Analyzing the Symmetry of a Polygon

The warm-up prompt for this lesson asks students to recall the properties of an isosceles triangle. The warm-up follows our Team Warm-up routine.  I choose students at random to write the team's answer on the board.  

Some students will no doubt recall that the base angles of an isosceles triangle are congruent. This is true, but it is not part of the definition.  I use the problem to make the distinction between the properties of a shape that are part of the definition and the properties that we can deduce using what we know about geometry.  Since deductive reasoning is part of what we hope to learn in a geometry course, the difference is important.

I display the agenda and learning targets for the lesson.  I tell the class that today we will use the symmetry of a polygon to make deductions about its properties.  As in the last two lessons, we will be using rigid motions to show that parts of a figure are congruent.  

We use the Triangles and Quadrilaterals Notes to review the definitions of familiar triangles and quadrilaterals. Students add congruence marks and parallel and perpendicular marks to the drawings. I have also included a general description of how I use guided notes in my classroom.

This activity uses the Team Jigsaw format.  I display the slide as I ask students to work together to practice describing symmetry in common polygons.   As students make deductions about the properties of the different shapes, I encourage them to write their observations in their notes on Triangles and Quadrilaterals.

As students work, I circulate.  From time to time, I display student work using a document camera and ask the class to take a critical look at the transformations.  As a class, we can refine the good ideas contributed by students so that we all can learn more (MP1).

A key feature of this activity is that students are asked to use congruence marks to show which sides, angles, and distances in the figure must be congruent.  This not only allows them to deduce the congruence of sides and angles of the polygon, but it helps them to think more precisely about the transformations they are using to show congruence (MP3, MP6, MP7). For example:

1. The line of symmetry of an isosceles triangle is the perpendicular bisector of the base and the angle bisector of the top angle. 
2. The center of symmetry of a parallelogram is the midpoint of a diagonal.
3. The center of symmetry of a regular polygon must be equidistant from the vertices. This can be seen by drawing auxiliary line segments from the vertices to the centroid of the shape, dividing it into congruent triangle.  (This strategy has other applications: to find the area or the sum of the interior angles, for example.)

Extensions Ideas  

The problems in the set have a lot of potential to encourage further inquiry, for students who are ready.  For example:

1. How do we know that the opposite sides of a parallelogram will coincide when the shape is rotated 180 degrees around the midpoint?  (Possible answers: Line Rotation Theorem, Converse of the Alternate Interior Angles Theorem)
2. How do we know that opposite vertices of a kite or rhombus will coincide when we reflect the shape over a diagonal?  (The diagonal can be shown to be the perpendicular bisector of the diagonal that joins the opposite vertices.)

I summarize this lesson with my students by completing the examples in the notes on symmetry that we began in the last lesson.  The examples provide students with a model of how to describe transformations precisely (MP6).

The lesson close follows our individual size-up routine. The prompt is a check for understanding.

Recognizing Good Work

While the class is completing the lesson close activity, I invite a student from each team to assign his or her team a score for the lesson.  Student scorekeepers write the score in a spot on the front board, and I write the team points scores I assign to each team next to them. 

Homework

For  homework, I assign problems #10-12 of Homework Set 2 for this unit.  Problems #10-11 are additional practice in the skills of this lesson.  Problem #12 reviews the definitions of common triangles and quadrilaterals.