SWBAT use a line to model possible locations in space. Students will understand the meaning of a line in terms of its properties.

Using lines to represent information about the position of an object. I know where your cell phone fell into the water!

5 minutes

**Team Warm-up**

Using the Slideshow, I display the Warm Up prompt for the lesson as the bell rings. As I take role and check for homework and required materials, I watch to see that students are following the Team Warm-up routine.

The warm-up asks students define a line. This will tell me if they have learned anything from previous lessons. Students will elaborate on or refine their answers at the end of the lesson.

**Goal-Setting**

Displaying the Agenda and Learning Targets, I tell the class that today we will learn another method of locating an object on a map. This one is based on the properties of lines, rather than of circles.

25 minutes

In this activity, students perform a triangulation to locate an object on a map. The activity is structured as a Team Jigsaw.

Displaying the Instructions, I distribute the Setup and ask for volunteers to read parts of the introduction out loud to the class. I explain that the team can only succeed if each student performs his or her part. Each student is a witness interviewer and must plot the information from his or her witness on the map. Together, the information from multiple interviews will allow the team to locate the last-seen-point where the cell phone went into the lake.

Each team will need a straight-edge and a copy of the Map.

As students get started, I distribute the Witness Interviews, which have been cut into strips. The activity works with 2-4 students in a team. If necessary, a student can plot the information from more than one witness interview.

As students work, I circulate. I am on the lookout for:

**Are students distracted by witness names, addresses, times, and other information provided in the witness interviews?**

I explain that dive rescue personnel ask witnesses information (description of people or boats, time observed) that allow them to be sure that witnesses are all describing the same event. but students can assume that is the case in this problem. It is a real-world skill to be able to extract the relevant information from all the data that is available. (**MP4**)

**Do students recognize that the location from which a witness observed and the location of an object observed in the distance can be represented as two points on the map (MP4), and that the pair of points determine the direction in which the witness was looking when he or she saw the man fall into the water?**

**Do they recognize that the point where the cell phone entered the water must be somewhere on the line between those two points?**

**Do students recognize that one line represents only what we know about the cell phone's location, given its direction from a location of a single witness?**

**Do students recognize that it takes directions from at least 2 witnesses to pinpoint the cell phone's location?**

**Do students wonder why the plotted lines make a triangle, rarther than intersecting at a point?**

I explain that this is the result in errors in plotting points (or in the witnesses' memories) since in theory the cell phone only entered the water at one point. Dealing with error is a normal part of using mathematics in the real world. A tight triangle usually means that the information was plotted precisely and accurately; a large triangle means that the points were not plotted precisely and might be inaccurate, as well. Precision in plotting improves with practice. (**MP6**)

10 minutes

Once each team finishes performing the modeling with lines activity, I ask them to consider a follow-on problem.

Displaying the slide, I ask teams to discuss the scenario and answer the questions.

The goal is to get students thinking about the meaning of an intersection and to confront the misconception that an intersection must always be a point. The scenario also gives them an opportunity to consider an axiom of geometry: that two lines which intersect at more than one point are in fact the same line.

Axioms of points, lines, and planes and the topic of intersections will be treated more fully in the next unit, Dimensions and Structure. The trilateration problem provides an opportunity to address the axioms in a real-world context in order to make them more meaningful to students.

I hold a class discussion when every team is ready. Alternately, I will address teams individually.

4 minutes

**Team Size-Up**

Displaying the Lesson Close prompt, I ask students to summarize what they learned from the lesson with their team-mates, then select the best answer to write on the board. This activity follows the Team Size-Up routine.

**Homework**

Homework Set 1 problems #14 and #16 ask students to review the properties of lines and circles. Problem #15 is a modeling problem in which students must justify their choice of geometric object to use to represent where to search for a missing person based on the information available.