Hand out the double-sided "Power Sprints" 3 & 4, with sprint 4 facing up.
On "Go!" students flip the sheet over and have 2 minutes to solve as many of the problems on Sprint 3 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 4. The point is for every student to make progress, not compare themselves to their classmates. 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. 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.
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.
Hand out the Exponents and nth Roots worksheet. Students work individually or in small groups without calculators.
Emphasis is on understanding a rational exponent as two operations. I will begin by doing one problem as an example at the whiteboard. For instance, with 4^(3/2) we are both taking the square root (b^(1/2)) and cubing (b^3). The two operations may be done in any order, so we should consider which order is simplest. In this case, it's clearly easier to take the square root of 4 and then cube the result. This kind of strategic thinking emphasizes both the concepts and skills of exponentiation. (MP 7)
In the middle of class, I may interrupt the students' work to "check their answers" on the first few sections. I'll read through the answers to sections I and II, and then let the everyone get back to work. This catches early errors and helps to build confidence moving forward.
I find that some students don't notice the connection between the radical forms in section II and the rational exponential forms in section III until it is pointed out explicitly. Some of the expressions are perfectly equivalent!
When students get to section III it will be very helpful for them to consider these three equivalent forms:
Generally, the expression will be easier to simplify in one form rather than another. This understanding is one of the primary goals of this problem set, so it's important to emphasize it over and over in one-on-one conversations, by writing it on the white board many times, and by ensuring that it makes its way into the students notes.
Note: All problems have integer solutions until section V. You might provide this as a scaffolding hint to some students.
After checking our progress, students may have some errors to correct, but they should quickly begin moving forward again. At this point, some students my begin to ask, "Do we have to finish by the end of class?" On the one hand, I don't want to stress them out by saying "yes, definitely", but I also don't want to affect their motivation by saying "no". So I will simply answer that they should try to get as far as they can correctly. At the very end of class I'll let them know how much needs to be done for homework, and this will typically mean completing sections I - IV.
I use these last 10 minutes for some formative assessment. As I make my rounds, I'll check to see how far most students have gotten, which problems are giving them trouble, and what sort of strategies students are using. For instance, I want to see students converting radicals to rational exponents and then using the properties of exponents to begin the simplification.
As class ends, I'll announce the homework: "Be sure to complete all of the problems in sections I - IV, and as many as you can in section V. We'll discuss the solutions tomorrow."