## Flipping Fractions

Yesterday, and today, I taught negative exponents to my Algebra 1 class.  Students started out with a “discovery” activity where they filled out two tables, one for $2^n$ and another for $3^n$, for n = 5 to -5 where they eventually figured out that each successive result was half of the prior result, for the $2^n$ table, and one-third of the prior result for the $3^n$ table, which made the answer for n=0 and any negative n more understandable.  I say “eventually” since most students did not recognize the pattern in the table until we covered it explicitly.

This activity led into an introduction of the following definition for a negative exponent.

For this definition, I mentioned that when any number a, a≠0, is raised to a negative power n, you took the reciprocal of the number a raised to the positive power n.  Students learned what reciprocal meant earlier in the year, as well as in prior math classes, so I did not explicitly mention how to write the reciprocal.  I also did not provide an explanation for why the definition was true.

If I had to explain why the definition is valid, I would show the following where any number a, a≠0, which is raised to a negative power n can be multiplied by its multiplicative inverse to yield 1.  The first step in the following started by multiplying the denominator in the definition above with the left hand side of the equation.

While I might have missed an important step from a true, mathematical standpoint, I believe students would understand why this is true if we talked about it further.

Likewise, when using fractions instead of whole numbers, the definition above is written as follows.

Since there is a fraction in the denominator in this case, we invert the fraction in the denominator and multiply it by the fraction in the numerator, 1 divided by 1.  The result is the reciprocal of the fraction raised to a positive power n.

The “proof” for this is the same as shown above for whole numbers as follows.

Hence, in my instruction, I took care to show students the first couple of steps each time, as in the following example, instead of directly proceeding to the solution using a slight of hand (e.g. just flipping the fraction and making the exponent positive).

My CT mentioned after class that she felt my method might be too confusing for students.  She mentioned that she simply taught students to write the reciprocal of a fraction to a negative exponent by flipping the fraction and making the exponent positive.  In essence, she skipped my intermediate steps two and three, above, and jumped straight to my step four.  While I mentioned to students that they will be able to do that in time, simply by recall, I wanted them to see the mathematical reason behind that “trick.”

After hearing my CT’s suggestion, however, I am not so sure my approach was the best.  Did I confuse students when I intended to help them?  Was my use of mathematical procedure for dividing fractions more confusing to students versus simply telling them they flip the numerator and denominator?

Did I overcomplicate my instruction?  I care deeply about helping my students learn math.  I will always show them tricks or simpler ways to solve problems.  However, I do not prefer to show the “memorize this” approach first.  I want to show them the mathematically correct way first, at least mostly correct, then show them shortcuts, tricks and the like.

What do others think?

## About Dave aka Mr. Math Teacher

Independent consultant and junior college adjunct instructor. Former secondary math teacher who taught math intervention, algebra 1, geometry, accelerated algebra 2, precalculus, honors precalculus, AP Calculus AB, and AP Statistics. Prior to teaching, I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.
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### 18 Responses to Flipping Fractions

1. David says:

I share proofs with my students all the time, even if I know that sometimes the actual steps of the proof are complicated. If you believe that one of the purposes of mathematics is to build logical reasoning into your students, then skipping proofs does not help this at all.

As an alternative though to starting with the algebra, I’d recommend starting with the numbers. Better yet, have the students calculate different exponents written in alternate forms (with a calculator for now) and see if they can develop the “rules” behind the exponents themselves. Of course you’ll have to choose your examples pretty carefully so that it is possible to construct all of the rules themselves.

Once they have this background, now you can introduce the proof, again starting with numbers and moving into abstraction.

If your students see you jump from something to something else and no explanation of the logic behind it, they will model you and do the same thing in their work, not understanding why it is unacceptable.

So either share a complete proof or pull the formula out of hat but definitely don’t do something in between.

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2. Gary Davis says:

Hi Dave,

nice post on a topic that is of perennial concern to students and teachers.

The reason you cannot give a proof of $a^{-n}=\frac{1}{a^n}$ is because, in most standard developments of a general index, this is a definition.

Why is it a definition? Why does it work this way?

Because it is the only definition consistent with the functional law $a^{m+n}=a^m\times a^n$, which holds good for positive m and n.

We use this functional law to GUIDE us as to how to define exponents for non positive integers (and fractions and other numbers).

Your CT is on the wrong track IMO, and you are taking steps in the right direction, namely toward understanding.

Good luck with your blog, and keep up the blogging!

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3. John Golden says:

You can just teach the students the procedure, but is that what you want them to know? It sounds like you care about the why.

The why here is really something mathematicians care about a lot, closure. We know what positive exponents do, can we extend that to zero? To fractions? To negative numbers? To complex numbers? That means, can we figure out what they do, if our rules stay true. We want exponents to add when multiplying like bases, so what does that mean?

Your CT cares about the skill issue, which is also important. But research (Skemp, etc) seems to show that retention and skill performance is better if it arises from understanding. Relational understanding, Skemp called it.

2^3=8
2^2=4
2^1=2 all solid
2^0=
2^-1=
This rule of negative exponents is not just true, it is right. It fits. It’s … pretty!

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I struggle with this kind of dilemma a lot too. I wanted to have students explore why the log properties work. However, because of time constraint and the pressure to move onto next units, I didn’t really carry through with this activity. I agree that if you just tell them that they flip the fraction, your students may be able to memorize but not really know why. I hate to see students thinking that there’s no reason behind why we flip the fraction and believe that it is just one of those things you need to memorize in math.

Perhaps we could first teach them the simple ways (flipping the fraction) so that they get the procedure/ rule down. On the next day, we could have them explore/ we could show them the proof. I feel that if I had shown them the proof when they first learned it, they might be trying too hard to grasp the proof that they don’t get the procedure down.

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• Valeri McGuire says:

So, I’m struggling with this for my algebra 2 students. I’ve come to explain the flipping fractions thing like this: If you have (1/2)^-1, I say that you can distribute the -1, and have 1^-1/2^-1. Then, you switch the negative exponents so that you then have 2^1/1^1. This makes sense to them, isn’t just telling them to switch the fraction, but doesn’t get into dividing by (1/2). That confuses them.

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5. I believe you should not be looking for an universal answer that would fit all students. Your CT might have been right, given the specific student audience. She may have been wrong, having an entrenched notion of students’ abilities and preparedness for certain kind of math reasoning. Also, you may have confused some students by declaring something a “proof” which is just a definition. You know that.

A teacher must be allowed more flexibility to adapt the materials and the teaching process to the student audience and individual student abilities.

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6. John Golden says:

Research suggests it’s harder to provide motivation once a rule is known. Not impossible, just harder.

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7. Rachael says:

I haven’t formally taught this yet, but when I’ve been needing to teach this to kids I’ve tutored one on one, I’ve done it by showing them the patterns. There are probably all sorts of reasons why this is a bad idea, but: I draw a number line and then pick a number (powers of 2 are simple to use), and go along the number line adding in the positive powers. I ask them what the pattern is (what are they doing to the number to get from one to the next ‘down’ the line). Easy to get them to see that a number raised to the power of zero is 1, and then you continue the pattern ie keep on halving it. I ask them how else you could show that and they generally get there on their own. I repeat it with another number if necessary (sometimes the penny drops really fast so it isn’t necessary) and then do the general case…

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8. eo says:

My personal opinion is that if you don’t show them why it is true, there is no hope (zero hope) that they will retain it. But hey, most adults can’t do those problems and they don’t need to.

So the question becomes why are we wasting resources teaching algebra to folks who will never us it except on a test?

In listening to the State of The Union the other day I just shook my head when all those politicians bobbed their heads at Obama’s pushing of math and science. Almost none of them know anything about what math and science are. They are not memorization.

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9. Cal says:

I’ve used the discovery activity you describe and it works well. Moving beyond that, your explanation runs smack into the real problem, which has nothing to do with exponents: all but the top kids don’t grok “reciprocal”, even though I’ve explained it and given exploration worksheets many, many (sob) times . That’s why I stop with the discovery that the 0 and negative exponents are logical–I want their last memory of the exercise to be that “aha, that makes sense”. If I add in too much explanation, they’ll lose that feeling.

I bought some fraction circles over Christmas to explain work problems to my top students. I’ve been thinking of using them in an exercise with reciprocals–again–before I revisit exponents.

It’s a good explanation, by the way–I just worry about overwhelming the kids with too much information. Struggling kids see exponents as utterly alien anyway.

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10. suehellman says:

Once we’ve reviewed reciprocals and worked through patterns and have moved on to solving questions, I like students to start by rewriting the original expression to have only positive exponents before simplifying. I use this ‘memory’ trick: Pop, Drop, and meet my cousin Flip. If the variable with the negative exponent is in the denominator, that variable ‘pops’ to the top of the fraction. If it’s in the denominator, the variable ‘drops’ below the fraction line. (If that leaves an empty space in the numerator, fill it with a 1.) If there’s one of each, that’s a ‘pop and drop’. If the fraction is in brackets with one negative exponent for all, that’s when you meet my cousin Flip. (Flip the fraction inside the brackets first and then distribute the exponent to numerator and denominator.) I get the kids to do the arm motions and dramatic actions when they explain their processes so there’s lots of laughter.

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11. Tsunoba says:

Thank you. I know how to do things with negative exponents, but I didn’t think about the “why” until a student I was tutoring asked me, and I had no answer.

In my experience, a lot of people seem to care more about the how than the why, which seems weird to me, because I understand better when I know why.

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12. Y UPSON says:

This is an old thread, but i just wanted to say (as an adult university student) i have always deplored shallow explanations and rote learning approaches. Far too few teachers take the time to provide comprehensive explanations; explanations I often require to really grip a concept and secure it in memory.

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13. Janelle Delas Armas says:

I love your method and I hope that my teacher did the same. I always want to know the mathematical reason behind “tricks” or shortcuts so I know why they happened and If I encounter a problem and do that thing I’d know why it was happening not just because it was said that it should be done.

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