In this lesson, students will investigate maxima and minima questions in which the perimeter is constant but the area is allowed to change. Students will detect patterns and the relationship between length and width, enabling them to solve area and perimeter problems. They will also resolve the prevalent misunderstanding that area determines perimeter. Students will: 
- understand that for rectangles the perimeter = (length + width) × 2, and area = length x width. 
- calculate the perimeter and area of a rectangle. 
- determine all possible whole-number rectangular dimensions based on a given perimeter. 
- organize rectangular dimensions in a table or list to detect patterns. 
- identify the maximum and minimum values. 
- understand that maximum area occurs when a rectangle is a square or the whole number dimensions is as close to square as possible. 
- understand that minimum occurs when a rectangle's whole number dimensions are the extreme values (e.g., 1 cm × 9 cm for a 20 cm perimeter). 
- solve real-world maximum and minimum problems with fixed perimeters and variable areas. 

- How can we use the relationship between area and volume to draw, construct, model, and represent real-world situations, as well as solve problems of area and volume?

- Area: The number of square units contained within a closed figure. 
- Fixed Value: A number that stays constant even when other variables related to the circumstance vary. 
- Maximum: The greatest value that can be obtained given a specific set of circumstances. 
- Minimum: The least value that can be obtained given a specific set of circumstances. 
- Perimeter: The distance around a closed figure.

- copies of Vocabulary Journal pages (M-6-2-1_Vocabulary Journal)
- student copies of Lesson 1 Entrance Ticket (M-6-2-1_Lesson 1 Entrance Ticket and KEY), copy page 1 and 2 back-to-back and cut into two tickets
- one rectangle card for each student (M-6-2-1_Rectangle Cards); cut cards apart and use each sheet of cards for 10 student
- copies of the Rectangle Table (M-6-2-1_Rectangle Table and KEY)
- Different Dimensions in Various Units of Measure reference sheet (M-6-2-1_Different Dimensions in Various Units)
- number cubes, three for each group of four students, or provide one or more spinners numbered 1–6 for the class to share
- copies of the Designer Dimensions Activity sheet (M-6-2-1_Designer Dimensions)
- 18 x 24-inch drawing paper or poster paper, one per group of four students
- copies of the Lesson 1 Exit Ticket (M-6-2-1_Lesson 1 Exit Ticket and KEY)
- one or more common rectangular prisms to display (such as a cereal or pasta box)
- copies of the Super Stationery Activity (M-6-2-1_Super Stationery and KEY)
- rulers (6-, 12-, or 18-inch length with metric markings on one side), one or two for each group of four students
- pattern blocks or tangram pieces (paper or plastic)
- centimeter grid paper (M-6-2-1_Centimeter Grid Paper)
- copies of Pattern Blocks Activity table (M-6-2-1_Pattern Blocks Activity)

- Use Lesson 1 Entrance Ticket (M-6-2-1_Lesson 1 Entrance Ticket and KEY) and the Think-Pair-Share strategy at the start of the lesson to review and assess the previous knowledge (perimeter and area) needed for this lesson. 
- Evaluate student comprehension throughout work time and presentations of the Designer Dimensions Activity (M-6-2-1_Designer Dimensions). 
- Lesson 1 Exit Ticket is a direct and individual assessment of student comprehension used at the end of the lesson (M-6-2-1_Lesson 1 Exit Ticket and KEY).

Scaffolding, Active Engagement, Modeling and Explicit Instruction 
W: Review the definitions of perimeter and area. Review finding area of triangles, quadrilaterals, and circles as needed. Compare one centimeter, one square centimeter and one cubic centimeter. Allow students to compare the areas of many rectangles with the same perimeter. 
H: Describe a situation where the perimeter is constant and maximizing or minimizing the area is important. You and the students will work together to develop a table to solve the problem. Generalizations will be created based on the table's results to help students solve similar situations. 
E: Students work in groups to solve a maximum or minimum problem with fixed perimeter and changing area. Each group of students presents its solution and reasons to the class. 
R: Student groups are encouraged to change their solutions while or after presenting to the class. You and students can ask questions to help presenters identify and correct inaccuracies in their reasoning. 
E: You evaluate students' grasp of the rectangle problem informally throughout work and presentations. Each student will complete an exit ticket to further evaluate their level of understanding. 
T: Use extension suggestions to personalize the lesson to students' needs. The small-group activity is appropriate for students who require further practice, while the expansion is for students who may be exceeding the standards. Additional exercises are suggested for classroom stations, as well as the use of technology. 
O: The lesson begins with a review of relevant vocabulary terms. Students learn that rectangles with a fixed perimeter can be used to generate a wide variety of areas. The use of an organized table is demonstrated as one approach for identifying all possible areas within a given fixed perimeter. Students investigate which rectangles have the smallest and largest areas, and are expected to develop a pattern or procedure for determining the least or greatest area for a rectangle with a fixed perimeter. 

To prepare, write the vocabulary terms area and perimeter on the board. Draw a segment, square, and cube with the same side lengths on the board. Label the length, width, and height as 1 centimeter each. As students enter the room, hand them an Entrance Ticket (M-6-2-1_Lesson 1 Entrance Ticket and KEY). Ask students to spend 3 to 5 minutes filling in their entrance ticket. Allow students 1 additional minute to share their responses with the student next to them. Randomly choose two or three students to share their perimeter responses. Compare and summarize correct thinking and responses, and correct any mistakes. Repeat the steps for area.

"Using our Entrance Tickets, we reviewed that the perimeter of a rectangle is calculated by multiplying the sum of the length and width by two (or 2L + 2W or adding up the measures of all the sides), and the area is calculated by multiplying length by width. I'll hand each of you a rectangle card (M-6-2-1_Rectangle Cards). First, check the dimensions on the front of your card. Then, turn over your card and write down the perimeter and area of your rectangle. Remember to identify your answers carefully. We'll share our answers in approximately a minute." Allow students 1-2 minutes to calculate and identify the perimeter and area. Place the rectangle table on the board or overhead (M-6-2-1_Rectangle Table and KEY), and then ask:

"Who can describe the difference between perimeter and area?" 
"What's the difference in the way we label them? Why?" 
Point out the 1 centimeter segment, 1 centimeter square, and 1 centimeter cube on the board. An optional activity can be used if desired: Distribute the Different Dimensions in Various Units of Measure Reference Sheet (M-6-2-1_Different Dimensions in Various Units). 

"The segment I have shown here has a length of 1 centimeter. When we talk about length or distance, whether it's a single straight segment or the distance around a figure, we use standard length labels like centimeter, inch, meter, and foot. When we measure the two-dimensional flat space inside a closed figure (area), we actually measuring how many unit squares like these (point to square on board) fit inside the figure. As a result, we label the area or surface area of any object using labels like square centimeters, square inches, square meters, or square feet (or ft² form). In both Lessons 1 and 2, we will utilize these types of measurements and labels to solve perimeter, circumference, and area problems. Now, observe that I have a centimeter cube, or cubic centimeter, on the board. When we measure the amount of space within a three-dimensional figure (as shown in the figure below), we are actually estimating how many unit cubes fit inside of it.

Ask, "As a result, we will use labels like cubic centimeters, cubic inches, cubic meters, and cubic feet (or ft³ form). In Lesson 3, we'll use labels like this to compute the volumes of three-dimensional solids." 

"Which of these labels should you have on your perimeter solutions?" (standard length labels, like centimeters or inches) 
"Which of these labels should you have on your area solutions?" (standard square labels like square centimeters or square inches) 
"I'd want to compare our answers. To do this, we'll arrange the perimeters and areas in a table. What do you observe about how I set up my table?" (The widths are ordered from least to greatest, beginning with 1.)
"This method is often used in math to help us uncover patterns and compare numbers in a systematic way. In addition to helping us in recognizing patterns, the table and order help to arrange the values and prevent us from missing combinations we should consider. I will call you up in the order listed in the table. More than one of you will hold each card. When I call on you, you will quickly compare your answers and reach an agreement before writing them into the table."

“Let's start with students who have a 1 cm × 10 cm rectangle. Please come to the front if you have this card." For the 1 cm × 10 cm rectangle, students will compare and agree on their perimeter and area answers. Instruct one student to fill in the perimeter and area values into the corresponding columns of the table, including labels. Repeat the process until all of the rectangle sizes have been added to the table.

Answer Key - Rectangle Table

"Now that we have completed the table, look closely at it. Take 2 minutes to write down as many observations as you can. Be ready to share one with the class." Ask each student to share one observation. Responses may vary, but should include:

To calculate perimeters, add length and width and double the result: (l + w) × 2.
To calculate the perimeter of a figure, use the formula l + w + l + w, since you go around the figure adding the sides, or 
2• l + 2• w, where opposite sides are equal and 2 of each length and width are included in the perimeter sum.

The perimeter divided by 2 was always equal to the sum of 1 length + 1 width.
The areas were identified using l × w.
The perimeters were all similar. (22 cm)
The areas differed, but each area was in the least twice.
The areas progressed from least to the greatest, and then gradually back to least.
As the width increased by 1, the length decreased by 1.
Each width and length pair was stated twice, but in the opposite order.
At the center of the list, where the width and length values reversed from 5 cm-6 cm to 6 cm-5 cm, the area values peaked and then began to fall. The maximum areas for a given perimeter occur when a figure becomes regular (square) or as close to regular as possible (as in this case, a rectangle with length and width very close but not quite square).
If widths or lengths are arranged from least to largest, the minimum areas appear at the top and bottom of the table.
Widths were listed from 1 to 10, and lengths from 10 to 1. 
Emphasize how easily the maximum and minimum values can be found when the data is organized in an ordered list or table. Also, make sure students understand that the maximum occurs halfway through the list, where the length and width dimensions reverse columns (i.e., 5-6 followed by 6-5), and that the second half of the table only contains the same length and width combinations in reverse order. This always happens when the figure is congruent on all sides (regular, square for quadrilaterals) or as close as possible regular (congruent on all sides).

"Identifying patterns is an important skill in a variety of real-world problems. There are two types of problems: maxima and minima problems. Here's an example of a problem with our data. Let's pretend I was making a bookmark. In this example, I already know I need a 22 cm perimeter to have enough space to make the specific design I intended for the border. I want to make a bookmark that takes the least amount of paper (area). From the dimensions we identified in our table, which bookmark size should I use? Why do you think so?"

Allow students to explain their answers and reasoning. Some may recommend 1 cm x 10 cm or 10 cm × 1 cm because they have the smallest areas (10 square centimeters). Students may ask whether these two rectangles are the same, or you may ask them. For this application, although they are two separate options, they are practically the same. One measures 10 cm vertically and the other 10 cm horizontally, which is unlikely to matter in a bookmark design. However, in some applications, particularly those involving printing of pictures, the orientation of the width and length orientation may need to be considered.

"If I were designing an address label rather than a bookmark, I might still choose 22 cm for the border design. However, this time I'd like to choose the rectangle with the largest space to write the address information. Which of our rectangle size should I choose in this case? Be ready to explain your reasoning." 

Again, choose a few students to share their responses. This time, the most likely choice is the 5 cm × 6 cm and 6 cm × 5 cm rectangles, which provide the highest amount of space (30 square centimeters). Tell students, "For an address label, it will make a difference if we choose to have 6 cm vertically or horizontally." Both will work, but students should evaluate which would be most useful. Students may argue that a height of 4 cm × width of 7 cm is a more useful size, as it offers a slightly smaller area but a longer horizontal space to write names or labels. The way in which a figure will be used may need to be incorporated into the decision. 

If your class has extra time for an enrichment exercise, assign students to work in groups of four for this activity. Give each group three number cubes (or a spinner marked with the numbers 1-6). Each group need a Designer Dimensions Activity sheet (M-6-2-1_Designer Dimensions), a ruler, and a sheet of 18 x 24 inch drawing paper. All groups will use a 1 inch width for their first rectangle. To calculate the length of the group's rectangle, have one member roll all three number cubes simultaneously (or spin the spinner three times). The sum of the three cubes will yield the first rectangle's length in inches. Instruct student groups to create a table that lists all possible rectangles with the same perimeter. You may want to remind students of the previously mentioned pattern, which states that the perimeter divided by 2 equals one length plus one width. In addition, students must determine the area of each rectangle.

"As you finish your table, remember to order the widths from least to greatest. Each rectangle below the first should have the same total length and width as the first rectangle. Once you've identified all of the rectangles, fill in their perimeters and areas. The back of the page includes questions to answer. You will need a ruler and drawing paper to display some of your answers. Be prepared to present your work in approximately 15 minutes. Do you have any questions? At this point, send one group member to the front to collect your group materials."

While students are working, move around the room to observe, ask questions, and provide guidance. Groups should use chart paper to create a display of their responses to share with the class. It would be great if they included their table on one side of the chart paper. After around 15 minutes, start calling on groups to present their results. Allow each group 2 to 5 minutes, including class discussion. During presentations, continue to ask questions to help students understand their work and reasoning. Encourage students to correct errors, provide detailed explanations, and potentially provide alternative solutions or methods of solving.

Summarize effective strategies and discuss the main patterns of reasoning used to determine maximum and minimum area values when the perimeter is fixed. 

At the end of the lesson, have each student complete a Lesson 1 Exit Ticket to assess their understanding (M-6-2-1_Lesson 1 Exit Ticket and KEY).

Extension: 

Routine: Discuss how important it is to comprehend and use the appropriate vocabulary terms while communicating mathematical ideas. During this lesson, students should record the following terms in their vocabulary journals: area, fixed perimeter, maximum, and minimum. Keep a supply of vocabulary journal pages on hand so that students can add them as needed (M-6-2-1_Vocabulary Journal). Bring up examples of area and perimeter from throughout the school year.
Ask students to bring examples of maxima and minima that they see outside the classroom and discuss their use and meaning in each particular context. As they are used throughout the year, distinguish between identifying lengths in standard units and areas in square units. Always require students to use appropriate labeling in their verbal and written responses.

Small Group: Super Stationery Activity. Students determine the possible dimensions of decorative note cards. Help students by reviewing rectangle properties, such as:
A = l × w
P = (l + w) × 2
\(1 \over 2\)  of the perimeter = l + w
The minimum size for the rectangle is the smallest number for l × w.
The maximum size for the rectangle is the largest value for l × w.
Students fill out the dimension table (M-6-2-1_Super Stationery and KEY). The table's data is used to answer questions about the maximum and minimum note card sizes, as well as to determine the best size to use for the note cards' specific purpose.

Station/Technology: Exploring Changing Area Station. Allow students to discover how the perimeters and areas of various rectangles are related. By selecting a perimeter for a rectangle or irregular figure to explore the changing areas at http://www.shodor.org/interactivate/activities/AreaExplorer/ 

Station: Pattern Block Activity. Supplies include centimeter grid paper (M-6-2-1_Centimeter Grid Paper), copies of the Pattern Blocks Activity table (M-6-2-1_Pattern Blocks Activity), rulers, and either pattern blocks or tangram pieces. 
Allow students time to create rectangles with pattern blocks or tangram pieces. Students should trace the figure on centimeter grid paper or whiteboards. Instruct students to use the Pattern Blocks Activity table to measure length, width, perimeter, and area. Challenge students to create additional rectangles with the same perimeter and measure their areas. 

Expansion: Toothpick Perimeter Activity. Allow students to work with a partner or small group on this activity. Give students a piece of chart paper and 20 toothpicks. The chart paper should be divided into four columns. Both the front and back can be used. Students should label the paper as described below for side 1. Side 2 is labeled similarly, except it includes a section for six sides. 
Begin by having students use four toothpicks to outline as many different quadrilaterals as possible. For four sides, neatly trace each one below the other in the "Figures" column. Next, calculate the perimeter and area of each using a toothpick's length as a unit. Finally, the steps are repeated for five- and six-sided figures. 

Expansion: Toothpick Perimeter Activity 

Front (use the entire front of the paper)