Our Youth Deserve Better – Computer Based Learning

There has been a push for computer-based learning in public education for about a decade or so now.  The thinking is that students can go at their own pace, have optimally focused and differentiated remediation and instruction, and thus, students will perform better.  That’s the sales pitch, anyway.

I teach remedial math courses part time at a community college (the observations made here pertain to all of education not just math), the shift was made so that 100% of these remedial math courses were taught on such computer programs.  Students take placement tests where their strengths and weaknesses are accurately identified and they then work their way through lessons and assignments, with help along that way that addresses their specific short-comings.  If students grasp something easily they can move quickly through the curriculum.  Students that need more time can go at their own pace.  At the end of the section (or chapter), students take a test and must show a predetermined level of accuracy before they’re allowed to move forward.

It sounds great, but it doesn’t work.  Even if it did work and students could pass these classes in a way that prepared them for higher level classes, it would be a failure.   The purpose of education is not future education.

The ugly truth here is that we’ve lost sight of the purpose of education.  Education has become a numbers game where schools receive funding based on graduation rates and percentages of students passing multiple choice tests that have mysterious grading schemes behind them (70 multiple choice questions will be graded on a scale of 450 points, for example).  We lull ourselves into believing we are servicing our students if they graduate or our school surpasses the state average on these tests.

The truth is that the quality of education is rapidly decreasing, seemingly in direct response to the remedies that seek to reverse this trend.

The question often asked by students, in minor rebellion to the tasks at hand in class, “When am I going to use this in my real life,” needs to be carefully considered, with honesty, by the public and by educators.

The particular skills and facts being tested are of little to no importance.  What is important is the ability to be teachable, the ability to learn, which requires a lot of maturation, determination, focus and effort.  The purpose of education is to create an adaptable person that can readily latch onto pertinent information and apply previous learning in new ways.  An educated person should have the skills to adapt to an unknown future, a future where they are empowered to make decisions about the direction of their own lives.

Absolutely none of that happens in a computer course.  The problems are static, scripted and the programs are full of basic “If-Then” commands.  If a student misses this question, send them here.  There’s no interpretation of why a student missed.  There’s no consideration of the student as a sentient being, but instead they are reduced to a right or a wrong response.

What do students gain from computer courses?  They gain those specific skills, the exact skills and knowledge that will serve little to no purpose at all in their lives after school.  But, they’ll gain those skills in a setting with a higher student-teacher ratio (fewer teachers, less students), and where the teachers need not know the subject or how to teach.  That’s right, it’s cheaper!

But the cost is enormous.  Students will be trained how to pass tests on the computer, but will not be receiving an education. They will not develop the interpersonal skills required to be successful in college or in the work place.  They will not develop as people.  They will miss the experiences that separate education from training.  They will be raised by computers that try to distill education down to right and wrong answers, where reward is offered for reciting facts and information without analysis, without learning to consider opposing points of view, without learning how to be challenged on what it is they think and believe.

Our youth deserve better.  They deserve more.

Not only that, our young teachers (and we have an increasingly inexperienced work force in education), deserve better support from within education.  Here in Arizona the attitude from the government is that the act of teaching has little to no value, certainly little to no skill, and that anybody can step in and perform the duties of teaching in a way that services the needs of young people.

And while those in education throw their hands up in disgust, they follow suit by finding quick, easy and cheap solutions to the ever-expanding problem of lack of quality education, especially here in Arizona.  Instead of providing meaningful professional development and support for teachers, teachers are blamed for their short comings.  Instead of being coached and developed, they are being replaced by something cheaper and quicker, something that is fully compliant.

I fully believe that a teacher that can be replaced by a computer should be.  I also believe that a computer cannot provide the inspiration, motivation, the example, mentorsing and support that young people need.

The objection to my point of view is that teachers aren’t being replaced, they are still in contact with students.  This is true, the contact exists, but in a different capacity.  Just like iPads haven’t replaced parents, the quality of parenting has suffered.  The appeal of having a child engaged, and not misbehaving, because they are on a computer, or iPad, is undeniable.  But the purpose of parenting is not to find ways for children to leave them alone.  Similar, the role of education is to to find ways to get kids to sit down and pass multiple tests.  Children are difficult to deal with.  Limiting that difficulty does not mean you are better fulfilling your duty to the young!

The role of a teacher in a computer-based course is far removed from the role of a teacher in a traditional classroom.  While students are “learning” from a computer, the role of the “teacher” is to monitor for cheating and to make sure students stay off of social media sites.  Sometimes policies are in place where teachers quantitatively evaluate the amount of notes a student has taken to help it seem like a student is performing student-like tasks.

Students learning on computer are policed by teachers.  The relationship becomes one of subjects being compliant with authority.

The most powerful tool a teacher has is the human connection with students.  That connection can help a student that sees no value in studying History appreciate the meaning behind those list of events in the textbook.  A teacher can contextualize and make relevant information inaccessible to young learners, opening up a new world of thinking and appreciation for them.  None of that is tested of course.

A teacher inspired me to become a math teacher, not because of her passion for math, but because of how she conducted her business as a teacher.  Before that I wished to work in the Game and Fish Department, perhaps as a game warden.  That would have been a wonderful career.  Consider though, over the last decade, I have had countless students express their appreciation of how I changed their thinking about math, how I made it something dynamic and fluid, something human.  Math went from a barrier, in the way of dreams, to a platform, upon which successful can be realized.  Those things happened because of human connection.

We owe our youth more.  They deserve better.

It is time to unplug.

Why Remediation Fails

Why Remediation Fails

Students that struggle unwittingly do two things that ensure they continue to struggle with concepts and procedures.  Students can go to tutoring over and again, and sometimes it works, but it’s a long and frustrating journey.

I’ve fallen victim to these two habits myself, we all have.  How students learn in school is not any different than how adults learn outside of school.  Learning is identifying something that’s wrong and replacing it with something that is right, or at least more efficient.

It is the act of identifying something that is wrong that is the hitch here, the hold up.  The first of the things students do when presented with remediation, that is review materials or a review of what went wrong before, is they morph what they’re seeing to fit what they know.  Of course if they did that the other direction, things would be great.  But that’s not how we learn.

It is imperative to recognize that we develop new learning by relating it to old knowledge.  We don’t just replace all that we’ve developed over time with this new thing.  Instead, we create connections between what’s already in our noggins and what is new.  The more connections we have, the stronger the new learning is and the more quickly it happens.

Consider someone learning to cook.  Say, they learned that Worcestershire sauce is yummy and delicious on steak.  Some spills over into potatoes and that’s not too bad either.  It’s not even unpleasant when it mixes with green beans or broccoli. With some experimentation we can learn that it’s good with chicken, rice and mushrooms.

What’s the thing we know?  Worcestershire sauce makes things taste good.  Not wrong, but not a very deep understanding, right?

Now let’s say this person want to make some desserts.  Someone hands them some cream and tells them to whip it up, so it can top a pie.  Why, they might ask.  Well, to make the pie better, of course.

This whipped cream is new information, it’s something different than what they know.  It’s fundamentally different than Worcestershire sauce.  Yet, whipped cream is supposed to make food better, just like Worcestershire sauce does.  So what students do, in effect, is say, oh, whipped cream is the same as Worcestershire sauce, and I’m used to Worcestershire, so let’s just use that instead.  Same thing after all, right?

A similar thing happens when trying to train someone to use the computer.  They know how to do a set of things and try to use those processes to manipulate this new software.

That is, instead of seeing the new protocol for interfacing with the software as completely new, they instead relate it to what they had done in the past.  They fail to replace old knowledge with new.  Instead, they see the new information as the same thing as what they already have at hand.

How do we, as teachers, combat that phenomenon?  Well, we have to expose what they believe as fundamentally different than what’s right.  We have to expose their misconception as being, well, a misconception that is not aligned with reality.

That’s a tricky thing to do, especially in math, for two reasons.  The first reason is that often in math we are dealing with abstractions.  We can’t have them taste Worcestershire topped cherry pie.  The second reason, especially for math, is that when students see a procedure performed, they feel they understand if they believe they’re able to follow that procedure. (That is not that they are able to perform the procedure themselves.)

That second reason that it is tricky to expose misconception is the second thing that students do, they latch onto procedure.  It makes them feel grounded, even if they are obviously off-base!

How many times has this happened?  You, as the teacher, review a quiz question with students.  They sit there, take notes as you work through a problem.  They all exclaim they can’t believe how dumb they are, how could they have missed that?  They get it now, right?

No.  They don’t.  They followed what you did, you doing all of the thinking along the way.  A large percentage of students will be no better off than before the review.  In some ways, some will be worse because they’ll now think they understand.  Before the review, they just knew they were wrong, probably had no idea why.

What can we do?

This is a tricky thing to answer, dependent on too many variables to articulate a clean protocol.  However, I think I have some ideas that will help in general.

First, when developing a review lesson, test or quiz review, or remediation lesson, you need to have students confront some mistakes.  Maybe they need to try a problem and get it wrong.

Once the misconception is exposed, address why it’s wrong, what’s wrong with it.  Don’t discuss what is right immediately, they’ll translate that to fit what they believe (and that is wrong).  Expose why the misconception is in fact wrong, on a fundamental level.

Next, if possible, arrive at the right conclusion without process or procedure.  Is there a way to think through the conception at play and arrive at what is right?  If so, that’s beautiful.

The last thing is that this new learning will be soft in their heads, a fragile thing.  They need to make a record of what they’ve learned, in their own writing, preferably on the old quiz or next to the thing they used to believe was true.  It’ll be a reminder, because they’ll go for that Worcestershire sauce again when they shouldn’t!  Old habits, they die hard!

I tried something along these lines in a video I prepared for a remedial math class at a community college.  The topic is fractions.  I tried to show how common denominators work without treating them like they were stupid, because they’re not, they just never had to learn fractions, and tried to do so without use of a process.

As I explored the inner workings, and why various things were wrong, I began describing what needed to be done, but the focus was conceptual.  The video is posted here at the end of this article.

This is a topic I hope to explore more in detail, how to help promote the efficacy of remediation and tutoring.  I am working on some experiments I’d like to try to determine more closely the behind the scenes workings here.  Until that time, thank you for reading, thank you for your time.

Philip Brown

Try to Solve This Problem … without Algebra

Can you solve the following, without doing any Algebraic manipulation?  Just by reading and thinking about what it says, can you figure out what x is?  (The numbers a, x, andare not zero.)

Given:  3ak

And:  ax = 4k

What must x be?

If you’re versed much at all in basic Algebra you will be tempted to substitute and solve.  After all, this is a system of equations.  But that will bypass the purpose and benefit of the exercise.

The intended benefit of this problem is that it promotes mathematical literacy, in particular, seeing relationships between terms.  It’s not a complicated relationship but it is of utmost importance to this problem.  Once you read and make sense of what the mathematical relationships are you can talk your way through the problem.

Once again, I believe the purpose of homework is learning.  Sure, sometimes it is practice and familiarity, but those are the only times that answer-getting is important.  Without understanding, having the right answer is often of little to no use.  If it were, then copying the answers from the back of the book would be sufficient for learning, right?

If you’re ready to see the solution, you can watch the video or read the text after the video.

I understand that sometimes it’s appropriate to read, but not listen or watch a video.  So here’s how this works.

Given:  3ak

And:  ax = 4k

The first statement says that the number k is three times bigger than the number a.  We don’t know what or k are but we know how they’re related and can think of lots of numbers that fit this relationship.  One number that’s three times as large as the other.

The number k is three times as big as the number a.

Think of this relationship one more way, for a moment.  The number k has two factors, 3 and a.  Whether is composite or prime is irrelevant really, it won’t change the fact that we could write k as the product of two numbers.  I mention this, not because it helps solve this problem but because it might.  Without knowing the path, sometimes it is a good idea to brain storm for a  moment and list as many things you know about the information given, before seeking an answer.  Sometimes, doing so, makes the answer apparent to you!

Let’s look at the second statement now.

Another number times a is four times as big as k.  This is perhaps a bit distracting, but the key information is there.  Remember, k is three times as big as a.  Now we have something four times larger than k.

Let’s look at this a different way.  The number 4k is not k at all, but instead, k and 4 are factors of new number.

If this new number is four times larger than k, and k is three times larger than a, how much larger is this new number than a?

You have three times as much money as me.  Bobert has four times as much as you do.  How much more money does Bob have than me?

For every dollar I have you have three.  For every dollar you have, Bobert has four.

Still don’t see it?  I know…picture good, word bad.  Here you go.

If 3ak, and ax = 4k, then is 12 because

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?

Now, let’s see some fractional exponents.  They mean the same thing with one change...instead of asking about repeated addition they’re asking about repeated multiplication.

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.

Why Does the Order of Operations Work?

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

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

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

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

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

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

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

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

For example:

4 baskets of 7 vials each

4 × 7 = 28

or

7 + 7 + 7 + 7 = 28

or

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

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

4 × 7 + 6

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

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

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

Exponents are repeated multiplication, of the same thing!

Consider:

3 + 6

4 × 7 = 7 + 7 + 7 + 7

74 = 7 × 7 × 7 × 7

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

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

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

So, let’s look at:

9 + 4 × 74

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

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

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

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

(9 + 4) × 74

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

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

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

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

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

Is Infinity Real?

How Many Primes are There Is Infinity Real Part 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The gaps between these prime numbers are below.

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

Another string would be:

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

The gaps here are listed below.

4, 8, 10, 8, 4, 6

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

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

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

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

Why all the fuss? What good are they?

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

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

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

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

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

The differences here are as follows.

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

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

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

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

Fact 1:  5! = 5×4×3×2 = 120

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

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

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

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

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

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

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

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

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

Then x! + a is composite.

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

Crazy, right?

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

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

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

3,756,801,695,685 × 2666,689 – 1
and

3,756,801,695,685 × 2666,689 +1

Those numbers are too large to be written out!

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

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

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

What are your thoughts?  Please share them in the comments below.

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

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

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

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

The largest prime number:  https://www.youtube.com/watch?v=lEvXcTYqtKU

Large Gaps Between Primes:  https://www.youtube.com/watch?v=BH1GMGDYndo

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

www.patreon.com/beardedmathman

Cube Roots and Higher Order Roots

other roots

Cube Roots
and

Square roots ask what squared is the radicand. A geometric explanation is that given the area of a square, what’s the side length? A geometric explanation of a cube root is given the volume of a cube, what’s the side length. The way you find the volume of a cube is multiply the length by itself three times (cube it).

The way we write cube root is similar to square roots, with one very big difference, the index.

$\begin{array}{l}\sqrt{a}\to \text{\hspace{0.17em}}\text{\hspace{0.17em}}square\text{\hspace{0.17em}}\text{\hspace{0.17em}}root\\ \sqrt[3]{a}\to \text{\hspace{0.17em}}\text{\hspace{0.17em}}cube\text{\hspace{0.17em}}\text{\hspace{0.17em}}root\end{array}$

There actually is an index for a square root, but we don’t write the two. It is just assumed to be there.

Warning: When writing cube roots, or other roots, be careful to write the index in the proper place. If not, what you will write will look like multiplication and you can confuse yourself. When writing by hand, this is an easy thing to do.

$3\sqrt{8}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt[3]{8}$

To simplify a square root you factor the radicand and look for the largest perfect square. To simplify a cubed root you factor the radicand and find the largest perfect cube. A perfect cube is a number times itself three times. The first ten are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000.

Let’s see an example:

Simplify:

$\sqrt[3]{16}$

Factor the radicand, 16, find the largest perfect cube, which is 8.

$\sqrt[3]{8}×\sqrt[3]{2}$

The cube root of eight is just two.

$2\sqrt[3]{2}$

The following is true,

$\sqrt[3]{16}=2\sqrt[3]{2}$,

only if

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

Arithmetic with other radicals, like cube roots, work the same as they do with square roots. We will multiply the rational numbers together, then the irrational numbers together, and then see if simplification can occur.

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

Two cubed is just eight and the cube root of two cubed is the cube root of eight.

${2}^{3}×{\left(\sqrt[3]{2}\right)}^{3}=8×\left(\sqrt[3]{2}\right)\left(\sqrt[3]{2}\right)\left(\sqrt[3]{2}\right)$

${2}^{3}×{\left(\sqrt[3]{2}\right)}^{3}=8×\sqrt[3]{2\cdot 2\cdot 2}$

${2}^{3}×{\left(\sqrt[3]{2}\right)}^{3}=8×\sqrt[3]{8}$

The cube root of eight is just two.

$8×\sqrt[3]{8}=8×2$

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

Negatives and cube roots: The square root of a negative number is imagery. There isn’t a real number times itself that is negative because, well a negative squared is positive. Cubed numbers, though, can be negative.

$-3×-3×-3=-27$

So the cube root of a negative number is, well, a negative number.

$\sqrt[3]{-27}=-3$

Other indices (plural of index): The index tells you what power of a base to look for. For example, the 6th root is looking for a perfect 6th number, like 64. Sixty four is two to the sixth power.

$\sqrt[6]{64}=2\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}because\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{2}^{6}=64.$

A few points to make clear.

·         If the index is even and the radicand is negative, the number is irrational.

·         If the radicand does not contain a factor that is a perfect power of the index, the number is irrational

·         All operations, including rationalizing the denominator, work just as they do with square roots.

Rationalizing the Denominator:

Consider the following:

$\frac{9}{\sqrt[3]{3}}$

If we multiply by the cube root of three, we get this:

$\frac{9}{\sqrt[3]{3}}\cdot \frac{\sqrt[3]{3}}{\sqrt[3]{3}}=\frac{9\sqrt[3]{3}}{\sqrt[3]{9}}$

Since 9 is not a perfect cube, the denominator is still irrational. Instead, we need to multiply by the cube root of nine.

$\frac{9}{\sqrt[3]{3}}\cdot \frac{\sqrt[3]{9}}{\sqrt[3]{9}}=\frac{9\sqrt[3]{9}}{\sqrt[3]{27}}$

Since twenty seven is a perfect cube, this can be simplified.

$\frac{9}{\sqrt[3]{3}}=\frac{9\sqrt[3]{9}}{3}$

And always make sure to reduce if possible.

$\frac{9}{\sqrt[3]{3}}=3\sqrt[3]{9}$

This is a bit tricky, to be sure. The way the math is written does not offer us a clear insight into how to manage the situation. However, the topic we will see next, rational exponents, will make this much clearer.

Practice Problems:

Simplify or perform the indicated operations:

$\begin{array}{l}1.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt[4]{64}\\ \\ \\ 2.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt[3]{9}+4\sqrt[3]{9}\\ \\ \\ \\ 3.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt[3]{9}×4\sqrt[3]{9}\\ \\ \\ 4.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt[5]{64}\\ \\ \\ 5.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\sqrt{7}}{\sqrt[3]{7}}\end{array}$

Exponents Part 2

two

Exponents Part 2

Division

In the previous section we learned that exponents are repeated multiplication, which on its own is not tricky. What makes exponents tricky is determining what is a base and what is not for a given exponent. It is imperative that you really understand the material from the previous section before tackling what’s next. If you did not attempt the practice problems, you need to. Also watch the video that review them.

In this section we are going to see why anything to the power of zero is one and how to handle negative exponents, and why they mean division.

What Happens with Division and Exponents?

Consider the following expression, keeping in mind that the base is arbitrary, could be any number (except zero, which will be explained soon).

${3}^{5}$

This equals three times itself five total times:

${3}^{5}=3\cdot 3\cdot 3\cdot 3\cdot 3$

Now let’s divide this by 3. Note that 3 is just 31.

$\frac{{3}^{5}}{{3}^{1}}$

If we write this out to seek a pattern that we can use for a short-cut, we see the following:

$\frac{{3}^{5}}{{3}^{1}}=\frac{3\cdot 3\cdot 3\cdot 3\cdot 3}{3}$

If you recall how we explored reducing Algebraic Fractions, the order of division and multiplication can be rearranged, provided the division is written as multiplication of the reciprocal. That is how division is written here.

$\frac{{3}^{5}}{{3}^{1}}=\frac{3}{3}\cdot \frac{3\cdot 3\cdot 3\cdot 3}{1}$

And of course 3/3 is 1, so this reduces to:

$\frac{{3}^{5}}{{3}^{1}}=3\cdot 3\cdot 3\cdot 3={3}^{4}$

The short-cut is:

$\frac{{3}^{5}}{{3}^{1}}={3}^{5-1}={3}^{4}$

That is, if the bases are the same you can reduce. Reducing eliminates one of the bases that is being multiplied by itself from both the numerator and the denominator. A general form of the third short-cut is here:

Short-Cut 3: $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$

This might seem like a worthless observation, but this will help articulate the very issue that is going to cause trouble with exponents and division.

$\frac{{3}^{5}}{{3}^{1}}={3}^{5}÷{3}^{1}$ .

But that is different than

${3}^{1}÷{3}^{5}$

The expression above is the same as

$\frac{{3}^{1}}{{3}^{5}}$

This comes into play because

$\frac{{3}^{1}}{{3}^{5}}={3}^{1-5}$,

and 1 $–$ 5 = -4.

Negative Exponents?

In one sense, negative means opposite. Exponents mean multiplication, so a negative exponent is repeated division. This is absolutely true, but sometimes difficult to write out. Division is not as easy to write as multiplication.

Consider that 3-4 is 1 divided by 3, four times. 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3. But if we rewrite each of those ÷ 3 as multiplication by the reciprocal (1/3), it’s must cleaner and what happens with a negative exponent is easier to see.

$1÷3÷3÷3÷3\to 1\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}$

This is classically repeated multiplication. While one times itself any number of times is still one, let’s go ahead and write it out this time.

$1\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}\to 1\cdot {\left(\frac{1}{3}\right)}^{4}$

This could also be written:

$1\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}\to 1\cdot \frac{{1}^{4}}{{3}^{4}}$

The second expression is easier, but both are shown here to make sure you see they are the same.

Since 1 times 14 is just one, we can simplify this further to:

$1\cdot \frac{{1}^{4}}{{3}^{4}}=\frac{1}{{3}^{4}}.$

Negative exponents are repeated division. Since division is hard to write and manipulate, we will write negative exponents as multiplication of the reciprocal. In fact, if instructions say to simplify, you cannot have a negative exponent in your final answer. You must rewrite it as multiplication of the reciprocal. Sometimes that can get ugly. Consider the following:

$\frac{b}{{a}^{-5}}$

To keep this clean, let us consider separating this single fraction as the product of two rational expressions.

$\frac{b}{{a}^{-5}}=\frac{b}{1}\cdot \frac{1}{{a}^{-5}}$

The b is not a problem here, but the other rational expression is problematic. We need to multiply by the reciprocal of $\frac{1}{{a}^{-5}}$, which is just a5.

$\frac{b}{{a}^{-5}}=\frac{b}{1}\cdot \frac{{a}^{5}}{1}={a}^{5}b$.

This can also be considered a complex fraction, the likes of which we will see very soon. Let’s see how that works.

$\frac{b}{{a}^{-5}}$

Note: ${a}^{-5}=\frac{1}{{a}^{5}}$

Substituting this we get:

$\frac{b}{\frac{1}{{a}^{5}}}$

This is b divided by 1/a5.

$b÷\frac{1}{{a}^{5}}$

Let’s multiply by the reciprocal:

$b\cdot {a}^{5}$

Now we will rewrite it in alphabetical order (a good habit, for sure).

${a}^{5}b$

Let us consider one more example before we make our fourth short-cut. With this example we could actually apply our second short-cut, but it will not offer much insight into how these exponents work with division.

This is the trickiest of all of the ways in which exponents are manipulated, so it is worth the extra exploration.

$\frac{2{x}^{-2}{y}^{-5}z}{{2}^{-2}x{y}^{3}{z}^{-5}}$

As you see we have four separate bases. In order to simplify this expression we need one of each base (2, x, y, z), and all positive exponents. So let’s separate this into the product of four rational expressions, then simplify each.

$\frac{2{x}^{-2}{y}^{-5}z}{{2}^{-2}x{y}^{3}{z}^{-5}}\to \frac{2}{{2}^{-2}}\cdot \frac{{x}^{-2}}{x}\cdot \frac{{y}^{-5}}{{y}^{3}}\cdot \frac{z}{{z}^{-5}}$

The base of two first:

$\frac{2}{{2}^{-2}}\to 2÷{2}^{-2}$

We wrote it as division. What we will see is dividing is multiplication by the reciprocal, and then the negative exponent is also dividing, which is multiplication by the reciprocal. The reciprocal of the reciprocal is just the original. But watch what happens with the sign of the exponent.

First we will rewrite the negative exponent as repeated division.

$2÷\frac{1}{{2}^{2}}$

Now we will rewrite division as multiplication by the reciprocal.

$2\cdot {2}^{2}={2}^{3}$

Keep in mind, this is the same as 23/1.

We will offer similar treatment to the other bases.

Consider first $\frac{{x}^{-2}}{x}=\frac{{x}^{-2}}{1}\cdot \frac{1}{x}$

Negative exponents are division, so:

$\frac{{x}^{-2}}{x}=\frac{{x}^{-2}}{1}\cdot \frac{1}{x}$

Notice the x that is already dividing (in the denominator) does not change. It has a positive exponent, which means it is already written as division.

$\frac{{x}^{-2}}{1}\cdot \frac{1}{x}\to \frac{1}{{x}^{2}}\cdot \frac{1}{x}=\frac{1}{{x}^{3}}$

This is exactly how simplifying the y and z will operation.

$\frac{{2}^{3}}{1}\cdot \frac{1}{{x}^{2}\cdot x}\cdot \frac{1}{{y}^{5}\cdot {y}^{3}}\cdot \frac{z\cdot {z}^{5}}{1}$

Putting it all together:

$\frac{2{x}^{-2}{y}^{-5}z}{{2}^{-2}x{y}^{3}{z}^{-5}}=\frac{{2}^{3}{z}^{6}}{{x}^{3}{y}^{8}}$.

Short-Cut 4: Negative exponents are division, so they need to be rewritten as multiplication by writing the reciprocal and changing the sign of the exponent. The last common question is what happens to the negative sign for the reciprocal? What happens to the division sign here: $3÷5=3×\frac{1}{5}$. When you rewrite division you are writing it as multiplication. Positive exponents are repeated multiplication.

${a}^{-m}=\frac{1}{{a}^{m}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{{a}^{-m}}={a}^{m}$

This is the second to last thing we need to learn about exponents. However, a lot of practice is required to master them fully.

To see why anything to the power of zero is one, let’s consider:

${3}^{5}$

This equals three times itself five total times:

${3}^{5}=3"#x22C5;3\cdot 3\cdot 3\cdot 3$

Now let’s divide this by 35.

$\frac{{3}^{5}}{{3}^{5}}$

Without using short-cut 3, we have this:

$\frac{{3}^{5}}{{3}^{5}}=\frac{3\cdot 3\cdot 3\cdot 3\cdot 3}{3\cdot 3\cdot 3\cdot 3\cdot 3}=1$

Using short-cut 3, we have this:

$\frac{{3}^{5}}{{3}^{5}}={3}^{5-5}$

Five minutes five is zero:

${3}^{5-5}={3}^{0}$

Then 30 = 1.

Τhe 3 was an arbitrary base. This would work with any number except zero. You cannot divide by zero, it does not give us a number.

The beautiful thing about this is that no matter how ugly the base is, if the exponent is zero, the answer is just one. No need to simplify or perform calculation.

${\left(\frac{{3}^{2x-1}\cdot {e}^{\pi i}}{\sum _{n=1}^{\infty }\frac{1}{{n}^{2}}}\right)}^{0}=1$

Let’s take a quick look at all of our rules so far.

 Short-Cut Example ${a}^{m}\cdot {a}^{n}={a}^{m+n}$ ${5}^{8}\cdot 5={5}^{8+1}={5}^{9}$ ${\left({a}^{m}\right)}^{n}={a}^{mn}$ ${\left({7}^{2}\right)}^{5}={7}^{10}$ $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$ $\frac{{5}^{7}}{{5}^{2}}={5}^{7-2}={5}^{5}$ ${4}^{-3}=\frac{1}{{4}^{3}}$ ${a}^{0}=1$ 50 = 1

Let’s try some practice problems.

Instructions: Simplify the following.

1. ${\left({2}^{8}\right)}^{1/3}$ 2. $3{x}^{2}\cdot {\left(3{x}^{2}\right)}^{3}$

3. $\frac{5}{{5}^{m}}$ 4. $\frac{{5}^{2}{x}^{-3}{y}^{5}}{{5}^{-3}{x}^{-4}{y}^{-5}}$

5. $7÷7÷7÷7÷7÷7÷7÷7$ 6. $9{x}^{2}y÷9{x}^{2}y$

7. $9{x}^{2}y÷\left(9{x}^{2}y\right)$ 8. ${\left({x}^{2}\cdot 2{x}^{6}\right)}^{2}\cdot {\left({x}^{2}\cdot 2{x}^{6}\right)}^{-2}$

9. ${\left({a}^{m}\right)}^{n}\cdot {a}^{m}$ 10. $\frac{{\left(3{x}^{2}+4\right)}^{2}}{{\left(3{x}^{2}+4\right)}^{3}}$

Exponents Part 1

exponents part 1

Exponents Part 1

One of the biggest things to understand about math is how it is written. The spatial arrangement of characters is syntax. Syntax, in English, refers to the arrangements of words to convey meaning.

Exponents are just a way of writing repeated multiplication. If we are multiplying a number by itself repeatedly, we can use an exponent to tell how many times the number is being multiplied. That’s it. Nothing tricky exists with exponents, no new operations or concepts to tackle. If you’re familiar with multiplication and its properties, exponents should be accessible.

That said, it is not without its pitfalls. A balance between conceptual understanding and procedural short-cuts is needed to avoid those pitfalls. The only way to strike that balance is through a careful progression of exercises and examples. An answer-getting mentality will lead to big troubles with exponents. People wishing to learn how exponents work must seek understanding.

Let’s establish some facts that will come into play with this first part of exponents.

1.      Exponents are repeated multiplication

2.      Multiplication is repeated addition

3.      Addition is “skip” counting

To simplify simple expressions with exponents you only need to know a few short-cuts, but to recall and understand, we need more. These facts are important.

With an exponential expression we have a base, the number being multiplied by itself, and the exponent, the small number on the top right of the base which describes how many times the base is being multiplied by itself.

${a}^{5}$

The number a is the base. We don’t know what a is other than it is a number. It’s not a big deal that we don’t know exactly what number it is, we still know things about this expression.

Five is the exponent, which means there are five a’s, all multiplying together, like this: $a\cdot a\cdot a\cdot a\cdot a.$

Something to keep in mind is that this expression equals another number. Since we don’t know what a is, we cannot find out exactly what it is, but we do know it’s a perfect 5th power number, like 32. See, 25 = 32.

What if we had another number multiplying with a5, like this:

${a}^{5}\cdot {b}^{3}$

If we write this out, without the exponents we see we have 5 a’s and 3 b’s, all multiplying together. We don’t know what a or b equals, but we do know they’re multiplying so we could change the order of multiplication (commutative property) or group them together in anyway we wish (associative property) without changing the value.

${a}^{5}\cdot {b}^{3}=a\cdot a\cdot a\cdot a\cdot a\cdot b\cdot b\cdot b$

And these would be the same:

$\left(a\cdot a\right)\left(a\cdot a\cdot a\right)\left(b\cdot b\cdot b\right)$

$\left(a\cdot a\right)\left[\left(a\cdot a\cdot a\right)\left(b\cdot b\cdot b\right)\right]$

$\left(a\cdot a\right){\left[ab\right]}^{3}$

${a}^{2}{\left[ab\right]}^{3}$

This is true because the brackets group together the a and b, making them both the base. The brackets put them together. The base is ab, and the exponent is 3. This means we have ab multiplied by itself three times.

Keep in mind, these are steps but exploring how exponents work to help you learn to read the math for the intended meaning behind the spatial arrangement of bases, parenthesis and exponents.

Now, the bracketed expression above is different than ab3, which is $a\cdot b\cdot b\cdot b$.

${\left(ab\right)}^{3}\ne a{b}^{3}$

Let’s expand these exponents and see why this is:

${\left(ab\right)}^{3}\ne a{b}^{3}$

Write out the base ab times itself three times:

$\left(ab\right)\left(ab\right)\left(ab\right)\ne a\cdot b\cdot b\cdot b$

The commutative property of multiplication allows us to rearrange the order in which we multiply the a’s and b’s.

$a\cdot a\cdot a\cdot b\cdot b\cdot b\ne a\cdot b\cdot b\cdot b$

Rewriting this repeated multiplication we get:

${a}^{3}{b}^{3}\ne a{b}^{3}$

The following, though, is true:

$\left(a{b}^{3}\right)=a{b}^{3}$

On the right, the a has only an exponent of 1. If you do not see an exponent written, it is one. If we write it out we see:

$\left(a\cdot b\cdot b\cdot b\right)=a\cdot b\cdot b\cdot b$

In summary of this first exploration, the base can be tricky to see. Parenthesis group things together. An exponent written outside the parenthesis creates all of the terms inside the parenthesis as the base. But if numbers are multiplying, but not grouped, and one has an exponent, the exponent only belongs to the number just below it on the left. For example, $4{x}^{3},$ the four has an exponent of just one, while the x is being cubed.

Consider: ${\left(x+5\right)}^{3}.$ This means the base is x + 5 and it is multiplied by itself three times.

Repeated Multiplication Allows Us Some Short-Cuts

Consider the expression:

${a}^{3}×{a}^{2}.$

If we wrote this out, we would have:

$a\cdot a\cdot a×a\cdot a$.

(Note: In math we don’t use colors to differentiate between two things. A red a and a blue a are the same. These are colored to help us keep of track of what’s happening with each part of the expression.)

This is three a’s multiplying with another two a’s. That means there are five a’s multiplying.

${a}^{3}×{a}^{2}={a}^{5}$

Before we generalize this to find the short-cut, let us see something similar, but is a potential pitfall.

${a}^{3}×{b}^{2}$

If we write this out we get:

$a\cdot a\cdot a×b\cdot b$

This would not be an exponent of 5, in anyway. An exponent of five means the base is being multiplied by itself five times. Here we have an a as a base, and three of those multiplying, and a b as a base, and two of those multiplying. Not five of anything.

The common language is that if the bases are the same we can add the exponents. This is a hand short-cut, but if you forget where it comes from and why it is true, you’ll undoubtedly confuse it with some of the other short-cuts that follow.

Short-Cut 1: If the bases are the same you can add the exponents. This is true because exponents are repeated multiplication and the associative property says that the order in which you group things does not matter (when multiplying).

${a}^{m}×{a}^{n}={a}^{m+n}$

The second short-cut comes from groups and exponents.

${\left({a}^{3}\right)}^{2}$

This means the base is a3, and it is being multiplied by itself.

${a}^{3}×{a}^{3}$

Our previous short cut said that if the bases are the same, we can add the exponents because we are just adding how many of the base is being multiplied by itself.

${a}^{3}×{a}^{3}={a}^{3+3}={a}^{6}$

But this is not much of a short cut. Let us look at the original expression and the outcome and look for a pattern.

${\left({a}^{3}\right)}^{2}={a}^{6}$

Short-Cut 2: A power raised to another is multiplied.

${\left({a}^{m}\right)}^{n}={a}^{m×n}$

Be careful here, though:

$a{\left({b}^{3}{c}^{2}\right)}^{5}$ = $a{b}^{15}{c}^{10}$

Practice Problems

 1.       ${x}^{4}\cdot {x}^{2}$ 8. ${\left(5xy\right)}^{3}$ 2.       ${y}^{9}\cdot y$ 9. ${\left(8{m}^{4}\right)}^{2}\cdot {m}^{3}$ 3.       ${z}^{2}\cdot z\cdot {z}^{3}$ 10. ${\left(3{x}^{5}\right)}^{3}{\left({3}^{2}{x}^{7}\right)}^{2}$ 4.       ${\left({x}^{5}\right)}^{2}$ 11. $7{\left({7}^{2}{x}^{4}\right)}^{5}\cdot {7}^{3}{x}^{5}$ 5.       ${\left({y}^{4}\right)}^{6}$ 12. ${5}^{3}+{5}^{3}+{5}^{3}+{5}^{3}+{5}^{3}$ 6.       ${x}^{3}+{x}^{3}+{x}^{3}+{x}^{8}+{x}^{8}$ 13. ${3}^{2}\cdot 9$ 7.       ${4}^{x}+{4}^{x}+{4}^{x}$ 14. ${4}^{x}\cdot {4}^{x}\cdot {4}^{x}$

The Purpose of Homework and My Response

The purpose of homework is to promote learning.  That’s it.  It’s not a way to earn a grade or something to keep kids busy.  It’s also not something that just must be completed in order to stay out of trouble.  Homework is a chance to try things independently, make mistakes and explore the nature of those mistakes in order to better learn the material at hand.

If students are not learning from the homework, it is a waste of time and effort.  There are a few things that could cause students not to learn from the homework.  Even if the assignments are of high quality, without the reflection and correction piece, students will not learn much from homework.

Reflection and correction go together.  It’s not about getting right answers, but thinking about what caused mistakes, identifying misconceptions or procedural inefficiencies and replacing those.  To reflect a student should NOT erase their incorrect working but instead should write on their homework, in pen, what went wrong and what would have been better.

It is quite possible more can be learned when reviewing homework than any other time.  It is certainly a powerful experience.

Textbooks and videos, tutors and peer help offer little appropriate support to help make homework, or practice, meaningful.  Textbooks only provide correct answers, YouTube videos usually do similar treatment to topics as textbooks offer.

I wish to help students learn and believe that reviewing work that has been done is too powerful of an opportunity to pass.  The trick is, how can I provide reflection and insight when to someone I am not sitting with and talking to?  I think I can help provide this reflection piece by doing all of the practice problems myself on a document camera and discussing pitfalls and mistakes, as well as sharing my thinking about the problems as I tackle them.  Further, I can share typical mistakes I see from students as they are learning topics.

So as I develop the Algebra 1 content I will be working on adding videos and short written responses to the assignments to help students think about what they’ve done, its appropriateness, correctness and their level of understanding.