Lesson 12

The purpose of this warm-up is to get students to think about a context that will be explored in the following activity and to reason about the speed, distance, and time each animal is traveling in relation to one another. In the next activity, students will write equations for the bugs and graph these relationships.

Tell students they will see a picture that shows a ladybug and ant traveling for 6 seconds. Tell students to think of at least one thing they notice and at least one thing they wonder about the picture. Display the problem for all to see and give 1 minute of quiet think time. Ask students to give a signal when they have noticed or wondered about something.

Ask students to share things they noticed and wondered. Record and display their responses for all to see. If any students remember a similar representation from an earlier unit where both the ladybug and the ant started on the same side, you may wish to include how this situation is similar and how it is different.

Important ideas to highlight during the discussion: 

• The bugs are moving in opposite directions and at some time in between \(t=2\) and \(t=4\), they pass each other.
• The bugs are moving at a constant speed.
• The ant is moving faster than the ladybug.

In this task, students find and graph a linear equation given only the graph of another equation, information about the slope, and the coordinates where the lines intersect. The purpose of this task is to check student understanding about the point of intersection in relationship to the context while applying previously learned skills of equation writing and graphing.

Identify students who use different strategies to answer the first problem to share during the whole-class discussion. For example, some students may reason about the equation from an algebraic perspective while others may start by drawing in the graph for the ant based on the provided information. Also, make note of what strategy is most common among students.

Display the graph from the task statement. Tell students that this activity is about a different ant and ladybug from the warm-up, and we are going to think about their distances using a coordinate plane. Give 4–6 minutes for students to complete the problems followed by a whole-class discussion.

The purpose of this discussion is to ensure all students understand both how the labeled point in the task statement relates to the context and how to write and graph an equation from the given information.

Select previously identified students to share their strategy for the first problem, starting with the most common strategy used in the class. Record and display in one place the equations students write. While students may come up with equations like \(\text-2=\frac{d-10}{t-2.5}\), let them know that this approach is valid, but that an equation of the form \(d=\text-2t+15\) will be easier to work with today. Ask each student who shares how they knew to use the point \((2.5, 10)\) when making the equation for the ant.

If students struggled to graph the ant’s path, you may wish to conclude the discussion by asking students for different ways to add the graph of the ant’s distance onto the coordinate plane. For example, some students may say to use the equation figured out in the first problem to plot points and then draw a line through them. Other students may suggest starting from the known point, \((2.5, 10)\), and “working backwards” to figure out that 1 second earlier at 1.5 seconds, the ant would have to be 12 centimeters away since \(10+2=12\).

In previous lessons, students encountered equations with a single variable that had infinitely many solutions. In this activity, students interpret a situation with infinitely many solutions. A race is described using different representations (a table and a description in words). Students graph the relationships given by the descriptions and notice that the lines overlap so that both relationships are true for any pair of values along the graphed line.

Allow students 7–10 minutes of silent work time followed by a whole-class discussion.

The key point for discussion is to connect what students observed about the graph they made to the concept of “infinitely many solutions” encountered in earlier lessons. Graphically, students see that there are situations where two lines align “on top of each other.” We can interpret each point on the line as representing a solution to both Elena’s and Jada’s equations. If \(t\) is the time from start and \(y\) is the distance from the start, the equation for Elena is \(y=6t\). The equation for Jada is also \(y = 6t\). Every solution to Elena’s equation is also a solution to Jada’s equation, and every solution to Jada’s equation is also a solution to Elena's equation. In this way, there are infinitely many points that are solutions to both equations at the same time.

Ensure students clearly understand that just because there are infinitely many points that are solutions, it does not mean that any pair of values will solve both Elena’s and Jada’s equations. In this example, the pair of values must still be related by the equation \(y = 6x\). So, pairs of values like \((1, 6)\), \((10, 60),\) and \(\left( \frac{1}{2}, 3\right)\) are all solutions, but \((1, 8)\) is not.