You will need meter sticks and/or rulers for this section of the lesson. I begin this lesson by reintroducing my students to rational numbers. I post this question on my board: "What is the relationship between the distance you are from an object and its apparent height?" Apparent height is the height an object seems to be when viewed from a specific distance away. I ask my students to pair-share their ideas then randomly call on students to share what they discussed with the class. I summarize their ideas on the board and when everyone has had a chance to speak, I ask how we can determine which ideas are correct. Usually someone suggests that we measure some distances and apparent heights at which point I suggest that each set of partners select one object for their apparent height, and make at least five distance and height measurements. While they're working I walk around offering encouragement and redirection as needed. For example, "What object have you selected to measure?" or "What units are you using to make your distance and height measurements. I also post a chart for each team on the board with one column for distance and one for height, and ask my students to post their results as their get them. (MP4) When everyone is done we review the charts for any obvious outliers and remove them without focusing on who might have made inaccurate measurements - that's not important to this work. I ask my students to review the remaining data and look for patterns.(MP7) After a few moments, I ask for volunteers to share what pattern(s) they've discovered, and if nobody mentions that the apparent height decreases with distance, I ask leading questions like "Is the height increasing or decreasing as the distances get larger?" and "Did every team get the same kind of pattern?" When my students recognize that the apparent height decreases as the distance increases for every team, I ask them to choose one set of data (it doesn't have to be the one they posted) and try to come up with an equation to model that data. (MP2, MP4) While they're working, I look for teams that are struggling and ask leading questions like "What variable have you selected to represent distance?" and "Is that the dependent or independent variable?" Some teams choose to work with a linear model initially and I let them work until they figure out that it doesn't fit the data very well, then help redirect if needed by suggesting they try looking for sums, products, differences and/or quotients of their variables. When everyone is done, I ask for one member of each team to post the equation they've created next to the data set they chose. I ask for volunteers to discuss any similarities in the equations and generally someone will observe that the equations either have distance multiplied by apparent height equaling some constant or they have apparent height equal to some constant divided by distance. I congratulate the class on creating rational functions!
Guided Practice 5 min: Now that we've identified what a rational function looks like I work through an example problem (or two) with my students. I often ask for a volunteer to "scribe" the problem on the board while his/her classmates walk him/her through the steps to solve the problem. This helps my students build confidence in their ability to figure things out for themselves and also encourages positive collaboration while allowing me the option of offering guiding questions and/or comments if needed. I take the example problems directly from the handout and leave them posted on the board as my students begin their independent work so they have them for reference.
Independent work 15 min: You will want copies of the Rational Problems handout for this section of the lesson. I begin this section by telling my students that they will be working independently for this next activity. I say that they now have the opportunity to practice solving rational equations and will be sharing at least one problem with their classmates. I distribute the Rational Problems handout and while my students are working I walk around offering encouragement and redirection for those students who are struggling. (MP1)
Class presentations 10 min: When everyone is done, I randomly select students to put the problems on the board (in order). I have room for three or four students at the board at one time, so nobody stands in front of the class alone. I invite the rest of the class to critique the solutions with opportunity for the presenters to respond. (MP3) We go through the remaining problems in the same manner with students checking their own work as they go. My video explains why I value this system of checking work.
To close this section I give my students a notecard and ask them to briefly explain what a rational equation is in their own words, not with a diagram. (MP2) This may be harder for some students than it sounds and you might have to remind them of the appropriate vocabulary requirement, for example using numerator and denominator instead of top number and bottom number! You probably will need to encourage some students to persevere since many are still not accustomed to putting their mathematical thinking into complete sentences.