I start with an example of Polynomial Long Division Example on the board and ask for volunteer to walk through all the steps. While the student is working I comment dramatically about how much of the board it takes up and how long it takes to complete. I praise my student for sticking it out (MP1) then ask if anyone knows an easier way? There are usually only a few suggestions (if any) but I accept all ideas and encourage the class to test them out to see if any are effective shortcuts. When we've exhausted those ideas I post an example of integer long division on the opposite side of my board, leaving the completed polynomial problem untouched. I work through the long division problem then ask if anyone knows a shortcut for this problem. I usually get several volunteers who are eager to show me that they don't need to write out all the steps because they can cancel as they go. I offer to demonstrate a similar shortcut for the polynomial problem if they're interested. By this time they are usually more than ready to find a shorter way to solve these problems because they've already worked through an entire lesson on them.
I walk my students through synthetic division and rectangular division of the same polynomial division example already on the board, carefully explaining each step as I go. I try to show the similarities between all three methods of division and also make connections to integer division as explained in my synthetic division video. This YouTube video from James Stanton illustrates the rectangular division method in more detail. When I'm done, I ask for a fist-to-five to check for understanding and work through another example if necessary, using student volunteers. Then I tell my students that for today they will be working in teams of two to complete more polynomial division problems. I explain that each student needs to work through each problem but that they can collaborate while they're working. (MP1) I distribute the polynomial division problem set then walk around while my students are working offering encouragement and redirection as needed.
When everyone is done, I randomly select students to post their work on the board. My board can fit about 4 students at a time, so it doesn't take long for all the problems to be posted. I ask my students to review all the work and see if they have any questions or critiques. I allow my students to discuss and critique the posted answers until they think they've corrected any errors. (MP3) If they miss any I use leading questions to help them complete the correction process, then have each student check and correct their own paper.
To close this lesson I have my students write a short compare/contrast piece about long division and synthetic division, (MP7) including a include discussion of which they prefer and why. By asking for their preference and reason I get some insight into the comfort level of each student with the less familiar and more abstract synthetic division. It also offers my students the opportunity to reflect on the two processes and write out their thinking so that they strengthen their understanding about both.