Previewing Congruence and Rigid Motions

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SWBAT use problem-solving strategies to begin to understand a problem. Students will understand that solving problems in the real world is a process that requires time and creativity.

Big Idea

Students learn what will be expected of them in this unit as they are encouraged to persevere in making sense of a pair of novel problems.

Lesson Open

5 minutes

The Team Warm-Up Prompt for today's lesson asks students to recall the meaning of congruence. I expect that most students will remember -- or with the help of their peers will remember -- learning that congruent means "same shape, same size".  So, we might as well get that out there. As I review the team responses, I comment on this definition of congruence.  I tell the class:

Same shape, same size is still true, but as a definition of a fundamental concept in geometry it leaves a lot to be desired. What does it mean for two figures to have the same shape, for example?  How can we tell...I mean, how can we tell for certain?  In this unit we will go beyond the old definition and learn to think about congruence in a more rigorous way (MP6).

Nevertheless, I expect most students will select the wrong answer to the first question on the unit pre-test. Learned misconceptions die hard.

I display the prompt using the Slide Show for the lesson.  The lesson opener follows our Team Warm-up routine, with students writing their answers in their Learning Journals.


Following the warm-up, I display the Agenda and Learning Targets.  Today is the first day of the new unit.

Unit Pre-Test and Portfolio Problems

30 minutes

As I pass out the Unit Pre-Test, I remind students of the purpose of Pre-Tests in this course. Students will revise their answers to pre-test questions and turn them in as part of their Learning Portfolios at the end of the unit. After I give instructions, I ask students to begin.  I encourage them to answer the questions they think they know first, then make educated guesses for the rest. 

Portfolio Problems

As students complete the pre-test (5-10 minutes), I collect it while distributing Portfolio Problem 2 - Lost Rover for the unit.  I begin with Problem 2, because it requires more work for students to begin.  (It is Problem #2, because students will learn what they need to fully understand this problem during the second half of the unit.)  

Once most students have finished the pre-test, I display instructions for the portfolio problems.  I circulate through the classroom, answering questions and offering encouragement.  If students have trouble starting, I ask them to try one of the problem-solving strategies listed in the instructions.  I encourage students to work alone at first.  They will have time to share ideas later, but it is important for every student to learn how to make a start on an unfamiliar problem on their own.  My goal is for every student to be engaged in working on each portfolio problem for 10 minutes. 

Most students will follow a similar path as they work to understand the Problem 2 (MP1): by comparing the rover's navigation instructions with the computer display of its intended route, they will spend time studying the route the rover shouldhave taken.  Only then will they turn their attention to reconstructing the route it actually did take, generally by following the navigation instructions but with a different starting direction.  As I circulate, I look for students who recognize a more efficient methods of solution: by recognizing that the route is a polygon, they may think of using tracing paper to rotate the entire route (MP5).  Or, they may recognize that all they have to rotate is a single side of the polygon: the segment between the rover's starting point and the point where it would be if it had started in the correct direction (MP7). 

As I walk around, I also take note of students whose work can be used as exemplars for others.  I am not looking for a complete or even a correct solution, especially if I suspect that the student will not be able to explain how they arrived at their answer.  Instead, I am looking for work that offers a good idea or effective stratetegy that others can emulate while feeling that it is something they could have thought of on their own. 

At the end of 10 minutes, I distribute Portfolio Problem 1 - Tricky Tiling.  Students normally find this problem engaging ("Hey!  There are dogs in this pattern!") and easy to begin, but it is deceptively tricky.  Since most students have encountered tessellations and symmetry before, they will usually recognize that the pattern has several types of symmetry.  Most will think that the tiling has reflection symmetry, until they are asked to identify corresponding parts of different dogs (noses, for example) and see that the dogs are not mirror images of one another.  This is an early opportunity to introduce vocabulary (reflections, rotations, translations) and to help students begin to think precisely about the properties of reflections and rotations (MP6MP7).  I  emphasize that we will be learning to describe these transformations with precision.  I encourage students to draw lines of reflection, and centers of rotation right in the drawing and to label them so that they can refer to them in their descriptions.

At the end of the time limit, I display the Standards for Mathematical Practice. I ask the class to tell me which standards they think came into play as they were working on the problem.  Students will certainly think of MP1.  I tell students that in this unit we will learn to think of some familiar concepts in a more precise and rigorous way (MP6).  This will allow us to solve problems like the two we have just encountered by reasoning about the properties of some very important operations in geometry - the rigid motions.  

In the time remaining, I ask students to share their ideas for solving the problem with their team-mates.  Or, I may ask the class to look at promising student work to help them get a start.

Today, I only expect students to make a beginning.  Nevertheless, the solutions to Tricky Tiling and Lost Rover are provided.  (NEED TO UPDATE WITH SOLUTIONS)

Unit Syllabus and Goal-Setting

15 minutes

I display instructions as I pass out the Unit Syllabus and Unit Learning Portfolio.  I give students 10 minutes to read over the learning goals, write the date of the unit test in their planners, and write a personal learning goal for the unit.  I emphasize that the goal should be about "understanding" or "being able to do" something specific, something related to the unit goals in which the student has a personal interest.  The goal should not be about getting a certain grade or turning in assignments.

Lesson Close

4 minutes

Individual Size-Up

The lesson close follows our Individual Size-Up Routine.  The Lesson Close prompt asks students to write their personal learning goal for the unit.


For homework, I assign problems #3-5 of Homework Set 1.  These problems review topics from the previous units.   Following the unit quiz--and before new material has been covered in the unit--is when I try to assign problems for review.  Problem #5 is important, because it asks students to recall the construction of a bisector; they will need this construction in the coming lessons.