“In today’s lesson, we are going to continue learning about perimeter, and we will look at how the perimeter of a rectangle relates to the area inside the rectangle. Let’s first review what perimeter means. It is the distance around a figure. How do you think we could use chairs to measure the perimeter of a table?” Lead students’ conversations to the conclusion that at home, in the school cafeteria, or in a restaurant, tables are often measured by the number of people who can comfortably sit around them. If the conversation does not go this direction, ask a follow-up question, “Have you ever heard someone in a restaurant say that s/he needs a table for eight? What does that mean? How many chairs must fit around that table?”
Read aloud Spaghetti and Meatballs for All! by Marilyn Burns to provide a context for investigating the relationship between area and perimeter. The focus of this book is to determine the fewest number of tables needed to seat 32 people. Different combinations of tables are explored as the number of people who can be seated is determined. In each example in the book, the perimeter of the rectangle, or the combined perimeters of several rectangles, remains constant at 32 units. However, the area of each arrangement is different, depending upon the number of small square tables needed. (If the specified book is not available, use one of the other books listed in the Related Resources section at the end of this lesson, or read the following similar scenario aloud that will focus on the same general concepts.)
“Suppose you invite 32 people to come to a dinner party. You set up 8 square tables and place 4 chairs at each table. As the earliest dinner guests arrive, they push 2 tables together so that 6 people can sit together. They set the 2 extra chairs out of their way. Now you don’t have enough places for people to sit. When more guests arrive, they start pushing tables together also. What are you going to do?”
Pass out the Square Tables resource (M-3-1-3_Square Tables.docx). Have students cut out the squares so that they can construct the different ways the guests in the story arranged the tables. Reread the story as small groups of students arrange and rearrange tables. Students can also use 32 dried beans, counters, or other counting manipulatives to keep up with where 32 people are able to sit. Students could act out a similar problem with classroom desks and chairs, although such an activity has the potential to be very noisy.
“Suppose there are going to be just 12 people at a dinner party. What different table arrangements are possible? Which arrangement would use the fewest tables? Which arrangement would use the most tables?” Give groups of students the More Tables resource (M-3-1-3_More Tables.docx) to cut apart and use for their explorations.
“So far, we have continued to discuss perimeter, and we have looked at how the shape, length, or width of a table affects the number of chairs that will fit around it. This would be a silly thing to do, but what if we put chairs on the tables instead of around the tables? Would the numbers be the same? Let’s find out.” Give each student two table models: one that is 4 squares long and 1 square wide, and one that is 4 squares long and 2 squares wide (see the More Tables resource). Also provide each student with 12 dried beans, counters, or other small counting manipulatives.
“Look first at the table that is just 1 square wide and 4 squares long. Put a bean in each place that a chair will sit. How many beans fit?” (10) “What is the perimeter of the table?” (10 beans, or 10 units) “The area of the whole table is the number of square units that make the table. We can count the number of squares by placing a bean in each square that you see on the table. How many beans fit on the table that way?” (only 4) “Four beans fit on the table. The units around the table make its perimeter, but the square units on the table make its area. What is the area of this table?” (4 beans, or 4 square units)
“The perimeter and area of this table are not the same. When we make a larger table out of the 1-square-wide tables, we can think of perimeter, the length around the table, as the number of chairs that fit around the table. We can think of the area as the number of
1-square-wide tables that make the bigger table.”
“Let’s look at the other table that I gave you. First place beans around that table. What is the perimeter?” (12 beans, or 12 units) “What is the area? Remember that area means the square units on the table.” (8 square units) “Again, the perimeter and the area are not the same number of units, and they are not the same type of measurement because they use different units. Now think about a square table that has 4 chairs around it. What are the perimeter and areas of that table?” (The perimeter is 4 units, but the area is only 1 unit.)
Encourage students to describe perimeter as a number of units of length and area as a number of square units. Students may notice patterns in finding area, such as length ´ width, but it is not important at this time to teach area as a formula.
Have students cut apart various table models from the More Tables resource for review and evaluation. Ask students to choose four different tables to glue onto art paper and record the perimeter and area of each. As students complete their work, review their papers, and provide immediate feedback to support or correct their thinking on perimeter and area concepts.
Extension:
Use the activities and strategies listed below to tailor the lesson to meet the needs of your students during the year.
- Routine: As students are seated for lunch, at work stations in the classroom, or in clusters of desks, have them take a few minutes to discuss the perimeter of the arrangement by the number of chairs or stools that fit around. Then ask them to note whether the perimeter and area of the arrangement are the same or different and discuss why.
- Small Group: Place a collection of pattern blocks and several dried beans at a workstation for students to explore perimeter. Have students measure the perimeters by the number of sides using triangles, squares, rhombi, trapezoids, and hexagons. As they count, students can place a dried bean at each side to keep up with the total number. Note that on a trapezoid, 2 beans will fit on the long side, but all sides of the other shapes will have only 1 bean per side. Encourage students to also place blocks together and discuss how doing so affects the number of beans that will fit around.
- Expansion: Provide students with 12 square 1-inch tiles and inch grid paper. Other manipulatives may be used if they will fit inside the squares of the grid paper. Tell students to make as many rectangles as they can with all 12 tiles, thus making various rectangles with an area of 12 squares. (Possible rectangles are 1 by 12, 2 by 6, and 3 by 4.) Have students record each rectangle by drawing or tracing it on the grid paper. When students have completed their rectangles, have them write the perimeter of each drawing and discuss how the perimeters change while the areas remain the same. Also ask students to note which arrangement created the greatest or fewest units of perimeter.