Favorite Technology

My Favorite Technology

With the invention of the radio came claims that, “This will revolutionize education, forever.”

Then came television, and more claims that, “This will revolutionize education, forever.”

Then came the VHS player.  You guessed it, more, “This will revolutionize education, forever.”

Then the internet came along, and louder than ever were the claims that, “This will revolutionize education, forever.”

In truth, all of those pieces of technology have revolutionized education.  Education is now, more than ever, about coming up with new ways to make information increasingly accessible and more engaging.  And, more kids than ever are starting college.  What’s not to love, right?

Well, there is plenty not to love.

The reason all of those pieces of technology were destined to change education forever was because they were going to allow experts in particular fields to communicate with students.  The thinking was that books and stuffy teachers were making learning unnecessarily difficult.  By allowing students to bypass the texts and teachers to gain access to the content to be learned, they’d learn better and faster.

It makes sense to me.  As an adult, if I want to learn about writing a blog, for example, I do a search on the internet and find some self-proclaimed blog expert.  I watch their videos, read their blogs for advice, and give it a shot!  Or, if I want to learn to change the air filter on a new car, and I can’t seem to figure it out myself, I look for videos on YouTube.  Technology like the internet has provided me with so much greater access to information that has enriched my life than was afforded before the internet.

That’s how it is supposed to work with students, too.  A kid might be stuck in Algebra 2; logarithms killing my grade, mister!  They look up “logarithms,” on the internet and there are tons of helpful videos.  The student learns how to do logarithms, and their grade is saved.

It sure seems like it is all on the up and up, right? Well…

With up to 60% of college freshmen needing remedial math classes, I’d say these revolutions have not had a positive outcome for students.

At this point you might be thinking, here’s another doomsday message: Kids these days are horrible, fear for the future.  I promise you, this is not a doomsday message.  Education needs to improve, and that’s what this blog is about.

Let’s take a step back and look at the example where I learned to change an air filter from a YouTube video.  Was I educated?  Was I trained?  What’s the difference?

There is a huge difference between training and education.  Training equips the trained with specific skills and knowledge that the trainer knows the learner will need, when they will need it, and how they will apply what they’ve been trained to do.  Training is what happens when you get a new job.

Training could be said to equip a person with a specialized tool.

Education is different.  People often complain why they weren’t taught certain practical skills in school.  The message is that education is worthless.

Education equips a person with the ability to find the specialized tool they need and then figure out how to use it.  While training prepares someone for a known task, education prepares someone for an unknown task.

When a student watched a video on the internet about logarithms are they being trained or educated?

If the intent of the video is to help a student complete homework and pass a quiz, then the person knows exactly what the student will need to be able to do, and when they’ll need to do it.

This is a seemingly subtle difference.  The difference between training and education is anything but subtle.  It is of massive consequence.  Why?

One attribute of an educated person is that they quickly incorporate new, more effective approaches.  By contrast, a trained person resists new methods, regardless of efficacy.  Education makes a person adaptable.

The reason that the radio, television, videos, and the internet have failed to improve educational outcomes is because they have not addressed the short-comings of a textbook.  All of these sources provide the same information, and use the same approach.  They disseminate information.

A good teacher entices curiosity, finds what motivates students to learn, and provides educational experiences for students.  That quality human connection is what makes education happen for students that are otherwise uninterested in being educated (which is an overwhelming majority).

There is such a massive push, with some much inertia behind it, to focus on comprehensible input, scaffolding, all of the components of teaching examined in isolation and treated with a leaning towards training a teacher instead of educating them about teaching, that it feels like quality teaching is becoming a lost art.  Maybe that’s a skewed perspective having only taught in Arizona, which by nearly every metric, is the worst state for education in the US.

What students need is a reconnection with their instructor.  The instructor needs to get in-tune with the needs, pace, and interests of the students.  PowerPoints, videos, SmartBoards, Chrome Books, and the like focus on the dissemination of information.

That is why my favorite piece of technology is the document camera.

Wait … hear me out.  I believe that it can be the most powerful piece of technology for a student in a math class.

The first reason why I love the document camera deals with how mathematics is a written, not spoken language.  The spatial arrangement of characters conveys meaning.  The way math is printed on paper, or a PowerPoint, and the way it is written on the board, is different than how math is written and performed on paper.  The physical parameters change the way we write.

In the image below you can see a lot of repeated information.  Some of the information is written mathematically, some of it is written in English, and there are arrows and annotation that connects the two.  These annotations are done in real time in response to questions from students and answers by students to my questions.

 

The way we write math greatly impacts how we perform the math.  This is an overt example, but I think it will make the point.  The first expression below is extremely difficult to deal with, while the second has the same meaning and is quite easily understood.

A bright student might realize to rewrite the first expression as the second.  But an average student will realize with the second that they only need to add the exponents, and they’re done.  This is not an example of how the interaction of math is different when writing on paper, versus typing.  What it does show is that how math is written greatly impacts the interpretation of the meaning.  That interpretation and translation occurs more naturally when written in real time compared to being typed.

What the example above does show is how writing mathematics drastically changes our interpretation of what is written. In effect, it rephrases the information.

What the document camera does is allow the teacher to show students, in real time, the mechanics of the mathematics, while allowing for discussion and annotation of the theory of the mathematics.  It does these things at a writing pace.

Here is a picture of a lesson in Algebra 1.

In this picture what you see is how a problem can be broken apart in response to what it is that the students in the classroom, at that very moment in time, are struggling with.

While this could have been addressed while writing on the board, or even in a PowerPoint lesson, it was more apparent to me, as the teacher, because I was going slower.  I was asking more questions, students were asking more questions.  Teaching with the document camera really can improve the dialogue between teachers and students, changing it from speaking to conversing.

While a conversation can be had over a YouTube video with students, or during a PowerPoint presentation, it is more difficult.  The pace is different; the engagement of the students is different.  When watching a video, or watching a PowerPoint, students are … watching.  If they begin writing, it is often dictation that’s being performed.

There is certainly a measure of dictation happening by students when engaged in a lesson delivered through a document camera.  However, the switch to addressing a question or point of confusion during a lesson in a way that students incorporate that response as a natural part of the lesson, happens naturally when using a document camera.

Consider a lesson about exponents.  No matter your teaching experience, you cannot anticipate all possible misconceptions, prior or actively developing, and dispel them pre-emptively.  Along the way there will be confusion and misunderstanding.  It is when the confusion is discussed, and properly addressed, that learning really takes place.

When that confusion is brought forward by the students, in a lesson delivered through a document camera, the question can be written, explored, answered and summarized in a way that feels natural for the students.  They’ll recognize this as part of the lesson, not a tangent.

In the picture below you will see a refocusing of a concept learned the day before.  In the day before this lesson, students really struggled to identify separate bases in one expression.  They could not distinguish between things like

Of course confusion is exposed and can be properly addressed in other delivery forms.  The message here is not that other methods are ineffective.  However, students typically view a diversion from the script as tangential to the lesson objectives.  They do not recognize that the diversion is the most important part of the learning.  How could it be when it doesn’t have pretty animations and bold, underlined font?

The last benefit of a document camera is pacing.  Students need think time.  The pace of delivering a message is slowed when you, the teacher, are essentially taking notes with the students.

This allows them to think about what is being written while they write it.  After all, you won’t be reading what you’re writing.  Instead, your writing will be a summary of what’s been said!

With the slower pace, which has a higher engagement because students are using the time to carefully take notes, comes better questions from students.  In response to these questions you can naturally annotate the notes throughout the lesson, highlighting the source of the confusion for the students continually.

What all of this means is that by using a document camera, a lot of the elements of quality teaching are naturally accessible.  The pace is naturally improved to match the needs of students, the dialogue is improved, the exploration of misunderstanding is seamlessly incorporated into the lesson itself, without feeling tangential to the learning.

And all of that, especially the exploration of misunderstanding, provides the teacher with opportunity to provide for students what technology cannot do.  It allows you to easily step into a role that you must carve out for yourself when using more advanced technology.  The most important function of the teacher is to entice interest in students, to discover their motivations and to teach them instead of cover material.

How many times has this happened:  You teach a lesson.  The lesson is organized, complete, you’re proud of how it is constructed and delivered.  The students seem okay.  But when they test, the results are horrible.

This is what happens when we focus too much on the material, too little on the students.  For me, anyway, the document camera really helps me to focus on the students.  This is especially true with low-achieving students.  They need more help, a slower pace, a more responsive teacher.  Low-achieving students are less adaptive, flexible, and have less inclination to explore and challenge their understanding independently.

I am not saying, of the document camera, “This will revolutionize education.”  The document camera, like all technology, is only as good as it is used.

What I have tried to show here is how the document camera naturally offers you opportunity to perform what cannot be scripted, what cannot be programmed into a computer, what need an expert on a video cannot fulfill.  Your role as a teacher is to teach students, not cover material.

Whatever technology you use will fail to be effective if it is not used in a way that furthers that connection between students and content.  If the technology only improves exposure to content, does not help students to engage with the content in a way that is challenging and builds conceptual understanding, then it, too, will be ineffective.

The take away is, there is not replacement for a good teacher.  Tools that are used to enhance what a good teacher provides for a student are great.  Tools that lose sight of what quality teaching is, ultimately, hamper the educational process and harm students.

 

This

Let’s talk about THE question in a teacher’s life … the baleful, “When am I going to use this in my real life?”

Yeah, that one.

The honest answer is probably, never… and they know it.  Why else ask, if not to subvert and diminish your role and purpose?  They don’t really want an answer.  What they want ... is just to watch you squirm, or to hear what B.S. you might spit out.  Either way, your class is now off the tracks!

However, it is a valid question, in its essence.  What is our purpose here?  It is a fair question, and one that needs to be answered.  And don’t be one of those people that thinks all content is applicable to daily life … it is NOT, nor should it be.  The purpose of education is not to train kids for every possible situation in life, but to equip them with an education so they can adapt to any possible situation.  The purpose of education is the development of the mind.  Sometimes we learn tricky things for the same reason some accountants lift heavy weights at the gym.

So let’s talk about how to change the answer to THE question.  Let’s turn it on its head.  Let’s answer the question: When am I going to use this in my real life, it in a way that swings the pendulum back in your direction.  Let’s answer the question in a way that stops the division and animosity that fosters the question in the first place.

See, the reality is, when a kid asks that question, they’re doing you a favor.  They’re providing you with insight that perhaps you’re serving the wrong this.  And if you’re not, whatever purpose you serve is not apparent to that student!

Let’s back up and take a larger scope view of the situation.  Especially in today’s educational climate, teachers stay in the classroom to be of service to students.  That’s it.  Teachers are blamed for all of the woes of society, for the failings of education, called lazy, and everybody seems to know what they’d do to fix it, if they were a teacher!

That sentiment, why teachers stay in the classroom, is the gateway to changing the answer to THE question.  Teachers are only in the classroom for the benefit of the students.  And surely, a teacher wouldn’t take on the sacrifices they do to stay teaching only to waste the time of their students, right?

Well … no, but kind of yes, too.

Let’s talk about job pressures … failing percentages in your classes, average scores on “high stakes tests.”   Those are big deals!  The test results are used to evaluate schools and teachers.  Administrators can be rewarded or fired on the basis of such things!

After all, good test results must be the sign of a great school.  Bad test results, well, that is really the teacher’s fault!  Yet, if a teacher holds a standard aligned with test results, the class failing rate will be too high, unacceptable, and a sign of bad teaching.  There will be unpleasant parent meetings; counselors, and administrators asking what’s being done to help the student, as if the student is hapless, a victim of the inevitable.

What am I doing to help this student?  I don’t know … showing up to work every day with lesson plans, a warm welcome and words of encouragement?  Oh, and I tell him to pick up his pencil fourteen times an hour, that’s a start, right?

Through either threat or blarney, bean-counters and pencil-pushers outside the classroom press hard to ensure that the teacher is performing due diligence to achieve high test grades.  Parents hover, students object, and through it all teachers are led to one inevitable observation: this is not why I teach.

Do those test scores really matter for students?  Sure, maybe ACT or SAT, AP, IGCSE or IB tests matter.  But those are the culmination of years of work.  Does it matter, to the student, if they pass their local state’s Common Core exam?  Not one bit.

In my real life, everybody will be impressed that I got a 3 on my Common Core State Test in English.

The purpose of an education is not to be able to pass a test.  The purpose of education is the development of the mind.  An educated person should be adaptable, thoughtful, able to communicate and appreciate various points of view other than their own.  An educated person should have perseverance and confidence, creativity and curiosity about the world around them.

A person that is educated should have an enriched life as a byproduct of their education.

When am I going to use this in my real life?

Well, that depends on what you mean by this.

Are you, as an educator, teaching this to help kids pass a test, or get a good grade in your class?  Are you teaching this to help them to know how to do their homework?  If so, there’s no judgement or blame.  Sometimes you have to make concessions just to get through the day.  We want kids to be successful.  The problem is, what are we using as markers of success?

But what if you could make this align with why you come back every year?

If you teach students about factoring polynomials so they can complete a worksheet, and maybe pass a quiz, your this is not powerful.  That is not why you teach.  Why make it what you teach?

The trick is to devise ways to teach kids how to think, to encourage creative problem solving and develop communication … to give them an education, while they learn how to factor a polynomial.

Personally, I never want a student to say to me, “Mr. Brown, you’re the only reason I got through math class.”  That’s too low of a standard.  That is exactly the this that makes THE question so damning to our efforts.  I don’t show up every day so my students can pass a quiz or test, or get a minimal passing grade in math class.

I show up every day to provide a challenge to my students, so they can test themselves and be better tomorrow than they were today.  And by better, I don’t mean greater proficiency at rationalizing the denominator. I mean of better mind.

To me, the best compliment a student can give a teacher is, “You taught me how to learn.”  In learning to learn, all of the pieces of an education are there.  To know how to learn you must be a problem solver, a critical thinker, be reflective, confident, and resourceful. A student that knows how to learn is prepared for an unknown future.

Ask yourself:  By teaching this, what’s being learned?  Are you just rehashing the same old lessons, just giving the same information the students could get on Khan Academy?  Are you asking them questions that can easily be answered by PhotoMath?  Are you printing another worksheet off of KutaSoft?

Challenge yourself to raise the bar.  Forget the bean-counters.  They’ll be happy when they see the results because when a student that knows how to learn takes a silly test, they do well!  Not only that, they’ll stand out when compared with students who were taught the content of the course only.

Unfortunately, if you’ve been dragged to the point where you realize, this is not why I got into teaching, and it consumes your day, you cannot answer THE student’s question honestly without using the word never.

You’re unlikely to find much guidance in the education industry that will change the this in THE question.  The industry sells books and professional development designed to get students to pass the test.  Their livelihood is generated from keeping the this we don’t want in THE question.

It is on us, educators dealing with students daily, to change the this.

The next time a student asks THE question … ask yourself, why?  Why did they ask?  Which this am I serving, the one designed for test scores, or the one educating students?

Confuse Them So They Learn

I recently did a lesson on the basics of reading and writing in Geometry.  You know, dry, dull stuff...what's a point, line, ray, segment, how do you write an angle, what types of angles are there, and so on.

While preparing all of this information I was thinking:

How can I expose misconceptions about such material so they learn it?

Remember, just seeing the facts is comfortable for students, but not only do they not learn, they somehow find confirmation that their held misconceptions are in fact correct.  It's not as wild as you think, and it's not limited to kids.  I took a psychology class in college and was unknowingly part of an experiment.  I was asked a question, a seemingly throw-away type.  But it's trickier than it looks and nearly everybody answers wrong.  But it was of such little consequence that I did not remember my answer (you weren't supposed to).  Then, I was shown the correct answer and asked if that's what I had said.

Turns out the vast majority of people mis-remember that they answered correctly.  That is, they answered it wrong, but it's hard for us to imagine we're wrong, and they latch on the to the idea they were right...even when it's quite obvious they weren't.

This is so powerful that to be wrong and be aware of it, being confronted with things we don't understand and such, is very uncomfortable and unpleasant.  Yet, that's what is needed for learning to occur.  (And I'm talking the type of knowledge where understanding is paramount to success.)

My assertions are that what Derek Muller has unconverted here goes beyond science and film.

Students are not void of knowledge in your content.  They have ideas.  Teaching them is more like part repair work on the frame of a house before roofing.  Presenting students with correct information will not shore up their misunderstandings.

Also, students need to experience some level of cognitive discord.  In education, nearly all of the "best practices" work hard to do the opposite of this.  There are things like Content/Language Objectives, or SWBAT, word walls and graphic organizers.  I'm not saying those things don't have their place, but that's it, they have a place when balanced with quality instruction that explores misconceptions and such.

When you can deliver a lesson that explores the misconceptions the students will be confused.  But if it is student lead, they won't be lost.  The amount of mental effort required is much higher than a typical delivery of information and note-taking style.  However, they'll learn!

So, how to create this tension and expose misconception over some pretty dull information?

Start by asking questions and exploring answers.  Do not use your authority in the subject to state if an answer is right or wrong, initially.  Instead, have students share their thinking on what other students are saying.

For example, a particularly nasty question that dealt with the boring definition-based lesson I just gave was, "What is an angle?"  To someone versed in geometry, this isn't a big deal.  But to a kid who hasn't taken geometry, this is monumentally difficult to describe.   The best response I received was, "Measuring the space between two lines."  So, of course, I drew to parallel lines and asked for explanation.

 

Now, this is just something I wonder, but is it possible that on these boring, just the facts, type lessons that exposing misconception is more important than ever?

Regardless of how that fleshes out, challenge yourself to challenge the thinking of students by exposing misconception through dialogue.  Be brave enough to explore misconception and encourage students to seek understanding by challenging the think of themselves and others.  If students understand the purpose of your methods, they'll play along.

Give it a shot, let me know how it goes.

Once again, thank you for your time.

The Most Important Component of Quality Teaching is …

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

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

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

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

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

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

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

Do I Really Need a Relationship with the Students?

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

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

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

How Do I Build a Relationship with So Many Students?

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

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

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

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

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

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

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

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

Let me know your thoughts.  Thanks for reading.

 

What Do Grades Really Mean?

What Do Grades Mean

The following is highly contentious.  Many of the situations discussed here should ultimately be considered on an individual basis.  The purpose of this is not to create a rubber-stamp solution to all problems that arise with grade assignment and student ability and or performance, but is to provide a general framework so that those individual decisions can be made in fairness and with respect to what is best for the student.

In a previous post I asked about a student in summer school that obviously knew Algebra 1 (he earned 100% on his quizzes and tests), but failed during the year because he didn’t do his classwork.  The question is, Does he deserve to fail Algebra 1?

When you flip the situation around it is equally interesting.  There are many kids who work hard, but do not really understand or learn the math.  Do they deserve to pass based on the merits of effort?

The real issue with both of these situations is what grades mean, or what should they mean.  When I worked at Cochise Community College I adopted their definition of letter grades which is described below:

A – Mastery

B – Fluency

C – Proficiency

D – Lacking Proficiency

Those are clean and inoffensive definitions of grades.  A student with an A has mastered the material.  To be fluent means you can navigate the materials but not without error.  To be proficient means you can get the job done, but there are some gaps in ability, but the student can demonstrate a measurable level of command of all of the objectives. Students who earn a D are not able to demonstrate proficiency.

A student who struggles with the material does not deserve an A, even if they worked harder than those who earned an A.  This might seem unfair, but unless the objective of the class is to teach the value of hard work, to reward the hardworking, but barely proficient, student with a label of mastery is to cheat the student and cheapen the merit of your class.

Do these definitions mean that a lazy kid that get 95% on the final exam deserves an A, but that a hard working kid that gets a 52% on the same final deserves an F?  I say, with a few qualifications, yes.

Is this really fair to the student who works hard but has not yet realized an appropriate level of mastery to be awarded a passing grade? (I used the phrase, “has not yet,” instead of, “cannot,” to acknowledge the belief that students can learn, and if they are motivated and working, the only question will be the time scale of when they learn the material.)  

I would say, for a math class, that the best thing that can happen is they are awarded the appropriate grade, an F.  Consider if this student is given a passing grade and the class is a prerequisite course?  They’re truly set up for failure in the subsequent class.

There is perhaps no worse example of bad teaching that remains within legals bounds than to inappropriately assign grades to students.  If a student deserves a C based on ability, but is given an A based on effort, they will believe they are doing everything right and do not need to improve in order to achieve similar success in subsequent courses.

But to give a student who possesses mastery a failing grade in a class because of lack of work ethic is to teach the student that passing classes is a matter of compliance.  Behave and you’ll be rewarded.  Those kids are taught that grades are not a reflection of knowledge or ability, and that means that education is not about learning.  To me, this is an injustice.

I do not believe in the efficacy of these objective lessons.  That would be, failing a student based on the notion that they do not deserve to pass because they are lazy. I believe that given meaningful and challenging opportunities, most of these highly intelligent, but seemingly lazy, students will show themselves to be hard working with amazing focus and direction and incredible capacity for quality work.

What about percentages.  Is it appropriate that an 80% is a B, if a B means fluency?  

When I first began teaching I would have said, absolutely, a student does not deserve an A if they scored an 87% on their test.  Since then I’ve changed my mind.  Some topics require higher than 90% accuracy to be awarded an A, while with other topics, mastery might be far below 90%.  

The level of complexity, variability of solutions and length of assessment all must be considered.  This is why sometimes a grading rubric is far superior to assigning grades based on a percentage of correctness or completion.  

I teach a curriculum that is designed and tested by Cambridge University, the IGCSE test is what students take.  They have a very different way of assigning and defining grades than we use here in the United States.  Without going into details about how they do the specifics, they assign large portions of credit based on evidence of appropriate thinking.  In other words, if a student demonstrates understanding they will receive passing credit.  But, to achieve a high grade, mastery is truly measured.  And yet, in math at least, the percentages of correctness for mastery are usually in the mid-70’s.  This is because the nature of the questions asked are often non-procedural and the method of solution is not clear, students cannot be trained on how to answer the questions they face on IGCSE exams.

How Do Students Earn Grades

How a student can earn a grade varies, or should, depending on subject and age, and perhaps even minor topic within the subject.  I believe that separating student work into weighted categories is an appropriate method of helping make transparent to the student how their grade will be assigned.  It also by-passes the tricky question of, “What is a point?”  For me, a homework assignment is worth 5 points, they’re assigned daily, except Fridays, for a total of 20 points for the week.  Yet, a quiz might only be worth 12 points, but will be a far more accurate representation of student’s ability on the topic.

By assigning weights to the categories, this can be easily balanced.  This begs the question, how do you weight the categories?  

But what about the student who works, performs all assigned tasks, but can only demonstrate a level of understanding best described as “Lacking Proficiency?”  Shouldn’t hard work be rewarded?

And whatever your beliefs on these questions, would your opinion change depending on the age of the student, or perhaps the subject?  Should a Chemistry student be rewarded for effort in the same way they’d be rewarded for effort in a Dance class?

At some point, nobody cares about potential or effort.  If a child’s mother wants his room clean, she knows he has the potential to clean it, but if he fails to do so, the potential matters not.  And if he’s really trying to get it done, but cannot master the discipline to carry through the task, does the effort really matter?

Here is how I set up my grades for high school.  It is nuanced and complicated, but I’ll give the outline.  Note that for college classes I use a different system.

In high school I weigh categories of grades and have changed the percentages and categories over time until I settled on what seems to work best.  These work for my students because it seems to motivate the lazy-smart students and also rewards the hardworking – low aptitude student, because if they remain persistent, they will learn.

Tests – 40%
Quizzes – 25%
Homework – 25%
Other – 10%

I believe extra credit should be awarded for students that perhaps help others, or for extraordinary performance.  However, a student should NOT be allowed to raise their grade through extra credit.  That is, at the end of the term a student is given a pile of work, that if performed, will raise their grade.  This is bad teaching!

The difference between a quiz and a test is similar to the difference between a doctor’s check-up versus an autopsy.  The quiz is a chance to see how things are going and adjust accordingly.  The test is final.  In high school I award credit for homework based on completion, but do not accept late homework.

Rewarding Effort?

While I wish that effort equaled success, it doesn’t always work that way…depending on how you define success.  For example, I can try as hard as possible to paint a world-famous landscape, but will likely fail if my measure of success is producing a world-famous piece of art. That said, I believe there is a reward beyond measure only discovered with true effort.  Our potential, our best, is not static, it changes.  It changes in respect to our current level of effort.  We can never fulfill our potential, you see.  It is always slightly above how hard we are trying.  So, if you’re not really trying, your potential decreases, but if you’re pushing your limits, the limits themselves stretch.  That is the real downfall of those with an inherent talent that never learn to push themselves.  Their potential decreases, dropping down to just higher than their level of effort.

I greatly reward effort, encourage it and makes positive examples of how effort promotes success.  However, I do not assign grades to effort.  How hard someone needs to try in a given subject to be successful varies entirely upon the student’s aptitude.  And suppose you have a truly gifted student, they could be great, if they learn to work hard, right?

Well, perhaps, but there’s more than work ethic involved in greatness.  What role does passion play?  Take a great young musician and over-structure their training and practice, they’ll burn out.  You’ll snuff their passion.

Grades

I asked the boy whose situation started this whole conversation if he felt he deserved to be in summer school.  Before he answered I explained that I didn’t have an expected answer, I didn’t really know if he belonged in summer school or not.  Without hesitation, he said he did deserve summer school, because, he said, he was lazy.

So maybe the kid will learn that if he’s lazy he gets punished.  But he also learns that grades are arbitrary, with respect to ability.  

I do not like objective lessons, do not believe them to be effective.  I prefer a punishment that fits the crime, but also one that redirects the offender, allows them to correct their action.

I cannot say in this child’s case specifically, I was not there and I am not judging his teacher, but perhaps a quicker punishment that redirected him could have also taught him that being lazy was unacceptable and at the same time also allowed him to see grades as a reflection of his abilities.  

All that said, this is highly contentious and varies incredibly depending on particular situations of students.  

Let me know what you think, agree or disagree.  Leave me a comment.  

 

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


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa
aaleqabaGaaGynaaaaaaa@37A0@


This equals three times itself five total times:




3
5

=33333

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa
aaleqabaGaaGynaaaakiabg2da9iaaiodacqGHflY1caaIZaGaeyyX
ICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaaaa@4589@


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






3
5




3
1




MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaaaaa@395E@


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






3
5




3
1



=

33333

3


MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpdaWcaaqaaiaaiodacqGHflY1caaIZaGaey
yXICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaabaGaaG4maaaa
aaa@4814@


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.






3
5




3
1



=
3
3



3333

1


MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpdaWcaaqaaiaaiodaaeaacaaIZaaaaiabgw
SixpaalaaabaGaaG4maiabgwSixlaaiodacqGHflY1caaIZaGaeyyX
ICTaaG4maaqaaiaaigdaaaaaaa@48DF@


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






3
5




3
1



=3333=
3
4


MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpcaaIZaGaeyyXICTaaG4maiabgwSixlaaio
dacqGHflY1caaIZaGaeyypa0JaaG4mamaaCaaaleqabaGaaGinaaaa
aaa@46EE@


The short-cut is:






3
5




3
1



=
3

51


=
3
4


MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaI1aGaeyOeI0
IaaGymaaaakiabg2da9iaaiodadaahaaWcbeqaaiaaisdaaaaaaa@4077@


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:






a
m




a
n



=
a

mn



MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WGHbWaaWbaaSqabeaacaWGTbaaaaGcbaGaamyyamaaCaaaleqabaGa
amOBaaaaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaeyOeI0
IaamOBaaaaaaa@3F10@


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.






3
5




3
1



=
3
5

÷
3
1


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGymaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaI1aaaaOGaey
49aGRaaG4mamaaCaaaleqabaGaaGymaaaaaaa@4002@


.

But that is different than




3
1

÷
3
5


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa
aaleqabaGaaGymaaaakiabgEpa4kaaiodadaahaaWcbeqaaiaaiwda
aaaaaa@3B8A@


The expression above is the same as






3
1




3
5




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaIXaaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaaaaa@395F@


This comes into play
because






3
1




3
5



=
3

15



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaIXaaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaIXaGaeyOeI0
IaaGynaaaaaaa@3DC0@


,

and 1



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa
aaaaaaaaWdbiaa=nbiaaa@37C3@


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÷31
1
3


1
3


1
3


1
3


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgE
pa4kaaiodacqGH3daUcaaIZaGaey49aGRaaG4maiabgEpa4kaaioda
cqGHsgIRcaaIXaGaeyyXIC9aaSaaaeaacaaIXaaabaGaaG4maaaacq
GHflY1daWcaaqaaiaaigdaaeaacaaIZaaaaiabgwSixpaalaaabaGa
aGymaaqaaiaaiodaaaGaeyyXIC9aaSaaaeaacaaIXaaabaGaaG4maa
aaaaa@5482@


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
1
3


1
3


1
3


1
3

1

(


1
3


)

4


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgw
SixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyyXIC9aaSaaaeaacaaI
XaaabaGaaG4maaaacqGHflY1daWcaaqaaiaaigdaaeaacaaIZaaaai
abgwSixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyOKH4QaaGymaiab
gwSixpaabmaabaWaaSaaaeaacaaIXaaabaGaaG4maaaaaiaawIcaca
GLPaaadaahaaWcbeqaaiaaisdaaaaaaa@4EE8@


This could also be written:



1
1
3


1
3


1
3


1
3

1


1
4




3
4




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgw
SixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyyXIC9aaSaaaeaacaaI
XaaabaGaaG4maaaacqGHflY1daWcaaqaaiaaigdaaeaacaaIZaaaai
abgwSixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyOKH4QaaGymaiab
gwSixpaalaaabaGaaGymamaaCaaaleqabaGaaGinaaaaaOqaaiaaio
dadaahaaWcbeqaaiaaisdaaaaaaaaa@4E54@


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


1
4




3
4



=
1


3
4



.

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgw
SixpaalaaabaGaaGymamaaCaaaleqabaGaaGinaaaaaOqaaiaaioda
daahaaWcbeqaaiaaisdaaaaaaOGaeyypa0ZaaSaaaeaacaaIXaaaba
GaaG4mamaaCaaaleqabaGaaGinaaaaaaGccaGGUaaaaa@40A3@


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:




b


a

5





MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WGIbaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaaaaa@39AD@


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




b


a

5




=
b
1


1


a

5





MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WGIbaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGH
9aqpdaWcaaqaaiaadkgaaeaacaaIXaaaaiabgwSixpaalaaabaGaaG
ymaaqaaiaadggadaahaaWcbeqaaiabgkHiTiaaiwdaaaaaaaaa@4243@


The b is
not a problem here, but the other rational expression is problematic. We need to multiply by the reciprocal of




1


a

5





MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaaaaa@3981@


, which is just a5.




b


a

5




=
b
1




a
5


1

=
a
5

b

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WGIbaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGH
9aqpdaWcaaqaaiaadkgaaeaacaaIXaaaaiabgwSixpaalaaabaGaam
yyamaaCaaaleqabaGaaGynaaaaaOqaaiaaigdaaaGaeyypa0Jaamyy
amaaCaaaleqabaGaaGynaaaakiaadkgaaaa@4529@


.

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




b


a

5





MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WGIbaaqqaaaaaaOpGqSvxza8qabaGaamyyamaaCaaaleqabaGaeyOe
I0IaaGynaaaaaaaaaa@3C44@


Note:




a

5


=
1


a
5




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeaaaaaa6di
eB1vgapeGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaak8aacqGH
9aqpqqa6daaaaaGuLrgapiWaaSaaaeaacaaIXaaabaGaamyyamaaCa
aaleqabaGaaGynaaaaaaaaaa@4134@


Substituting this we get:




b


1


a
5






MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WGIbaaqqa6daaaaaGuLrgapeqaamaalaaabaGaaGymaaqaaiaadgga
daahaaWcbeqaaiaaiwdaaaaaaaaaaaa@3BB5@


This is b
divided by 1/a5.



b÷
1


a
5




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgE
pa4oaalaaabaGaaGymaaqaaiaadggadaahaaWcbeqaaiaaiwdaaaaa
aaaa@3BB6@


Let’s multiply by the reciprocal:



b
a
5


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgw
SixlaadggadaahaaWcbeqaaiaaiwdaaaaaaa@3AFA@


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




a
5

b

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa
aaleqabaGaaGynaaaakiaadkgaaaa@38BA@


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.





2
x

2



y

5


z



2

2


x
y
3


z

5





MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIYaGaamiEamaaCaaaleqabaGaeyOeI0IaaGOmaaaakiaadMhadaah
aaWcbeqaaiabgkHiTiaaiwdaaaGccaWG6baabaGaaGOmamaaCaaale
qabaGaeyOeI0IaaGOmaaaakiaadIhacaWG5bWaaWbaaSqabeaacaaI
ZaaaaOGaamOEamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaaaaa@45E3@


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.





2
x

2



y

5


z



2

2


x
y
3


z

5





2


2

2







x

2



x




y

5





y
3




z


z

5





MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIYaGaamiEamaaCaaaleqabaGaeyOeI0IaaGOmaaaakiaadMhadaah
aaWcbeqaaiabgkHiTiaaiwdaaaGccaWG6baabaGaaGOmamaaCaaale
qabaGaeyOeI0IaaGOmaaaakiaadIhacaWG5bWaaWbaaSqabeaacaaI
ZaaaaOGaamOEamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGHsg
IRdaWcaaqaaiaaikdaaeaacaaIYaWaaWbaaSqabeaacqGHsislcaaI
YaaaaaaakiabgwSixpaalaaabaGaamiEamaaCaaaleqabaGaeyOeI0
IaaGOmaaaaaOqaaiaadIhaaaGaeyyXIC9aaSaaaeaacaWG5bWaaWba
aSqabeaacqGHsislcaaI1aaaaaGcbaGaamyEamaaCaaaleqabaGaaG
4maaaaaaGccqGHflY1daWcaaqaaiaadQhaaeaacaWG6bWaaWbaaSqa
beaacqGHsislcaaI1aaaaaaaaaa@5ED4@


The base of two first:




2


2

2




2÷
2

2



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIYaaabaGaaGOmamaaCaaaleqabaGaeyOeI0IaaGOmaaaaaaGccqGH
sgIRcaaIYaGaey49aGRaaGOmamaaCaaaleqabaGaeyOeI0IaaGOmaa
aaaaa@40D5@


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÷
1


2
2




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgE
pa4oaalaaabaGaaGymaaqaaiaaikdadaahaaWcbeqaaiaaikdaaaaa
aaaa@3B5E@


Now we will rewrite
division as multiplication by the reciprocal.



2
2
2

=
2
3


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgw
SixlaaikdadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIYaWaaWba
aSqabeaacaaIZaaaaaaa@3D58@


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

We will offer similar
treatment to the other bases.

Consider first






x

2



x

=


x

2



1


1
x


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WG4bWaaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaamiEaaaacqGH
9aqpdaWcaaqaaiaadIhadaahaaWcbeqaaiabgkHiTiaaikdaaaaake
aacaaIXaaaaiabgwSixpaalaaabaGaaGymaaqaaiaadIhaaaaaaa@42A1@


Negative exponents are division, so:






x

2



x

=


x

2



1


1
x


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WG4bWaaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaamiEaaaacqGH
9aqpdaWcaaqaaiaadIhadaahaaWcbeqaaiabgkHiTiaaikdaaaaake
aacaaIXaaaaiabgwSixpaalaaabaGaaGymaaqaaiaadIhaaaaaaa@42A1@


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.






x

2



1


1
x


1


x
2




1
x

=
1


x
3




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WG4bWaaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaaGymaaaacqGH
flY1daWcaaqaaiaaigdaaeaacaWG4baaaiabgkziUoaalaaabaGaaG
ymaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaaaOGaeyyXIC9aaSaa
aeaacaaIXaaabaGaamiEaaaacqGH9aqpdaWcaaqaaiaaigdaaeaaca
WG4bWaaWbaaSqabeaacaaIZaaaaaaaaaa@4A23@


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






2
3


1


1


x
2

x



1


y
5


y
3





z
z
5


1


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIYaWaaWbaaSqabeaacaaIZaaaaaGcbaGaaGymaaaacqGHflY1daWc
aaqaaiaaigdaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyyXIC
TaamiEaaaacqGHflY1daWcaaqaaiaaigdaaeaacaWG5bWaaWbaaSqa
beaacaaI1aaaaOGaeyyXICTaamyEamaaCaaaleqabaGaaG4maaaaaa
GccqGHflY1daWcaaqaaiaadQhacqGHflY1caWG6bWaaWbaaSqabeaa
caaI1aaaaaGcbaGaaGymaaaaaaa@5256@


Putting it all together:





2
x

2



y

5


z



2

2


x
y
3


z

5




=


2
3


z
6




x
3


y
8




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIYaGaamiEamaaCaaaleqabaGaeyOeI0IaaGOmaaaakiaadMhadaah
aaWcbeqaaiabgkHiTiaaiwdaaaGccaWG6baabaGaaGOmamaaCaaale
qabaGaeyOeI0IaaGOmaaaakiaadIhacaWG5bWaaWbaaSqabeaacaaI
ZaaaaOGaamOEamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGH9a
qpdaWcaaqaaiaaikdadaahaaWcbeqaaiaaiodaaaGccaWG6bWaaWba
aSqabeaacaaI2aaaaaGcbaGaamiEamaaCaaaleqabaGaaG4maaaaki
aadMhadaahaaWcbeqaaiaaiIdaaaaaaaaa@4E87@


.

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×
1
5


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgE
pa4kaaiwdacqGH9aqpcaaIZaGaey41aq7aaSaaaeaacaaIXaaabaGa
aGynaaaaaaa@3F12@


.
When you rewrite division you are writing it as multiplication. Positive exponents are repeated
multiplication.




a

m


=
1


a
m



,
1


a

m




=
a
m


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa
aaleqabaGaeyOeI0IaamyBaaaakiabg2da9maalaaabaGaaGymaaqa
aiaadggadaahaaWcbeqaaiaad2gaaaaaaOGaaiilaiaaykW7caaMc8
UaaGPaVlaaykW7daWcaaqaaiaaigdaaeaacaWGHbWaaWbaaSqabeaa
cqGHsislcaWGTbaaaaaakiabg2da9iaadggadaahaaWcbeqaaiaad2
gaaaaaaa@4A81@


What about Zero?

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


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa
aaleqabaGaaGynaaaaaaa@37A0@


This equals three times itself five total times:




3
5

=3"#x22C5;3333

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa
aaleqabaGaaGynaaaakiabg2da9iaaiodacqGHflY1caaIZaGaeyyX
ICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaaaa@4589@


Now let’s divide this by 35.






3
5




3
5




MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaaaaa@3962@


Without using short-cut 3, we have this:






3
5




3
5



=

33333


33333


=1

MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaGccqGH9aqpdaWcaaqaaiaaiodacqGHflY1caaIZaGaey
yXICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaabaGaaG4maiab
gwSixlaaiodacqGHflY1caaIZaGaeyyXICTaaG4maiabgwSixlaaio
daaaGaeyypa0JaaGymaaaa@55F5@


Using short-cut 3, we have this:






3
5




3
5



=
3

55



MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa
aGynaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaI1aGaeyOeI0
IaaGynaaaaaaa@3DC7@


Five minutes five is zero:




3

55


=
3
0


MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa
aaleqabaGaaGynaiabgkHiTiaaiwdaaaGccqGH9aqpcaaIZaWaaWba
aSqabeaacaaIWaaaaaaa@3BFF@


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.





(




3

2x1



e

πi








n=1





1


n
2







)

0

=1

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada
WcaaqaaiaaiodadaahaaWcbeqaaiaaikdacaWG4bGaeyOeI0IaaGym
aaaakiabgwSixlaadwgadaahaaWcbeqaaiabec8aWjaadMgaaaaake
aadaaeWbqaamaalaaabaGaaGymaaqaaiaad6gadaahaaWcbeqaaiaa
ikdaaaaaaaqaaiaad6gacqGH9aqpcaaIXaaabaGaeyOhIukaniabgg
HiLdaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIWaaaaOGaeyyp
a0JaaGymaaaa@4DBA@


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

Short-Cut

Example




a
m


a
n

=
a

m+n



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa
aaleqabaGaamyBaaaakiabgwSixlaadggadaahaaWcbeqaaiaad6ga
aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaey4kaSIaamOBaa
aaaaa@4140@





5
8

5=
5

8+1


=
5
9


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa
aaleqabaGaaGioaaaakiabgwSixlaaiwdacqGH9aqpcaaI1aWaaWba
aSqabeaacaaI4aGaey4kaSIaaGymaaaakiabg2da9iaaiwdadaahaa
WcbeqaaiaaiMdaaaaaaa@41C8@






(


a
m


)

n

=
a

mn



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca
WGHbWaaWbaaSqabeaacaWGTbaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaWGUbaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaiaad6
gaaaaaaa@3EB7@






(


7
2


)

5

=
7

10



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca
aI3aWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaaI1aaaaOGaeyypa0JaaG4namaaCaaaleqabaGaaGymaiaaic
daaaaaaa@3D93@







a
m




a
n



=
a

mn



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WGHbWaaWbaaSqabeaacaWGTbaaaaGcbaGaamyyamaaCaaaleqabaGa
amOBaaaaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaeyOeI0
IaamOBaaaaaaa@3F11@







5
7




5
2



=
5

72


=
5
5


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aI1aWaaWbaaSqabeaacaaI3aaaaaGcbaGaaGynamaaCaaaleqabaGa
aGOmaaaaaaGccqGH9aqpcaaI1aWaaWbaaSqabeaacaaI3aGaeyOeI0
IaaGOmaaaakiabg2da9iaaiwdadaahaaWcbeqaaiaaiwdaaaaaaa@4087@





a

m


=
1


a
m



 &  
1


a

m




=
a
m


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa
aaleqabaGaeyOeI0IaamyBaaaakiabg2da9maalaaabaGaaGymaaqa
aiaadggadaahaaWcbeqaaiaad2gaaaaaaOGaaeiiaiaabAcacaqGGa
GaaeiiamaalaaabaGaaGymaaqaaiaadggadaahaaWcbeqaaiabgkHi
Tiaad2gaaaaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaaaaaa
a@4637@





4

3


=
1


4
3




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa
aaleqabaGaeyOeI0IaaG4maaaakiabg2da9maalaaabaGaaGymaaqa
aiaaisdadaahaaWcbeqaaiaaiodaaaaaaaaa@3C0F@





a
0

=1

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa
aaleqabaGaaGimaaaakiabg2da9iaaigdaaaa@398F@


50 =
1

 

Let’s try some practice
problems.

Instructions: Simplify the following.

1.





(


2
8


)


1/3



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca
aIYaWaaWbaaSqabeaacaaI4aaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaaIXaGaai4laiaaiodaaaaaaa@3B8D@


2.



3
x
2



(

3
x
2


)

3


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI
hadaahaaWcbeqaaiaaikdaaaGccqGHflY1daqadaqaaiaaiodacaWG
4bWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabe
aacaaIZaaaaaaa@400E@


 

 

3.




5


5
m




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aI1aaabaGaaGynamaaCaaaleqabaGaamyBaaaaaaaaaa@38A4@


4.






5
2


x

3



y
5




5

3



x

4



y

5





MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aI1aWaaWbaaSqabeaacaaIYaaaaOGaamiEamaaCaaaleqabaGaeyOe
I0IaaG4maaaakiaadMhadaahaaWcbeqaaiaaiwdaaaaakeaacaaI1a
WaaWbaaSqabeaacqGHsislcaaIZaaaaOGaamiEamaaCaaaleqabaGa
eyOeI0IaaGinaaaakiaadMhadaahaaWcbeqaaiabgkHiTiaaiwdaaa
aaaaaa@44E1@


 

5.



7÷7÷7÷7÷7÷7÷7÷7

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiabgE
pa4kaaiEdacqGH3daUcaaI3aGaey49aGRaaG4naiabgEpa4kaaiEda
cqGH3daUcaaI3aGaey49aGRaaG4naiabgEpa4kaaiEdaaaa@4B9C@


6.



9
x
2

y÷9
x
2

y

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadI
hadaahaaWcbeqaaiaaikdaaaGccaWG5bGaey49aGRaaGyoaiaadIha
daahaaWcbeqaaiaaikdaaaGccaWG5baaaa@3F94@


 

 

 

 

7.



9
x
2

y÷(

9
x
2

y

)

MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadI
hadaahaaWcbeqaaiaaikdaaaGccaWG5bGaey49aG7aaeWaaeaacaaI
5aGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhaaiaawIcacaGLPa
aaaaa@411D@


8.





(


x
2

2
x
6


)

2



(


x
2

2
x
6


)


2



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca
WG4bWaaWbaaSqabeaacaaIYaaaaOGaeyyXICTaaGOmaiaadIhadaah
aaWcbeqaaiaaiAdaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaik
daaaGccqGHflY1daqadaqaaiaadIhadaahaaWcbeqaaiaaikdaaaGc
cqGHflY1caaIYaGaamiEamaaCaaaleqabaGaaGOnaaaaaOGaayjkai
aawMcaamaaCaaaleqabaGaeyOeI0IaaGOmaaaaaaa@4BF0@


 

 

9.





(


a
m


)

n


a
m


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca
WGHbWaaWbaaSqabeaacaWGTbaaaaGccaGLOaGaayzkaaWaaWbaaSqa
beaacaWGUbaaaOGaeyyXICTaamyyamaaCaaaleqabaGaamyBaaaaaa
a@3F08@


10.







(

3
x
2

+4

)

2





(

3
x
2

+4

)

3




MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada
qadaqaaiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIa
aGinaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaamaabm
aabaGaaG4maiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI
0aaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaaaaa@4390@


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
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb
WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaG4maiaadIha
cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa
qaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaaaaaa@43D9@


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
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb
WaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaeyypa0JaaG4mamaabmaa
baGaaGOmaaGaayjkaiaawMcaaiabgkHiTiaaikdadaahaaWcbeqaai
aaikdaaaGccqGHRaWkcaaIXaaabaGaamOzamaabmaabaGaaGOmaaGa
ayjkaiaawMcaaiabg2da9iaaiAdacqGHRaWkcaaI0aGaey4kaSIaaG
ymaaqaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaacqGH9aqp
caaIXaGaaGymaaaaaa@4F4E@


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

Here’s what the
students actually read:





f(
x
)
=3x
x
2

+1




f(
2
)
=3(
2
)
+
(

2

)

2

+1


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb
WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaG4maiaadIha
cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa
qaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaacqGH9aqpcaaI
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
answers, but instead to learn from them.

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

 

How to Save Time Grading

 

How to Grade Efficiently

and Promote Assignment Completion

 

Grading papers is one of the most time-consuming responsibilities of teaching.  Hours upon hours can be, I argue, wasted, pouring over daily homework assignments.  This article will discuss how to integrate awarding credit for daily assignments in a way that saves hours of time while increasing your awareness of student progress, increases student completion rates and better informs students regarding their progress in the subject.

This routine described here is a daily variety, not how I grade quizzes, tests or projects.  However, there are some tips that apply to recording those grades later in this article.

Let’s begin with a question: What is the purpose of homework?  For me, it’s practice needed for students to gain proficiency.  Homework is about trying things, working out how to struggle through difficult problems and making, and learning from, mistakes.  Without effective homework, students will not integrate their learning into a body of knowledge that they can draw upon for application or just recall.

The breadth of the purpose of homework and how that purpose is best served is beyond the scope of this article, but I would like to suggest that homework is something done in their notes, whenever possible.  The reason being is that notes are a receipt of their learning, to be reviewed in the future to help remember observations and important facts.

Overview of How It Works:

At the beginning of class, often before the bell rings, I begin walking around the classroom stamping homework that deserves full credit.  (What merits full credit is up to your discretion, but it should be a clear and consistent expectation, known to students.)  As I circle the room, I look for common mistakes, ask kids if they have questions or difficulties and make small talk.

Students that didn’t do, or complete, the homework have to answer for it on the spot!

Then, I simply mark those that did not receive credit for the homework on a student roster I keep on a clip board.  (For a video of how this works, visit the link here: )

Quick Notes:  This method has students ready for class because they have their notes.  They’ve also asked me questions if they had any, so I can begin with meaningful review.  I also have forced students that are remiss to account for their actions and done so in a way that applies positive peer pressure.  The scores are recorded by leaving blanks for completion and only marking those that do not get credit (which will be very few).

Credit:  I award full credit or zero credit when checking homework.  If a student attempted all problems, with evidence of attempt demonstrated by work shown and questions written, they get full credit.  Those that fail to receive full credit have the opportunity to reclaim 80% (the percentage is arbitrary but again needs to be consistent, clear and known by all), the students must see me during tutoring time by the Friday of the week of the assignment to show that they’ve fulfilled the expectation.  Students that did not attempt the homework can also see me during tutoring time (before or after school, not between class times or lunch), and receive partial credit.

But the rule of being due the Friday of the week assigned is big.  The purpose of homework is practice.  Without proper practice skills and knowledge are not developed.  Homework is not about compliance and fulfilling an expectation with a grade as a reward.  Students that are hustling to complete homework from two months prior are likely not promoting their understanding of current materials.  Plus, by having the time requirement applied to the homework policy, students are not enabled to fall too far behind.

The added bonus is that you will not be buried with make-up work the last week before grades are due to be reported!

Work to be Turned In:  If the nature of the work is not something that can be kept and must be turned in, have the students pass their work forward by row.  As you collect each row’s stack, count them.  If a row’s stack is incomplete, ask who in the row didn’t turn in the work.

If students can NOT fulfill the expectation and only receive a bad grade from it, and that reprimand comes well after the unwanted behavior, they will quite happily go along thinking nothing bad is going to happen.  Having to answer, publically, for their lack of work, especially when the vast majority will work, is a powerful deterrent!  Just as when checking the work of students and asking those who failed to complete for an explanation, this keeps them accountable and will increase the amount of students completing their work.

When collecting the papers, alternate the direction of the stacks and do not mix them up when grading.  This will allow you to quick return the papers after you’ve been done.  If it is a daily practice type of work turned in, I’d suggest awarding full or no credit and only recording, again on the printed class roster, those that were awarded no credit.

Recording Grades:  Whether you’ve collected daily practice or are carefully grading quizzes and tests, how you record those grades can either waste your time, or greatly reduce the amount of time spent.

By recording each grade as it is calculated by hand on the student roster it is quick and easy to transfer them to the computer.  This is a huge time-saving practice.  You don’t need to hunt on the computer screen for each student, and do so for each assignment.  When they’re recorded by hand, you can simply enter the column of numbers in the computer.  When the last name lines up with the last number that you entered, you know they’re all entered correctly.

By following this method, the data entry side of grading is done in a few moments of time instead of over hours, working through those stacks of papers, again!

Final Thoughts:  By looking at, and discussing, homework with students on an individual basis, very briefly, you gain insight into their progress.  They get a chance to ask questions.  Students that need a little bit of motivation receive it as an immediate consequence for poor behavior, rather than waiting until the end of the quarter, when a lot of pressure will be placed on you to help them bring up their grades.

This routine has proven to be a cornerstone of my classroom management.  It gives me a way to set the expectation that we are here to learn and that learning is done through work and reflection.  Students that need discipline receive it immediately and in a way they find uncomfortable, but it is done so with an invitation that guides them to the desired behavior (of completing their work).

 

Things NOT Taught in Teaching College

What College Should Teach You About Teaching

As a salty veteran teacher it is almost sweet seeing the hopeful expectation in the eyes of new teachers.  They've just graduated college and they are ready to fix education.  Thing is, there is much to learn that's not covered in college.  I'd like to share some of those things with you.  Whether you're a salty veteran or wet-behind-the-ears, I think there's something here for you.

Number 1:  The Most Important Skill for Teachers

There is no better skill for a teacher than the ability to get along with others.  This is especially true for those teaching high school.  In high school you'll be navigating around 150 students a day, all with blossoming personalities, body odor, love-interests, extravagant behavior and mood-swings.  If you can't find it in yourself to be gracious for the outrageous behaviors, you'll be in for an unpleasant career.

The thing I always try to remember is that I would NOT want to be judged today for the person I was when I was 15 years old.

Number 2:  Say NO to Your Boss

This is probably the most powerful for new teachers, but all can be victims of being over-worked.  It's true, there's a great need for man-power at a high school.  Class sponsors, club sponsors, coaches, curriculum projects, prom, after school activities and so on are all roles that need to be filled.  The eager, the new, the young and energetic ... well, they're the group most likely to say yes when asked to take on these tasks, so they'll likely be asked first.

But new teachers are the last who should be taking on these additional duties.

You have a limited bandwidth and the more you try to do with that bandwidth, the lower the quality.  Plus, there's a STEEP learning curve to teaching.  The first year should be spent doing nothing but learning how to teach, refining your procedures and practices.  Seriously, spend a lot of energy focusing on how to be efficient and effective.

Saying No to your boss isn't easy, but you can manage.  You won't get fired, they need you.  Just explain that you don't want to take on more than you can handle.  Once you've got a strong grip on the teaching side of things you'll explore taking on other duties.

Number 3:  Don't Grade Everything

Just because students did it doesn't mean you have to grade it.  Sometimes participation or completion is all that needs to be noted.  Think of it this way...the purpose of them working is to promote their learning.  If grading doesn't inform students about their progress (are they even going to consider why they were marked wrong?), and if it doesn't provide meaningful insight for you regarding their progress, then why grade?

And often reviewing the materials completed by students as a class is far more informative to both you and the students than sitting at a desk looking through each problem, making notes for the students and recording all of the scores.

Number 4:  Don't Try and Pacify Parents

If a parent is upset, let them be upset.  If you have a good structure for how their child earns their grade, stick with it.  "Johnny is failing because Johnny hasn't done homework.  Because he hasn't done homework he hasn't learned and he fails the quizzes.  Johnny fails to take advantage of the remediation offered for his quizzes and then fails the tests.  At the end of each class I can get Johnny to understand what he needs to understand.  But then he is responsible for performing the assignments to make his learning permanent."

Again, if parents are upset about grades, stick to your guns.  Whatever your late policy is, stick with it.  I personally do NOT allow late homework past the Friday of the week it was assigned.  End of story, not open for discussion.

Use this line:  "We can't fix the past, can only use the lessons learned from those mistakes to inform our future actions."

Number 5:  You Only Need 2 Pairs of Pants (men)

Monday wear pair 1.  Tuesday wear pair 2.  Wednesday wear pair 1.  Thursday wear pair 2.  Friday is usually casual day, wear jeans.  DONE!

 

How to Limit Tardiness Without Losing Your Mind

How to Limit Tardies Without Losing Your Mind

Mr. Goodie-Two-Shoes here...sorry...but I might have just a handful of tardies a month.  I teach 4 honors classes now, but switching from the "regular" classes to honors classes didn't impact the number of tardy students I have. I'd like to share with you a few things that I do that I believe contribute to students being on time.

  1.  Fake It 'til You Make It

A few years ago, or so I heard on an episode of Freakonomics (awesome podcast), Stanford University made a video featuring upperclassmen sharing what they did as freshmen when they started to struggle academically.  They said they studied, made friends with good students in their classes, went to tutoring and office hours, went to bed instead of going to parties and so on.  The video was shown to incoming freshmen and there was a significant academic improvement.  The thing is...the video was fake, the students lied.

You just have to tell the lie once, if you follow through with the rest of it.  Just say, "Students are on time to my class!  We start class on time here, that's how we roll."

Now you may be wondering if I'm telling the truth about the low number of tardies in my room, and I am.  However, I have used this Fake It 'til You Make It method for other things, like improving homework completion, but that's another story.

Part of the lore I establish in the students minds is done by planting the seed in their head that being the last person to class is embarrassing.  I tell them to notice that it is generally the case that the kids who show up last have the lowest grades and are the lowest achieving.  (Not always the case depending on the location of the previous class, I explain.  However, it is particularly true for first period and just after lunch.)  The reason this is, I explain, is because their heads aren't in the game.  They're preoccupied with the silly teen-drama that transpires in the hall.

"It's embarrassing to be last."

2. Establish the Expectation

With clarity and direct communication, make sure the students know what is expected of them, and do so with both the positive and negative statements (Be on time, don't be late).  This should be an early-year focus, when you establish your procedures and expectations.

Being specific here is incredibly important.  I spend quite a bit of time explaining that to be on time the students needs to be seated and ready when the bell rings.  On the board, before every class, I have instructions regarding what they need to have ready for the day.  On occasion I ask them to do "bell work," but that is not my routine. Running in the door, standing around the room, or anything else short of being ready when the bell rings is a failure to meet the expectation and they're tardy.

Signs on the wall about tardy policy, marking students tardy and even complaining to and yelling at them will not help.  In fact, if a student is always late and you fail to address them directly and clearly, they have successfully lowered your expectation of their behavior.  Further, they're now in control of establishing classroom norms!  But the classroom norms and group behavior are another topic, and a huge one at that.  More on that in the future.

3.  Start Class on Time

If you're talking with your neighbor teacher, checking email, or watching reading this on the internet when the tardy bell rings you are failing to fulfill your own expectation!

While it irks me when people say, "Teachers are just as bad as students," the sentiment is applicable here.  You are the leader of the classroom.  As leader of the classroom, you set the expectations through example!

So, to help with this be visible before class starts, have clear expectations for what is expected for the day on the board (if it is something that varies from the norm), and welcome them to class.  When the tardy bell rings, don't sit down immediately and take attendance.  Instead, start class by introducing the schedule for the day, reviewing homework, or with a welcoming conversation.  You'll find a moment of time when you can record attendance early in class, and I often ask the students to help me identify who is absent when that moment comes. (Sometimes I have the students practice something brief, retry a homework problem, check an answer with someone, or something along those lines and take attendance then.)

4.  Incentives for Punctuality

I don't believe students should be given candy for doing a basic thing that they should do anyway.  However, a show of gratitude can go a long way!  Just thanking a student who is always ready on time is a positive reinforcement that really helps kids be on time.  Or, perhaps mid-year, try explaining to students that it is important to you that class begins on time and thanking them for being punctual reinforces that the expectation is to be on time.

If you have a class that is made up of students that love to be late, use some positive peer pressure.  Offer to add 5% to the week's quiz if all students can be on time for a certain number of days.  Write it on the board before class, make it a production.  If the extra credit thing doesn't work for you offer something else, (I believe it is a scourge in education because it is used to help students get grades in place of learning).  The idea is, do not take their good behavior for granted.  If they're playing ball with you, acknowledge that.

5.  Consistent Reprimand

A friend of mine has students write a one page tardy essay every time they are tardy.  Another used to make students stand at the back of the classroom for the entire period if they were late.  Another still assigns lunch detention. They're all good if they're done consistently.  However, I prefer to use peer pressure and a bit of embarrassment.

First off, if a student is late they do not get to walk into the classroom as though it is the passing period.  I stop them at the door.  Now, this is a bit difficult to articulate in writing, but it is important you consider the nature of the student.  It matters not to me why they're late, they need to be on time, but sometimes they have a reasonable excuse.  After hearing the reason for tardiness from a student that is usually punctual I will reiterate the expectation and importance of punctuality.

Sometimes, if it is a kid who has a boyfriend or girlfriend that causes them to be late, I won't ask for a reason but will tell them they need a new boy/girl friend, one that cares about their future, "...not one who is selfish and only interested in their own entertainment at your expense.  What are you, a play thing?  What happens when they're tired of you and then you're lonely and uneducated?"  And again, this is said for all to hear.

Yet, a little grace upon occasion can go a long way to promoting your desired end result, students being on time.  Sometimes a student is obviously having a BAD day ... and you don't want to start a fight, you want to teach class and have them be on time so you can better do that.  So, sometimes a gentler approach is better.  After hearing why a student is late (which is done with the full class as audience), I often ask them if they can do me a favor and show up on time tomorrow.

I hope you found this helpful.  Thank you for reading.  If you have questions or comments, please leave them below.  And subscribe for updates on the blog.

For a PDF of this post, click here:  How to Limit Tardies Without Losing Your Mind.