.  If we tried to write out 35, we would have a page-long monstrosity.  We can perform this calculation readily, but often write the expression instead of the calculation because it is cleaner.

The order of operations respects level of nesting, or compaction, of exponents, multiplication and addition, as they related to counting individual things.  The higher the order of arithmetic we go, up to exponents, the higher the level of compaction.

Exponents are repeated multiplication, and multiplication is repeated addition, and addition is repeated, or skip counting.  We group things together with all of these operations, but how that grouping is done must be done in order when perform the calculations.

## The Toaster Problem in Education

It’s easy to talk about shifting education towards a more concept based approach.  But it’s hard to see what that really means in practice.  I’m not a betting man, but would be willing to bet that upon inspection there are many things you think you understand conceptually in your topic, but you just feel that way because you understand procedure well enough to always arrive at correct answers.

I can offer an example in math:  Why does the order of operations work?  Why does the structure in the order of operations guide us to the correct calculation?

Let’s use a mathematical way of thinking to approach this problem of understanding what conceptual approach education looks like, compared to our current procedure based approach.  (Tell-tale sign that you’re procedurally based is if your students cannot remember how to do something big a year later.  Or, do you consider the work assigned before your lesson?)

Imagine you want your students to know how to make toast.  You could introduce them to a toaster.  Then, demonstrate how the bread-item is dropped in the slots on the top, the little knobby is turned to select the desired level of darkness, the button with the picture of the type of bready-material being toasted is pushed, and the lever is depressed.

If it’s an advanced class, maybe some discussion is given to what to do if the toast gets stuck, and why you should always unplug the toaster when finished with it because toasters have notoriously cheap circuits that short out, causing a fire.

Oh, one last thing.  All toasters are good toasters.  There are no bad toasters.  Some make light toast, some dark toast.  If you show preference to one kind of toaster, you’re then the exception to the rule because we tolerate everything except intolerance.

That’s a very typical American style of teaching something.  We cover how to use a tool and throw in a little social justice message to boot.  (That is not a comment on the need for awareness of social issues except to say that math textbooks are inappropriate platforms for them.)

Imagine that instead of wanting your students to know how to make toast, you wanted them to know about toast.   You teach them what it is, previously cooked bread that is now slightly, but evenly, burned on the cut-faces making a slightly stiffer, crunchier piece of substrate for the delicious spreadable material of your choosing.

For the sake of this thought experiment, let’s say you also show them a toaster, but that’s it.

Now consider a pair of students.  One who learned the first method, and the second learned about toast, but spent little time with a toaster.

Which student could make toast if the toaster broke?  Which understands what a toaster really does?

Teaching how to use a toaster is procedural, while teaching what toast is would be conceptual.

Education is a HUGE industry with an enormous amount of inertia to overcome before change is realized.  There are jobs at stake if responses to changes go wrong.  Companies invest millions to supply the desires of schools.  And what do schools want?  They want to be like everybody else, because it’s safe!

We have these methods, that if not effective, are at least safe because we have used them for a long time, so has everybody else.  So if we’re close to the average, we’re okay.

But don’t get me wrong, things in education will change.  Pretty soon curriculum will be all conceptual.  Kids will be reinventing the wheel at every turn.  We see some of that in the elementary levels right now.  That’s truly a shame because it’s harmful.  Young kids do not yet possess the faculties for abstraction!  They need to know how to use a 3rd grade toaster, if you will.

I am NOT a doomsday preacher here, but I do not believe education cannot fix itself.  It is so established in the way it operates that the path we are on will remain until something really big from way up high changes.  The likelihood of that being a good change is slim because politicians aren’t educators.  Even if the idea is good, from above, the execution will be poor because it’s ideas, not how they play out, that gets people elected.

But, the change from teaching how to use a toaster to teaching what toast is, well, is needed.  Even for students to pass the new style of standardized testing they need to know what toast is.

Beyond that, for them to be successful in college, the nature of toast must be understood.  To change math from a hurdle to an opportunity, they’ve got to know all about toast, not just how to use the toaster.

It is these last two things, the belief that the education system cannot right itself, and the need for conceptual understand, that has motivated me to step outside of education for my project.

## Why Good Lessons Fail

Ever had a lesson you were THRILLED about?  You loved it, it was fantastic, interesting, crisp, approachable and ... wonderful in every possible fashion.  And yet, when you delivered that lesson, it flopped!

What gives?  What was wrong with the lesson?

In reality, there was probably nothing wrong with the lesson.  Sure, all can be improved, but the lesson wasn't the problem, the delivery was.  It seems there exists an inverse relationship between how much I love a lesson and how well received it is.  The more I love it, the more students hate it!

What it really boils down to is engagement.  We are so sure that what we have to say will blow minds, that we forget our number one task ... making sure we are teaching students, not just covering material.  We assume that because we find it interesting and fascinating, and because we had such a grand time putting the lesson together that they'll gravitate towards it.

But gravitate towards it in favor of what?  What captures the attention of students?  Drama at lunch, fights with family members, changes in weather, they might be tired from staying up and watching the new season of Stranger Things on Netflix ... we don't know.  But whatever has their attention, we must wrestle it away.

In a normal lesson we are usually vigilant and on top of distractions and such.  We work hard to make the lesson itself interesting.  But in a lesson that needs no such adornments, we fail to sell it.

So regardless of whether you think it's great, they need to be sold on the fact!

There are a couple things that you can do, at any point in time, if they're not engaged.  These work for average and poor lessons, not just the great ones that we hope will inspire a future generation of (whatever it is you teach).

Before I share with you three ways to quickly grab their attention, let me say that once you have it, you can just jump right back into the lesson.  You'll have their attention, they'll not even notice that suddenly they're learning stuff!

My favorite, go-to, method of grabbing attention is with a quick, cheesy, usually Dad-Joke.  I sometimes look up a bunch of them, print them off and have them at the ready.  There are a few that I have on the ready at any given moment, but since I don't often tell them outside of the classroom, I forget.

Make it short and dumb, they'll be captured, even if they complain.  Then, back to the lesson.

And with all of these, you just jump right into the attention getting performance, you can do it mid-sentence if you please.

The second method is with a quick story about something interesting.  It can be that you wanted some cereal for breakfast and there was only a splash of milk left in the fridge!  So you couldn't even have dry cereal, just slightly less than soggy junk -- How FRUSTRATING!?!?!  Get some feedback and jump right back in.

The last method I use is direct.  I simply tell them they're distracted and that they need to do their best to focus.  I'll sell why (perhaps the material is dry but will be very important and interesting in context later, or some other reason).  I'll share that I feel the same way, burned out and tired, but explain that we all have a job to do.  "Let's just get through these next few parts and we're done for the day, if we do them well.  If not, we'll have to revisit this again in the near future."

Whatever methods you use, mix it up.  If you become too predictable with these they'll fail to gather attention.  So, "Stay frosty," like the line in Aliens suggests.

Anyhow, I hope these are helpful tips.  Just remember, no how great your lesson is, engagement is still the most important part of the lesson.  Without it, they'll not learn anything!

## What is Algebra?

This past month has been very busy here for The Bearded Math Man.  I’ve learned a lot about things I have merely taken for granted and have shared most of them with you here on my site.  And while I have a goal and a mission, the methods of achieving that goal are still forming.  I’m learning what works best and what doesn’t work.  One such thing I’ve discovered is the purpose of this blog.

This blog is meant for two audiences.  Those interested in math and those teaching math.  Now that I have that defined, I’ll keep a more focused range of topics.  I just thought that was worth mentioning.

Now, for today’s topic, Algebra.  I do not intend to teach you Algebra, but would like to share something I did not know about the subject.  Algebra means to make complete, or to resolve.  I knew it was named after a Persian mathematician in the early 9th century, but that the branch of mathematics goes farther back in time than the name itself, even the Babylonians used Algebriac concepts.  But I thought the name was just that, a name.

It is stunningly powerful to recognize what Algebra means.  Everything operation we perform in Algebra is to meet this end, to complete or resolve an equation!  That’s what we do when we’re solving an Algebraic Equation.

One other thing you may not have known about Algebra is the equal sign.  The symbol itself never appeared until the 16th century and it traveled the entire width of the page.  It is hard to imagine how this would be a more efficient way of describing the equality present between two things, but it was.  Over time it was shortened to what we have today.  This is more than just an interesting factoid, too.  It goes to show that sometimes great ideas are so revolutionary that they seem obvious in hindsight.  First, we have a symbol that means equals, then we have, over time, an easier way of writing that symbol.

In many ways, isn’t that what makes mathematics so difficult, the jargon and abstraction?  That’s why one of my main points of focus is instilling mathematical literacy in students.  If they can read the math for what it says, not just as a funky collection of shapes and symbols, the mathematical ideas present themselves in a sensible and approachable fashion.

That’s what I’ve tried to do with my introduction to Algebra as a branch of mathematics, which is taught in Algebra 1, the class.  Here’s the link to the page.

As always, I thank you for reading and hope I’ve stirred some curiosity in you.

PS:  If you are interested in some of the history behind Algebra, the following book is highly recommended.  If you purchase it through this link you will help support the mission here of changing math from a hurdle in the way of young peoples’ dreams to a platform upon which success is built, and at no additional cost to you.

## Is Infinity Real?

### How Many Primes are There Is Infinity Real Part 1

Teachers: The following is a discussion that can be had with students to create interest in mathematics by discussing two very easy to understand, but perplexing problems in mathematics.  First, the nature of infinity.  The second is the lack of pattern and order in the prime numbers.

The number of primes is infinite.  Euclid proved it in a beautiful, easily understood proof by contradiction.  Paraphrasing, he said that there are either infinitely many primes, or a finite number of primes.  So let’s pick one and explore it.  Say there are a finite number of prime numbers.  If you were to list them all, then take their product you would have a very large number.  But if you just add one to that number, it would be prime because none of the other prime numbers would be a factor of it.  It would have exactly two factors, one and itself.

In case you don’t believe this works, let’s say we can list all of the primes, but there are only four.  Let’s say the entire list of primes was 2, 3, 5, and 7.  Their product, 2 × 3 × 5 × 7 = 210.  This number is composite because all of the primes are factors of it.  Add one to it, arriving and 211 and none of the prime numbers are a factor of it…making it have the factors of 211 and 1.  That means it is prime.

So it is false that there are a finite number of primes. Therefore, the are infinitely many prime numbers.

Beautiful, right?  Case closed. … or is it?

The case is closed, if you believe infinity exists.  To be clear, infinity is not a number, it’s a concept.  A set can only approach infinity, nothing ever equals infinity because it’s an idea.  The idea behind infinity is that the collection of things just keeps growing and growing.

We, as humans, have a very big problem with very big numbers, even large groups of things.  For example, there are some things that we only have a plural word for, we do not possess a singular word for these things.  A few examples are rice, sand, hair, shrimp and fish.  You can have a single hair, a grain of sand (or rice), and so on.  They are so vast in quantity they become indistinguishable.

And yet, they’re finite. You could conceivably collect all of the sand in the world and count every grain.  More sand does not magically appear once it is all collected.

What about stars in the sky?  What we call the observable universe is how far we can see.  We don’t know if it goes on forever, or if it is somehow contained.  Perhaps the word, universe, is misleading.  Perhaps there are multiples of it, maybe as many as there are grains of sand on the earth.

Before we chase that rabbit down its hole, let’s get back to earth.  Euclid’s proof that there are infinitely many prime numbers is beautiful.  But is he right?  Surely his proof is flawless, but what about infinity.  We have no examples of infinity, it might just be a human construction.  Now, if mathematics can discover things that are real and applicable from such a thing, that’s all the more powerful the tool it is, but what if we’re wrong about infinity?  There are two things I want you to consider as we explore prime numbers and their relationship with infinity.

The first thing is:  There’s an axiom (a statement we just accept as truth), called the Axiom of Infinity.  It basically says that there are infinite sets of things, like natural numbers.  We just say it’s true and roll with it until we discover a problem.  Then, we either adjust our axiom or start a new one.

The second thing is:  In the early 20th century a man named Kurt Gödel showed that we cannot actually prove any system of mathematics is true without assuming some supporting evidence is true.  We have to assume something is true in order to know if other things are true, roughly speaking.  In order to know if the thing we assumed to be true is actually true or not (like infinity), we have to assume that something else, more basic, is true.  So, and I’m taking some liberties here to make my point, but a conclusion, like the number of primes being infinite, is only as worthy as the presupposition (infinities exist).

Let’s look at a few strings of prime numbers and see if we can’t get our heads around this whole infinity thing.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

The gaps between these prime numbers are below.

1, 2, 2, 4, 2, 4, 2, 4, 6

Another string would be:

907, 911, 919, 929, 937, 941, 947

The gaps here are listed below.

4, 8, 10, 8, 4, 6

They are still relatively close.  Many mathematicians have tried to find a pattern in prime numbers.  After all, if you can find a pattern, then you can find the next one.  How cool would that be, right?

You might be thinking, uh, why would that be cool?

Well, there’s big money being paid if you can find the next prime number.  There is a project called GIMPS (Great Internet Mersenne Prime Search), where you can participate in the search.  And if your computer finds the next prime, you get some cash!

The last prime found with GIMPS was in 2013.  (At the time of this being written, it is 2017.) The number is massive.  The text file of the digits in the number is 7.7 MB.  That’s more data that a song and this is just a list of numbers.  The number is 257,885,161 – 1.  The number is huge that to verify that it is prime takes massive super computers days upon days to perform the calculation.  Finding the next prime number is a huge undertaking, very complicated and difficult, requiring computers all over the world working together before one is discovered.

Why all the fuss? What good are they?

Well, they keep you from being robbed, for one.  Internet security uses prime numbers to encrypt (code) your banking information.  The merchant will have a huge number that they multiply your card number by (kind of).  The huge number is the product of two of these gigantic prime numbers.  It’s so big that even though everybody (would be thieves) know it’s the product of two primes, they can’t figure out which two numbers.  The encrypted number is sent to your financial institution, who knows which two primes were used, which is basically like a key.

It’s also weird, and cool, that some bugs have a life cycle that only occurs in prime numbers!  Cicadas only come out and breed, and then die, in prime number years.  Incredible.

Back on track, forgive me.  It feels there are infinitely many tangents I can follow with math!  We have not been able to find a pattern in the prime numbers yet and let’s take a look at why.  You see, as these primes get huge, the gaps get larger and larger…approaching infinity!

Let’s take a look at one more string of prime numbers.

10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099

The differences here are as follows.

28, 2, 22, 6, 10, 12, 2, 6

No discernable pattern, right?  If you can find one, you stand to make significant history, no one has found one yet.  We have some approximations that work within certain constraints but they all break down eventually.

But, to be clear, if you could find a pattern in the gaps between the primes a formula could be created that would generate prime numbers.  We can generate natural numbers by just adding one to the largest we have come up with so far.  But primes, as you’ve seen with the GIMPS project, aren’t so easily discovered.

And here’s one of the issues.  The gaps between prime numbers can get huge, perhaps infinitely huge.  Consider this.

Fact 1:  5! = 5×4×3×2 = 120

Fact 2:  120 is not prime because it is divisible by 5 and 4 and 3 and 2.

Fact 3:  5! + 5 is not prime because it is divisible by 5.  (When we add another 5, it’s like skip counting when you first learned multiplication.)

The same is true for 5! + 4 being divisible by 4, because 120/4 = 30.  5! + 4 is 4 × 31, there’s one more four.

The same holds true for 5! + 3 being divisible by 3 and 5! + 2 being divisible by 2.

Fact 4:  What all this means is that there after 5! + 1 there are four consecutive numbers that are composite.

This would also work for 100!  The number 100! + 100 would be composite.  For that matter, 100! + 37 would be composite also.  100! Plus all of the numbers up to and including 100 would be composite, (except possibly adding 1).

This means there is a gap of 99 after 100! + 1.

This goes on forever, arbitrarily large numbers, like 1,000,000,000,000!  There would be a gap of 1,000,000,000,000 – 1 numbers after this number that are composite.

We could write this in a general sense.  Let a and x be a whole numbers such that a is less than or equal to x.  (a x).

Then x! + a is composite.

Since x is a whole number and whole numbers are infinite, then there are infinitely large gaps between the large prime numbers, themselves being infinite.

Crazy, right?

So if the gaps between primes gets infinitely large, how can there be infinitely many prime numbers?

Well, there’s one more piece of information to be considered.  Twin primes are prime numbers that are just two numbers apart.  The primes 2 and 3 are only one apart, but all others are an even number apart, the smallest gap being a gap of two, like 5 and 7, or 11 and 13.

There’s a conjecture (not as strong as an axiom), that is yet unproven, but we’re getting closer, that states that there are an infinite number of twin primes.  The largest known pair of twin primes is below:

3,756,801,695,685 × 2666,689 – 1
and

3,756,801,695,685 × 2666,689 +1

Those numbers are too large to be written out!

While we do not yet know, with a proof, that there are infinitely many twin primes, we do know that there are infinitely many primes that have a maximum distance between them and it might be as low as a difference of sixteen.  This is all being discovered and explored and fought over at the moment.

So on one hand we have infinitely large gaps between prime numbers, but when they do pop up, they will do so in clumps and groups?

If all of this makes your head spin, then I have succeeded.  I am not trying to convince you that infinities do not exist, or that they do.  I am trying to show that math is contentious and changing.  As we learn and discover new things math is changing.  Math is just a language we use to describe the world around us.  So powerful is math that we are not even sure if it is a human invention at all or rather a discovery!

As always, thank you for your time. I hope this has stirred some thought, maybe even sparked a passion for mathematics!

At the time of the making of this video the world’s largest prime number is not the last one found by the GIMPS project.  However, they’re likely to find another even larger one, sometime soon.  There’s a video below (Largest prime number) that discusses that number and prints it out … it takes up as much paper as three large books!

For some fascinating and approachable treatment of prime numbers, consider the following videos:

Gaps between prime numbers: https://youtu.be/vkMXdShDdtY

If you found this helpful and would like to help make these videos possible, to help break down the obstacle that math presents itself as to young people, please consider visiting my patreon site:

www.patreon.com/beardedmathman

## The Problem with PEMDAS

The problem with PEMDAS

This problem has really stirred a lot of interest and created a buzz on the internet. I can see why, it’s an easy one to miss.  And yet, PEMDAS is such an easy thing to remember, the mnemonic devices offered make for a strong memory.  So people passionately defend their answers.

6 ÷ 2(2 + 1)

I am going to tell you the answer in just a moment, but before I do, please listen to why I think this is a worthy problem to explore.

There are two fundamental misconceptions with math that make math into a monster for so many people, and this problem touches on both.  In a sense, neither has anything to do with the order of operations specifically.

The first issue is understanding that spatial arrangements in math mean something.  The way we write the numbers and symbols has a meaning, very specific at that.  In this video by Mind Your Decisions, https://youtu.be/URcUvFIUIhQ, he shares where there was a moment in time when we used different conventions to write math.

And while math may or may not be a human invention, the symbols and arrangements and their meanings certainly are.  Just like the letter A is only a letter and with a specific sound because we all agree.  Just like a red light means stop, a green light means go and a yellow light means HURRY HURRY HURRY!

The second, and more over-arching issue here, is the misconception that addition and subtraction are different.  They are fundamentally the same thing.  Subtraction is really addition of opposite numbers.  Perhaps to shore this misconception negative numbers should be introduced instead of subtraction.

Now you might argue and say, Wait, addition has properties that subtraction lacks, like the commutative property.

You’re correct, 5 + 3 = 3 + 5, while 5 – 3 does not equal 3 – 5.  However, 5 – 3 is really five plus the opposite of three, like written below.

5 + - 3

And that is the same as this expression below.

-3 + 5

So the AS at the end of PEMDAS is really just A, or S, whichever leads to the better nursey rhyme type device to improve recall.

Since we believe that addition and subtraction are different, we also come away with the belief that multiplication and division are different.  Sorry, they’re not.  Division is multiplication of the reciprocal.  Remember that whole phrase from your school days? (How was that for a mnemonic device?)

And while division does not have the commutative property, that again is a consequence of the way we write math.  If we only wrote division as multiplication of the reciprocal, we would see that multiplication and division are in fact the same.

So, back to the problem.  The most common wrong answer is 1.  The correct answer is 9.  Here’s a great video on the order of operations, super catchy and articulates the importance of left to right as written for multiplication and addition.

Last thing:  Now, in creative writing the intent of the author must be considered, should it also be considered here?

Let me know what about this you like, dislike or disagree with.  Let me know what is helpful.  I really want to promote success through making math transparent.  It’s my mission.  You can help support my mission by just sharing and liking this.  Subscribe to my blog if you’re a teacher as I will be populating it with lots of teacher advice, not all math related.

## The Square Root Club

If you’re a teacher, I have a short story that you can share, adapted to fit your own style, that you can use to address the biggest issue with teaching … students learn what they want to learn.  Creating interest in mathematics for teenagers can sometimes be a challenge.  One of the easiest ways to do so is with humor.  The following story is actually true, but humorous, and I think will create some curiosity and thus learning opportunities for students.

I believe the appropriate audience would be pre-algebra students learning about square roots up to algebra students learning about square roots.  Anyhow, if you find this helpful, please let me know.

# The Square Root Club

My daughter, a senior at the University of Arizona, called and said she’d uncovered an issue in math that is both absolutely impossible and yet, true.  My interest piqued, I listened attentively as she asked if I’d ever heard of the square root club.

The square root club, I was informed, is a club of dubious membership.  To become a member the square root of your GPA must exceed your GPA.  What a delightful treat this was…and to think, I’d never heard of such a thing!

She continues to tell me that she met someone who was a member.  I asked her how she knew, because certainly her friend would be ignorant of his membership.  Surely, someone in the club would not be smart enough to be aware of the fact, right?

That’s what she said was the funniest part, the part that was seemingly impossible!  He knew about it, even made up the name of the club himself.  He was no longer a member, just graduated with his bachelor’s degree with a 3.0 GPA.

Note:  GPA (grade point average) is calculated by assigning a numerical value to letter grades.  An A is 4, B is 3, C is 2, D is 1 and an F is zero.

The moral of the story is that grades don’t reflect potential, they reflect what you show you know.  Many high school students get by with intelligence but never work.  Upon arriving in college they are overwhelmed, never having had to work hard or apply themselves.  Before they know it, they’re buried and there’s no quick fix like there can be in high school.

To that point, nobody cares about someone’s potential, not even your mother. Imagine your mom told you to clean your room.  Because she told you to do it, she believes you have the potential.  However, if you do not clean it, she will be satisfied by the fact that you could have cleaned it.

Now of course, the question being begged here is, what could his GPA have been?

## How Math Fixed Music

### Rational Exponents Sound GREAT

Before we dive in, music is primarily defined by what we hear, not by the analysis and insight provided by math.  For example, an octave is a note whose frequency is double that of its parent note.  The mathematical relationship was discovered after the fact.  The following is an exploration of how math is used in music, but I don’t want to put the cart before the horse here.  The math supports the music, makes it work.  But the math is really fine-tuning what we hear.

Pythagoras developed a musical system that over the years evolved into what we have today. (At least Pythagoras is often credited for it.)  Not until “recently,” however, has one of the major problems with music been resolved (see what I did there with resolve?).

The problem the ancients had is that their octaves didn’t line up.  An octave, as I mentioned early, is a note that has twice (or half) the frequency of another note.  Octaves, in modern western music, share the same names, too.  The note A, at 440 Hz, has an octave at 880 Hz, and also 220 Hz.   (There are infinitely many octaves, in theory, though our ears have a limited range of things we can hear.)  The ancients, however, had a problem because after a few octaves, well, they were no longer octaves.

In western music we have 12 semi-tones, A, A# (or B-flat), B, C, C#, D, D#, E, F, F#, G, G# and then A again.  It’s cyclic, repeated infinitely both higher and lower. Each semi-tone in the next series of 12 notes is an octave of our first series of notes.  And the relationship between notes is what makes them, well, musical, not just sounds.

The problem is defining that relationship.  You see, because each note is slightly higher (has a higher frequency), and each note’s octave is double that frequency, what happens is the notes get further and further apart (the differences in their frequencies increases).

Let’s take a look at the frequencies:

 Note Frequency A 220.00 A# 233.08 B 246.94 C 261.63 C# 277.18 D 293.66 D# 311.13 E 329.63 F 349.23 F# 369.99 G 392.00 G# 415.30 A 440.00 A# 466.16 B 493.88 C 523.25 C# 554.37 D 587.33 D# 622.25 E 659.25 F 698.46 F# 739.99 G 783.99 G# 830.61 A 880.00

As you can see, the differences between consecutive notes is increasing, at an increasing rate!  This is not a linear relationship.  Because of this, the ancients had a very hard time defining what was an A and what was a D, especially when you started moving around between octaves.  Things got jumbled, and out of tune.

It is tricky to find the proportion and rate of change between consecutive notes, any two consecutive notes that is.  That’s where math comes in to save the day.  Let’s build the rate of change, shall we.

First, note that the rate is increasing, at an increasing rate, so we cannot add.  I show that in the video below.  We have to multiply.  When we repeatedly multiply, we can use exponents.  Since we need a note and it’s octave to be doubles, our base number is 2.

Since there are twelve notes between a note and its octave, we need to break the multiple of two into twelve equal, multiplicative parts.  That’s a rational exponent, 1/12.

The number we need to multiply each note by is 21/12.  Each note is one-twelfth of the way to the octave.  It is pretty cool indeed.