Students will compare the relationship between perimeter and area as they proceed with their study of perimeter. Students are going to: 
- Examine multiple rectangular representations of specified boundaries. 
- Examine rectangular spaces with varying perimeters and the same quantity of square units. 
- Examine and arrange two and three rectangular objects according to their respective areas. 
- Create and reorganize models to evaluate the relationships between area and perimeter.

- What does it mean to evaluate or estimate a numerical quantity? 
- For what reason does "what" we measure affect "how" we measure? 
- How are the mathematical characteristics of things or procedures quantified, computed, and/or understood?

- Area: The measure, in square units, of the inside of a plane figure. 
- Perimeter: The distance around a figure. 
- Square Unit: The base unit for measuring the area of an object. A square with each side measuring one unit.

- Square Tables (M-3-1-3_Square Tables)
- More Tables (M-3-1-3_More Tables)
- Grid paper (M-3-1-2_Grid Paper)
- Perimeter and Area Checklist (M-3-1-3_Checklist)
- dried beans or small counting manipulatives
- inch tiles or centimeter cubes
- base-ten blocks
- related storybooks (if available)

- As students experiment with different table and chair configurations during small-group time and in the subsequent classroom discussions, keep an eye on them using the Perimeter and Area Checklist (M-3-1-3_Checklist). Take particular note of whether students perform well or require additional practice; give feedback to reassure learners that they are on the right path and reroute tasks as necessary.
- Throughout the lesson's reflection and evaluation phases, keep using observational techniques to gauge student engagement and comprehension. Ask students to illustrate their answers to the additional perimeter problems you pose on blank grid paper.

Scaffolding, Active Participation, Modeling, Clear Instruction 
W: Go over the perimeter definition, and discuss the area and its relationship to it.
H: Study Marilyn Burns' "Spaghetti and Meatballs for Everyone!" Talk about the boundaries and spaces described in the book's context.
E: Introduce students to the concept of accommodating different seating arrangements by having them arrange model tables. This will engage students in the assignment.
R: Utilize the table models going forward, instructing students to calculate area rather than perimeter by placing manipulatives on the models. Although it's not necessary to introduce the concept just yet, some students might notice that the area is length × width.
E: Give the students an assortment of rectangles and instruct them to determine each one's area and perimeter. 
T: Have students talk about the approximate size of the table and the number of seats available when they are seated at a lunch table or in a classroom. Instruct students to measure the edges of various shapes, like trapezoids, hexagons, and triangles.
O: The purpose of this lesson is to help students gain a deeper understanding of perimeter, investigate area, and make conclusions about how perimeter and area relate to one another. Students don't need to be able to define area or perimeter using a formula as long as they are aware of the distinction between the two. Students discover that shifting arrangements lead to shifting perimeters through experiments involving pushing and pulling tables apart. Students define area as the square units on (or inside) a table model through additional activities, as opposed to the units surrounding (perimeter). Lastly, a lesson extension exercise asks students to design multiple arrangements for a specific space.

"In today's lesson, we will continue our study of the perimeter and learn about the relationship between a rectangle's perimeter and its interior area. First, let's go over what perimeter means. It's the circumference of a figure. How might we measure a table's circumference using chairs, in your opinion?" Motivate your students to discuss how tables are frequently measured by how many people can fit around them comfortably, whether at home, in a restaurant, or the cafeteria at school. If the conversation does not progress in this direction, ask a follow-up question, such as, "Have you ever heard someone in a restaurant say that he or she needs a table for eight? What does that mean? How many chairs must fit around that table?"

It is recommended to read Marilyn Burns' book, "Spaghetti and Meatballs for All!" aloud to explore the relationship between area and perimeter. The book's main objective is to determine the minimum number of tables required to seat 32 individuals. The book examines various table configurations as the maximum number of people who can be seated is determined. In all the examples provided in the book, the perimeter of the rectangle, or the total perimeter of multiple rectangles, always remains at 32 units. However, the size of each configuration will differ based on the number of small square tables needed. (In case the specified book is not accessible, you can read a similar scenario that focuses on the same general concepts or refer to one of the other books listed in the Related Resources section at the end of this lesson.)

"Let's say you invite 32 people to come to your dinner party. Place four chairs at each of the eight square tables that you have set up. To accommodate six people at a table, the first diners arrive and push two tables together. They moved the two additional chairs out of their path. There are currently not enough seats available. As more guests show up, they also begin pushing tables closer together. How will you proceed?"

Distribute the resource named "M-3-1-3_Square Tables" among the students to learn about square tables. Provide the students with squares to cut out and let them build the various table arrangements made by the story's guests. As small groups of students reorganize the tables, read the story again to make it more interactive. Students can also keep track of where 32 people can sit by using counters, 32 dried beans, or other counting tools. In terms of desks and chairs in the classroom, students could role-play a similar issue, but please make sure to keep the noise level under control.

"Assume that the dinner party will only have 12 guests. Which possible arrangments of tables are there? Which arrangement would use the fewest tables? Which arrangement would use the most tables?" Assign students to divide and utilize the More Tables resource (M-3-1-3_More Tables) in groups for their research.

"Up until now, we have continued to discuss perimeter and examined how a table's length, width, or form influences how many chairs can fit around it. I know it sounds ridiculous, but what if we arranged the chairs on the tables rather than around them? Would the totals remain the same? Let's check." Provide two table models per student: one measuring four squares long by one square wide, and another measuring four squares long by two squares wide (refer to the More Tables resource for dimensions). Give each student 12 dried beans, counters, or other tiny counting tools in addition.

"First, take a look at the table, which is only one square wide and four squares long. Place a bean in each spot where a chair is supposed to sit. How many beans will fit?" (10) "What is the table's circumference?" (10 units, or 10 beans) "The total square units that comprise the table are its area. We can ascertain the quantity of squares by inserting a bean into every square visible on the table. That way, how many beans could fit on the table?" (just 4) "The table could fit four beans. The square units on the table define its area, but the units surrounding it form its perimeter. What is the size of this table?" (4 beans or 4 square units)

"This table's perimeter and area are not identical. The perimeter, or the length around the table, can be thought of as the number of chairs that can fit around the table when we construct a larger table out of the 1-square-wide tables. We can think of the area as the number of 1-square-wide tables that make the bigger table.”

"Now let's examine the other table I provided you with. Go around that table with beans first. What's the perimeter? (12 units, or beans) "What's the area? Keep in mind that the area refers to the table's square units. (8 square units) Once more, because they employ different units, the perimeter, and the area are not the same quantity of units and are also not the same kind of measurement. Let's now envision a square table with four chairs surrounding it. What is the area and perimeter of that table?" (The area is only one unit, but the perimeter is four units.)

Students should be taught to define area as the number of square units and perimeter as the number of length units. While students may notice patterns when calculating area—such as length multiplied by width—it is not necessary to introduce area as a formula at this point. 

To review and assess their understanding, provide students with four different table models from the More Tables resource. Ask them to stick each table onto art paper and note down the respective area and perimeter of each table. Check the students' work as they complete the task and promptly offer feedback to help them improve their grasp of the perimeter and area concepts.

Extension:

You can modify the lesson to fit the needs of your students throughout the year by using the strategies and activities listed below.

Routine: Have students spend a few minutes discussing the perimeter of the arrangement based on how many chairs or stools fit around it while they are seated for lunch, at workstations in the classroom, or in groups of desks. Next, have them observe if the arrangement's area and perimeter are the same or different, and if so, why.

Small Group: Give students a set of pattern blocks and a few dried beans at a workstation so they can investigate the perimeter. Ask students to use trapezoids, hexagons, squares, rhombi, triangles, and hexagons to measure the perimeters based on the number of sides. Students can keep track of the total number by placing a dried bean on each side as they count. Keep in mind that only one bean will fit on each side of the other shapes, whereas two beans will fit on the long side of a trapezoid. Students should be encouraged to group blocks together and talk about how this changes how many beans can fit in each one.

Expansion: Give students inch-grid paper and twelve square, one-inch tiles. If they can fit inside the grid paper's squares, other manipulatives can be employed. Instruct students to use all 12 tiles to create as many different rectangles that each have a surface area of 12 squares. The following rectangles are possible: 1 by 12, 2 by 6, and 3 by 4. Let the students draw or trace each rectangle on the grid paper to mark it. Once they have finished drawing their rectangles, have the students write the perimeter of each one and talk about how the perimeters vary but the areas stay the same. Students should also be asked to write down which configuration produced the most or least number of perimeter units.