Lesson 11

The purpose of this warm-up is to elicit the idea that points on the graph of the equation are solutions to the equation, which will be useful when students explore graphs of linear inequalities in their Algebra 1 class. While students may notice and wonder many things about these images, the marked points above, below, and on the graph of the equation are the important discussion points.

Through articulating things they notice and things they wonder about the graph, 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.

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice 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 points on, above, and below the line do not come up during the conversation, ask students to discuss this idea. Ask students to describe those points in terms of the graph (e.g., above, on, or below the line, or to the left, on, and right of the line) as well as in terms of the context (more or less sugar for the same amount of flour).

In this activity, students take turns with a partner describing graphs, and then use that reasoning to match graphs with equations. Students trade roles, explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3), as well as attending to precision in their mathematical language (MP6).

Monitor for students who match only one equation with each graph. Monitor for disagreements about which equation goes with which graph.

Arrange students in groups of 2. Give each group a set of cut-up slips.

Students will take turns in two different rounds. In the first round, students will take turns identifying the slope, \(x\)-intercept, and \(y\)-intercept on each graph. The first partner says out loud what the slope, \(x\)-intercept, and \(y\)-intercept are, while the other checks their thinking and records the answers on the card. Then they switch roles.

In the second round, students will add the equation cards, face up, to the graph cards that are already on their desks. Students will take turns. The first partner identifies a match and explains why they think it is a match, while the other listens and works to understand. Then, they switch roles.

The goal of this synthesis is to make sense of how equations that look different can have the same graph. Ask previously identified students to share which equation they matched with Graph A. If students matched a different equation with Graph A than those previously shared, invite them to share their thinking and allow students to clarify each other's thinking.

• ''Are students who matched more than one equation to Graph A wrong?" (No, more than one equation fit with Graph A.)
• "What does it mean for equations to be equivalent and how does this relate to this task. (Equations are equivalent if they have the same solutions, which means they will have the same graph.)
• How are the multiple equations that match Graph D similar to and different from the multiple equations that match Graph A." (The equations that matched Graph A were in different forms; one was in \(ax + by = c\) form, and one was in \(y = mx + b\) form. Both equations that match Graph D are in \(ax + by = c\) form. The equations that match Graph D are multiples of each other.)

The purpose of this activity is for students to practice identifying the parts of a linear equation that affect specific features of its graph, specifically the slope and \(y\)-intercept. This will be useful when students make connections between two-variable linear equations and their graphs and situations that they represent in an associated Algebra 1 lesson.

Discuss the slopes and \(y\)-intercepts and how they can be used. Here are discussion questions:

• "How do you find the slope and \(y\)-intercept for the graph of a linear equation written in standard form like \(\text-3x+y=\text-8\)?" (Substitute 0 for \(x\) and solve for \(y\).)
• "The rate of change in a situation corresponds to what part of the graph?" (The rate of change corresponds to the slope of a graph.)
• "Why is it useful to know the \(y\)-intercept and slope?" (The \(y\)-intercept and slope can make it very easy to create a graph from an equation because we know the key features. Also, they are useful to know in understanding a situation and the relationships between each quantity involved.)