Teaching Negative Exponents
to Promote Student Mathematical Literacy

On Teaching Math Podcast
Episode Four

 

Introduction:  Hello, and welcome to the On Teaching Math podcast.  This is the podcast where we discuss all things related to the development of mathematical literacy in your students.  Thank you for joining us again.  I’m you’re host, Philip Brown. 

Before we jump into today’s topic, a quick word from our sponsor.

Sponsor:  Today’s episode is brought to you by a question.  Is 1% of 2 the same as 2% of 1?  When viewed in written form, the question is a little more perplexing.  1% × 2 = 2% × 1?  The answer is easy discovered to be true.  That may or may not be intuitive to you, or the students.  However, students might not be so quick to discover it.  They’re often reluctant to actually perform a calculation in this kind of situation … the burden of over-confidence is a shackle I too know. 

How I’d suggesting using this in class is a bit different than the previous questions.  Is 5% of 2 the same as 2% of 5?  Why does it work? 

As always, the conclusion or answer to the problem is not what will benefit students.  The benefit is found when they devise a strategy and try it.  The benefit is from having something to think about mathematically, and then doing so.

 

Story Time:  This week’s story actually has to do with last week’s episode on teaching square roots.  Let me set the stage so that the story makes sense.  I teach a two-year course called Cambridge IGCSE.  This takes up four of the five classes I teach daily.  I teach two sections of the first year of the Cambridge, two sections of the second year of IGCSE, and smashed in the middle is a single section of Algebra 1. 

Those Cambridge classes, especially second year, are packed with aggressive learning, engaged, intellectually challenging students.  Sometimes, that Algebra 1 class feels like racing along in a sports car and the car just ran out of gas! 

The difference is not average intelligence.  It is world view, how they see education fitting in with their futures.  It is interest and drive, not aptitude, generally speaking.

This current cohort is the lowest I’ve seen since I’ve been teaching.  Other teachers in other subjects echo the observation. 

A few weeks ago we went through the series of lessons on square roots.  The results were a disaster.  There is no perfect lesson or approach, sometimes things just don’t work.

Of the 35 students only five got all of these square roots simplified correctly.  Here they are, ready:

Square root of 4, square root of 25, square root of 100, square root of 36 and square root of 64.

Five students.  Most students got all of them wrong!  And don’t even ask about the square root of a forty-eight!

As a class we moved on.  The standard for that topic had been established by the students.  They weren’t getting it and were not really interested.  So they got their bad grades and we pushed on.  The message from me was one of disappointment, with encouragement.  The truth is that I was disappointed…I do not believe that a human being is incapable of the thinking and understanding required to master square root numbers.

We moved onto exponents, which went much better.

Last week, on the heels of producing the episode on square roots, I told the students that I was going to potentially get in trouble.  I explained that I am supposed to stay on track with the curriculum calendar and other teachers.  But, because I believe the students are far more capable than they showed, and I don’t want to give up on them, I was going to give them one more chance with square roots.  We were going to do a crash course on the topic and then take another quiz.

So I gave them the five key concepts, again.  This time my delivery was better…it fit them, grabbed their attention.  Also, because they had failed, and I stuck them with the result of their efforts, they now knew the rules of the game in my classroom.

The lesson was delivered, homework assigned.  I made it explicitly clear that there would be hell to pay if a single student showed up the next day in class and said they didn’t do the homework because they didn’t get it.  The message was that if they didn’t get it, they better do something about it!

One single kid showed up the next morning for tutoring.  She was really flummoxed by … the square root of one!  I had her pull out her 5 key concepts reference sheet she made and read the first two concepts.  After she read them and thought for a moment, I asked her, “What does the square root of one mean?”

She said, “It is a number.  If you multiply it by itself the answer is one.”

I asked, “Is there another number that you could square and get one as the product?”

She said, as if I was dumb, “Yeah … one.”  There was a “duh, you’re stupid mister,” tone for sure. 

I just smiled.  I asked her to read the second fact again. 

She got all of the square numbers down at that point.

She asked about the square root of twelve.  She was having trouble factoring to find a square product.  So I had her refer to the list of square numbers less than one-hundred.  I explained that most likely, because they problems are written to be simplified, the radicand has a factor that is one of these. 

We went through how to rewrite the square roots of non-square numbers as the product of a square root of a square number and the square root of a non-square number, and then just finish it off with what she learned with the square root of one.

Guess who got a 100% on her quiz later that day! 

The class average on the first quiz, three week previously, was 32%.  Seriously.  The second go-around, with a 45-minute crash course and a handful of direct, and a lot of veiled threats, was 87%. 

There are two reasons I wanted to share this story.  The first is to show an example of how when students root their thinking in concepts (see what I did there with square roots), they perform better.  The second, and more important idea is that the lesson delivery is of top priority.  Without buy-in from students, a great idea and lesson is worthless.

That said, let’s get on with today’s topic.

Episode Topic Outcomes:  Today we are going to discuss negative exponents and ways to help our students master these.   Before we get into this, a quick word on when to get into this.  I am not sure about state standards in other states, but in Arizona, students are taught exponents in middle school, short of negative exponents.  They are they taught negative exponents in Algebra 2.  This is a disaster of an idea, in my opinion.  Students cannot understand scientific notation (they can manipulate it with rules, but not understand), simplify rational expressions or algebraic fractions well, and will be lacking a fundamental element of mathematical literacy required for Algebra. 

I think this is something that should be taught in 9^th grade Algebra 1.  If the students haven’t seen it before, and learn it in Algebra 2, not all is lost, but it is not ideal.  The approach outlined here is appropriate for whenever students are learning negative exponents.  I am suggesting that they learn negative exponents at the same time they learn the topic of exponents. 

 

Anyhow, that said:  By the end of the episode you should be well equipped with some new ways to introduce negative exponents to students so that they are actually reading and understanding.  The idea is that when they see a negative exponent they read the intended meaning of the notation, not just think of some short-cut to follow. 

It might be the case that you walk away with a new understanding of what negative means in math, too.  It is because of this development of understanding of the meaning of the notations involved with negative exponents that I believe this is a key concept that can really develop mathematical literacy!

Episode Methods:   Here’s the take-away, then we’ll dive into how to develop this understanding in students.  Negative exponents are repeated division.  Division is multiplication by the reciprocal.  The reason we “flip and change the sign,” is because we are rewriting division as multiplication by the reciprocal.  The sign is changed for the same reason we change division signs into multiplication signs when we rewrite division, written with the division symbol, as multiplication … we have rewritten the information in a new way. 

Now, we cannot just tell them what’s right and expect perfection.  We have to expose misconception and get them to confront the misconceptions.  It isn’t different than learning to ride a bike at all.  Demonstration and explanation aren’t much good really. 

To kick this off, let’s start with two numbers written in scientific notation and ask students to compare them, by size. 

3.28 × 10³  _________ 3.28 × 10^-3

Have students place less than, less than or equal, great than, greater than or equal, or just an equal sign between the numbers.

Students will be correct here, most likely.  Verify their correctness and then explain that the second number 3.28 times ten to the negative third, is 0.00328. 

Ask the students why, and do not accept that the decimal is moved three places to the left.  That’s a description of a procedure, not the reason.  What we are trying to get here is for students to understand that this is 3.28 divided by 1,000. 

How is it division by a thousand if there is clearly a multiplication symbol? 

Our objective is to get them to explicitly understand what the negative sign means.  We want them to understand why that is true.

In the show notes you can see some examples of what how I write this, but I show them the various ways of writing division, which we will in this setting, consider as multiplication by the reciprocal.

$\frac{1}{2}\to 1÷2\to 1\times 2^{-1}\to 1\times \frac{1}{2}$

 

What does negative mean anyway?

Perhaps the soundest way to think of negative in math is as opposite.  That is easily seen as holding water with negative numbers.  Negative 5 is the opposite of five.  The reason that the absolute value of negative five is the same as the absolute value of five is because they share the same distance from zero.

What about 3 $–$ 2.  This is the opposite operation of addition.  And negative exponents? 

Well, since exponents are repeated multiplication, negative exponents are repeated division!

The issue students have here is not with understanding that, but applying the knowledge.  They are not skilled at writing in math.  So once we get them to understand that negative exponents are repeated division, we have to make sure they know how to write division.

A good start on this is to ask them to evaluate:

$8÷2,\text{and}8÷\frac{1}{2}$ 

In the first example, they’ll rewrite the divided by 2 as times one-half.  In the second example they will rewrite the division by $\mathrm{½}$ as times two. 

Our job is to help them transfer this level of literacy to algebraic expressions like:

$3x^{-1}\text{ and }\frac{3}{x^{-1}}$ 

It is tempting to just follow the thinking process that a well-experienced person would use to address problems like these.  Many students can follow it, but they’ll lose sight of what they understand and focus on the procedure.

It is best to go slow here, incorporate their understanding into the procedure carefully, and encourage them to do so until it becomes automatic.  This is similar to practicing form in a sport.  With slow, precision focus on a fundamental, we can perform better over time.

Let’s look at the first one, three times x to the power of negative two.  There are two bases and two exponents here.  The three has an exponent of one, and the x has an exponent of negative one. 

That means if we unpacked this notation it would be three divided by x.  Rewriting as multiplication by the reciprocal, we get 3 times one divided by x.  That calculation is easily performed and we end up with a fraction, 3 over x. 

 

The second example is a little trickier.  Let’s unpack this.

$\frac{3}{x^{-1}}=3\times \frac{1}{x^{-1}}$

The second part, one over x to the power of negative one is a compound fraction … a fraction within a fraction.   With the sequence of topics that I use in teaching freshmen math, they’ll not have seen these yet.    But we can handle it with care anyway. 

Let’s just dive in, shall we.  Rewriting x^-1 we get one over x. 

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

This is one divided by one over x. 

$\frac{1}{\frac{1}{x}}=1÷\frac{1}{x}$

And we rewrite division as multiplication by the reciprocal.

$1÷\frac{1}{x}=1\times x$

That is tricky, ugly and inefficient.  That’s division for you though, isn’t it?  Now we don’t want students wasting a ton of time doing this … but we need them to understand that the reciprocal is written because of division.  Negative exponents mean division.  The reason we change the sign of the exponent is because we rewrote it as multiplication by the reciprocal.

To this point there aren’t usually any massive hiccups.  Where the trouble comes is applying this understanding to expressions with multiple bases, like the one that follows.

$\frac{a^3{b}^{-4}{c}^{-1}d}{a^{-5}{b}^4{c}^5{d}^{-3}}$ 

One thing that you can do to help students approach this type of problem is to help them unpack it.  Having them rewrite each base as a separate problem, all being multiplied can help. 

 

$\frac{a^3}{a^{-5}}\times \frac{b^{-4}}{b^4}\times \frac{c^{-1}}{c^5}\times \frac{d}{d^{-3}}$

There’s one more exercise that I like to use.  It is basically a matching game, though the students don’t know it initially.  In the activity there are 8 total expressions, all very similar, but with negative exponents, different numerators and denominators and the like.  In the end, there are four pairs of identical expressions.  Through the activity students are walked through multiply ways of approaching the simplification of each expression.  The idea is that one “rule,” is insufficient. Depending on the expression a rule can be very easy or very confusing and difficult.  Another valid approach may be much more appropriate when that taught “rule,” runs them into trouble.  It is a powerful lesson that would serve the student’s development of mathematical literacy wonderfully following a treatment of negative exponents as has been described in this podcast.  I’ll leave a link to the PowerPoint below that uses this exercise. 

 

Summary:  Let’s recap what we’ve discussed here today.  Negative exponents are another way of writing division.  That can be seen because negative means opposite and exponents are repeated multiplication.  The opposite of repeated multiplication is repeated division.  The tricky part comes in manipulating the expressions in a concise and correct fashion.  The reason that it is tricky to handle is because division is multiplication of the reciprocal.

The rule students are taught is that when they see a negative exponent, they flip the base and change the sign of the exponent.  The goal here is for students to be able to read the math and understand that set of directions. 

By exploring, carefully, the notation and its meaning, we hope to develop a mathematical literacy that in our students.  With that literacy will come improved retention and hastened acquisition of future topics in math. 

Closing:  Thank you for your time, once again.  I am humbled by your participation with the podcast.  It is my sincerest desire that you find what has been shared here thought provoking and that you will carry ideas gained here into the classroom to the benefit of your students.

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