The goal of this Warm-Up is for students to discuss the ideas of the lesson with their partners in as much depth as possible.
The front (page 1) of the warm-up is the essential page. The second page is included for students who master the ideas of the first page more quickly and are ready for more depth. During this long opening, I circulate and ask students the following questions about the three_key_problems.
During this time and this entire lesson, it is way more important to take the time to talk about the real world example than it is to write the equations. The abstraction of the equations is not essential today.
Students will now work on the Applications of Rational Functions Problems. The goal of this time is to solidify the key ideas of the warm-up. In today’s class, the emphasis is not on equations. I am okay if students do not work abstractly. In fact, it is almost better if students don’t use equations, but rather think about the real-world situation in order to complete the data table. If students don’t figure out the equations today, I won’t provide them. They can learn them from peers who do figure them out, but as the teacher I will not ask students for or give them equations. If they come up with equations, I'll ask them: “Does this equation make sense based on the situation? If you use the equation, do you get the same outputs as your partner?”
It is important that all students understand the approach statements. One great question to ask is: “Will your total time ever be exactly 3 hours? Will it ever be less than 3 hours?”
Instructional Note: Your decision is how much to do as a whole class versus in small groups. I spend time discussing the context of each problem and making sure that students know how to fill out the data tables without using equations. It might make sense to briefly discuss each situation as a whole class and make sure that students understand the real world context of the problem (with actually doing any math.) Another option would be to assign different students the 3 key problems and do these in a jigsaw format.
The appliance situation requires the most explanation because it is the least intuitive, especially for inputs between 0 and 1 year. I finally figured out that the best way to explain it is to say that the refrigerator only lasts for a certain amount of time. For instance, if the refrigerator only lasts 0.5 years, you will have to buy two refrigerators in the one year in order to have a refrigerator for the year. Alternatively, if your refrigerator lasts for 1,000 years, the initial price of buying the refrigerator will be spread out over all those years.
While students work through these three problems, my main focus will be on the behavior of the function and on making connections between the graph, the data table, and the real world situation. For instance, a student may say, “As the speed increases, your times gets closer and closer to 0 because you are going faster.” This is a great example of a partial understanding. It is true that the time gets closer and closer to 0, but really the time is not getting that close to 0. It is getting closer to 3 hours. Why would this be true? The graph shows this using the horizontal asymptote. The data table shows this because if you use a huge number for the speed, the time will be close to 3. This makes sense based on the situation because no matter how fast you go, you will still be sitting still for 3 hours, so even if you go a million mph, you will still be stopped for 3 hours. These are the types of conversations I will try to facilitate during this time.
If I notice students getting bogged down in the details of the calculations or the graphing (especially setting up the axes), this is when I provide them with more scaffolding, like setting up the axes or using a calculator for the calculations. I think it is worthwhile for a student to be able to determine that if your speed is 0.5 mph, you will take 160 each way because it will take you 2 hours to go one mile. I do want students to understand that calculation without a calculator, but they don’t need to do all calculations without a calculator. Also, students can copy each other’s data tables. The goal is to make sure that they don’t spend the whole class period doing calculations and plotting points, because they will then miss the whole point of the lesson.