Lesson 12

In the warm-up, students are given a situation and asked to describe the graph without actually graphing the lines. Identify students who correctly use mathematically correct terminology such as \(y\)-intercept, slope, \(x\)-intercept, and intersection to describe the graph.

Arrange students in groups of 2. Give 3 minutes quiet work time followed by brief partner discussion for the last question.

The purpose of the discussion is for students to practice describing graphs in words using correct mathematical terminology.

Select previously identified students to share their descriptions of the graphs. After each student shares, ask the class if the description is clear and to identify any vocabulary they heard that made the description precise or any vocabulary that is unclear. If any vocabulary needs to be reinforced for student understanding, this is a good time to discuss these words.

In this activity, students start with an equation relating distance and time for Han’s hike and enough information to write a second equation relating distance and time for Jada’s hike. After writing Jada’s equation and graphing both lines, students then use the lines to identify the point of intersection and make sense of the point’s meaning in the context.

This activity is a culmination of student’s work writing, solving, and graphing equations along with the thinking they have done on what it means for an equation to be true. From this foundation, students are ready to understand solving systems of equations from an algebraic standpoint in the following lessons. Fluently solving systems algebraically is not expected at this time.

The purpose of this discussion is to strengthen the connection between graphs and equations and formally introduce the vocabulary for systems of equations.

Begin the discussion by asking groups to share their responses for the last three questions. Ask students how they could check to make sure the coordinates they found with the graph are correct and give a brief quiet think time before selecting students to share their strategies. While students may suggest ideas like checking to make sure the lines are graphed correctly, it is important to point out that graphs are not always perfect. If not brought up by students, mention that a better strategy would be to substitute the coordinate values in for the variables to see if those values make both equations true.

Display a graph of the two equations for all to see alongside the system of equations:

\(\displaystyle \begin{cases} d = \text-2.4 t + 4.8 \\ d = 3.2 t + 0.6 \end{cases}\)

Explain to students that this is called a system of equations, and “solving a system of equations” means to find the values of the variables that make both equations true at the same time. Point out that in this problem, the solution to the system of equations is the point where Han and Jada are at the same distance from the parking lot at the same time (point to \((0.75, 3)\)). Tell students that it is also possible to solve a system of equations without graphing. To find when Jada and Han are the same distance away from the parking lot, we can set the expression for Jada’s distance equal to the expression for Han’s distance to get the equation \(\text-2.4 t + 4.8 = 3.2 t + 0.6\), which is an equation that can be solved for \(t\). Ask students to solve this equation and confirm that \(t=0.75\), the is the same value they found earlier by carefully graphing the lines of each equation. Emphasize that the intersection point gave a value of both \(t\) and \(d\), so it is important when solving algebraically to substitute \(t\) back into one of the equations to find the value for \(d\). Since Han and Jada are at the same distance from the parking lot when \(t=0.75\), it doesn't matter which equation is used to find the value of \(d\).

Students explore a system of equations with no solutions in the familiar context of cup stacking. The context reinforces a discussion about what it means for a system of equations to have no solutions, both in terms of a graph and in terms of the equations (MP2). Over the next few lessons, the concept of one solution, no solutions, and infinitely many solutions will be abstracted to problems without context. In those situations, it may be useful to refer back to the context in this activity and others as a way to guide students towards abstraction.

The key point for discussion is to connect what students observed about the graph to the concept of “no solutions” from earlier lessons. Graphically, students see that the lines are parallel and always separated by a distance of 3 cm. This means the stack of \(n\) large cups will always be 3 cm taller than a stack of \(n\) small cups. Connecting this to the equations, this means that there is no value of \(n\) that is a solution to both \(1.5n+6\) and \(1.5n+9\) at the same time. By the end of the discussion, students should understand that the following are equivalent:

• The lines don't intersect.
• The lines are parallel.
• There is no value of \(n\) for which the stacks have the same height.
• There is no value of \(n\) that makes \(1.5n+6 = 1.5n +9\) true.

Invite students to explain how they used the graph or equations to answer the second question. Ask other students if they answered the question with a different line of reasoning. If not brought up by students, demonstrate that setting the expression for the height of the large cup \(1.5n + 9\) equal to the expression for the height of the small cup \(1.5n + 6\) and subtracting \(1.5n\) from both sides gives \(6 = 9\), which is false no matter what value of \(n\) is used.