In this activity, students express what it means for two shapes to be the same by considering carefully chosen examples. During the whole-class discussion, students come to a consensus about what it means for two shapes to be the same and are introduced to the word "congruent" to describe this relationship.

There may be discussion where a reflection is required to match one shape with the other. Students may disagree about whether or not these should be considered the same and discussion should be encouraged. As students discuss if they believe a pair of figures is the same or is not the same, they construct arguments and critique the reasoning of others (MP3). 

Monitor for students who use these strategies to decide whether or not the shapes are the same and invite them to share during the discussion, ordered from less precise to more precise:

• Observing: This is often sufficient to decide that they are not the same. Encourage students to articulate what feature(s) of the shapes help them to decide that they are not the same.
• Measuring side lengths using a ruler or angles using a protractor: These students then use differences among these measurements to argue that two shapes are not the same.
• Using tracing paper to trace one figure and slide it to match with another figure: This is a version of applying a rigid transformation.

Providing access to tracing paper, rulers, and protractors in the geometry toolkit allows students the opportunity to choose appropriate tools strategically (MP5) and is necessary for students to generate multiple strategies in this activity. 

The routine of Anticipate, Monitor, Select, Sequence, Connect (5 Practices) requires a balance of planning and flexibility. The anticipated approaches might not surface in every class, and there may be reason to change the order in which strategies are presented. While monitoring, keep in mind the learning goal and adjust the order to ensure all students have access to the first idea presented (whether that be a common misconception or a different approach).
 

The purpose of this discussion is to build an understanding of congruence. 

Invite previously selected students to share their reasoning. Sequence the discussion of the methods in the order listed in the Activity Narrative: making general observations, taking measurements, and applying rigid transformations with the aid of tracing paper. If possible, record and display their work for all to see.

The most general and precise of these criteria is the third which is the foundation for the mathematical definition of congruence: The other two are consequences. The moves allowed by rigid transformations do not change the shape, size, side lengths, or angle measures.

Connect the different responses to the learning goals by asking questions, such as the following:

• "When looking at the figures to determine if they are the same, what are you looking for?"
• "How does having the same side lengths and angle measures show 2 figures are the same?"
• "Do reflections keep figures the same?"

There may be disagreement about whether or not to include reflections when deciding if two shapes are the same. Here are some reasons to include reflections:

• A shape and its reflected image can be matched up perfectly using a reflection.
• Corresponding angles and side lengths of a shape and its reflected image are the same.

Explain to students that people in the world can mean many things when they say two things are "the same." In mathematics there is often a need to be more precise, and this need can be met with the term "congruent".

We say that Figure A is congruent to Figure B if there is a rigid transformation that takes Figure A exactly to Figure B. This can be any sequence of translations, rotations, and reflections.

We can connect this definition of congruent with properties of rigid motions to conclude that: 

• Corresponding sides of congruent figures are congruent.
• Corresponding angles of congruent figures are congruent.

The purpose of this activity is for students to identify features that must be the same for congruent figures.

All of the figures in this activity have the same shape because they are all rectangles, but they are not all congruent.  Students examine a set of rectangles and classify them according to their area and perimeter. Then they identify which ones are congruent. Because congruent shapes have the same side lengths, congruent rectangles have the same perimeter. But rectangles with the same perimeter are not always congruent. Congruent shapes, including rectangles, also have the same area. But rectangles with the same area are not always congruent. This allows students to highlight important features, like perimeter and area, which can be used to construct the argument that two shapes are not congruent (MP3).  

Watch for students who think about the final question in terms of “same shape and size.” Remind them of the definition of congruence introduced in the last activity.

Invite students who used the language of transformations to answer the final question to describe how they determined that a pair of rectangles are congruent.

Perimeter and area are two different ways to measure the size of a shape. Ask the students:

• "Do congruent rectangles have the same perimeter? Explain your reasoning." (Yes. Rigid motions do not change distances, and so congruent rectangles have the same perimeter.)
• "Do congruent rectangles have the same area? Explain your reasoning." (Yes. Rigid motions do not change area or distances. Therefore, they do not change the length times the width in a rectangle.)
• "Are rectangles with the same perimeter always congruent?" (No. Rectangles D and F have the same perimeter but they are not congruent.)
• "Are rectangles with the same area always congruent?" (No. Rectangles B and C have the same area but are not congruent.)

One important takeaway from this lesson is that measuring perimeter and area is a good method to show that two shapes are not congruent if these measurements differ. When the measurements are the same, more work is needed to decide whether or not two shapes are congruent.

A risk of using rectangles is that students may reach the erroneous conclusion that if two figures have both the same area and the same perimeter, then they are congruent. If this comes up, challenge students to think of two shapes that have the same area and the same perimeter, but are not congruent. Here is an example: