We'll begin class by reviewing the homework from the previous lesson. First, I'll ask students to compare their answers with their peers and try to resolve any differences. This should catch most of the simple mistakes like flawed arithmetic or easily identified algebraic errors. After a few minutes, we'll pick one or two of the more challenging problems to solve as a class. See this video for details.
Along the way, it's important to continually emphasize the ways in which we can make use of the structure of the given equation. Also, reiterate the rules we've developed for efficiently graphing rational functions.
This first problem is a modeling problem in the sense that students are using rational functions to model something simpler - arithmetic with unit fractions. It is grounded in a historical context and harkens back to things that students have been familiar with since elementary school. (MP 4)
Before handing out the problem set, you might want to set the stage by showing the class the fraction 87/110. The ancient Egyptians preferred to think only in terms of unit fractions, so instead of writing 87/110, they would write the sum 1/2 + 1/5 + 1/11. What do your students think? What advantages or disadvantages are there to the Egyptian system?
After this brief introduction, hand out Egyptian Fractions, and have the students begin working either individually or in small groups to solve the problems. As they work, circulate around the room offering encouragement, quietly assessing progress, and answering questions.
This problem set is based on this one from Illustrative_Mathematics.
Certain patterns and rules are pointed out in the problem set, but it's quite possible that your students will discover some other ones on their own. If that's the case, I like to set aside 5 - 10 minutes at the end of class for these things. As I notice students making these discoveries, I'll suggest that they write down them down clearly on a separate sheet of paper for sharing with the class. Now, I'll call these students up to use the document camera to show their discoveries to their peers. (MP 8)
The main thing is that everyone has been able to make use of rational functions to make generalizations about fractions. This sort of algebraic proof is very important in mathematics, but it isn't always easy for students to understand.