Cell Phone Towers
Lesson 4 of 14
Objective: SWBAT describe the properties of a (segment) bisector. Students will understand how lines and circles can represent possible locations in space.
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 and that scribes write a High Quality Answer.
The warm-up asks students to draw and label a point equidistant from two points, which they draw in their learning journals. Most students will probably draw a point midway between the two points, but any point on the bisector of the segment defined by the given endpoints will be equidistant. If students are finished quickly, I may challenge them to come up with a second point. I do not spend much time on the warm-up, only promise that it has something to do with a problem we will solve today. I make sure that every student understands the meaning of the term 'equidistant'.
Displaying the Agenda and Learning Targets, I tell the class that today we will be learning about one of the properties of a bisector, an object that comes in handy in many geometric modeling situations. In particular, it can be used to narrow down the location of your cell phone.
Your Phone Knows...
To motivate the next activity, I show a 1 minute news clip about cell phones and privacy. The hyperlink to the CNBC website is in the slide. At the end of the clip, I ask if anyone knows anything about cell-phone tracking. Students will have various experiences. One student told me a sad story about being in a car accident on a rural road: the 9-1-1 dispatcher kept asking the callers to turn their cell phones on and off in order to get a better fix on their location.
Displaying the instructions, I tell the class that they will be working together to find out more about cell phone tracking: why it is done and, especially, how it is done. I ask them to begin by having one student read the instructions in the handout. Once they have done that, they may retrieve the articles and decide how they will divide them up.
I give students 10 minutes to read the articles and answer questions 1 and 2. I tell them that I will cold call on students to share their team's answers with the class. No one should be concerned, since everyone should make sure that all their team-mates are prepared. If they finish early and have answered those questions well, I ask them to go on to the remaining questions.
Placing a Call
At the end of the time limit, I review the answers to questions 1 and 2 with the class. I tell the class that we are going to focus today on one of the most common methods of tracking a person's location, which was introduced in an earlier lesson.
I ask students to get out Portfolio Problem 2- Placing a Call, which they started working on during the second lesson of the unit. My goal now is to get students started on a productive plan of solution which will lead them to discover that the points on the map which are equidistant from any two cell phone towers form a line: a segment bisector.
I start with a whole-class discussion. If one or more students have made a promising start on this problem, I want to recognize it publicly. Solutions to math problems should come from students whenever possible, so that students will realize that good ideas for solving problems are within their own grasp. I will suggest that those students continue what they were doing and see where it leads them. I will try to guide the rest of the class using questions and suggestions like the following (MP1):
- How can we make this problem simpler by just focusing on one part of the problem?
- Suppose we just looked at the region of the map that was closer to Tower 002 than to Tower 001. How could we begin to figure out the boundaries of that region?
- Can you find a location on the map that is definitely on the interior of the region we are interested in? A point that is definitely on the outside of our region? What about a point that is not obviously closer to Tower 002 than Tower 001?
- How can we check for certain, whether a particular point is inside the region or not?
- What do you suspect might be true about the points that define the boundaries of our region?
One way to engage the whole class in the discussion would be to display a copy of the map on the front whiteboard using an interactive whiteboard and ask for volunteers to come up to mark locations using the stylus. The strategy might evolve into "guess and check", with students picking points on the map and labeling them 'X' or 'O' depending on whether they are closer to Tower 002 or Tower 001 (or perhaps, they will mark only the points which prove to be equidistant from the two towers). This will eventually lead students to see that the points which are equidistant form a line: a bisector. I show what this might look like in the accompanying Video Demonstration.
Once the class has made a start, I ask students to continue to work on their own for about 5 minutes, then share the work of one or two students who have made the most progress.
I am interested in seeing how students measure distances (MP5). I expect some to measure with a inch- or centimeter- scale and transfer their measurements to the map scale. I am looking to see if students will realize that a compass is a very efficient tool for measuring or comparing distances. I might try to plant the suggestion. My goal is for students to realize that they do not actually have to measure any distances in miles or kilometers, just use the compass to compare distances. This is a powerful strategy that may not occur to students: they often feel that they have to give you a number.
I am also looking to see whether students put the point of the compass on a location on the map and measure to either tower, or they put the point of the compass on the tower locations and strike arcs of equal radius from each tower. I do not try to influence this now, as we will address it in the next lesson.
If at all possible, I want to leave time in this lesson for students to finish the activity we began in the last lesson, Mystery of the Nazca Lines, and construct a bisector by swinging arcs using ropes of equal length. I have found that students find this activity engaging--and kind of a surprise. It shouldn't take too long for the arc teams to perform their function, but time is required to move the class outdoors and to get things started. The kinesthetic component is important: for some students, this is the best way for them to understand the concept.
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, which is basically like the team warm-up we used to begin the lesson.
Homework Set 1 problem #9 provides practice in measuring with a compass and map scale. Problem #10 asks students to recall the properties of a bisector. Problem #11 is algebra review.