• Use Lesson 1 Entrance Ticket (M-6-2-1_Lesson 1 Entrance Ticket and KEY.docx) combined with the Think-Pair-Share strategy at the beginning of the lesson to review and assess the prior knowledge (perimeter and area) required for this lesson.
• Evaluate student understanding during work time and presentations of the Designer Dimensions Activity (M-6-2-1_Designer Dimensions.docx).
• Lesson 1 Exit Ticket is a direct and individual assessment of student understanding to be used at the end of the lesson (M-6-2-1_Lesson 1 Exit Ticket and KEY.docx).

To prepare, place the vocabulary terms area and perimeter on the board. Also draw a segment, square, and cube of the same side length on the board. Label length, width, and height as 1 centimeter each. As students enter the room, hand each student an Entrance Ticket (M-6-2-1_Lesson 1 Entrance Ticket and KEY.docx). [IS.4 - Struggling Learners] Ask students to spend 3 to 5 minutes filling in their entrance ticket. Give students 1 additional minute to share their responses with the student next to them. Randomly select two or three students to share their perimeter responses. Compare and summarize correct thinking and responses and correct any misconceptions. Repeat these steps for area.

“Using our Entrance Tickets, we have reviewed that perimeter of a rectangle is found by multiplying the sum of the length and width by two (or 2L + 2W or add up the measures of all the sides) and area is found by multiplying length by width. I am going to hand each of you a rectangle card (M-6-2-1_Rectangle Cards.docx). First look at the dimensions on the front of your card. Then, turn over your card and write both the perimeter and area of your rectangle. Remember to label your answers carefully. We will share our answers in about one minute.” Give students 1–2 minutes to calculate and label the perimeter and area. [IS.5 - Struggling Learners] Place the rectangle table on the board or overhead (M-6-2-1_Rectangle Table and KEY.docx), and then ask,

• “Who can describe the difference between perimeter and area?”
• “What is the difference in how 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: Hand out the Different Dimensions in Various Units of Measure reference sheet (M-6-2-1_Different Dimensions in Various Units.docx).

“The segment I have shown here represents a length of 1 centimeter. Any time we discuss length or distance, whether it is a single straight segment or the distance around a figure, we use standard length labels such as centimeter, inch, meter, and foot. If we are measuring the two-dimensional flat space inside of a closed figure (area), we are actually measuring how many unit squares like these (point to square on board) fit inside the figure. Therefore, we label the area or surface area of any object using labels such as square centimeters, square inches, square meters, or square feet (or ft² form). We will use both of these types of measurements and labels in Lesson 1 and in Lesson 2 for perimeter, circumference, and area problems. Now, notice I also have a centimeter cube, or cubic centimeter on the board. When we measure the amount of space inside of a three-dimensional figure (see figure below), we are actually calculating how many unit cubes fit inside of it. [IS.6 - Struggling Learners]

Therefore, we will use labels such as cubic centimeters, cubic inches, cubic meter, and cubic feet (or ft³ form). We will use labels like this in Lesson 3 when we calculate volumes of three-dimensional solids.” Ask,

• “Which of these labels should you have on your perimeter solutions?” (standard length labels such as centimeter or inch)
• “Which of these labels should you have on your area solutions?” (standard square labels such as square centimeters or square inches)
• “I would like to compare our answers. To do this, we will organize the perimeters and areas in a table. What do you notice about how I set up my table?” (The widths are listed from least to greatest beginning with 1.)

“This strategy is used often in math to help us find patterns or compare numbers in a systematic way. In addition to helping us see the patterns, the table and order help organize the values and prevent us from missing combinations we should consider. I will call you up in the order listed here in the table. More than one of you will have each card. When I call on you, you will quickly compare your answers and come to agreement before writing them into the table.”

“Let’s begin with students who have the 1 cm × 10 cm rectangle. Please come to the front if this is your card.” Students called up for the 1 cm × 10 cm rectangle will compare their perimeter and area answers and must agree on their pair of answers. Instruct one student to fill the perimeter and area values into the appropriate columns in the table, including labels. Repeat this process until the different rectangle sizes have all been added to the table.

Answer Key—Rectangle Table

“Now that we have the table completed, look closely at it. Take 2 minutes to write as many observations as you can. Be prepared to share one with the class.” Ask each student to state one observation. [IS.7 - Struggling Learners] Responses will vary, but should include:

• The perimeters could all be found by adding length and width, and doubling the sum: (l + w) × 2.
• The perimeter can be found by using l + w + l + w, since you go around the figure adding the sides, or

2• l + 2• w, since 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 all found using l × w.
• The perimeters were all the same. (22 cm)
• The areas were different, but each area was in the list twice.
• The areas went from least to greatest and gradually back to least.
• As the width increased by 1, the length decreased by 1.
• Each width and length pair was listed twice, but in 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 reached the maximum and then began to decrease. Maximum areas for a given perimeter happen 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).
• The minimum areas are at the top and bottom of the table if widths or lengths are ordered least to greatest.
• Widths were listed in order from 1 to 10, while lengths were listed 10 to 1.

Emphasize how the maximum and minimum values are found easily when the data is placed in an ordered list or table. Also be sure students can see that the maximum occurs halfway through the list, at the point where the length and width dimensions reverse columns (i.e., 5–6 followed by 6–5), and the same combinations of length and width in reverse order are all that remain in the second half of the table. This always occurs at the point where the figure is congruent on all sides (regular, square for quadrilaterals) or as close as possible to regular (congruent on all sides).

“Identifying patterns is an important skill in many types of real-life problems. Two of these types are maxima and minima problems. [IS.8 - Struggling Learners] Here is an example of such a problem using our data. Let’s say I was making a bookmark. In this case I already know I need a 22 cm perimeter to have enough room to make the specific design I planned for around the border. I want to create the bookmark that uses the least amount of paper (area). From the dimensions we identified in our table, which size of bookmark should I choose? Why do you think so?” [IS.9 - Struggling Learners]

Let students share their answers and reasoning. Some may suggest the 1 cm × 10 cm, and others the 10 cm × 1 cm because they have the smallest areas (10 square centimeters each). Students may ask if these two rectangles are the same, or you may want to ask them. For this application, although they are two separate options, they are essentially the same. One is 10 cm vertically and the other is 10 cm horizontally, which is not likely to matter for a bookmark design. However, for some applications, especially those involving printing of pictures, the orientation of the width and length may need to be a consideration.

“If I were designing an address label instead of a bookmark, I might still want to use 22 cm for the border design. However, this time I want to choose the rectangle that provides the most space to write the address information. Which of our rectangle dimensions should I choose in this case? Be prepared to explain your reasoning.”

Again, select several students to share their responses. [IS.10 - Struggling Learners] The most likely choice this time is the 5 cm × 6 cm and the 6 cm × 5 cm rectangles because they provide the maximum amount of space (area of 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 ask students to consider which may be most useful. Students may also argue that a height of 4 cm × width of 7 cm would be a more useful size because it has only a slightly smaller area, but a longer horizontal space to write names or other 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 activity have students work in groups of four for this activity. Give each group three number cubes (or have a spinner available, marked with numbers 1–6). Each group also needs a Designer Dimensions Activity sheet (M-6-2-1_Designer Dimensions.docx), a ruler, and a sheet of 18 × 24-inch drawing paper. All groups will use a width of 1 inch for their first rectangle. Have one group member roll the three number cubes all at once (or spin the spinner three times) to determine the length of the group’s rectangle. The sum of the three cubes will be the length, in inches, of their first rectangle. Instruct student groups to complete a table identifying all possible rectangles with the same perimeter. You may want to remind students of the pattern noted earlier in which the perimeter divided by 2 equals one length plus one width. Students also need to calculate the area for each rectangle. [IS.11 - Struggling Learners]

“As you complete your table, remember to arrange your widths from least to greatest. Each rectangle below the first should have the same sum for length and width as the first rectangle. Once you have all of the rectangles listed, fill in the perimeter and area for each one. There are questions to answer on the back of the page. You need a ruler and drawing paper to display some parts of your answers. Be prepared to share your work in about 15 minutes. [IS.12 - Struggling Learners] Are there any questions? At this point, send one group member to the front to pick up your group materials.”

While students work, move about the room observing, asking questions, and providing guidance. Groups should use the chart paper to make a display of their answers to share with the class. It would be helpful for them to include their table on one side of the chart paper. After approximately 15 minutes, begin calling on groups to share their results. Give each group 2 to 5 minutes including class discussion. During presentations, continue to ask questions in which students clarify their work and reasoning. Encourage students to correct errors, include detailed explanations, and possibly offer additional solutions or methods of solving.

Summarize good strategies and review the main patterns and thinking used to find maximum and minimum values for area when the perimeter is fixed.

At the end of the lesson, have each student complete a Lesson 1 Exit Ticket to evaluate level of understanding (M-6-2-1_Lesson 1 Exit Ticket and KEY.docx). [IS.13 - Struggling Learners]

Extension: [IS.14 - Struggling Learners]

• Routine: Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. During this lesson the following terms should be entered into students’ vocabulary journals: area, fixed perimeter, maximum, and minimum. Keep a supply of vocabulary journal pages on hand so students can add pages as needed (M-6-2-1_Vocabulary Journal.docx). Bring up instances of area and perimeter as seen throughout the school year. Ask students to bring up maxima and minima examples that they see outside of class and discuss the use and meaning in each particular context. As they are used throughout the year, distinguish the difference in labeling lengths with standard units and areas with square units. Always require students to use appropriate labeling in both verbal and written responses.
• Small Group: Super Stationery Activity.Students find the possible dimensions for decorative note cards. Assist students by reviewing rectangle properties, such as:
  • A = l × w
  • P = (l + w) × 2
  •  of the perimeter = l + w
  • Minimum size for the rectangle is the smallest value for l × w
  • Maximum size for the rectangle is the largest value for l × w

Students complete the dimensions table (M-6-2-1_Super Stationery and KEY.docx). The data in the table is used to answer questions regarding the maximum and minimum note card size and determining a good size to select for the specific purpose of the note cards.

• Station/Technology: Exploring Changing Area Station. Allow students to explore how the perimeters and areas of a variety of rectangles are related. By selecting a perimeter for a rectangle or irregular figure, the changing areas can be explored at http://www.shodor.org/interactivate/activities/AreaExplorer/ .
• Station: Pattern Block Activity. Provide centimeter grid paper (M-6-2-1_Centimeter Grid Paper.doc), copies of the Pattern Blocks Activity table (M-6-2-1_Pattern Blocks Activity.docx), rulers, and either pattern blocks or tangram pieces.

Allow students time to form rectangles using pattern blocks or tangram pieces. Have students use centimeter grid paper or white boards to trace the figure. Instruct students to use the Pattern Blocks Activity table to record length, width, perimeter, and area. Challenge students to form additional rectangles with the same perimeter and record their areas.

• Expansion: Toothpick Perimeters Activity. Allow students to work with a partner or small group for this activity. Provide students with a piece of chart paper and 20 toothpicks. The chart paper should be separated into four columns. Both the front and back can be used. Have students label the paper as shown below for side 1. Side 2 is labeled similarly but with a section for six sides.

Ask students to begin by using four toothpicks to make the outline of as many different quadrilaterals as possible. Each should be neatly traced one below the other in the “Figures” column for four sides. Next, the perimeter and area of each should be calculated using the length of a toothpick as one unit. Last, these steps are repeated for five- and six-sided figures.

Expansion: Toothpick Perimeters Activity

Front (use entire front of paper)