Congruence and Coincidence
Lesson 2 of 17
Objective: SWBAT use a rigorous definition of congruence--aided by tracing paper, compass, etc.--to identify congruent figures. Students will understand the meaning of congruence in terms of superimposing figures with rigid motions.
We review the team answers as a class. Most of my students will probably define congruent as "same shape, same size". I tell students that that definition is still true, but it is only part of the story. For example, what does it mean for objects to have the same shape? Students will probably agree that objects have the same shape and size if--and only if--one can be superimposed on the other so that all points coincide--line up. In geometry, that means we can describe a sequence of reflections, rotations, and translations that carries one object onto the other.
I display the agenda and learning targets for the lesson. Today we will focus on re-examining the concept of congruence in terms of the rigid motions.
Matching Congruent Objects
Displaying the slide, I tell students that I want them to think about how they can use transformations to tell whether objects are congruent. A range of tools is available for them to use, for example: compass, rulers. (I am looking for students to find their own methods of deciding whether objects are congruent (MP5). I make tracing paper available, but only after students have tried using other methods to identify congruent polygons in problem #3 of the set. I like tracing paper, because students can use it to represent rigid motions in a very concrete way. However, I want students to make other connections, as well. Among these, I want students to see that a compass performs transformations on a line segment. Also, I want students to make the connection between the congruence of segments and the equality of their lengths, and between the congruence of angles and the equality of their measures.)
I am on the lookout for:
- Do students see that a compass is a very efficient tool for comparing the length of segments (MP5)? I do not discourage students from using a scale to measure length, but I want to make sure that they are aware of the other tools in their toolbox.
- In problem #2, do students see that angles may be congruent regardless of the apparent "length" of their sides? To highlight this point, I use an animated slide.
- Are students using protractors correctly to measure angles? They should be extending the rays of the angles or sides of the polygons to measure accurately (MP6).
Matching Congruent Parts
Displaying the slide , I tell students that now we will practice writing congruence statements and using congruence marks to show that objects are congruent. We will also practice matching up corresponding sides and angles of congruent polygons.
Be On the Lookout For: Do students see that they can use the congruence statement to identify pairs of corresponding vertices and sides (MP7)?
Corresponding vertices are given in the same order when naming congruent polygons to make it easier for the reader to follow. I will expect students to follow this convention, as well.
For homework, I assign problems #6-8 of Homework Set 1 for this unit. These problems provide additional practice in matching congruent segments, angles, and polygons and in identifying corresponding parts of congruent polygons.