Solving for Independent and Dependent Variables

Students work in pairs on the Think About It problem.  I have students share out the table and the graph.  I ask students how they answered Parts C and D.  Some might say they looked at the table, others the graph, some might have substituted into the equation.  I want all three of these strategies to come out in our discussion.  Each method is equally valid.  

The Intro to New Material section contains one example.  In this lesson, students are putting together what they've learned in the first three lessons of this unit.  They'll work today to use the equations, tables, and graphs to find the value of one variable when given the value of the other variable.  

Graphical Representation

1. Find the given value for the independent or dependent variable on the corresponding axis
2. Identify the corresponding value for the other variable

Table Representation

1. Find the given value for the independent or dependent variable in the table
2. Identify the corresponding value for the other variable

Symbolic Representation (Equation)

1. Rewrite the equation with the substituted value for the independent or dependent
2. Use reasoning skills to solve the equation

Students work in pairs on the Partner Practice problem set.  As they work, I circulate around the classroom and check in with teach pair.  I am looking for:

• Are students correctly identifying the independent and dependent variable?
• Are students correctly using the graph/ table/ or equation to solve for a value?
• Are students explaining their answers and discussing the reasonableness of their answers?

I'm asking:

• How did you know you were solving for the value of the independent/dependent variable?
• How did you know to use the graph, table, or equation to solve for a value?
• How did you know that your value is correct?

After 10 minutes of partner work time, I display one student's work for Problem 2 on the document camera.  I ask the same questions of this one student that I asked of pairs during partner time (see above).

Today, I will ask the students to complete the Check for Understanding problem independently. Before moving on to the independent practice, I have students turn and tell their partners how they'd determine how long it'd take to brew 95 pounds of coffee. I want my students to compare strategies and learn from each other. Taking this extra step helps some students to learn new strategies. Other students are developing their ability to explain their mathematical practices to others.

Students work on the Independent Practice problem set.  

As I circulate, I ask individual students how they'd solve for a variable with a given tool.  I also ask them which is most efficient, given the situation.  I want students to be able to use all three methods for solving for an unknown, but also recognize that there are times when one strategy is more appropriate than the others.  

Problem 6 can be difficult for students to graph, because of the decimal values and the scale of the y-axis.  Part D is a good problem that reviews the meaning of remainders in division.

After independent work time, I bring students back together for some whole-class conversation.  First, we talk about Problem B.  I call on a student to explain why extending the graph would not be the best way to solve for the unknown value in this situation.  After the explanation, I ask 'If we wanted to, could we do it?'  It is possible to extend the graph to a value of 24, but we'd need more graph paper.  

We also talk about Problem 7.  I share out student work that has graphed the relationship incorrectly (namely, the student did not include the initial value of $3 when creating the ordered pairs).  We talk about why this is wrong, and also how we can look at the graph and tell that something is amiss.  If a student graphs (0,0), (1, 5.50), and (2, 8), there will not be a constant rate of change.

Students independently complete the Exit Ticket to close the lesson.