Graphing Unit Rates
Lesson 1 of 7
Objective: SWBAT graph proportional relationships, interpreting the unit rate as the slope of the graph.
Because this lesson builds on concepts previously learned, I have provided students four quick unit rate problems to compute as a Warm Up. As my students work, I circulate through the classroom and clarify misconceptions (What is your unit? Pound per cost or cost per pound?) and/or provide support as needed (What operation will you need to use to get unit rate?).
Once the timer sounds, I randomly select students to share their answers with the class. After they respond I ask for class agreement. Once all are in consensus about the four Warm Up problems, I introduce today's Learning Objective.
During our previous unit on linear equations, I introduced the Rule of 5 (story, picture, table, equation, graph) to my students which helped them to conceptualize the interrelationships between tables, equations, and graphs that come from real-world (or fictitious) stories (see Turtle and Snail Part 1). As some of my students are apt to visualize the situations, I also include sketching the story before representing it mathematically. To remind my students of this process today, I took the first scenario from the warm up and prompted students to help me apply the Rule of 5 to it.
Once we have completed applying the Rule of 5 to the scenario, I explain today's Work Time Task.
For today's Work Time Task, I paired like-skilled students to apply the Rule of 5 to one of four rate problems (A-D) that I supplied at random.
- Graphing Unit Rates- Story A
- Graphing Unit Rates- Story B
- Graphing Unit Rates- Story C
- Graphing Unit Rates- Story D
By doing this, several different groups were working on a given scenario so that during presentations, students had three to four representations to compare for each situation. I set the timer for 14 minutes, but I will adjust it according to my students' pace.
When the timer sounds to end Work Time, I randomly select a letter group to come forward and present their work using the document camera (e.g., all those with Story B). I ask the rest of the students in the class to compare and contrast the representations (tables, equations, then graphs). I encourage them to ask questions about the work (MP3). Once the class exhausts its supply of comments and questions, I call the next group to the front.
As students present, I am intentional about asking particular questions:
- Do their tables match? They typically do.
- Do their equations match? These typically do, as well.
- Do their graphs match? Typically, they do not, so I ask guiding questions to lead students to the reason why: "What can you tell me about the scales of each graph? How does this affect the graph?"
After each group has presented, I want to make sure that my students leave with the understanding that regardless of representation, the rate of change (or slope) for each scenario is the same because of proportionality. So, for each group, I take the table from one group, the equation from another, and the graph from a third and display them on the document camera. I ask students to discuss what they notice for 30 seconds. I eavesdrop on the groups, making note of points made that I would like shared in the larger group. I then call on those groups to wrap up the lesson.
Ticket Out the Door
For the Ticket Out the Door, I have provided a scenario and two different graphs and ask the students to choose the one that represents the given rate. My students must also justify their answers. The graphs differ in scale as well as y-intercept and slope. However, I have intentionally included (9, 450) as a point on both graphs. This common point should be helpful in making a comparison. I am interested to see what other information students will use to justify their answers.
For extra practice with today's work, my students will complete the activity Buying Bulk Candy for homework. The exercise asks students to find a unit rate, then represent that rate in a table and a graph.