Lesson 10

The purpose of this warm-up is to elicit ways students can describe different characteristics that arise when more than one line is graphed in a coordinate plane. 

Arrange students in groups of 2–4. Display the image of all four graphs for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask the groups to offer at least one reason why each graph doesn’t belong.

After students have conferred in groups, invite each group to share one reason why a particular graph might not belong. Record and display the responses for all to see. After each response, ask the rest of the class if they agree or disagree. Since there is no single correct answer to the question of which graph does not belong, attend to students’ explanations and ensure the reasons given are correct. Encourage students to use concepts and language introduced in previous lessons about lines such as slope and intercepts. In particular, draw students’ attention to any intersections of the lines.

In previous 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 investigate the mathematical structure with two stated facts using familiar representations and context while reasoning about what must be true.

Monitor for students using different representations, such as a graph, table, or words, as they solve the final problem to share during the whole-class discussion.

Before looking at the task, tell students, “I have $2 in my pocket. What might be in my pocket?” Students will likely guess that you have two $1 bills, but ask what else it might be. Some answers could be 8 quarters; 200 pennies; a $2 bill; or 20 nickels, 2 quarters, and 5 dimes.

Read the problem context together. Ensure students understand that we know that Jada has exactly $2 in her pocket, that she only has quarters and dimes, and that she has exactly 17 coins. Give 1–2 minutes for students to read and complete the first problem. Display the table for all to see and ask students for values to fill in the table.

Give students 5–7 minutes of quiet work time to finish the remaining problems followed by a whole-class discussion.

Students may be confused about the last row of the table. Tell them they can enter any values that make sense in the context and are not already on the table.

The goal of this discussion is to focus students on what must be true based on the two facts they know: that the coins total $2 and that there are 17 coins. Begin the discussion by asking students about some things 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.

Select previously identified students to share how they answered the last problem. If possible, begin a sequence with a student who added on to the table representation to figure out the solution, followed by one who added onto the graph. Include any students who wrote out their reasoning in words last. After each student shares, connect to the idea 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 only provided in the graph. Students have worked with lines that represent a context before. 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.

Arrange students in groups of 2. Give students 1 minute to read the problem and answer any questions they have about the context. Tell students to complete the table one row at a time with one person responding for Clare and the other responding for Andre. Give students 2–3 minutes to finish the table followed by whole-class discussion.

Display the graphs from the task statement. The goal of this discussion is for students to realize that points that lie on one line can be interpreted as statements that are true for Clare, and points that lie on the other line as statements that are true for Andre. Ask students:

• “What is true for Clare and Andre after 20 minutes?” (Clare has 15 signs completed and Andre has 20 signs completed.)
• “What, then, do you know about the point \((20, 15)\) and the equation for Clare's graph?” (The point \((20, 15)\) is a solution to the equation for Clare's graph.)
• “What do you know about the point \((20, 20)\) and the equation for Andre's graph?” (The point \((20, 20)\) is a solution to the equation for Andre's graph.)

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.