## How Habits and Education Collide

The best definition I have come across for a habit is, “action without thought.”  A quick search on the internet says that a habit is, a settled or regular tendency or practice, especially on that is hard to give up …

We certainly need habits, especially in education.  Students, in order to be successful, need to be in the habit of being on time, having their homework done properly, whatever the classroom norms and expectations are need to habitual.  In other words, the day to day activities of school should be done automatically, without thought or the student needing to be reminded.

And certainly we want students to be in habits when it comes to performance.  For example, putting their name on their paper, showing appropriate work, employing effective questioning strategies and the like all end in higher levels of academic performance.

But what about what they’re learning.  Are we teaching them habits, that is, the action without the thought?  I say that we definitely are, and that is in direct conflict with the purpose of education.  That purpose is to give people the opportunity to learn how to think in a safe environment where the messiness that comes from the process of learning to think does not have major consequences.

As with most thinks related to teaching, this is highly nuanced and subjective, and there are certainly times where teaching a kid a habit that leads to a right answer or desired outcome is best.  That’s part of what makes education so powerful is that you can learn from what others before have done and take the next step, right?

What makes this double tricky is that we grade the results of habits.  Can a student see a prompt and spit out an appropriate output?  If so, they’ve obviously learned, right?

If you’re an expert in the field you’re teaching, you most likely approach problems at the level you’re teaching habitually.  Little reference to the ideas at play is required for you to arrive at a solution.

If you’re not an expert but have enough background to teach the topic, you’ve probably brushed up with some Khan Academy videos or the like, where you were shown those efficient methods and techniques that are the ways the expert acting habitually would do.

If a student is able to pass a standardized test they must also possess these habits.  However, if they’re taught the actions without thought, the process alone, they have no way to connect what they’re doing to other things.

Let’s consider how thinking and problem solving really works.  After all, learning to think is the purpose of education, right?  It’s highly unlikely that any student will have a practical use for 90% of the materials learned in your class.  But the learning that takes place, that is entirely useful and practical!

In thinking and problem solving the issue at hand must have a level of novelty.  If not, a habitual approach will be successful and little thinking will take place.  The problem must first be grappled with and understood and then the person dealing with this task can generate some ideas.  These ideas are the conceptual understanding of the task at hand.  From these ideas come the actions, the steps taken.  Upon completion review of the entire undertaking is performed and if the outcome was desirable, success can be claimed.

Often it is the case that not only is success claimed, but all similar problems now have a heuristic background.  Upon further review and generalization and actual procedure can be articulated.

Since the procedure is the measured and share-able portion of this entire development, that is what is written in books and what is measured on tests.

Yet, it all came from a conceptual understanding, an idea.  The idea initiates the procedure.

To not allow students access to the time and level of involvement required to explore ideas and develop heuristic approaches to problems is to rob them of the very purpose of education.  They do not learn how to learn when they are trained to follow steps given a particular input.  That’s training.  Sit Ubu, sit. Good dog!

It is certainly a challenge and uncomfortable for all parties involved to have students develop this level of understanding and explore without explicit direction.  However, it is the absence of such things that has education in the United States in such a terrible predicament.

My challenge to you, the reader, is to pick an overarching, big idea in your topic, something that is coming up next, and develop an activity/problem that will require a lot of thinking and little direction from you.  Make it something where the student result can be assessed as correct or incorrect based on the concepts at play, or by reverting back to the original question itself.

What you’ll find is that the students uncover connections that you have forgotten or taken for granted, or maybe never realized at all.  Over time, with regular activities/lessons like this they will begin to adjust to what is expected of them and they’ll increasingly enjoy actual learning!

Let me know what you think by leaving me a comment.

Thank you once again for reading.

Philip Brown

## Vestiges of the Past Making Math Confusing

Something in Math HAS to Change

Convention is a beautiful thing.  It allows us to use symbols to convey little things like direction or a sound.  We can piece those things together to make larger things, and eventually use it to create something like what you’re reading now.  There are no inherent meanings to these shapes we call letters, or the sounds we use when speaking.  It all works because we agree, somehow, upon what they mean.  Of course, over generations and cultures, and between even different languages, some things get crossed up in translation, but it’s still pretty powerful.

The structure of writing, punctuation, and the Oxford comma, they all work because we agree.  We can look back and try to see the history of how the conventions have changed and sometimes find interesting connections.  Sometimes, there are artifacts from our past that just don’t really make sense anymore.  Either the language has evolved passed the usefulness, or the language adopted other conventions that conflict.

One example of this is the difference between its and it’s.  An apostrophe can be used in a conjunction and can also be used to show ownership.  Pretty simple rule to keep straight with its and it’s, but whose and who’s.  Why is it whose, with an e at the end?

According to my friendly neighborhood English teacher there was a great vowel shift, which can be read about here, where basically, people in around the 15th century wanted to sound fancy and wanted their words to look fancy when written.  So the letters e and b were added to words like whose and thumb.

Maybe we should take this one step further, and use thumbe.  Sounds good, right?

But then, there’s the old rule, i before e except after c, except in words like neighbor and weight, and in the month of May, or on a Tuesday.  Weird, er, wierd, right?

All said, not a big deal because those tricks of language will not cause a student to be illiterate.  A student can mix those things up and still have access to symbolism and writing and higher level understanding of language.

There are some conventions in math that work this way, too.  There are things that simply are a hold-over of how things were done a long time ago.  The convention carries with it a history, that’s what makes it powerful.  But sometimes the convention needs to change because it no longer is useful at helping making clear the intentions of the author.

One of the issues with changing this convention is that the people who would be able to make such changes are so well versed in the topic, they don’t see it as an issue.  Or, maybe they do, but they believe that since they got it right, figured it out, so could anybody else.

There is one particular thing in math that stands out as particularly problematic.  The radical symbol, it must go!  There’s a much more elegant method of writing that is intuitive and makes sense because it ties into other, already established ways of writing mathematics.

But, before I get into that exactly, let me say there’s an ancillary issue at hand. It starts somewhere in 3rd or 4th grade here in the US and causes problems that are manifested all the way through Calculus.  Yup, it’s multiplication.

Let me take just a moment to reframe multiplication by whole numbers and then by fractions for you so that the connection between those things and rational exponents will be more clear.

Consider first, 3 × 5, which is of course 15.  But this means we start with a group that has three and add it to itself five times.

Much like exponents are repeated multiplication, multiplication is repeated addition.  A key idea here is that with both we are using the same number over and again, the number written first.  The second number describes how many times we are using that first number.

Now of course 3 × 5 is the same as 5 × 3, but that doesn’t change the meaning of the grouping as I described.

3 + 3 + 3 + 3 + 3 = 3 × 5

Now let’s consider how this works with a fraction.

15 × ⅕.  The denominator describes how many times a number has been added to itself to arrive at fifteen.  We know that’s three.  So 15 × ⅕ = 3.

3 + 3 + 3 + 3 + 3 = 15

Three is added to itself five times to arrive at fifteen.

Let’s consider 15 × ⅖, where the five in the denominator is saying we are looking for a number that’s been repeatedly added to get to 15, but exactly added to itself 5 times.

In other words, what number can you add to itself to arrive at 15 in five equal steps?  That’s ⅕.

The two in the numerator is asking, how far are you after the 2nd step?

3 + 3 + 3 + 3 + 3 = 15

The second step is six.

Another way to see this is shown below:

3 →6→9→12→15

Step 1: 3 → Step 2: 6 → Step 3: 9 → Step 4: 12 → Step 5: 15

Thinking of it this way we can easily see that 15 × ⅘ is 12 and 15 × 5/5 is 15.  All of this holds true and consistent with the other ways we thinking about fractions.

So we see how multiplication is repeated addition of the same number and how fractions ask questions about the number of repeats taken to arrive at an end result.

Exponents are very similar, except instead of repeated addition they are repeated multiplication.

Multiplication:  3 × 5 = 3 + 3 + 3 + 3 + 3

Exponents:  3⁵ = 3 × 3 × 3 × 3 × 3

Do you see how the trailing numbers describe how many of the previous number there exists, but the way the trailing number is written, as normal text or a superscript (tiny little number up above), informs the reader of the operation?

Pretty cool, eh?

Just FYI, 3 times itself 5 times is 243.

15 × ⅕ = 3, because 3 + 3 + 3 + 3 + 3 = 15.  That is, three plus itself five times is fifteen.

2431/5= 3 because 3 × 3 × 3 × 3 × 3 = 243.  That is, three times itself five times is two hundred and forty three.

You might be thinking, big deal... but watch how much simpler this way of thinking about rational exponents is with something like an exponent of ⅗.  Let’s look at this like steps:

3 × 3 × 3 × 3 × 3 = 243

3→9→27→81→243

Step one is three, step two is nine, step three is twenty-seven, the fourth step is eighty one, and the fifth step is 243.  So, 2433/5is asking, looking at the denominator first, what number multiplied by itself five times is 243, and the numerator says, what’s the third step?  Twenty-seven, do you see?

Connecting the notation this way makes it simple and easy to read.  The only tricky parts would be the multiplication facts.

## Rational Exponents and Logarithmic Counting …

rational exponents

Rational Exponents

In the last section we looked at some expressions like, “What is the third root of twenty-seven, squared?” The math is kind of ugly looking.

$\sqrt[3]{{27}^{2}}$

The procedures are clunky and it is very easy to lose sight of the objective. What this expression is asking is what number cubed is twenty-seven squared. You could always square the 27, to arrive at 729 and see if that is a perfect cube.

There is a much more elegant way to go about this type of calculation. Turns out if we rewrite this expression with a rational exponent, life gets easier.

$\sqrt[3]{{27}^{2}}={27}^{2/3}$

These two statements are the same. They ask the same question, what number cubed is twenty-seven squared?

By now you should be familiar with perfect cubes and squares. Hopefully you’re also familiar with higher powers of 2 and 3, as well as a few others. For example, you should recognize that 625 is ${5}^{4}.$ If you don’t know that yet, a cheat sheet might be helpful.

Let’s look at our expression again. If you notice that 27 is a perfect cube, then you can rewrite it like this:

${27}^{2/3}\to {\left({3}^{3}\right)}^{2/3}$

Maybe you see what’s going to happen next, but if not, we have a power raised to another here, we can multiply those exponents. Three times two-thirds is two. This becomes three squared.

${\left({3}^{3}\right)}^{2/3}\to {3}^{2}=9$

Not too bad! We factor, writing the base of twenty-seven as an exponent with a power that matches the denominator of the other exponent, multiply, reduce, done!

Let’s look at another.

Simplify:

${625}^{3/4}$

We mentioned earlier that 625 was a power of 5, the fourth power of five. That’s the key to making these simple. Let’s rewrite 625 as a power of five.

${\left({5}^{4}\right)}^{3/4}$

We can multiply those exponents, giving us five-cubed, or 125. Much cleaner than finding the fourth root of six hundred and twenty-five cubed.

What about something that doesn’t work out so, well, pretty? Something where the base cannot be rewritten as an exponent that matches the denominator?

${32}^{3/4}$

This is where proficiency and familiarity with powers of two comes to play. Thirty-two is a power of two, just not the fourth power, but the fifth.

${\left({2}^{5}\right)}^{3/4}$

If we multiplied these exponents together we end up with something that isn’t so pretty, ${2}^{15/4}.$ We could rewrite this by simplifying the exponent, but there’s a better way. Consider the following, and note that we broke the five twos into a group of four and another group of one.

${\left({2}^{5}\right)}^{3/4}={\left({2}^{1}\cdot {2}^{4}\right)}^{3/4}$

Now we’d have to multiply the exponents inside the parenthesis by $¾$, and will arrive at:

${2}^{3/4}\cdot {2}^{3}$

Notice that ${2}^{3/4}$ is irrational, so not much we can do with it, but two cubed is eight. Let’s write the rational number first, and rewrite that irrational number as a radical expression:

$8\sqrt[4]{{2}^{3}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}or\text{\hspace{0.17em}}\text{\hspace{0.17em}}8\sqrt[4]{8}$.

There’s an even easier way to think about these rational exponents. I'd like to introduce something called Logarithmic Counting.  For those who don't know what logarithms are, that might sound scary.

Do you remember learning how to multiply by 5s...how you'd skip count?  (5, 10, 15, 20, ...)  Logarithmic counting is the same way, except with exponents.  For example, by 2:  2, 4, 8, 16, 32, ... Well, what’s the fourth step of 2 when logarithmically counting? It’s 16, right? 

Let’s look at ${16}^{3/4}$. See the denominator of four? That means we’re looking for a fourth root, a number times itself four times that equals 16. The three, in the numerator, it says, what number is three of the four steps on the way to sixteen?

2 4 8 16

Above is how we get to sixteen by multiplying a number by itself four times. Do you see the third step is eight?

Let’s see how our procedure looks:

Procedure 1:

${16}^{3/4}={\left({2}^{4}\right)}^{3/4}$

${\left({2}^{4}\right)}^{3/4}={2}^{\frac{4}{1}×\frac{3}{4}}={2}^{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}or\text{\hspace{0.17em}}\text{\hspace{0.17em}}8.$

Procedure 2:

${16}^{3/4}=\sqrt[4]{{16}^{3}}$

$\sqrt[4]{{16}^{3}}=\sqrt[4]{{\left({2}^{4}\right)}^{3}}$

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

$\sqrt[4]{{2}^{4}}×\sqrt[4]{{2}^{4}}×\sqrt[4]{{2}^{4}}=2×2×2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}or\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}8.$

The most elegant way is to realize the 16 is the fourth power of 2, and the fraction $¾$ is asking us for the third entry. What is 3/4s of the way to 16 when multiplying (exponents)?

Let’s look at ${625}^{2/3}.$ Let’s do this three ways, first with radical notation, then by evaluating the base and simplifying the exponents, and then by thinking about what is two thirds of the way to 625.

Now this is going to be a tricky problem because 625 is NOT a perfect cube. It is the fourth power of 5, though, which means that 125 (which is five-cubed) times five is 625.

${625}^{2/3}=\sqrt[3]{{625}^{2}}$

$\sqrt[3]{{625}^{2}}=\sqrt[3]{{\left({5}^{4}\right)}^{2}}\to \sqrt[3]{{5}^{8}}$

$\sqrt[3]{{5}^{8}}=\sqrt[3]{{5}^{3}×{5}^{3}×{5}^{2}}$

$\sqrt[3]{{5}^{3}×{5}^{3}×{5}^{2}}=5×5\sqrt[3]{25},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}or\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}25\sqrt[3]{25}$

Pretty ugly!

Exponential Notation:

${625}^{2/3}={\left({5}^{4}\right)}^{2/3}$

${\left({5}^{4}\right)}^{2/3}={\left({5}^{3}×{5}^{1}\right)}^{2/3}$

${\left({5}^{3}×{5}^{1}\right)}^{2/3}={5}^{2}×{5}^{2/3}$

${5}^{2}×{5}^{2/3}=25×{5}^{2/3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}or\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}25\sqrt[3]{25}$

A little better, but still a few sticky points.

Now our third method.

${625}^{2/3}$ asks, “What is two thirds of the way to 625, for a cubed number?”

This 625 isn’t cubed, but a factor of it is.

${625}^{2/3}={\left(125×5\right)}^{2/3}$
This could also be written as:

${125}^{2/3}×{5}^{2/3}$

I am certain that 5 to the two-thirds power is irrational because, well, five is a prime number. Let’s deal with the other portion.

The steps to 125 are: 5 25 125

The second step is 25.

${125}^{2/3}×{5}^{2/3}=25×{5}^{2/3}$

To summarize the denominator of the rational exponent is the index of a radical expression. The numerator is an exponent for the base. How you tackle the expressions is entirely up to you, but I would suggest proficiency in multiple methods as sometimes the math lends itself nicely to one method but not another.

Practice Problems:



## The Smallest Things Can Cause Huge Problems for Students

preemptive

Pre-Emptive Explanation

It is often the case, for the mathematically-insecure, that the slightest point of confusion can completely undermine their determination. Consider a beginning Algebra student that is learning how to evaluate functions like:

$\begin{array}{l}f\left(x\right)=3x-{x}^{2}+1\\ f\left(2\right)\end{array}$

A confident student is likely to make the same error as the insecure student, but their reactions will be totally different. Below would be a typical incorrect answer that students will make:

$\begin{array}{l}f\left(2\right)=3\left(2\right)-{2}^{2}+1\\ f\left(2\right)=6+4+1\\ f\left(2\right)=11\end{array}$

The correct answer is 3, and the mistake is that -22 = -4, because it is really subtract two-squared. And when students make this mistake it provides a great chance to help them learn to read math, especially how exponents are written and what they mean.

Here’s what the students actually read:

$\begin{array}{l}f\left(x\right)=3x-{x}^{2}+1\\ f\left(2\right)=3\left(2\right)+{\left(-2\right)}^{2}+1\end{array}$

A confident student will be receptive to this without much encouragement from you. However, the insecure student will completely shut down, having found validation of their worst fears about their future in mathematics.

There are times when leaving traps for students is a great way to expose a misconception, and in those cases, preemptively trying to prevent them from making the mistake would actually, in the long run, be counter-productive. Students would likely be mimicking what’s being taught, but would never uncover their misconception through correct answer getting. Mistakes are a huge part of learning and good math teaching is not about getting kids to avoid wrong answers, but instead to learn from them.

But there are times when explaining a common mistake, rooted in some prerequisite knowledge, is worth uncovering ahead of time. This -22 squared is one of those things, in my opinion, that is appropriately explained before the mistakes are made.

## Why Teaching Properties of Real Numbers is Important

If you are going to do a fraction review, the lesson here might be of some help.  I believe things are best reviewed in context, but this is a decent set of information that also introduces the real numbers and some other basics of math.

The PDF icon to the left has a lesson outline you can feel free to use with the PowerPoints of in any way you see fit.

The structure is all there in the lessons, but they're not over scripted.  Remember, I believe the majority of a lesson should be spontaneous.  It should be anticipated and prepared for, but how the lesson really unfolds depends on the audience.

Below you will find an overview of how and why I teach real numbers as well as two PowerPoint icons you can download and use as your own.  I only ask that you share where you found them.

Anything you purchase from Amazon.com through the banner below goes to producing more materials, and at no cost to you.

### What Good Is It?

The Real Number Line has always been one of the dullest lessons I have to teach.

Natural Numbers are the set of numbers you can count on your fingers, beginning with one.  The Whole Numbers are the Natural Numbers and Zero...Integers are ...

Blah Blah Blah

I have to teach it because it's in the curriculum.  And I always wonder, what use is it if a student knows the difference between a whole number and a natural number?

It is hypocritical of me to complain in such a fashion because I laud the virtues of education being greater than a set of skills or a body of knowledge.  Education is about learning to think, uncovering something previously unknown that ignites excitement and interest.  Education should change how you see yourself, how you think about the world.  It should enrich our lives.

Teaching the Real Number Line can be a huge first step in that direction, if done properly.

### Math is About Ideas, Not Just Computation

There are some rich, yet entirely approachable, mathematical ideas that can be introduced with the Real Number Line (RNL).  For example, a series of questions to be posed to students could be:

1.  The Natural Numbers are infinite, meaning, they cannot be counted entirely.  How do we know that?
2. The Integers are also infinite.  How do we know that?
3.  Is infinity a number?
4. Which are there more of, Natural Numbers of Integers?  How can you know, if they're both infinite?

The idea of an axiom can be introduced.  Most likely, students assume math is true, or entirely made up, but correct or incorrect, because it is written in a book and claimed to be such by a teacher.  The idea of how we know what we know and if math is an invention or a discovery can be introduced by talking about axioms.  For example:

1. Is it true that 5 + 4 = 4 + 5 ?
2. If a and are Real Numbers, would it always be true that b = b?  (What if they were negative?)
3. Is it also true that b = b - a?  How do we know that?
4. Is the following also true:  If a = b, and b = c, then a = c?  How do we know?

The idea here is not to teach students the difference between the Associative Property and the Commutative Property, but to use these properties to introduce students to math as a topic that can be discussed, and that it is not about answer getting, but instead about ideas.

For more on this topic and a few other related items, visit this page.

### Why Are Some Rational Numbers Non-Terminating Decimals?

If you had a particularly smart group of students, you could pose this question.  I mean, after all, 1/3 = 0.3333333333333333...  And yet, we are told rational numbers include decimals that can be written as a fraction (the ratio of two integers).

How it works is sometimes very clear and clean.  For example, 0.7 is said, "Seven tenths." And "Seven tenths," can also be written as the ratio of seven and ten.  And the number seven tenths is of course equal to itself, regardless of how it is written.  The number 0.27 is said, "twenty seven hundredths," which can easily be written as the ratio of twenty seven, for the numerator, and one hundred, as the denominator.  And this can continue so long as the decimal terminates.  But try the same thing with the a repeating decimal and you do not end up with things that are equal.

The algorithm to convert a repeating, but non-terminator decimal into a fraction is pretty straight forward.

But that does not address why a rational number would be a non-terminating decimal.

Click the PPT Icon to the left to download a lesson on converting repeating decimals into fractions for honors students.  It includes a proof of why the square root of two is irrational.

The simple reason that some rational numbers cannot be expressed as terminating decimals has to do with our numbering system.  We use base 10 numbers.  Our decimal system provides us an easy way to write fractions with denominators that are powers of 10.

That means that we have ten numbers, including zero, that fill up one column, like a car’s odometer.  When you travel 9 miles the odometer will read 000009.  When you travel the tenth mile the odometer will read 000010.

Not all of our number systems are base ten. While the metric system is base 10, or at least translates into powers of ten, the Imperial system is base 12 from inches to feet, but 5,280 feet to the next unit of miles, and so on.

Time is another great example of bases other than ten.  Seconds and minutes are base sixty.  You need sixty seconds before you have an hour, not ten.  But hours are base 24 because 24 hours are needed to make one of the next category, which is days.

In time, 25 minutes of an hour is the ratio:

But in base ten this is 0.4166666666666666... Our decimal system does math in base ten, not base sixty.  This is not 41 minutes!  A typical mistake would be two say 25 minutes is 0.25 of an hour.

Back to our original example of 1/3.  Not all numbers can be cleanly divided into groups of ten, like 3.  If we had a base 3 numbering system, where after the third number we moved, then 1/3 would just be 0.1.  But in our numbering system, 0.1 is one tenth.

Other numbers, like four, translate into ten more easily.  Consider the following:

The only issue remaining is that 2.5/10 is not a rational number because 2.5 is not an integer and rational numbers are ratios of two integers.  This can be resolved as follows:

Let's try the same process with 1/3.

As you can see, we will keep getting ten divided by three, forever.

This is a great example of how exploring a question can uncover many topics within the scope of the course being taught.

I hope this has caused you to pause and think of how exploring questions, relationships and properties in mathematics can lead to greater understanding than just teaching process and answer getting.

The video below is a fun way to explore some of the attributes of prime numbers in a way that provides insight into the nature of infinity.   All of the math involved is approachable to your average HS math student.

Here is a link to the blog post that goes into a little more detail than offered in the video:  Click Here.

If you find these materials valuable, you could help me create more.

## Sets of Numbers and the Problem with Zero Chapter 1 – Section 1

1.1 3

Sets of Numbers
and the
Problem with Zero and Division

We will begin with the various types of numbers called Real Numbers. Together, these numbers can be ordered and create a solid line, without gaps.

Ø  Natural Numbers: These are counting numbers, the smallest of which is 1. There is not a largest Natural Number.

Ø  Whole Numbers: All of the natural numbers and zero. Zero is the only number that is a Whole Number but not a Natural Number.

Ø  Integers: The integers are all of the Whole Numbers and their opposites. For example, the opposite of 11 is -11.

Ø  Rational Numbers: A Rational Number is a ratio of two integers. All of the integers, whole and natural numbers are rational.

o   Decimals that terminate or repeat (have patterns) are rational as they can be written as a ratio of integers.

Ø  Irrational Numbers: A number that cannot be written as a ratio of two integers is irrational. Famous examples are π, and the square root of a prime number (which will be discussed next).

Together these make up the Real Numbers. The name, Real, is a misnomer, leading people to conclude that the word real in this context has the same definition as used in daily language. That misconception is only strengthened when the Imaginary numbers are introduced, as the word imaginary here harkens back to a day when the nature of these numbers, and their practical use, was unknown.

Is zero rational?

A rational number is a number that is the ratio of two integers. Before we tackle the issues that arise from zero, let’s reframe how we think about rational numbers (fractions) and develop a different language for these to promote greater proficiency in Algebra and allow for greater ease in understanding how zero causes real problems with rational numbers. (If you understand the nature of what follows you do not have to memorize or remember the tricks, you just understand.)

Consider the fraction $\frac{8}{2}$ . You were likely taught to think of this fraction as division and would also likely be taught to ask the question, “How many times does two go into eight?” That is sufficient for this level of mathematics, but the Algebra ahead is seemingly more complicated, but by simply rephrasing the language we use to talk about fractions, we can expose the seemingly more complex as being the same level of difficulty.

Instead of asking, “How many times does two go into eight,” the better question is, “Two times what is eight?”

It is true that $\frac{8}{2}=4,$ because two times four is eight. Simply answer the question “Two times what is eight,” and you’ve found the answer.

This will come into play with Algebra when we begin reducing Algebraic Fractions (also called Rational Expressions) like:

$\frac{9{x}^{2}}{3x}$.

If you ask the question, “How many times does three x going into nine x squared,” you’ll likely be stuck, especially when the expressions become more complicated.

But asking, “three x times what is nine x squared,” is a little easier to answer.

$\frac{9{x}^{2}}{3x}=3x,$ because $3x\cdot 3x=9{x}^{2}$.

There will be much more on reducing Algebraic Expressions later in this chapter. Let’s turn our attention to zero and how it “behaves” in with rational numbers.

Zero is an integer, and again, a rational number is a ratio of two integers. Consider the following:

$\frac{5}{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{0}{5}$

The first expression asks, “Zero times what is five?”

The second expressions asks, “Five times what is zero?” (Again, phrase the question in this fashion to provide easier insight into the math.)

The product of zero and any number is zero. So, the answer to, “zero times what is five,” is … well, there is no answer. There is no number times zero that is five. There is not a number times zero that equals anything except zero. We say this is undefined, meaning, there is no definition for such a thing.

The second expression, “five times what is zero,” is zero. Five times zero is zero.

One of these two expressions is rational, the other is not a number at all. It does not just fail to fit within the Real Numbers, it fails to fit in with any number.

Repeating Decimals Written as Fractions

Consider the fraction $\frac{1}{3}.$ This is a rational number because it is the ratio of two integers, 1 and 3. Yet, the decimal approximation of one-third is $0.\overline{3}$ (the bar above the three means it is repeating infinitely).

Here is how to express a repeating decimal as a fraction. Let us begin with the number $0.\overline{27}$ .

 We don’t know what number, as a fraction is $0.\overline{27}$, so we will write the unknown x. $x=0.\overline{27}$ Since $0.\overline{27}$ is repeating after the hundredths place, we will multiply both sides of the equation by 100. (note, for 0.333333… we would multiply by 10, since the decimal repeats after the 10ths place, but we would multiply 0.457457457457…by 1,000 since it repeats after the thousandths place.) $100×x=0.\overline{27}×100$   $100x=27.\overline{27}$ The following step is done by a procedure learned with solving systems of equations, which will be covered later. (In fact, this procedure would be a great topic to review when systems of equations is learned.) Subtract the first equation from the second.       Note: $27.\overline{27}-0.\overline{27}=27$ $\begin{array}{l}\underset{_}{\begin{array}{l}100x=27.\overline{27}\\ -\left(x=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.\overline{27}\right)\end{array}}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}99x=27\end{array}$ Divide both sides by 99 to solve for x.   Recall that x was originally defined as the fractional equivalent of the repeating decimal. $x=\frac{27}{99}$

Practice Problems.

1.      Change the following into rational numbers:

a.       $5$

b.      0

c.       $\frac{3}{0.4}$

d.      $0.\overline{23}$

2.      Why is a the following called undefined: $\frac{a}{0}$ ?

3.      List all of the sets of numbers to which the following numbers belong:

a.       0 b. 9 c. -5 d. $5.37\overline{9}$ e. $\frac{5}{\pi }$ f. 5.47281…

4.      Can a rational number also be a whole number?

5.      What number is whole but not natural?

## How To Make Formative Assessments Powerful Learning Tools

I’d like to share with you one way to get students to engage in metacognition.  But before I do, let me explain why I believe this is one of the most powerful learning experiences a student can experience.

It is my opinion that if a formative assessment does not provide feedback to the students it is of little use.  In educational training, teacher-evaluation and professional development too much focus is placed on the teacher.  And yet, it is clear, that a motivated student will learn without a teacher. Sometimes, groups of students learn DESPITE bad teaching.  Students that seek understanding, in any topic, are successful.  A quiz or test is more than an evaluation, and in fact, rarely does a student perform in a way that is a surprise to the teacher.  We know who will get an A and who will be middle of the road and so on.

A quiz should be a learning opportunity. I didn’t say, “A teaching moment,” with intent because learning doesn’t always come from teaching.  We can set up an opportunity before handing back graded quizzes and tests that will be a powerful learning experience for the students.  We just need to get them to think about what they did.

The issue here is, how do you get kids to learn from mistakes on their quizzes and tests?  I mean, they have years of experience doing the following:

1. Quizzes are handed back.
2. Kid says to another: What did you get?
3. Then they say:  Let me see yours … they hold them up side by side seeing if all of the teacher’s marks match.
4. “Mister, he got one marked wrong that I got right!”
1. And of course you could ask, How do you know you are right?  Maybe I didn’t mark yours wrong.  Maybe I marked hers wrong on accident.  Maybe you’re both wrong!

Reviewing their own work and thinking about their understanding and performance is perhaps the single biggest learning opportunity that students have at their disposal.  We are remiss, terribly so, if we do not make full use of these opportunities.

But HOW do you get someone to think about their thinking?  You can’t really make them, can you?  It can’t be coerced, tricked or done under threat of punishment.  For someone to be willing to engage in metacongition, they must be intrinsically motivated.

Here’s one way I get students to engage in this.

Before handing back a quiz or test, I review a problem that was largely misunderstood.  Then, I have the students practice a similar problem on their own (from the quiz or test, if possible).  I walk around the room to check for understanding and then after a few moments, I allow them to help each other on the problem briefly.  I continue this as long as I feel is appropriate for the level of competency displayed on the test.

It is absolutely critical that this review is done when the students do not yet have their quizzes or tests back in their possession.

When this method of review is finished, I back their tests and instruct them to find the problems we reviewed as a class.  I ask them to figure out what they could’ve done differently on the test to improve their score.

Without such a discussion they will often fail to realize how simple sign errors wreck their grades, or how a simple conceptual misunderstanding is causing all of their work to just … well, collapse.  You could encourage students to highlight mistakes and annotate their new understanding by the mistakes, or have them use small sticky notes to write down questions that are still confusing to them.

This is a powerful technique, but like all methods, their effectiveness diminishes if done too frequently or too infrequently.  I would not try this method on a test where the class average was high.  I believe it is best reserved for those topics students generally struggle with.  I would advise against using this method for a summative assessment (end of unit, where the class will be moving on).

## Success Develops Confidence

Education isn’t really about the subject being learned, or the specifics of the topic being practiced.  No, education is about changing who we are for the better by learning how to get more out of ourselves.  An education should change you, change you think, how you see the world and it should change how you carry yourself, for the better.

The following is a story of how facing challenges and experiencing success did just that.

Like a typical freshman girl, Cristina was sometimes awkward, shy, and on occasion, over-reacted to situations. And like most kids, she had more going on in her life than just school.  But, she had a great desire to be successful in the honors math class I taught, Cambridge Math.  However, the challenge was great, and likelihood of success…well, not so great.

See, the previous year, the first our school participated in the Cambridge program, not a single student passed the end of course examination (IGCSE).  In fact, only 8% of students in Arizona passed that year.  To further complicate things, I was now appointed as the new teacher for Cambridge and there was a huge learning curve ahead of me.

So there we were, Cristina and I, facing a difficult situation together.  I’d never taught any honors program and know the teacher that taught Cambridge before me is a quality teacher.  And Cristina, as well as the other students, had never faced a course like this.  At the end of their sophomore year they would take a pair of hand-written tests.  To prepare for the tests the students had two school years to learn everything we teach in Algebra 1 and Geometry, most of what is taught in Algebra 2, Probability and Trigonometry, as well as large portions of Statistics and a handful of other topics not usually taught in the US.  To make it more complicated, most of the test required complex thought and application of concepts in unpredictable, unteachable ways.  To have just 8% of students, and these are honors students, pass in the state of Arizona was alarming to all of us!

Cristina did not stand out as a particularly strong math student.  In fact, when speaking with her mother one day, her mother said, “Cristina’s going to do what Cristina is going to do.  However the day strikes her will determine how the day goes.”

She passed the first year, but not without tears and heartache as she received far lower grades than she ever got in middle school.  There was a lot of frustration and the decision to stay in Cambridge her sophomore year, or move to an easier regular class, was considered at some length.

During her sophomore year she became pretty inconsistent, often sabotaging her own efforts.  I believe she saw herself as a weak math student with little to no chance of success.  Often when we see ourselves in a particular way we unknowingly take steps to fulfill that expectation.  This was unfortunate in Cristina’s case as she’d sometimes lack the discipline to complete homework, and often when it was completed, it was done so in low quality…just to get it done, not to promote learning and to practice.

On one occasion in particular Cristina was become very frustrated with her lack of progress.  Her performance had been suffering, grade dropping and agitation was on the rise!  During class that day we were working on a complex problem, the type they’d see on the difficult portion of the end of course Cambridge exam.  Cristina wasn’t participating, not even working on her own.  I asked her to work and eventually she snapped at me, “Why do I have to do this?!?!”

She’s not a bad kid, and as I mentioned, she had some things going on outside of school that added to her stress.  But, the fact remains, she was sabotaging her own efforts with inconsistent work, poor work ethic and sometimes bad attitude.

Cristina, like many students, would often say things like, “Let’s just get this over with, I’m going to fail anyway.”  I believe those are defense mechanisms, designed to take the sting out of the potential failure.

I encouraged all of the students to try, without reservation.  If they try their best and fail, it’s a win because our best is plastic…it improves or diminishes depending on what we demand of ourselves.  I believe that students that try their best and come up short will over-come, they will succeed!

When test day finally arrived Cristina was a nervous wreck.  I was a nervous wreck.  The students took the tests and we mailed them overseas to Cambridge University to be graded.

And then we waited … and waited… May to August we waited.

The night before the grades we to be released I had nightmares about having to tell students like Cristina that she did not pass.

Cristina passed the Cambridge exams … she did something that over 90% of participating honors students in the state failed to do!  Not only did she pass, she smashed it!  Maybe I’m wrong, but the experience seemed to change her. I believe the success she experienced, not just my class, but her other Cambridge classes (equally difficult) gave her the background to KNOW, not just hope, that she was capable of such difficult tasks.

Cristina just graduated High School.  I spoke with her about writing this blog post about her, and she consented, hoping that sharing her story would embolden others to try their best to achieve their goals, with reckless abandon…swing for the fences, as it were.

## Wednesday’s Why

There are many things in math that are just memorized, with little or no understanding of the meaning behind the scenes.  To help promote greater understanding, promote recall and accuracy, but most importantly, to empower people to be able to glean this deeper understanding behind the math by learning how to read the math, I am starting a video series on my YouTube channel called Wednesday's Why.   Every Wednesday I'll take a topic that is either largely misunderstood or just assumed to be true without any questioning and unpack it for you, show you how to read it so that you see what factors are at play.

This last week was the first episode and we tackled a tricky property of exponents.  The video is here below.

## How to Study!

Can you think of a thing that more often praised, promoted as virtuous and imperative to success, yet largely remains undefined than studying?  Well, I have four easy steps you can follow to structure your studying and give you great use of your time and will produce excellent outcomes both in the short term (for your quiz), but also improve retention of difficult and elusive topics.

1.  Set aside time.  You can't rush learning.  I'd suggest starting at least a week before your big test.  Pick a time every night where the only thing you're doing is studying, but the time period doesn't need to be long.  Twenty to thirty minutes should be long enough, provided you study with focus and for several days consecutively.

The time is required because learning is developing your brain...can't compress the time needed to make learning permanent.

2. Create a study guide.  List all topics that you've covered in class that might be tested.  When finished, notate which are most difficult for you.  Briefly review the topics you feel are mastered and research the rest.

To research use the internet, your book, notes, friends and teacher.  But be focused in your questions.  Know what it is you don't know so that your teacher can actually help you!

3. Practice Problems:  From old tests, quizzes, and homework, create a practice test of your own.  Perform the problems under testing conditions.  If notes are not allowed, don't take practice with notes.

Pay particular attention to problems you missed on old quizzes and tests.  Learn them, figure them out!  That's the point of studying anyway, right, to learn what you don't know?  Keep practicing until you're solid!

4.  Nail the test!

If you set aside time and follow through you'll be rock solid.  You can't control what grade you'll get, but you will have taken care of the things within your control.  You'll be confident on test day and things will go well for you.