Mapping and Congruence

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Objective

SWBAT establish congruence using rigid motions

Big Idea

Let's Map it Out. In this lesson, students show that figures are congruent by mapping one onto the other.

Cooperative Activity

20 minutes

To start this section I organize the class into groups of 4 students each. Then I give each group Congruent Shapes and More Figures copied single-sided on non-transparent card-stock, and the Congruence Mapping Collaborative Activity Task Sheet.  The task is for each group to develop a process for determining if two figures are congruent.

Before that, though, I want to make sure to set the ground rules for working collaboratively in this lesson. We read through the Congruence Mapping Collaborative Activity Task Sheet together and I emphasize the point about how each group member has a responsibility to enrich the group experience. I also stress the importance of knowing and adhering to the role assignments. 

Next students start on the task by identifying the four pairs of figures that need to be compared. This is a nice exercise in combinatorics and common sense. 

Next I draw random numbers to determine the order of performers. E.g., 4, 1, 3, 2 would mean that the person the group had identified as 4 would be assigned to perform the task for pair 1, student 1 for pair 2, and so forth.

Finally, I have the students get started trying to figure out if the figures are congruent. 

When each group has finished this part they are to discuss how they will respond to the prompt at the bottom of the handout. I stress the importance of developing a definition that is inherent in the method(s) they used to determine congruence. Many students will just regurgitate the definition of congruence they have learned previously, i.e., all corresponding angles and sides are congruent.

 

When the groups have finished I have one or two groups come up to share their results. Once we agree as a class on which figures are congruent, and which are not, it's time to move on.

 

Concept Development

10 minutes

Now that students have experienced showing congruence by mapping, and they have seen the corollary that all corresponding lengths and angle measures are equal, it's a good time to give some language to this experience.

I start with the concept of "mapping", which I haven't used at this point. So I explain to students that both in the  case of the transparency and the case of the cutout, we showed that the figures were congruent by showing that one could be "mapped" onto the other by a sequence of rigid motions.

 

Then I highlight the fact that we took one of the figures and, without changing its lengths or angle measures, flipped it, turned it, and/or slid it until we found a way to map it onto the other figure. Therefore we established that the figures were congruent. 

So I tell students we can use this as a definition for congruence in this class [which they put into their notes]: Two plane figures are congruent if and only if they can be mapped onto each other by a series of rigid transformations (aka isometries: reflections, rotations, and translations).

Secondly, we know that isometries preserve length and angle so the corresponding lengths and angle measures of congruent figures are equal.

Finally, I explain to students that in this lesson, we'll be showing that figures are congruent by demonstrating how one can be mapped onto the other by a series of rigid transformations.

Guided Practice

20 minutes

For this section, I give students the GP_Mapping and Congruence resource. The resource consists of four pairs of figures. For each pair, our goal is to come up with a series of rigid transformations that will map the first figure onto the second. This entails some detective work, and possibly some trial and error.

My goal is for students to come away from this section (1) having strategies to get them through the independent practice and (2) knowing the expectations for documenting and keeping an account of the mapping sequence.

Before we start, I provide a Transformations as Functions Reference Sheet that shows the rules for the "canonical" transformations (rotation 90/180 cw/ccw; reflection across y-axis, x-axis, y=x, y=-x). Students will need to refer to this handout throughout this phase of the lesson.

For each pair of figures, I run through the following steps to create an interactive discussion:

1. I ask students to form a hypothesis as to what combination of transformations took place. The first time out, I give students two options: rotation/translation or reflection/translation, and then I take a vote. For the second and third example, I use think-pair-share. The final example I ask students to think independently. This is a good example of SMP 2 which asks students to reason concretely and abstractly (or, in this case, spatially).

2. I weigh in with my thinking on what combination of transformations took place.

3. I show my process for documenting and keeping an account of the transformations that map figure 1 onto figure. This is relevant to SMP 6: Attending to precision. See the Demonstration for Mapping and Congruence Guided Practice video for an illustration of this process.

 

Independent Practice

30 minutes

For this section, I organize the students into groups of 3 to 4. I hand each group the Independent Practice_Mapping and Congruence resource. As the students are working, I walk around making sure that students are documenting their solutions properly and that their series of transformations actually map figure 1 onto figure 2.

 

When students are stuck, I urge them to just try one of the reflection or rotation rules, whichever makes sense based on the figures, and then see if they can translate from wherever that reflection or rotation takes them.

 

Other than that, I just make sure that students don't stop trying things.