In this activity, students apply what they know about systems of linear equations to solve a problem about a real-world situation, but they do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.

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

• “Did you write 2 equations for each system and solve?” (I did, but didn’t need to. For the race, I could’ve tried different numbers for to find when the distances are equal.)
• “Did you graph anything? If so, what was it useful for?” (I graphed the equations to check my answers.)
• “Did you check your answers? How?” (I could check by graphing, but I could also substitute into the original equations to check that both equations are true for the values.)

In this activity, students are presented with a number of scenarios that could be solved using a system of equations. Students are not asked to solve the systems of equations, because the focus at this time is for students to understand how to set up the equations for the system and to understand what the solution represents in context. 

The focus of the discussion should be on making sense of the context and interpreting the solutions within the context of the problems.

Invite groups to share their systems of equations and interpretation of the solution for each problem. As groups share, record their systems of equations for all to see. When necessary, ask students to explain the meaning of the variables they used. For example, represents the number of minutes the family rides after Lin’s dad starts riding again after taking the picture.

To highlight the connections between the situations and the equations that represent them, ask:

• “How many solutions will each of these systems of equations have?” (Each system has exactly one solution. I can tell this because the slopes of each pair of equations are different.)
• “If Lin’s dad biked 0.17 miles per minute instead of 0.24 miles per minute, how would that change the system of equations?” (The first equation would be .)
  • “How many solutions would there be for this new system where Lin’s dad rides slower?” (Based on the equations there should still be one solution.)

  • “Would Lin’s dad ever catch up with the family?” (He would not. He started farther back and rides slower than the family. The solution to the system would have a negative value for time, which does not make sense in the context of the problem.)

If students disagree that there is a solution to the modified first problem in which Lin’s dad rides slower than the family, you can display the graph of the modified system and point out the point where the lines intersect. So, although the system has a solution, it is disregarded in this context because it does not make sense.

In this activity, students solve a variety of systems of equations, some involving fractions, some involving substitution, and some involving inspection. This gives students a chance to solidify that learning by practicing the methods that they have learned for solving systems of equations.. Some of the systems listed are ones that students could have used in an earlier activity in this lesson, to describe the situations there. In the discussion, students compare the systems here to the ones that they wrote in the earlier activity, and they interpret the answer in context.

There are two key takeaways from this discussion. The first is to reinforce that for some systems, students can determine—by reasoning—whether it’s possible for a solution to one equation to also be a solution to another equation. The second takeaway is that there are some systems that students will be able to solve only after learning techniques in future grades.

For each problem, ask students to indicate if they identified the system as least or most difficult. Record the responses for all to see.

Bring students' attention to this system:

Ask students what the two equations in the system have in common with each other and to think about whether a solution to the first equation could also be a solution to the second. One can reason that there is no solution because cannot be equal to both 6 and -5.

Ask students to return to the earlier activity and see if they can find any of those systems in these problems. (Lin's family ride is the sixth system. Noah's kayaking trip is the seventh system. Diego's crafting is the eighth system.) After students notice the connection, invite students who chose to solve those systems to interpret the numerical solution of each system in the context of the earlier activity.

Students may be tempted to develop the false impression that all systems where both equations are given as linear combinations can be solved by inspection. Conclude the discussion by displaying this system that students defined in the last activity about birdseed:

Tell students that this system has one solution and they will learn more sophisticated techniques for solving systems of equations like this in future grades.