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.

Math Can Not Be Taught, Only Learned

Math is something that cannot be taught, but can be learned.  Yet, math is taught in a top-down style, as if access to information will make a student successful, and remediation is rehearsal of that same information.  Earnest students copy down everything, exactly like the teacher has written on the board, but often still struggle and fail to comprehend what is happening.  I argue that if copying things down was a worthy exercise, why not just copy the textbook, cover to cover.  Of course such an activity would yield little benefit at all because math is a thing you do more than it is a thing you know.  Math is only partly knowledge based and the facts are rarely the issue that causes trouble for students.  I’d like to propose that you, either parent, student, administrator or teacher, considers math in a different light and perhaps with some adjustment the subject that caused such frustration will be a source of celebration.

There are many things that cannot be taught but can be learned.  A few examples are riding a bike, playing an instrument, creative writing and teaching.  Without question knowledge is a key component to all of these things, but it is rarely the limiting factor to success or performance.  Instead, the skill involved is usually the greatest limiting factor.  I argue that to learn these things a series of mistakes, incrementally increasing in complexity, must be made in order to learn.  Let’s see if this will make more sense with a pair of analogies.

First, watching someone perform something that is largely skill-based is of little use.  Consider driving a car.  A fifteen year old child has spent their entire life observing other people drive.  And yet, when they get behind the wheel for the first time, they’re hopelessly dangerous to themselves and all others on, or just near, the roads!

Second considering learning to ride a bike.  Sure, the parts of the bike are explained to the child, but they have to get on and try on their own.  The actually learning doesn’t really occur until the parent lets go (letting go is huge!) and the child rolls along for a few feet until they fall over.  Eventually they get the hang of the balance but then crash because they don’t know how to stop.  After they master braking they crash because they don’t know how to turn.  And then speed, terrain, and other obstacles get thrown in the mix.  Each skill must be mastered in order.  Preemptively explaining the skills, or practicing them out of context does not help the child learn to ride a bike.  They must make the mistakes, reflect, adjust and try again.

What a math teacher can provide is the information required, but more importantly feedback, direction and encouragement.  If a student understands that making mistakes isn’t just part of learning, but that a mistake is the opportunity to learn (and without it only imitation has occurred), and a teacher helps provide guidance, encouragement and feedback, then both parties will experience far greater success.  When a math teacher completes a problem for a student it is similar to an adult taking the bicycle away from the child and riding it for them.  When a student gives up on a problem, it’s as if they stopped the car and got out, allowing the adult to drive them home.

The job of math teacher is perhaps a bad arrangement of words.  Coach, mentor or sponsor is perhaps more appropriate.  There is no magic series of words, chanted under any circumstance, that will enlighten a struggling student.  The frustration making mistakes should be cast in a different light, a positive light.  The responsibility of learning is entirely on the student.  They cannot look to teachers, friends or tutors for much beyond explanation of facts.

In a future post I will explain how too much direction and top-down teaching of math promotes failure of retention and inability to apply skills in new applications.  But for now, please consider that math cannot be taught.  A teacher cannot teach it, but can help a student to learn.
Thank you for reading,

The Bearded Math Man