This Warm-up prompts students to compare four figures. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.

During the discussion, prompt students to explain the meaning of any terminology they use, such as “right angle,” “trapezoid,” “rectangle,” “square,” “parallelogram,” “area,” or “perimeter,” and to clarify their reasoning as needed. Consider asking:

• “How do you know ?”
• “What do you mean by ?”
• “Can you say that in another way?”

The purpose of this activity is to revisit and internalize the meanings of perimeter and area. To accomplish this, students play around with locking in one attribute (like area), coming up with lengths and widths that produce that area, and considering the resulting perimeter of each option. As students generalize the relationship between the side lengths of rectangles and their perimeter and area, they look for and express regularity in repeated reasoning (MP8).

The goal of this discussion is to generalize how the dimensions of a rectangle affect its perimeter and area.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their answers about Rectangles D and E. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Focus discussion on any consistencies that students noticed when perimeter was held constant and they considered side lengths that resulted in different areas, and vice versa. Students may recall from earlier courses how the shape of the rectangle generally affects the perimeter and area. If they do not bring up this concept, draw their attention to it. Here are some questions for discussion:

• “What do you notice about the shape of the rectangles that have larger or smaller perimeters when the area is 36?” (Long skinny rectangles have the larger perimeters, and the more square-shaped rectangles have the smaller perimeters.)
• “What do you notice about the shape of the rectangles that have larger or smaller areas when the perimeter is 32?” (The more square-shaped rectangles had larger areas.)

The purpose of this activity is for students to, by the end, plot points that represent a discrete graph of and to differentiate this graph from linear and exponential graphs. When students make repeated numerical calculations and then generalize by expressing perimeter or area in terms of a variable, they are expressing regularity in repeated reasoning (MP8).

The purpose is to describe the relationships explored in the tables and graphed. Here are some questions for discussion:

• “Describe how the perimeter of the square changes as the width increases by 1.” (It’s a linear relationship, and every time the side length increases by 1, the perimeter increases by 4.)
• “Describe how the area of the square changes as the width increases by 1.” (Every time the side length increases by 1, the area increases by a greater and greater amount.)
• “Is the relationship between the width, , and the area of the square exponential? How do you know?” (The relationship is not exponential, because each time increases by 1, the area doesn’t increase by a common factor.)