Using Graphs and Tables to Determine Relationships
Lesson 9 of 11
Objective: SWBAT analyze relationships between dependent and independent variables using graphs and tables, and relate these to an equation.
Think About It
After 2 minutes of work time, the class discusses this problem. First, I ask a student who went the furthest, and have him/her share how (s)he knew. Then, I ask a student who walked faster, and have him/her share how (s)he knew. I'll then ask students if anyone thought about this part differently. Some students will use the graph to pull out the unit rate for each girl. Some will use the graph to reason about speed, given the distances traveled at 5 hours for each. Students may comment that Monica's line is always higher than Dylan's line.
I ask students to identify the independent and dependent variables for each walker. In this scenario, the independent variable is the same for both. It is the dependent variable (total distance) that is different. I ask students to explain how that can be true.
Intro to New Material
In this lesson, students are using what they learned in the previous lesson about independent and dependent variables, and representing the given relationships using tables, graphs, and words. For each problem, students will identify the independent and dependent variable, create a table of data points, and then graph each coordinate pair in their table.
When students are creating the tables, I have students use the values given on the x-axis as a guide. I tell students that we can use any value we'd like for our independent variable when making a table, but it makes the most sense to use the values we know we'll have to plot. When there are fewer spots in the table than there are x-values on the graph, we talk about which numbers might be the easiest for us to substitute in to the equation.
Teaching Note: In this unit, we do not talk explicitly about slope or linear functions. However, students will see that all of our relationships are linear (without accessing that specific word). If they have a point on the graph that is not a part of the line, they'll recognize that something has gone wrong. They'll also recognize and articulate that there is a constant rate of change in each of the scenarios.
Students work in pairs on the Partner Practice problem set. As they work, I circulate around the classroom and check in with each group. I am looking for:
- Are students explaining their thinking to their partner?
- Are students explaining the relationship between the two variables?
- Are students discussing the change in the two variables?
- Are students understanding what the numbers and variables in the equation represent?
- Are students making connections between the graph, table and equation?
- Are students placing the independent variable first in their tables?
- Are students using substitution to find the dependent variable values?
I am asking:
- How did you know this is what the graph should look like?
- How did you know that this explanation correctly describes the change that is happening?
- How can you use the graph to explain the relationship between the two variables?
- How do you know that the graph, table and equation all represent the same situation?
- Explain you determined this answer.
- Tell me how you found the value of...
The Check for Understanding problem may initially confuse students, because the coefficient on the independent variable is 1. Students are not used to seeing a table with the x- and y- values being the same. As we discuss this problem, I ask them what the coefficient would be, if we wanted to write a equation to represent the relationship between touchdowns and a football score. I ask students what might make this difficult (the score is influenced by more than just touchdowns). I then ask why basketball wouldn't be an easy sport for us to write a similar equation for.
Students work on the Independent Practice.
As I circulate, I watch to see that students are carefully graphing Problem 1. The values on the y-axis are not the values that they'll have in the table, so students will need to plot in between the grid marks.
When I check in with individual students, I ask them questions about the particular equation they're working on. I'll ask about the meaning of the coefficient, how they decided on the values for the table, what the relationship is, how we could use this line to predict other values, etc.
Closing and Exit Ticket
At the end of Independent Practice work time, I bring the class back together to discuss Problem 7. As a whole class, we discuss parts B and C. I have students share out their thoughts. I then have students turn-and-talk with their partners about what they chose for Part A, and why they chose that response. Writing an equation will come in a later lesson, but this problem gives students the opportunity to make sense of a graph.
After students have talked with their partners, I have 2-3 students share out how they thought about Part A of this problem.
Students then complete the Exit Ticket independently to close the lesson.