Rational Exponents

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SWBAT articulate properties of exponents. SWBAT use properties of exponents to evaluate rational powers.

Big Idea

Mathematics is all about patterns, and exponents reveal some "power"-ful ones!

Power Sprints

10 minutes

Class will begin with the first set of sprints

The idea behind the sprints is simple: students improve their understanding of exponentiation though repeated practice.  Just as multiplication facts in elementary school are helpful when learning division, my students are learning exponentiation facts that will be extremely helpful when they study logarithms.

Hand out the double-sided "Power Sprints" 1 & 2, with sprint 2 facing up.

On "Go!" students flip the sheet over and have 2 minutes to solve as many of the problems on Sprint 1 as they can.  I use the two minutes to walk up and down the rows to get a quick feel for how everyone is doing.  Based on this, I may quietly add or subtract 10 - 15 seconds from the allotted time.  Shhh, don't tell!

When time's up, call "Pencils down!".  Everyone breathes a sigh of relief, and I begin reading the answers.  It's important to establish the norm that I will read the answers exactly once.  Students are simply supposed to mark the ones they answered correctly, not write down the answers for the ones they missed.  When I'm done, the students are supposed to put their score at the top of the sheet and circle it.

Now, give the class about 2 minutes to consider & discuss strategies for increasing their efficiency.

Time for the second sprint!  On "Go!" the students flip the sheet and race through Sprint 2.  The point is for every student to make progress, not compare themselves to their classmates.  If I feel like my students really need a confidence boost, I may quietly give them a little more time on the second sprint (5 or 10 seconds), so they can answer one more question. 

After the sprint is over and we've marked the correct answers, I may ask how many students did better on the second sprint, or whether they felt the second one was easier or harder than the first. 

When we're all done, I collect the sprints, but I make a point of saying that they're not being graded.  I just want to know where everyone is so that I can keep track of their progress.  I generally use a very simple spreadsheet for this purpose.


Large Powers

5 minutes

With student input, write down the three well-known properties of exponents.

We'll be adding to this list as the class progresses, so be sure not to erase it!

Ask students or groups to share one of the ways they found to evaluate 3^16, 3^27, and 3^64 on Exponents.  For example,

The goal here is to see that these powers can be thought of in a variety of ways by making use of the properties of exponents listed above.  Variety is the goal, so don't worry about finding a "simplest" expression.  Rather, focus on the equivalence of the various expressions.

Since much of this should have been covered in Algebra 1 according to the CCSS, this lesson may be accelerated or abbreviated in the future.  For now, I think all of my students need it.

Negative Powers

10 minutes

Moving to a new section of the board, I will list the first four or five powers of 3, and I'll ask the class what we have to do to calculate the next one.  The answer I'm looking for is to "multiply the previous one by 3."  Great, I'll say, so we can keep going down the line simply by multiplying again and again by 3.  That makes perfect sense because that's comes from the definition of exponentiation that we all know and love. (MP 8)

Now, I'll call on a student to come to the board and explain to the class how we can make sense of powers that are non-positive integers.  Depending on what I saw during the previous lesson, I may ask for a volunteer or I may call on a particular student.  (It might have been a good idea to have warned this student yesterday!)

As the student gives his or her explanation, I'll move to the back of the room where I'll be out of sight.  This encourages the rest of the class to pay careful attention to their classmate and prevents them from looking to me for confirmation.  Even if the student at the board makes a mistake, I'll keep quiet.  I'm sure some of the other students will have noticed it, and I'll encourage questions at the end.  (Of course, if no one noticed the mistake, I'll ask a gentle question myself.) (MP 3)

The white board is going to be used like one of Vi Hart's notebooks, and may end up looking just as cluttered.  While the final image may be hard to make sense of, the process was the important thing.  (Check out her video below, if you aren't familiar with her work! )

Finally, I'll return to the front of the class and summarize the presentation/discussion by adding this new pattern to our list of the "Properties of Exponents" that is still on the board from earlier.

Rational Powers

15 minutes

Before I call another student to the board to explain rational exponents, it's worth noticing what we have so far.  If we were to graph the powers of 3, we would see a scatter plot of discontinuous points.  Would we be justified in "connecting the dots" on our graph?  Certainly not!  That would imply that we could raise 3 to any power we like!  Since we all "know" that exponentiation is simply "repeated multiplication", how could we make any sense of something like 3^0.5 or 3^1.25?! (MP 2)

After a minute or two for students to discuss this question with a neighbor, it's time to call a one of them to the board to explain just how to make sense of such rational powers.  In this case, I'm very careful to call on a student who I am confident can offer a good explanation (it would have been good to have prompted this student yesterday!)

Again, the student will offer an explanation from the front of the room, while I stand quietly in the back.  I do my best to keep a straight face and withhold my judgement until the class has had a chance to ask questions and come to their own conclusions.  "What do you think," I ask, "does this explanation make sense?  Is this the way we should evaluate all rational powers of all bases?"  Hopefully the answer is YES, and hopefully they're correct!  (If not, I'll need to intervene much more explicitly.)  Also, be sure to bring up the fact that we'll run into problems if the base is not positive!

Finally, we will make another entry in our list of "Properties of Exponents" to formalize the meaning of rational exponents.

What is Exponentiation?

10 minutes

Now, I'd like to use the last 10 minutes of class to revisit the definition of exponentiation.  We all "know" that exponentiation is simply repeated multiplication, don't we?  Doesn't 3^x mean "3 multiplied by itself x times" or "x three's multiplied together"?  Then how on Earth can we say that 3^1.5 or 3^-2 has any meaning whatsoever?!

This question invariably generates a lot of conversation and debate, but I have to be careful.  I want to lead my students to the realization that mathematicians are free to modify the definition of exponentiation, but I do NOT want them to come away thinking that these changes are arbitrary or capricious.  Rather, the meaning of exponentiation has developed and mathematicians have come to recognize that the former definition was too narrow.

Now, we can acknowledge that "repeated multiplication" is sufficient for explaining positive integer powers.  From these powers, we discover the properties of exponents.  And the properties of exponents allow us to extend the operation of exponentiation to all of the integers and to any positive rational power.  (We'll leave negative rational powers, irrational powers, and complex numbers for some other time.)

If all goes well, class will end with students carrying the debate out the door and into the halls.