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).

• “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).

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).