I start with two examples of rational expressions on my front board and ask my students to pair-share ways to simplify each example. (MP1) As I walk around I expect most of them to recognize that the numerator of the first expression can be factored and then common factors can be crossed out to give a simpler expression. However, it's always interesting to listen to the discussions about the second expression. There are usually a few students who are determined to factor the numerator of that expression as well, so I try to redirect those students before they become too frustrated and shut down for the day. After a few minutes I ask for volunteers to share their ideas with the class. My emphasis for this discussion is that some rational expressions can be simplified fairly easily by looking for common terms or factors, while others take additional work or cannot be simplified further. To give another example of this I put two fractions on the board and ask my students for ways to simplify them. I intentionally choose one fraction that reduces to an integer and one that doesn't. These examples help my students connect what we're working on today to something they're familiar with and already know how to do. Rather than simplifying either of these integer examples yet, I just have my students talk about how the process for simplifying a rational expression is similar and different from the process of simplifying an integer fraction. (MP7)
I walk my students through side-by-side long division to simplify problems #2 (rational algebraic expression) and #4 (rational numerical expression) from the lesson launch to demonstrate similarities. I want to help my students make a connection to something they already understand to make today's work easier and to make it more understandable, as I discuss in my be rational video. While I'm working, starting with the numerical example, I ask for suggestions about next steps (MP1, MP2). The reason I work through both the numerical and algebraic myself is that I don't have any really strong students to work alongside me this year and I want to make the example problems very clear. If you have some strong students, try letting them work the numeric while you work the algebraic. As I do this I'll throw in a question challenging them as to why they think that's a good idea or how they knew what to do next. Once we've completed the examples and discussed the algebraic-numeric connections (MP2) I check for understanding (fist-to-five)before they move to the BE RATIONAL practice problems (MP1, MP2) I have students work independently to develop confidence in their skill and so that I can identify those students who may need additional support and/or scaffolding. Typically students who struggle with this never really understood integer long division. Often I will schedule extra time outside of class to support them.
When everyone has completed the problems, I let them choose a person to work with and ask them to critique each other's answers. (MP3) My students like the privilege of choosing their own partners, but I advise them to be careful in comparing answers because just because two people got the same result doesn't necessarily mean it is correct. While they're working I again walk around observing and assisting as needed.
I often have students correct their own work or classmates work in class and occasionally if I want to review their work more closely I collect it. That's my choice for this assignment because it is the first work for a new unit, and can serve as a formative "pre-test".
To close this lesson I give each student a notecard and ask them to work out a polynomial long division and integer long division problem of their choice on one side of the card. On the other side I ask them to compare/contrast in writing polynomial long division and integer long division. (MP6) This gives those students who think verbally an opportunity to articulate what they've learned today and it pushes those students who don't like to write to put their actions into words.