## 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?

## The Most Important Component of Quality Teaching is …

What do you think the single most important part of effective teaching, in high school, is?

Breaking down classroom management and teaching into a lock and step routine is impossible.  People are too variable.  And, especially in high school, we are talking about the interactions of 150 – plus people a day!

It is because of the nature of how people behave and interact, how our motivations to fit in and get along guide a lot of our decisions that I claim establishing relationships is the single most important aspect of effective teaching, in high school.

I didn’t always feel this way.  I believed that discipline, structure, and content were king.  They’re certainly first tier, but they’re not king: Relationships are.

For me the light first clicked on when I watched an episode of Undercover Boss.  Here's a clip of the episode.

In this episode the corporate offices wanted to see why one location, that was not geographically or demographically different than the other stores, outsold the other stores.  Was it management, something on the retail side?

It turned out this woman, Dolores, had worked there for 18 years and she knew EVERY single customer by name and knew about them.  People just kept coming back because she knew them, took care of their needs because she knew them, and also, because she knew them, they felt welcome.

Do I Really Need a Relationship with the Students?

In high school students don’t have much choice.  They have to come see you daily.  But that alone will not make them respectful, engaged, and willing participants.  Dolores showed me that if you just get to know people, and are warm and welcoming, they’ll be willing and eager to show up.  This translates nicely to high school.

When you have a relationship with students that are far more compliant out of genuine respect.  They’re willing to participate and enjoy being in your class, even if they don’t like your subject (happens to me a lot with math).

By having relationships with students your day is also a lot nicer.  If you’re down, or off, for whatever reason, instead of taking advantage of you, like sharks smelling blood in the water, they’re on their best behavior – if you have a good relationship with them.

How Do I Build a Relationship with So Many Students?

So how can we build relationships with students when you have 35 per class shuffling in and out every 55 minutes or so?  I mean, there’s teaching, testing, checking homework, discipline, interruptions from the office, … the list goes on and on.  How can we develop relationships with students with all of that going on?

The first way is just small talk.  Not everybody is good at that, but it is easy with kids.  Ask them simple things like if they have pets, and then about their pets or if they wish they could have a pet.  Ask them about the nature of their family, how many siblings they have, where they fit in (birth order).

Another way to build this relationship is to have a “Pet Wall” where students can bring pictures of their pets and place them on that part of the wall.  It generates conversation, which is what’s needed to establish these relationships.

Giving sincere compliments is a great way to build relationships.  But, they must be sincere.  There’s almost nothing more insulting than an insincere compliment, there’s certainly nothing more condescending.  When students see you treating others with kindness and generosity it endears you to them.  They gauge a lot of their relationship with you on how you treat others.

How you handle discipline is very important, too.  If you berate a child in an unprofessional manner, you lose a lot of that hard earned relationship with other students.  They may not like the kid who is always a distraction, however, again, they gauge their relationship with you by how they see you treating others.

The last thing I’ll share here is that you can share things about yourself with them.  It can be funny stories or minor conflicts in your life, nothing that crosses a professional boundary, but things to which they can relate.  A story about how your toast fell and landed jelly side up (or down as the case may be), and so on.

It is incredibly difficult to site one thing as most important because no one factor of teaching stands on its own.  If too much focus is placed on one thing, at the expense of others, an imbalance will lead to poor teaching.

All that said, I believe that establishing relationships is the most important thing you can do as a high school teacher.  It will not only make the students more willing, it will also greatly improve the quality of your day!

## Square Roots Part 1

Note:  Square roots are pretty tricky to teach and learn because the tendency is to seek answer-getting methods.  Patience at the onset, allowing for full development of conceptual understanding is key.  Do not revert to tricks and quick "gets" when first learning square roots.  Always revert back to the question they ask and how you know if you've answered that question.

square roots part 1

Square Roots

Part 1

Introduction: Square roots are consistently among the most misunderstood topics in developmental math. Similar to exponents, students must possess both procedural fluency but also a solid conceptual foundation and the ability to read and understand what square roots mean, in order to be proficient with them. It is often the case that problems with square roots do not lend themselves to a correct first step, but rather, offer many equally viable methods of approach.

Square Roots Ask a Question: What number squared is equal to the radicand? The radicand is the number inside the square root symbol (radical). This expression asks, what number times itself (squared) is 11?

$\sqrt{11}$

This is a number. It is not 11. It turns out this number is irrational and we can never actually write what it is more accurately than this.

Big Idea: The area of a square is calculated by squaring a side (multiplying it by itself). Since all sides of a square are equal, this about as easy of an area to calculate as possible. A square root is giving us the area of a square and asking us to find out how long a side is.

$\overline{)42}$

For example, this square has an area of forty-two. Instead of writing out the question, “How long is the side of a square whose area is forty-two?” we simply write, $\sqrt{42}.$

The majority of the confusion with square roots comes back to this definition of what a square root is. To make it as clear as possible, please consider the following table.

 English Math How long is the side of a square that has an area of 100? $\sqrt{100}$ How long is the side of a square that has an area of 10? $\sqrt{10}$

These two numbers were chosen because students inevitably write $\sqrt{100}=\sqrt{10}.$

 English Math Answer How long is the side of a square that has an area of 100? $\sqrt{100}$ 10 How long is the side of a square that has an area of 10? $\sqrt{10}$ 3.162277660168379…

Key Knowledge: In order to be proficient with square roots we need to know about perfect squares. A perfect square is a number that is the product of a number squared. Sixteen is a perfect because four times four is sixteen.

The reason you need to know perfect squares is because square roots are asking for numbers squared that equal the radicand. So if the radicand is a perfect square, we have an easy ‘get,’ that is, simplification.

For example, since 42 = 16, and the square root of sixteen $\left(\sqrt{16}\right)$ asks for what number squared is 16, the answer is just four.

Let’s take a look at the first twenty perfect squares and what number has been squared to arrive at the perfect square, which we will call the parent.

 Perfect Square “Parent” 1 1 4 2 9 3 16 4 25 5 36 6 49 7 64 8 81 9 100 10 121 11 144 12 169 13 196 14 225 15 256 16 289 17 324 18 361 19 400 20

You should recognize these numbers as perfect squares as that is a key piece of knowledge required!

Pro-Tip: When dealing with square roots it is wise to have a list of perfect squares handy to help you familiarize yourself with them.

How to Simplify a Square Root: To simplify a square root all you do is answer the question it is asking.

The best way to go about that is to see if the radicand is a perfect square. If so, then just answer the question. For example:

Simplify $\sqrt{256}$

Since this is asking, “What number squared is 256?” and 256 is a perfect square, 162, the answer to the question is just 16.

$\sqrt{256}=16$

What if we had something like this:

$\sqrt{{x}^{2}}$

If you’re confused by this, revert back to the question it is asking. This is asking, “What squared is x $–$ squared?” All you have to do is answer it.

$\sqrt{{x}^{2}}=x$

What if the radicand was not a perfect square?

If you end up with an ugly square root, like $\sqrt{48},$ all you have to do is factor the radicand to find the largest perfect square.

List all factors, not just the prime factors. In fact, the prime factors are of little use because prime numbers are not perfect squares. And again, we are looking for perfect squares because they help us answer the question posed by the square root.

48

1, 48

2, 24

3, 16

4, 12

6, 8

Pro-Tip: When factoring, do not skip around. Check divisibility by all of the numbers in order until you get a turn around. For example, after 6, check 7. Seven doesn’t divide into 48, but 8 does. Eight times six is forty eight, but you already have that pair. That’s how you know you’re done!

In our list we need to find the largest perfect square. While four is a perfect square, sixteen is larger. So we need to use three and sixteen like shown below.

$\sqrt{48}=\sqrt{16}\cdot \sqrt{3}$

The square root of three is irrational (square roots of prime numbers are all irrational), but the square root of sixteen is four. So rewriting this we get:

$\begin{array}{l}\sqrt{48}=\sqrt{16}\cdot \sqrt{3}\\ \sqrt{48}=4\cdot \sqrt{3}\end{array}$

See Note 1 and Note 2 below for an explanation of why the above works.

Fact:

That means that Let’s see if it is true.

${\left(4\sqrt{3}\right)}^{2}=4\sqrt{3}×4\sqrt{3}$

Because we can change the order in which we multiply, we can rearrange this and multiply the rational numbers together first and the irrational numbers together first.

$4\sqrt{3}×4\sqrt{3}=4×4×\sqrt{3}×\sqrt{3}$

The square root of three times itself is the square root of nine.

$4×4×\sqrt{3}×\sqrt{3}=16×\sqrt{9}$

The square root of nine asks, what squared is nine. The answer to that is three.

$16×\sqrt{9}=16×3$

So,

Note 1: $4\sqrt{3}$ cannot be simplified because the square root of three is irrational. That means we cannot write it more accurately than this. Also, the product of rational number and an irrational number is irrational. So, $4\sqrt{3}$ is just written as “four root three.”

Note 2: We can separate square roots into the product of two different square roots like this:

$\sqrt{75}=\sqrt{25\cdot 3}$ figure a.

or

$\sqrt{75}=\sqrt{25}\cdot \sqrt{3}$ figure b.

If we consider the question being asked, what number squared is seventy five, we can see why this works. What number squared is seventy five is the same as what number squared is twenty five times three,” (figure a). The number squared that is twenty five times the number squared that is three is the same as the number times itself that is twenty five times three.

For example:

But also: $\sqrt{64}=\sqrt{4}\cdot \sqrt{16}$

And this simplifies to:

$\sqrt{64}=2\cdot 4=8$

What we will see in a future section is that square roots are actually exponents, exponents are repeated multiplication and the order in which you multiply does not matter. This allows us to manipulate square root expressions in such a fashion.

Let us work through two examples. Before we do, let us define what simplify means in the context of square roots. Simplify with square roots means that the radicand does not contain a factor that is a perfect square and that all terms are multiplied together.

Simplify: $9\sqrt{8}$

What is the nine doing with the square root of eight? It is multiplying by it. We cannot carry out that operation. However, eight, the radicand, does contain a perfect square, four. Do not allow the fact that 9 is also a perfect square confuse you. This is just 9, as in 1, 2, 3, 4, 5, 6, 7, 8, 9. The square root of eight cannot be counted. It is asking a question, remember?

$9\sqrt{8}=9\sqrt{4}\cdot \sqrt{2}$

Pro-Tip: When rewriting radical expressions (square roots), write the perfect square first as it is easier to manipulate (you won’t mess up as easily).

$\begin{array}{l}9\sqrt{4}\cdot \sqrt{2}\\ 9\cdot 2\cdot \sqrt{2}\\ 18\sqrt{2}\end{array}$

Example 2:

Simplify $8\cdot \sqrt{\frac{32}{{x}^{4}}}$

The eight is multiplying with the radical expression. Just like we could separate the multiplication of square roots, we can also separate the division, provided it is written as multiplication by the reciprocal. So, let’s consider these separately, to break this down into smaller pieces that are easier to manage.

$8\cdot \sqrt{\frac{32}{{x}^{4}}}=8×\frac{\sqrt{32}}{\sqrt{{x}^{4}}}$

Let’s factor each square root, looking for a perfect square. Note that x2 times x2 is x4.

$8\cdot \sqrt{\frac{32}{{x}^{4}}}=8×\frac{\sqrt{16}×\sqrt{2}}{\sqrt{{x}^{4}}}$

$8\cdot \sqrt{\frac{32}{{x}^{4}}}=8×\frac{4×\sqrt{2}}{{x}^{2}}$

Notice that 8 is a fraction 8/1.

$8\cdot \sqrt{\frac{32}{{x}^{4}}}=\frac{8}{1}×\frac{4×\sqrt{2}}{{x}^{2}}$

Multiplication of fractions is easy as π.

$8\cdot \sqrt{\frac{32}{{x}^{4}}}=\frac{32\sqrt{2}}{{x}^{2}}$

Summary: Square roots ask a question: What number squared is the radicand? This comes from the area of a square. Given the area of a square, how long is the side?

To answer the question you factor the radicand and find the largest perfect square.

Time for some practice problems:

1.7 Square Roots Part 1 Practice Set 1

$\begin{array}{l}1.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{125}=\\ 2.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{27}=\\ 3.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{162}=\\ 4.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{75}=\\ 5.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{45}=\end{array}$ $\begin{array}{l}6.\text{\hspace{0.17em}}\sqrt{4{x}^{2}}=\\ 7.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{4{x}^{4}}=\\ 8.\text{\hspace{0.17em}}\sqrt{\frac{4}{25}}=\\ 9.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{98}=\\ 10.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{48}=\end{array}$ $\begin{array}{l}11.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{4}\cdot \sqrt{8}=\\ 12.\text{\hspace{0.17em}}\text{\hspace{0.17em}}4\sqrt{8}=\\ 13.\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\sqrt{9}=\\ 14.\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\sqrt{\frac{1}{4}}=\\ 15.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{300}=\end{array}$

1.7 Square Roots Part 1, Practice 2

Simplify problems 1 through 8.

1.      $\sqrt{24}$ 2. $\sqrt{8{x}^{3}}$

3. $\sqrt{200}$ 4. $\sqrt{27}$

5. $\sqrt{7x}$ 6. $4\sqrt{{a}^{2}b}$

7. $3x\sqrt{98{x}^{2}}$ 8. $\frac{1}{3}\cdot \sqrt{\frac{9}{{x}^{2}}}$

9. Show that ${a}^{2}b=\sqrt{{a}^{4}{b}^{2}}$ 10. Why is finding perfect squares appropriate

when simplifying square roots?

## Where the First “Law” of Logarithms Originates – Wednesday’s Why E6

rule 1

Wednesday’s Why $–$ Episode 6

In this week’s episode of Wednesday’s Why we will tackle why the following is “law” of exponents is true in a way that hopefully will promote mathematical fluency and confidence. It is my hope that through these Wednesday’s Why episodes that you are empowered to seek deeper understanding by seeing that math is a written language and that by substituting equivalent expressions we can manipulate things to find truths.

${\mathrm{log}}_{2}\left(M\cdot N\right)={\mathrm{log}}_{2}M+{\mathrm{log}}_{2}N$

Now of course the base of 2 is arbitrary, but we will use a base of two to explore this.

The first thing to be aware of is that exponents and logarithms deal with the same issue of repeated multiplication. There connection between the properties of each are tightly related. What we will see here is that the property of logarithms above and this property of exponents below are both at play here. But it is not so easy to see, so let’s do a little exploration.

Just to be sure of how exponents and logarithms are written with the same meaning, consider the following.

${a}^{b}=c↔{\mathrm{log}}_{a}c=b$

Let us begin with statement 1: ${2}^{A}=M$

We can rewrite this as a logarithm, statement 1.1: ${\mathrm{log}}_{2}M=A$

Statement 2:
${2}^{B}=N$

We can rewrite this as a logarithm, statement 2.1: ${\mathrm{log}}_{2}N=B$

If we take the product of M and N, we would get
2A ·2B. Since exponents are repeated multiplication,

2A ·2B = 2A + B

This gives us statement 3: $M\cdot N={2}^{A+B}$

Let us rewrite statement 3 as a logarithmic equation.

$M\cdot N={2}^{A+B}\to {\mathrm{log}}_{2}\left(M\cdot N\right)=A+B$

In statements 1.1 and 2.1 we see what A and B equal. So let’s substitute those now.

$\begin{array}{l}{\mathrm{log}}_{2}\left(M\cdot N\right)=A+B\\ {\mathrm{log}}_{2}\left(M\cdot N\right)={\mathrm{log}}_{2}M+{\mathrm{log}}_{2}N\end{array}$

It took a little algebraic-juggling to get it done, but hopefully you can now see that this is not a law or a rule, but a property of repeated multiplication, just like all of the properties of exponents are consequences of repeated multiplication.

Let me know what worked for you here and what did not. Leave me a comment.

Thank you again.