# Teaching Square Roots Conceptually

teaching square roots

How to Teach Square Roots Conceptually

If you have taught for any length of time, you’ll surely have seen one of these two things below.

Sure, this can be corrected procedurally.  But, over time, they’ll forget the procedure and revert back to following whatever misconception they possess that has them make these mistakes in the first place.

I’d like to share with you a few approaches that can help.   Keep in mind, there is no way to have students seamlessly integrate new information with their existing body of knowledge.  There will always be confusion and misunderstanding.  By focusing in on the very nature of the issues here, and that is lack of conceptual understanding and lack of mathematical literacy, we can make things smoother, quicker, and improve retention.

Step one is to teach students to properly read square roots.  Sure, a square root can be an operation, but it is also the best way to write a lot of irrational numbers.  Make sure you students understand these two ways of reading a square root number.

Students are quick studies when it comes to getting out of responsibility and side-stepping expectations.  Very quickly, when asked “What does the square root of 11 ask?” students will say, “What squared is the radicand?”

When pressed on the radicand, they may or may not understand it is 11.  But, they’ll be unlikely to have really considered the question for what it asks.  Do not be satisfied with students that are just repeating what they’ve heard.  Make them demonstrate what they know.  A good way to do so is by asking a question like the one below.

Another way to test their knowledge is to ask them to evaluate the following:

$\sqrt{2}×\sqrt{2}.$

We do not want students saying it is the square root of four at this point.  To do so means they have not made sense of the second fact listed about the number.  An alternative to using a Natural Number as the radicand is to use an unknown.  For example:

$\sqrt{m}×\sqrt{m}.$

Step two requires them to understand why the square root of nine, for example, is three.  The reason why it is true has nothing to do with steps.  Instead, the square root of nine asks, “What squared is 9?”  The answer is three.  There is no other reason.

Once again, students make excellent pull-toy dolls, saying random things when prompted.  Once in a while they recite the correct phrase, even though they don’t understand it, and we get fooled.  It is imperative to be creative and access their knowledge in a new way.

Before I show you how that can be done with something like the square root of a square number, let’s consider the objections of students here.  Students will complain that we’re making it complicated, or that we are confusing them.

First, we’re not making the math complicated.  Anything being learned for the first time is complicated.  Things only become simple with the development of expertise.  How complicated is it to teach a small child to tie their shoes?  But once the skill is mastered, it is done without thought.

The second point is that we are not confusing them, they are already confused.  They just don’t know it yet.   They will not move from being ignorant to knowledgeable without first working through the confusion.  If we want them to understand so they can develop related, more advanced skills, and we want them to retain what they’re learning, they have to understand.  They must grasp the concept.

So how can we really determine if they know why the square root of twenty-five is really five?  We do so by asking the same question in a new way.

Another way to get at the knowledge is by asking why the square root of 25 is not 6.  Students will say, “Because it’s five.”  While they’re right, that does not explain why the square root of 25 is not six.  Only when they demonstrate that 62 = 36, not 25, will they have shown their correct thinking.  But, as is the case with the other questions, students will soon learn to mimic this response while not possessing the knowledge.  So, you have to be clever and on your toes.  This point is worth laboring!

Step three involves verifying square root simplification of non-perfect squares.  This uncovers a slew of misconceptions, which will address. Before we get into that, here is exactly what I mean.

Have students explain what is true about the square root of twenty-four.  There are two ways they should be able to think of this number (and one of them is not as an operation, yet).

1.      What squared is 24?

2.      This number squared is 24.

The statement is true if “two times the square root of six, squared, is twenty-four.”  Just like the square root of 9 is three only because 32 = 9.

The first hurdle here is that students do not really understand irrational numbers like the square root of six.  They’ve learned how to approximate and do calculation with the approximations. Here is how they see it.

$\sqrt{2}=1.4$

$3+\sqrt{2}=4.4$

$3×\sqrt{2}=4.2$

What this means is that students believe:

1.      Addition of a rational number and an irrational number is rational.

2.      The product of a rational and irrational number is also rational.

a.       This can be true if the rational number is zero.

This misunderstanding, which naturally occurs as a byproduct of learning to approximate without understanding what approximation means, is a major hurdle for students.  It must be addressed at this time.

To do so, students need to be made to understand that irrational numbers cannot be written with our decimal or fraction system.  We use special symbols in the place of the number itself, because we quite literally have no other way to write the number.

A good place to start is with π.  This number is the ratio of a circle’s diameter and its circumference.  The number cannot be written as a decimal.  It is not 3.14, 22/7, or anything we can write with a decimal or as a fraction.  The square root of two is similar.  The picture below shows probably over 1,000 decimal places, but it is not complete.  This is only close, but not it.

Students will know the Pythagorean Theorem.  It is a good idea to show them how an isosceles right triangle, with side lengths of one, will have a hypotenuse of the square root of two.  So while we cannot write the number, we can draw it!

The other piece of new information here is how square roots can be irrational.  If the radicand is not a perfect square, the number is irrational.  At this point, we cannot pursue this too far because we’ll lose sight of our goal, which is to get them to understand irrational and rational arithmetic.

This point, and all others, will be novel concepts.  You will need to circle back and revisit each of them periodically.  Students only will latch on to correct understanding when they fully realize that their previously held believes are incorrect.  What typically happens is they pervert new information to fit what they already believed, creating new misconceptions.  So be patient, light-hearted and consistent.

Once students see that the square root of two is irrational, they can see how they cannot carry out and write with our number system, either of these two arithmetic operations:

This will likely be the first time they will understand one of the standards for the Number Unit in High School level mathematics.

Students must demonstrate that the product of a non-zero rational and irrational number is irrational.

Students must demonstrate the sum of a rational and an irrational number is irrational.

Keep in mind, this may seem like a huge investment of time at this point, and they don’t even know how to simplify a square root number yet.  However, we have uncovered many misconceptions and taught them what the math really means!  This will pay off as we move forward.  It will also help establish an expectation and introduce a new way to learn.  Math, eventually, will not be thought of as steps, but instead consequences of ideas and facts.

Back to our question:

Just like the square root of nine being three because 32 = 9, this is true if:

${\left(2\sqrt{6}\right)}^{2}=24.$

Make sure students understand that there is an unwritten operation at play between the two and the irrational number.  We don’t write the multiplication, which is confusing because 26 is just considered differently.  It isn’t 12 at all (2 times 6)!

Once that is established, because of the commutative property of multiplication,

$2\sqrt{6}×2\sqrt{6}=2×2×\sqrt{6}×\sqrt{6}.$

There should be no talk of cancelling.  The property of the square root of six is that if you square it, you get six.  That’s the first thing they learned about square root numbers.

$2×2×\sqrt{6}×\sqrt{6}=4×6.$

As mentioned before, students are quick studies.  They learn to mimic and get right answers without developing understanding. This may seem like a superficial and easy task, but do not allow them to trick themselves or you regarding their understanding.

A good type of question to ask is:

To do this, we students to square the expression on the left of the equal sign to verify it equals the radicand.  This addresses the very meaning of square root numbers.

Last step is to teach them what the word simplify means in the context of square roots.  It means to rewrite the number so that the radicand does not contain a perfect square.

The way to coach students to do this is to factor the radicand to find the largest square number.  This is aligned with the meaning of square roots because square roots ask about square numbers.  When they find the LARGEST perfect square that is a factor of the radicand, the rewrite the expression as a product and then simply answer the question asked by the square roots.  Here’s what it looks like.

$48=2×24,\text{\hspace{0.17em}}3×16,\text{\hspace{0.17em}}4×12,\text{\hspace{0.17em}}6×8.$

$\sqrt{48}=\sqrt{16}×\sqrt{3}$
Write the square root of the perfect square first so that you do not end up with
$\sqrt{3}4,$ which looks like $\sqrt{34}.$

$\sqrt{48}=4×\sqrt{3}$.

At this point, students should be ready to simplify square roots.  However, be warned about a common misconception developed at this point.  They’ll easily run the two procedures into one.  They often write things like:

$\sqrt{18}=\sqrt{9}×\sqrt{2}$

$\sqrt{18}=3\sqrt{2}$

${\left(3\sqrt{2}\right)}^{2}=9×2$

$9×2=18$

$\sqrt{18}=18.$

The moral of the story here is that to teach students conceptually means that you must be devoted, diligent and consistent with reverting back to the foundational facts, #1 and #2 at the beginning of this discussion.

This approach in no way promises to prevent silly mistakes or misconceptions.  But what it does do is create a common understanding that can be used to easily explain why $\sqrt{12}$ is not $3\sqrt{2}.$  It is not “three root two,” because

This referring to the conceptual facts and understanding is powerful for students. Over time they will start referring to what they know to be true for validation instead of examination of steps.  There is not a step in getting $\sqrt{12}=3\sqrt{2},$ that is wrong.  What is wrong is that their work is not mathematically consistent and their answer does not answer the question, what squared is twelve?

If a student really understands square roots, how to multiply them with other roots, and how arithmetic works irrational and rational numbers, the topics that follow go much more quickly.  After this will be square root arithmetic, like $5\sqrt{2}-3\sqrt{8},$ and then cube roots and the like.  Each topic that you can use to dig deep into the mathematical meaning will, over time, quicken the pace of the class.

In summary:

1.      Square roots have a meaning.  The meaning can be considered a question or a statement, and both need to be understood by students.

a.       This meaning is why the square root of 16 is 4.

2.      Square roots of non-square numbers are irrational.  Arithmetic with rational and irrational numbers is irrational (except with zero).

3.      To simplify a square root is to rewrite any factor of the radicand that is a perfect square.

a.       When rewriting, place the square root of the square number first.

4.      The simplification of a square root number is only right if that number squared is the radicand.

I hope you find this informative, thought-provoking, and are encouraged to take up the challenge of teaching conceptually!  It is well worth the initial struggles.

For lessons, assignments, and further exploration with this topic, please visit: https://thebeardedmathman.com/squareroots/

## Best Practices

Philosophy and Best Practices

Cart Before the Horse

The easy part of teaching any curriculum is the curriculum.  The hard part of learning to teach a new curriculum is figuring out how to teach it most effectively.  For the sake of clarity, effective teaching will develop student conceptual understanding and problem solving.  Earmarks of quality teaching include retention, connection, engagement, and thoughtful discussion among students, as well as high test scores of course.

One of the problems with modern education and education research is the need to quantify everything.  The adage, If it can’t be measured it cannot be improved, has bound us with conclusions based on extremely soft data, nonsensical results, and has engaged us in an insidious pattern of behavior.  That same pattern is easily seen in students, and it drives teachers crazy.

If a student is in a class, then that class should cover new material for them.  If a student gets a good grade, they will have learned the material.  Grades measure learning and gained ability.  Students that focus on the grade in lieu of learning struggle to get good grades.  They have confused the marker of success with success.

Is there a more frustrating question to a teacher, from a student, than, “What can I do to bring my grade up?”  Isn’t the answer almost always, “… just about anything will work, so long as it involves learning.”

That is exactly what is happening today in education.  In order to conduct “scientific” research we quantify markers of success and measure them.  We lose sight of what is really important and chase what has been measured.  Teachers jump through hoops to create word walls, daily and in-depth content and language objectives, the good old SWBAT, and perform remediation that teaches students how to get quiz questions correct.

All of these actions are designed to show an increase in the markers of success but potentially do so in direct conflict of the true goal of education.  Our job as educators is to train the minds of young people.  Many best practices ease the cognitive strain to promote short-term positive results.

The end result is that between 40% to 60% of first year college students need to take remedial courses in English or Mathematics, or both.  Let us explore how this happens.

Use retention as an example.  Suppose a teacher is being coached that good teaching requires that students remember the material in say, four months.  That teacher will take steps to make sure that students remember.  There will be tricks, rhymes, and rewards for getting good test results on retention.

The problem is, this will not be the type of retention we are trying to develop.  The retention our students need is a consequence of conceptual understanding, not brute-force memorization.  With conceptual understanding comes the ability to reconstruct lost memories.  That is a great tool!  No matter what, the likelihood of losing information over time is astronomically likely.  Besides that, if a student only remembers the fact or procedure, their ability to use that fact or procedure will be bound by how they’ve been taught to use it.

Retention is just an earmark of conceptual understanding.  There’s more that comes along with conceptual understanding, like the increasing pace of integration of new material due to connections made, and improved engagement and problem solving skills.  And, perhaps most importantly, when students conceptually understand, they’re not bound by what we have taught them, they can take what they know and go further than we did!

Not everything in education has artificial restrictions placed upon it by inappropriate data seeking.  Climate and culture, for example, are two areas of focus for administrators.  Yet, climate and culture would be incredibly difficult to measure … but you can feel it the moment you walk onto a campus!  The same goes for relationship building between a teacher and the students.

Take a minute and imagine how artificial markers of determining the existence of a quality relationship between a teacher and student could be made.  Now if the efficacy of your teaching was being evaluated on this false metric, and if that metric had been engrained in the culture of education, it is likely you would be engaged in the best practices of the day that would increase the frequency of that marker.  Yet, would you really be building relationships?  What if you believed you were and you were not, and the system told you that you were doing well?  How would you know you were wrong if the system told you that you were right?

Baby and Bath Water

Now it is likely that quality outcomes are realized by chasing faulty markers of success.  For example, if you learn how many siblings each of your students has, you’ll likely open the door to conversation and find connection.  If a student attends tutoring to bring up their grades and they participate, they’ll likely learn.  By learning their grades will improve, even if they believe the act of attending tutoring is what brought about the improvement.

There is nothing inherently wrong with a Content and Language Object, or Learning Target, or SWBAT, or whatever it may be called at your school.  In fact, objectives are really needed when designing and implementing a lesson.  Without the target the lesson will meander.

Before deciding that a particular best practice is a false metric and deciding to abandon it or treating it as a matter of compliance, first see if that false metric can be employed to foster the desired outcome.  What should this best practice really build and why?  Can you make it serve that end?

A great example is allowing students to re-take a test.  If a student takes the same exactly test, especially after teacher-led remediation, of course the student will improve their grade.  But, did they really learn the content, did they develop a conceptual understanding?  Did they improve their problem solving skills?

Most likely, the conceptual understanding remained unchanged but the problem solving skills would have damaged.  It is a bad outcome when students learn that the way they solve problems is by seeking someone else to solve the problem (unless they’re bound for politics).

That does not mean that allowing students to re-test is a bad practice.  But it is a bad practice when done as described.  What if, in order for a student to re-test, the student had to demonstrate their conceptual understanding and problem solving before they re-tested, and then the second test was different than the first? (Changing the numbers on a test does not make it different.)  This would serve the student’s needs and encourage them to perform well.  It would provide you with data that could be used to more accurately assign a grade.  It is all around a great outcome!

How and Why

It is how and why we do what we do that makes it worth-while.  That adds an additional layer of complexity to the profession of teaching the United States today.  A teacher can follow all of the directives and engage in all the best practices as assigned by their leadership team.  They can have wonderful reteach and differentiation strategies, the best posted objectives, standards referenced lesson plans informed by classroom data, and formative assessments published by respected companies.  In their classroom you may see varied questioning techniques, students getting up and performing tasks, and the teacher “owning the room.”

I was that high school math teacher.  I was energetic, my students loved my class and proclaimed I made math fun for them.

I was also teaching remedial courses at the local community college.  When I began seeing some of those same students in my classes, I realized I was not effective.  The reason I wasn’t effective is I was putting on a show.  Sure, I followed the best practices of the day, but I failed to see what was important.

In the next few blogs, I will break down what I have discovered to be best practices that promote student conceptual understanding and improve problem solving skills.  I will try to explain why I believe they work so that you can adapt them to your needs.

## Student Skills and Tools

The biggest hurdle in transitioning from Middle School to High School is the lacking set of student skills possessed by incoming Freshmen.  Students come in failing to appreciate the importance of homework, struggle to think independently, cannot communicate mathematical thinking, and are easily frustrated to the point of quitting.

This observation is not a knock on the students’ experiences in Middle School.  It is entirely likely that the brain of a 12 to 14 year old cannot develop these skills.

In the upcoming 2019/20 school year I will be running an experimental program where I use SMART Goals focused on student skills to hasten the development of those lacking student skills.  The pay-off could be huge…the development of quality student skills would transcend the classroom, even school.  Ultimately, student skills are goal-oriented problem solving and personal management skills.

Here’s how it is going to work.  During the first week of school I will teach students what SMART Goals are (read about them here if you don’t know: https://www.yourcoach.be/en/coaching-tools/smart-goal-setting.php).  We will practice setting small SMART Goals in order to learn what is required, and how to foster them.

During the first week I will also teach students what quality student skills are.  I’ve made a reference sheet of what they are, what they look like in action, and how they’re beneficial.

Perhaps the most important thing taught in the first week will be how motivation drives engagement.  If a student is deeply engaged in their studies, they’ll persevere and be successful.  The two types of motivation, intrinsic and extrinsic, are directly related to the quality of engagement.  A student motivated by reward, or fear, from grades is extrinsically motivated.  They’ll easily give up and will engage in their work at a shallow level.  Their mindset is to complete required work.  A student that is intrinsically motivated is motivated to learn.  They engage deeply and seek learning.  They persevere and find learning opportunities in their work.

At the end of the first week of school students will draft individual SMART Goals that focus on student skills.  I have created a four-week long form where students will be guided through the reflection, monitoring and fostering required to have those goals come to fruition.

If you’d like to see the documents I’ve created, they are here.  Here is the Student Skill Sheet:  https://drive.google.com/file/d/1TCDKiwWrU-Ycoc1JILbOVd_y0f5_VkxR/view?usp=sharing

Here is the Smart Goal Planner:  https://drive.google.com/file/d/1mrZtEM3sAUcYkU5pxLdatnyn9DjVqT4k/view?usp=sharing

If you’d like to follow along with how this goes, you can read my blog:  https://thebeardedmathman.com/home/blog/

## Thoughts on Teaching

Foundation

1. The goal: If the question, "When am I going to use this in my real life," derails your class, there's a problem with your purpose and goal. The truth is, almost nothing after taught 5th grade is knowledge used daily. The purpose of education is not to teach MLA formatting or how to factor a polynomial.

The goal is to develop a careful, thoughtful and resourceful young person that is adaptable, a problem solver and has perseverance. That's the destination. The particular subject serves as (1) the vehicle to arrive at the destination, and (2) an exploration into potential aptitude and interest, (3) as well as a foundation of reference knowledge.

2. Autonomy: When students understand they're in charge of education outcomes and find value and validation from their efforts, they'll perform.

In other words, when they do it for themselves and receive appropriate praise and feedback for progress, their potential and performance will increase.

3. Letting Go: Some kids aren't ready. I barely passed Algebra 1 as a freshman in HS...in fact, I'm sure that 60% final semester grade was rounded generously. Yet, I ended up with a BS in Math.

You, the teacher, cannot reach them all. Leave the door open, realize every misstep is a chance to teach them, but learning is done on their end, not ours.

If a kid fails, let them. Work with them to succeed, but hold firm to the standard. If you falter, and pass a student that didn't deserve it, the value of the accomplishments of other students will be discounted.

Why I'm sharing this is to color this short story:

The last three years I had 100% passing rate by all takers, not cherry picking, on IGCSE, around 10% passing rate in AZ. This year I'm pretty sure at least one student will fail. They earned the first F grade I have assigned in six years in that class.

That student just wasn't ready. At the end, the student came begging to get a passing grade. I explained to the student that while they were close to passing, to change their grade would be a grand insult...it would say that I did not believe they were capable of performing as well as their peers.

The next day the student approached me. I thought, ut oh, more grade grabbing negotiation...but to the student's credit, they just thanked me, said they're glad for the F and will do better in the future. No more crying, no hang-dog look...but instead a confidence because the student was capable and will be in the future. Perhaps now, the student is ready.

I don't want students to say, I only got through math because of you, Mr. Brown. That would make math the destination, not the vehicle. Best compliment a teacher can get is, you taught me to learn.

## 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.



27

2

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIYaGaaG4namaaCaaaleqabaGaaGOmaaaaaeaacaaIZaaaaaaa@392A@

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.

of calculation. Turns out if we rewrite
this expression with a rational exponent, life gets easier.



27

2

3

=

27

2/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIYaGaaG4namaaCaaaleqabaGaaGOmaaaaaeaacaaIZaaaaOGaeyyp
a@3E10@

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

.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaGinaaaakiaac6caaaa@385D@

If you don’t know that yet, a cheat sheet

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

(

3
3

)

2/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
baGaaG4mamaaCaaaleqabaGaaG4maaaaaOGaayjkaiaawMcaamaaCa
aaleqabaGaaGOmaiaac+cacaaIZaaaaaaa@4157@

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.



(

3
3

)

2/3

3
2

=9

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaaIYaGaai4laiaaiodaaaGccqGHsgIRcaaIZaWaaWbaaSqabe
aacaaIYaaaaOGaeyypa0JaaGyoaaaa@40FA@

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

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
dacaaI1aWaaWbaaSqabeaacaaIZaGaai4laiaaisdaaaaaaa@3A8D@

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.



(

5
4

)

3/4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaaIZaGaai4laiaaisdaaaaaaa@3B8F@

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

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

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.



(

2
5

)

3/4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIYaWaaWbaaSqabeaacaaI1aaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaaIZaGaai4laiaaisdaaaaaaa@3B8D@

If we multiplied these exponents together we end up
with something that isn’t so pretty,



2

15/4

.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaGymaiaaiwdacaGGVaGaaGinaaaakiaac6caaaa@3A87@

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.



(

2
5

)

3/4

=

(

2
1

2
4

)

3/4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIYaWaaWbaaSqabeaacaaI1aaaaaGccaGLOaGaayzkaaWaaWbaaSqa
ahaaWcbeqaaiaaigdaaaGccqGHflY1caaIYaWaaWbaaSqabeaacaaI
0aaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaGaai4laiaais
daaaaaaa@462A@

Now we’d have to multiply the exponents inside the
parenthesis by


¾
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaaaaaaWdbiaa=5laaaa@384E@

,
and will arrive at:



2

3/4

2
3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaG4maiaac+cacaaI0aaaaOGaeyyXICTaaGOmamaaCaaa
leqabaGaaG4maaaaaaa@3D08@

Notice that



2

3/4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaleqabaGaaG4maiaac+cacaaI0aaaaaaa@390E@

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

2
3

4

,or8
8
4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aabaGaaGOmamaaCaaaleqabaGaaG4maaaaaeaacaaI0aaaaOGaaiil
qaaiaaiIdaaSqaaiaaisdaaaaaaa@445B@

.

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?



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

Let’s look at



16

3/4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

. 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

=

(

2
4

)

3/4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
baGaaGOmamaaCaaaleqabaGaaGinaaaaaOGaayjkaiaawMcaamaaCa
aaleqabaGaaG4maiaac+cacaaI0aaaaaaa@4072@



(

2
4

)

3/4

=
2

4
1

×
3
4

=
2
3

,or8.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIYaWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaaIZaGaai4laiaaisdaaaGccqGH9aqpcaaIYaWaaWbaaSqabe
aaqaaiaaisdaaaaaaOGaeyypa0JaaGOmamaaCaaaleqabaGaaG4maa
4aGaaiOlaaaa@4FAB@

Procedure 2:



16

3/4

=

16

3

4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
a@3E10@



16

3

4

=

(

2
4

)

3

4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIXaGaaGOnamaaCaaaleqabaGaaG4maaaaaeaacaaI0aaaaOGaeyyp



(

2
4

)

3

4

=

2
4

4

×

2
4

4

×

2
4

4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
daahaaWcbeqaaiaaiodaaaaabaGaaGinaaaakiabg2da9maakeaaba
GaaGOmamaaCaaaleqabaGaaGinaaaaaeaacaaI0aaaaOGaey41aq7a
aOqaaeaacaaIYaWaaWbaaSqabeaacaaI0aaaaaqaaiaaisdaaaGccq
aaaaaaa@479A@



2
4

4

×

2
4

4

×

2
4

4

=2×2×2,or8.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIYaWaaWbaaSqabeaacaaI0aaaaaqaaiaaisdaaaGccqGHxdaTdaGc
na0oaakeaabaGaaGOmamaaCaaaleqabaGaaGinaaaaaeaacaaI0aaa
aOGaeyypa0JaaGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGaaiilai
aiOlaaaa@54D0@

The most elegant way is to realize the 16 is the
fourth power of 2, and the fraction


¾
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aaaaaaaaWdbiaa=5laaaa@384E@

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

.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccaGGUaaa
aa@3B47@

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

=

625

2

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqp
aacaaIZaaaaaaa@3F8C@



625

2

3

=

(

5
4

)

2

3

5
8

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
kiabg2da9maakeaabaWaaeWaaeaacaaI1aWaaWbaaSqabeaacaaI0a
aaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaqaaiaaioda
GaaG4maaaaaaa@445D@



5
8

3

=

5
3

×
5
3

×
5
2

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aWaaWbaaSqabeaacaaI4aaaaaqaaiaaiodaaaGccqGH9aqpdaGc
baaSqabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqabaGaaGOm
aaaaaeaacaaIZaaaaaaa@438B@



5
3

×
5
3

×
5
2

3

=5×5

25

3

,or25

25

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqa
GaaG4maaaakiabg2da9iaaiwdacqGHxdaTcaaI1aWaaOqaaeaacaaI
YaGaaGynaaWcbaGaaG4maaaakiaacYcacaaMc8UaaGPaVlaaykW7ca
WGVbGaamOCaiaaykW7caaMc8UaaGPaVlaaikdacaaI1aWaaOqaaeaa

Pretty
ugly!

Exponential Notation:



625

2/3

=

(

5
4

)

2/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqp



(

5
4

)

2/3

=

(

5
3

×
5
1

)

2/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa
ahaaWcbeqaaiaaiodaaaGccqGHxdaTcaaI1aWaaWbaaSqabeaacaaI
XaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaGaai4laiaaio
daaaaaaa@45FA@



(

5
3

×
5
1

)

2/3

=
5
2

×
5

2/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqa
baGaaGymaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaiaac+
cacaaIZaaaaOGaeyypa0JaaGynamaaCaaaleqabaGaaGOmaaaakiab



5
2

×
5

2/3

=25×
5

2/3

,or25

25

3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
caGGVaGaaG4maaaakiabg2da9iaaikdacaaI1aGaey41aqRaaGynam
aaCaaaleqabaGaaGOmaiaac+cacaaIZaaaaOGaaiilaiaaykW7caaM

A little better, but still a few sticky points.

Now our third method.



625

2/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaaaaa@3A8B@

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

=

(

125×5

)

2/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqp

This could also be written as:



125

2/3

×
5

2/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGHxdaT
caaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaaaaa@3FBF@

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

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGHxdaT
caaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqpca
aG4maaaaaaa@4779@

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:



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x



Simplify the following:

1.
(

16
x

16

)

3/4

2.
128

5/6

3.

125

3

5

4.
32

3/5

5.
(

81
x

27

)

2/3

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
GaaeyAaiaab2gacaqGWbGaaeiBaiaabMgacaqGMbGaaeyEaiaabcca
caqG0bGaaeiAaiaabwgacaqGGaGaaeOzaiaab+gacaqGSbGaaeiBai
aab+gacaqG3bGaaeyAaiaab6gacaqGNbGaaeOoaaqaaiaabgdacaqG
iodacaGGVaGaaGinaaaaaOqaaaqaaaqaaiaaikdacaGGUaGaaGPaVl
aaykW7caaMc8UaaGymaiaaikdacaaI4aWaaWbaaSqabeaacaaI1aGa
aMc8UaaGPaVpaakeaabaGaaGymaiaaikdacaaI1aWaaWbaaSqabeaa
caaIZaaaaaqaaiaaiwdaaaaakeaaaeaaaeaaaeaacaaI0aGaaiOlai
aaykW7caaMc8UaaGPaVlaaiodacaaIYaWaaWbaaSqabeaacaaIZaGa
ai4laiaaiwdaaaaakeaaaeaaaeaaaeaacaaI1aGaaiOlaiaaykW7ca
caaI3aaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaGaai4lai
aaiodaaaaaaaa@80AF@

## 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:



f(
x
)
=3x
x
2

+1

f(
2
)

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa

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:



f(
2
)
=3(
2
)

2
2

+1

f(
2
)
=6+4+1

f(
2
)
=11

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaeyypa0JaaG4mamaabmaa
aaikdaaaGccqGHRaWkcaaIXaaabaGaamOzamaabmaabaGaaGOmaaGa
caaIXaGaaGymaaaaaa@4F4E@

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



f(
x
)
=3x
x
2

+1

f(
2
)
=3(
2
)
+
(

2

)

2

+1

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa
ZaWaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacq
GHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa
ey4kaSIaaGymaaaaaa@4E85@

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

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

## 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



8
2

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI4aaabaGaaGOmaaaaaaa@3785@

. 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.

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

It is
true that



8
2

=4,

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI4aaabaGaaGOmaaaacqGH9aqpcaaI0aGaaiilaaaa@39F9@

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:



9
x
2

3x

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI5aGaamiEamaaCaaaleqabaGaaGOmaaaaaOqaaiaaiodacaWG4baa
aaaa@3A74@

.

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



9
x
2

3x

=3x,

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI5aGaamiEamaaCaaaleqabaGaaGOmaaaaaOqaaiaaiodacaWG4baa
aiabg2da9iaaiodacaWG4bGaaiilaaaa@3DE4@

because



3x3x=9
x
2

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
hacqGHflY1caaIZaGaamiEaiabg2da9iaaiMdacaWG4bWaaWbaaSqa
beaacaaIYaaaaaaa@3F64@

.

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:



5
0

0
5

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aaabaGaaGimaaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa
ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG
PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM
c8+aaSaaaeaacaaIWaaabaGaaGynaaaaaaa@5AFB@

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.



5
0

Not a Number
0
5

Rational

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aaabaGaaGimaaaacaaMc8UaaGPaVlaaykW7cqGHsgIRcaqGGaGa
aeOtaiaab+gacaqG0bGaaeiiaiaabggacaqGGaGaaeOtaiaabwhaca
qGTbGaaeOyaiaabwgacaqGYbGaaGPaVlaaykW7caaMc8UaaGPaVlaa
ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG
PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaalaaabaGa
aGimaaqaaiaaiwdaaaGaeyOKH4QaaeOuaiaabggacaqG0bGaaeyAai
aab+gacaqGUbGaaeyyaiaabYgaaaa@7129@

Repeating Decimals
Written as Fractions

Consider
the fraction



1
3

.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIXaaabaGaaG4maaaacaGGUaaaaa@3831@

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.
3
¯

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

(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.

27

¯

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

.

 We don’t know what number, as a fraction is  0. 27 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cadaqdaaqaaiaaikdacaaI3aaaaaaa@38F1@ , so we will write the unknown x.  x=0. 27 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaicdacaGGUaWaa0aaaeaacaaIYaGaaG4naaaaaaa@3AF4@ Since  0. 27 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cadaqdaaqaaiaaikdacaaI3aaaaaaa@38F1@ 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. 27 ¯ ×100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeG+aaaaaai vzKbWdbiaaigdacaaIWaGaaGimaiabgEna0+aacaWG4bGaeyypa0Ja aGimaiaac6cadaqdaaqaaiaaikdacaaI3aaaa8qacqGHxdaTcaaIXa GaaGimaiaaicdaaaa@45C9@    100x=27. 27 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaaIWaGaamiEaiabg2da9iaaikdacaaI3aGaaiOlamaanaaabaGa aGOmaiaaiEdaaaaaaa@3DE6@ 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. 27 ¯ −0. 27 ¯ =27 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaiE dacaGGUaWaa0aaaeaacaaIYaGaaG4naaaacqGHsislcaaIWaGaaiOl amaanaaabaGaaGOmaiaaiEdaaaGaeyypa0JaaGOmaiaaiEdaaaa@401E@  100x=27. 27 ¯ −( x=     0. 27 ¯ ) _      99x=27 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaadaa abaeqabaGaaGymaiaaicdacaaIWaGaamiEaiabg2da9iaaikdacaaI 3aGaaiOlamaanaaabaGaaGOmaiaaiEdaaaaabaGaeyOeI0YaaeWaae aacaWG4bGaeyypa0JaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaI WaGaaiOlamaanaaabaGaaGOmaiaaiEdaaaaacaGLOaGaayzkaaaaaa GaaGPaVdqaaiaaykW7caaMc8UaaGPaVlaaiMdacaaI5aGaamiEaiab g2da9iaaikdacaaI3aaaaaa@585F@ Divide both sides by 99 to solve for x.   Recall that x was originally defined as the fractional equivalent of the repeating decimal.  x= 27 99 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9maalaaabaGaaGOmaiaaiEdaaeaacaaI5aGaaGyoaaaaaaa@3B0D@

Practice
Problems.

1.
Change the following into rational numbers:

a.


5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

b.
0

c.



3

0.4

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aIZaaabaGaaGimaiaac6cacaaI0aaaaaaa@38EE@

d.



0.

23

¯

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x

2.
Why
is a the following called undefined:



a
0

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
WGHbaabaGaaGimaaaaaaa@37A7@

?

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

a.
0 b. 9
c. -5 d.



5.37
9
¯

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
cacaaIZaGaaG4namaanaaabaGaaGyoaaaaaaa@39BA@

e.



5
π

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
aI1aaabaGaeqiWdahaaaaa@3883@

f.
5.47281…

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

5.
What
number is whole but not natural?