Mission Impossible

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SWBAT describe the ways in which it is possible for geometric objects to intersect. Students will understand the meaning of conjectures about the intersections of points, lines, and planes.

Big Idea

Your mission, should you choose to accept it: How many pieces of linguini can pass through two points at the same time?

Lesson Open

5 minutes

I display the Warm-up prompt using the slideshow for the lesson.  The prompt asks students to identify the intersection of two rays, using correct geometric notation, of course.  This task is fairly straight-forward, so I warn students that I will hold them to the 3 minute time limit.  The purpose of the question is to counter a common misconception and reinforce the idea that an intersection can be more than just a point (MP6).

The lesson opener follows our Team Warm-up routine.


Following the warm-up, I display the agenda and learning targets for today's lesson.  

I explain to the class: We have been learning to describe the intersections of different geometric objects.  Up to now, we have largely relied on our ability to imagine the way objects interact in 2 or 3 dimensions.  This is sometimes called spacial visualization ability, and it is an important skill if you want to be, say...an architect, or jet pilot. 

By now, most students will have discovered that their "mind's eye" can sometimes lead them astray when they try to predict the way in which objects will intersect.  It seems that your individual life experiences have a lot to do with how strong you are in this area.  For example, people who build things, or play sports, or play computer games often do better on tests of spacial visualization ability.  It is a skill that can improve with practice.

We have already had several opportunities to get this practice, and we will have a few more before the end of the unit.  Today, though, we will be taking a different approach.  We will be learning some basic rules that govern the way points, lines, and planes intersect.  In the coming lessons, students should see that a little factual knowledge goes a long way to help out our imaginations.


Investigating Intersections

30 minutes

In this activity, students use physical models to experiment with the properties of points, lines, and planes.  I ask students to get out their guided notes on Intersections, which we began in the previous lesson, and turn to the reverse side.  I encourage students to match the tasks they are asked to perform--with, for example, linguini and toothpicks--to the conjectures involving the intersections of these objects.

This is a stations activity in which students perform tasks as a team.  As described in the instructions, they take turns filling different roles.  (This is just for fun, and to encourage all students to participate.)  I describe the activity in the video

I use a variety of materials to represent points, lines and planes in the activity.  I start with an adequate supply to allow for breakage:

- A box of toothpicks

- A box of dry linguini

- Two balls of silly putty, play dough, or modeling clay

- 2 clear plastic discs, such as those used in the Intersection Derby lesson.  I prefer discs to rectangular cards, because their edges do not have corners or other features.  You may have to remind students that the discs represent planes, so they do not have edges.

- 2 containers half-filled with water, such as the serving savers used in the Flatland Encounters lesson.  The container should be large enough to allow the plastic disc to lie flat on the surface of the water.

- 5 Manila Envelopes or Folders, to which I affix Dossier Labels

To prevent bottlenecks from forming, it is important for there to be as many stations as you have student groups in your classroom.  Since my sections may be organized into as many as 6 or as few as 3 cooperative learning teams, I will often have the teams remain in place while passing the materials for each station (mission dossiers) around the classroom.  This allows me to "park" extra station materials on a counter until a team is ready for them.  For a section of more than 5 teams, I make an extra copy of each set of materials and rotate them into the mix, so that during any turn there can be two copies a set of materials in use at the same time.

The hyperlink at the bottom of the slide is a link to the Mission Impossible theme song on You-Tube.  The song runs for 3.5 minutes, after which I give teams 2 more minutes to write their reports and move to the next station.

These videos show Students Receiving Their Missions and some Classroom Action.



Summarizing Intersections

15 minutes

I ask students to take out their Guided Notes on Intersections, which we began in the previous lesson.  We complete the reverse side, which outlines conjectures about the ways in which points, lines, and planes can intersect in space. 

I encourage student questions, which I answer by referring to the tasks students tried to perform in the previous section.  We will gain more experience with these conjectures in the next lesson, when we use them to justify claims.

More on how I use Guided Notes can be found in my strategies folder. 

Lesson Close and Homework

9 minutes

The lesson close follows our Team Size-Up routine.  The Lesson Close prompt asks students how many different ways a plane can intersect with three points in space.  (This corresponds with the activity in which students touch the tips of three toothpicks to a solid surface)  Will students notice that the plane can intersect three collinear points in an infinite number of ways?


For homework, I assign problems #35-37 of Homework Set 2.  Problems #35 and #36 ask students to use a straight-edge and compass to revisit some of the experiments they tried in class.  Problem #37 reviews vocabulary and geometric notation.  It also includes a pair of tricky questions to see if students have learned that 2 points are always collinear and 3 points are always coplanar.