This Math Talk focuses on finding the slope of a line from a linear equation. It encourages students to think about how to rearrange equations to find slope and to rely on the structure of slope-intercept form to mentally solve problems. The strategy elicited here will be helpful later in the lesson when students determine the number of solutions for a system of equations.

To find the slope, students need to look for and make use of structure (MP7).

To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone use the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

This optional activity gives students another opportunity to apply what they learned about the features of systems of linear equations with one solution, zero solutions, and many solutions. Plan to use this activity if students need additional time to recognize the importance of slope and intercept in determining the number of solutions for a system.

To answer the first question (a system with one solution), students could write a second equation with randomly chosen parameters. Answering the second and third questions, however, relies on an understanding of what "zero solutions" and "infinitely many solutions" mean and how these conditions are visible in a pair of equations and graphically.

For example, students could reason that in a system with no solutions:

• The two equations have the same variable expressions on one side but different numbers on the other side, and then write a second equation accordingly. For instance, if is equal to 10, it cannot also be equal to 4, so and would form a system with no solutions.
• The graphs of the equations have the same slope, but they cross the vertical axis at different points. Rewriting in the form of gives . A second equation with a coefficient of for but a different constant would have a graph that is parallel to the graph of the first equation, forming a system with no solutions.

Regardless of the form that students use to write the second equation, they need to choose the parameters strategically to achieve a system with the desired number of solutions. The work here prompts students to look for and make use of structure (MP7).

Invite students to share their equations. If possible, use graphing technology to graph each equation that students share and to verify that the resulting system indeed has the specified number of solutions. Display the graphs for all to see.

Focus the discussion on how students wrote an equation that would produce a system with no solutions and a system with infinitely many solutions. Highlight strategies that show an understanding of equivalent equations, of the meaning of solutions to equations in two variables, and of the graphical features of these systems.

This activity gives students an opportunity to determine and request the information needed to write and solve a system of linear equations.

The Information Gap structure requires students to make sense of a problem by determining what information is necessary, and then to ask for information that they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).

After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “Were there any questions you asked that could’ve helped solve the problem, but were not on the data card?” (I asked if the lines were parallel, but that was not on the card.)
• “What information do you think is most helpful when trying to find the number of solutions that a system has?” (The slopes of the equations are very helpful for narrowing down the number of solutions. Knowing that there is at least one different point that is a solution for each equation can further narrow down the options.)

Highlight for students that slopes and intercepts are important to consider when analyzing systems of equations. Remind students of how they wrote equivalent equations to put them in other forms.