In earlier lessons, students have set two expressions equal to one another to find a common value where both expressions are true (if it exists). A system of two equations asks a similar question: At what common pair of values are both equations true? In this activity, students focus on a context involving coins and use multiple representations to think about the context in different ways. The goal of this activity is not for students to write equations or learn the language “system of equations,” but rather to investigate the mathematical structure with two stated facts using familiar representations and context while reasoning about what must be true.

Monitor for students who use these representations to solve the last problem:

• Graphs 
• Tables 
• Words

Students may wonder why the last row of the table is blank. Tell them they can enter any values that make sense in the context and that are not already in the table.

The goal of this discussion is to focus students on the last problem and what must be true based on the two facts that they know: The coins total \$2 and there are exactly 17 coins. Begin the discussion by asking students about some things that they know cannot be true about the coins in Jada’s pocket and how they know. Students may respond that Jada cannot have 20 dimes and 0 quarters because that is not 17 coins or other variations where either the coins do not total \$2, there are not exactly 17 coins, or neither are true.

Display 2–3 approaches/representations from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different representations. Here are some questions for discussion:

• “What do the representations have in common? How are they different?”
• “How does the solution show up in each representation?”

Ensure that students understand that their solution is one where both facts are true.

In the previous activity, the system of equations was represented in words, a table, and a graph. In this activity, the system of equations is partially given in words, but key elements are provided only in the graph. Students have worked before with lines that represent a context. Now they must work with two lines at the same time to determine whether a point lies on one line, both lines, or neither line.

Display the graphs from the task statement. The goal of this discussion is for students to realize that points that lie on a line make that situation true. If a point is on more than 1 line, such as at an intersection, then it makes all of those situations true. Ask students:

• “Is the point on either line? What does that mean in this situation?” (It is on Clare’s line. It means that, 20 minutes after they worked today, Clare has made a total of 20 signs.)
• “Imagine a point that is not on either line. What does that information tell us about this point?” (It means that it represents a situation that is not true for either Andre or Clare.)
• “If a point is on a line, what does it mean? If the point is the intersection of 2 lines, what does that mean?” (A point on a line means that it represents a situation that is true for the situation represented by the line. The intersection point represents a situation that is true for both situations represented by the lines.)

Invite groups to share their reasoning about points A–D. Conclude by pointing out to students that, in this context, there are many points true for Clare and many points true for Andre but only one point true for both of them. Future lessons will be about how to figure out that point.

Math Community

Conclude the discussion by inviting 2–3 students to share a norm that they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm “” in action today, and it really helped me/my group because .”