How to Begin

Imagine someone taught you how to play Brahm’s Lullaby on the piano but told you the name of the song was Go to Sleep, Baby.  You can completely master the song.   However, your ability to apply the song is limited to the request, “Play Go to Sleep, Baby.” 

This is analogous to what is often a best case scenario in math.  A student completely masters a skill that is only repeated under a particular contextualization.  The problem in learning math is exacerbated by students not mastering the skills they’ve been taught.

To get everybody on the same page a teacher gives a pre-test and a book has a prerequisite section.  Using our previous example, both methods seek to rename the song appropriately, and get students to become proficient playing it.  The problem is that the focus here is too narrow, too skill based.  It only addresses how to do something, not what it is that is being done, or why it works.

An example in math is,

Simplify the following expressions: 

1.      a³× a²

2.      (a³)²

If a student is stuck they will ask for help and someone will tell them, “If the bases are the same you add the exponents but when a power is raised to another power, you multiply them.”  Maybe, if the student has a diligent teacher, they will be presented with a list of rules for exponents.  If the student has previously been taught these things they perform the task and all involved are happy. 

Unfortunately the student has just been taught Lullaby by the wrong name, so to speak.   Even more unfortunately, this will likely earn both the student and teacher high marks, and a student will likely do well on standardized tests, and all without understanding. 

When the student takes a college entrance exam they’ll see a problem like:

Simplify the following (without a calculator)

4⁵ + 4⁵ + 4⁵ + 4⁵

With that, I will only assume that you know basic arithmetic facts.  That doesn’t mean that the pace will be slow, but the prerequisite knowledge will be covered in embedded contexts. 

One last note to be made is that some math is convention.  That is, we simply agree upon some things, it’s not that the nature of all things in math are discoverable.  For example, the features of a coordinate plane, x being the horizontal axis and y being vertical are so because they are defined to be so.   But the fact that there are infinitely many prime numbers is a discoverable fact.  (The discoverable facts are a lot more fun to learn.)

Reading and Number

Reading and Number

When a Thing isn’t Itself

I apologize ahead of time because much of the following is rote information.  They are not discoverable facts about math, but rather conventions and language that give us a way to communicate about math.  That may make it less interesting than a lot of things we will discuss later, but it is very important because without it communication is difficult.

Math is all about numbers and their relationships.  The numbers can represent tangible things (applied mathematics) or nothing at all (pure mathematics).  The beauty in mathematics is that when truths are found abstractly there is often an application where the relationship provides great insight, even though the application may not yet have existed when the relationship was discovered. 

An often sited example exists in the Fibonacci sequence, which is 1, 1, 2, 3, 5, 8, 13, 21 ... and the next number is 34 because that is the sum of the previous two entries.   The sequence was invented as a way to practice arithmetic but there are some fascinating things found in it, like the Golden Ratio, which we’ll leave for another time.  However, the sequence of numbers is often how things are found to grow naturally.  An example is how the seeds of a sunflower are laid out, or pedals on a flower.  Most flowers have a number of pedals that is in the Fibonacci sequence.   The neatest thing about this is that the sequence was developed to help people learn arithmetic, not to articulate how flower pedals are formed or to describe how things look beautiful to us (which involves the Golden Ratio).

Without chasing that rabbit any further, let’s begin with what a number is.  The symbol we use for the number five is just a symbol, like any other.  It has no inherent meaning, it’s just a series of lines and curves carefully joined.  Some examples would be 5,aK5.  These are all just symbols.  Some make sense to us, some do not.   The distinction between the symbol 5 and the thing 5 is important to make because math is a written language.  Sometimes 5 is just five things, like five zombies.  Sometimes a 5 means you will perform some operation 5 times, like x⁵ means you’re multiplying x by itself 5 times.

The point is that symbols represent things.  People often complain that math was fine until the alphabet was introduced.  Those people failed to recognize that the alphabet is just a series of symbols that we attach sounds to, but they’re perfectly valid for other uses, like X marks the spot! 

What I will attempt to explain is how the number line is a line, not a discrete collection of points.  It’s continuous, meaning that we can find a number that exists between any two other numbers, no matter how close. 

Let’s start at the beginning, with the number one.  One is called the identity because if you multiply or divide by it, the value of the other number does not change.   It is also strange because it is neither prime or composite (which we will discuss later). 

If we began to build our number line with 1, we would have just one point.

The first set of numbers are the Natural Numbers which start at 1 and continue infinitely by increments of one.    That is, 1, 2, 3, 4, …  They continue in this fashion infinity.  Please note that infinity is not a number, just a concept of something that continues forever.    

Now we can create a new point for each one of the natural numbers, for as long as we desire, making sure to place them in order as numbers respect size.  But there are gaps between each.

Start at zero, then count the natural numbers and you have the Whole Numbers.  The number zero means absence of value, nothing. 

The next set of numbers are called Integers, and it is with these that many people first encounter trouble with math.  Integers are signed numbers, positive or negative.  If no symbol (+  - ) is assigned, the number is assumed to be positive.  (We are just introducing the types of numbers, not discussing each in this section.  There will be more on integers later.) 

The integers are just a set of numbers made of the whole numbers and their negatives, which we call their opposite numbers.  Note, zero has no opposite as it has no value.  It is neither positive of negative.

So now we have a zero in the middle and a string of positive numbers going to the right and a string of negative numbers going to the left.  But still, there are gaps between consecutive integers.

The next set of numbers are the numbers between the integers, the fractions, repeated or terminating decimals, called the Rational Numbers.  A rational number is any number that can be written as a fraction.  So, 11 is rational, because 11/1, which can be treated as 11 divided by 1, is 11.  The number 3.1 is also rational because it is 3 and one tenth.

If we were to list all of the numbers we have so far between any two consecutive integers, like 1 and 2, writing a dot for each, we could zoom in and find gaps.  We do not yet have a continuous string of numbers.   We can zoom in very close, as close as we wanted really, and find gaps.  For this example we will only look at the second decimal place, between 1.41 and 1.42.  Between these two numbers is the square root of two.  That means if you had a box whose area was two, the side length would be between 1.41 and 1.42. 

The number itself cannot be written because it is a non-terminating, non-repeating decimal.   In fact, the following link displays the first 1,000,000 digits, http://apod.nasa.gov/htmltest/gifcity/sqrt2.1mil.  In case you’re wondering, this is still just an approximation of the number times itself that is 2. 

These numbers are called Irrational Numbers, because they cannot be written as a fraction (rational is from ratio, and irrational means not a ratio). 

There are more irrational numbers than rational numbers and there are irrational numbers that are called transcendental numbers which are not solutions to polynomial equations.  It gets weird.  Some of these even got people killed!  (More on that later.)

Now we have a complete number line.  The irrational numbers fill the gaps between consecutive rational numbers.  The list of numbers we have so far comprise what is called the Real Numbers.  It is an unfortunate name causing many people to believe that the next set of numbers, called Imaginary Numbers, are just that, of fiction. 

The imaginary numbers are numbers that involve the square root of a negative.  Since we have not discussed square roots in detail yet, you may just have to take this one on faith!

In summary:

The Real Numbers are:

• ·         Natural Numbers are counting numbers starting with 1.
• ·         Whole Numbers are the natural numbers and zero.
• ·         Integers are the whole numbers and their opposites.  (Zero has no opposite.)
• ·         Rational Numbers include all of the integers and the fractions between them (which include terminating or repeating decimals).
• ·         Irrational Numbers are non-terminating and non-repeating decimals

There are other numbers that involve the square root of negative one which we will not be addressing now.

The Why, The What, The How

and

The Switcheroo

Math isn’t about remembering everything. 
It’s about knowing a few things and figuring out everything else.

Allow me to introduce you to the three main players in the math scene.   They are The How, The What and the Why.  The How is the low level thug on the street and The What is his manager.  But The Why is the king pin, the one calling the shots!

Most of teaching and learning math revolves around The How.  That is, how do you find the vertex of a parabola, or how do you solve for x, steps one, two three, weee!   I get it.  Math is the least favorite, and often most terrifying, subject for many students.  They have to take math, it’s required.   The How is easy.  It’s easy to “learn,” looks great for the teacher if they are being observed by an administrator, and it’s very comfortable for student and teacher.  Sometimes The How is all the teacher knows.  I know I have been in that situation before.

An example of The How follows.  I will show you how to calculate products of numbers close to 100, in your head.  Not only that, with a little practice, you’ll be able to do it more quickly than you could using a calculator.  Ready?

Let’s look at the product of two numbers close to, but greater than 100.  For example, the product of 102 and 107, which is 10,914.  Without doing column multiplication, here’s how you can calculate that product quickly, in your head.

The answer will be a one followed by  four digits which we will refer to by WXYZ.  It will look like 1WXYZ.  The WX will be the two digit sum of the 2 and 7, which is 09.  The YZ will be the two digit product of 2 and 7, which is 14. 

So all you do is add and then multiply the two numbers at the end and place them properly. 

Let’s try 104 and 108. 

104 X 108 = 1WXYZ

WX =  12 (because 4 + 8 = 12).

YZ = 32 (because 4 X 8 = 32).

So 104 X 108 = 11,232

Why don’t you try a few and then check using a calculator:

1.        101 X 103 = ________________

2.       109 X 102 = _________________

3.       105 X 107 = _________________

There are similar methods for finding precise square roots of prime numbers and products of 4 digit numbers and so on.  But I say, big whoop.  First off, if all a student knows is The How, they’ll forget.  Without The What and The Why, they’ll never be able to recover their lost “learning.”  That takes this trick out of the realm of learning, in my book (haha), and into the realm of training. 

In math classes there is some, but not enough of, The What.  That is, what does it mean to find the vertex, or what does it mean to solve for x?  What does it mean to factor a polynomial? 

The What, in this example you’ve just seen is so obvious that there’s little to explain.  The What here, is finding the product.  You already know how to do that, although in a different way.  Usually you need to know the what so that if The How you know doesn’t work, you can perhaps find another.  Beyond that, if you know The What, you can use the good ole internet to help you! 

I often ask my students, What does it mean to solve for x?  Most do not know the first time they’ve been asked.  These same students are those following lists of steps, the same that will fail to recall how to solve for x after the briefest of vacations from the topic.   To be good at math a student must know what it is they’re being asked to do, then they must know how to do it. 

The Why refers to, Why does this work?  The Why is the conceptual foundation.  Without it you cannot connect ideas or apply previous knowledge in a new way.  All three parts How, What and Why, are needed, and sometimes just knowing The How is acceptable.  That said, The Why is the granddaddy of them all.  What it is you’re doing in math, and how it is you’re doing it, both come from The Why.  The Why explains the inner workings, it justifies that what you are doing is valid and true.

The Why in this trick you’ve just learned involves the order of operations, distributing and some other things that I will not draw out with great detail as the point is to explain how it is The Why drives The How. 

·         In the product of 104 and 108, the first digit will always be a 1 because 100 x 100 is 10,000. 

·         The reason the next two numbers are the sum of 4 and 8 is because we are adding 4 to 100 (to get 104) and multiplying that by 8 added to 100 (to get 108), as shown below here:

104 X 108 = (100 + 4)(100 + 8)

·         By distributing, which you may not yet know (it’s okay if you don’t), we get:

(100 + 4)(100 + 8) = 100(100 + 8)  + 4(100 + 8)

·         Which gives us the following sum:

10000 + 800 + 400 + 32

·         If we write this in a column it’s easy to see that we add the 8 and 4 to create the second two digits, and of course the last two digits are the product of 8 and 4. 

10000 800 400     +         32

11232

Now, if you really understand The Why (it works), then perhaps you can come up with a similar short cut for multiplying two numbers slightly less than 100, like 97 and 98.  If you can, then you’re so much better equipped than had you just learned the simple trick I showed at first.

On a side note, I once started the school year off with this, and a few other tricks (Just The How, nothing else).  Word got out that this amazing teaching was taking place in my room and that students were doing great math on the first day of school.  The principals heard and soon all of the mucky-mucks in the administration office stopped by my room.  They witnessed and participated and were wowed…and I was ashamed because it is simply a parlor trick.  The point of it was to get students to engaged in class, to do a little math and have fun and to know that every day, even on the first day of school, we will work.  I was awarded the school teacher of the year award that year.  I hope it had nothing to do with this trick!

The How (to do it), without understanding The What (you’re doing) and The Why (it works), is just a trick.  A computer could be programmed to do it and computers understand nothing. 

In this course, while still remaining practical, I will try to address The Why far more than I address The How.  That’s the switcheroo, if you were wondering.  It is my sincere desire that you know why it is that you do certain things, why it works.  If you also understand why things work and what it is you’re being asked to do then you can always figure out how to do it.  That would make you a powerful math student and I suggest it will change how you think about problems outside of math.

I hope you are convinced of the value, and appropriate places, of The How, The What and The Why.  If so, you will no longer settle for just getting an answer.  Someone telling you how to do it, without explaining why, is just as useful as someone telling you the answer.  If you don’t first understand what it is you’re doing and second why it works, then you simply do not understand.