Lesson 12

The purpose of this warm-up is to elicit the idea that looking at two graphs simultaneously can yield information about solutions that satisfy both constraints in a situation simultaneously, which will be useful when students solve systems of equations graphically in their Algebra 1 class. While students may notice and wonder many things about these images, intersection points and solutions are the important discussion points.

Through articulating things they notice and things they wonder about the point of intersection and what it means in the situation, students have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Display the graph for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the meaning of the point of intersection does not come up during the conversation, ask students to discuss this idea.

The purpose of this activity is to give students contexts to help connect graphical representations to descriptions of situations.

This will be useful when students study solving systems of linear equations in their Algebra class.

The purpose of this discussion is to elicit from students the idea that graphs can be used to answer questions about a situation represented by a system of equations.

Display the graphs for all to see. Select students to share the their matches. If any students wrote equations, record the equations next to each graphed line.

Here are some questions for discussion.

• "What was some information you got from a graph that helped you figure out which situation it belonged to?" (You could use the intercepts to figure some things out. For example, the \(y\)- intercept of one line is \((0,20)\), which made me think it had to do with the problem with 20 tables. You could also think about what the variables would mean and what the solution would mean, and which situation this would be reasonable for. For example, the solution to the equations represented by the graphs in D would be two values that are less than 1. The variables in the situation about buying in bulk would represent the cost of 1 ounce of almonds or raisins.)
• "Why is only the first quadrant (with positive \(x\) and \(y\) values) shown in the graph?" (In all of these situations, it does not make sense to use negative values.)

The purpose of this activity is for students to practice graphing and making sense of the graphs of systems of equations. Students construct equations from descriptions of situations, graph those equations, then consider points on the graph in context. This will be useful in their Algebra class when students write their own system of equations to represent situations.

If students are proficient at graphing lines, consider providing access to graphing technology to focus student work on the creation of the equations and interpretation of the results.

The purpose of the discussion is to understand the connection between the situation and the equations as well as understanding the meaning of points on one of the lines, on both of the lines, or on neither of the line in context.

Display the graph showing both lines as well as the points from the activity.

Ask students, "How can the graph help you determine whether a point will fit one of the situations, both of the situations, or neither? How can the equations help you determine the same thing?" (On the graph, we can see if a point is on one of the lines, on both of the lines, or on neither of the lines. If it is on a line, it fits with that situation. From the equations, we can substitute in the \(x\)- and \(y\)-values to determine whether the values make the equation true or not. If the equation is true, the values fit with that situation.)