• Informally evaluate student understanding through your observation during the Reviewing Areas of Polygons and Investigating Compound Figures practice.
• Evaluate students’ understanding using the Perplexing Polygons activity and presentations.
• Assess student understanding using the Partner Quiz at the end of the lesson.

As students enter the room, hand out a Lesson 2 Entrance Ticket (M-7-6-2_Lesson 2 Entrance Ticket and KEY.docx). Make sure students have access to a PSSA or other formula sheet. Ask students to spend 4–8 minutes working individually on filling in their Entrance Ticket. Reassure students that you are simply trying to find out what they already know, and that they may not know how to do everything. Use student responses on the entrance ticket to guide the pace and depth at which you teach or review perimeter and area concepts later in this lesson. It may be helpful to have students score the perimeter and area calculations, so you can make a quick evaluation by scanning through the Entrance Tickets while students have small spans of work time during the early part of the lesson.

An optional refresher on how to graph using the coordinate plane is available (M-7-6-2_Optional Refresher-Graphing on Coordinate Grid.docx).

Introducing Perimeter and Area of Graphed Figures Activity

“Let’s graph some more points together.” Hand out the worksheet on coordinate planes (M-7-6-2_Blank Coordinate Plane.docx). Say, “Start by graphing (1, 1), (6, 1), (6, 4), and (1, 4). Connect the points in this order as you graph them.”

Give students time to graph. Then, ask: [IS.3 - Struggling Learners]

• “What figure was created?” (rectangle)
• “How many units is it around the outside of our rectangle?” (16 cm)
• “You found the perimeter of our rectangle. Your label should be centimeters since perimeter is a distance. How did you determine your answers?” (counted the grids around the outside; added the length of each side; added two lengths plus two widths; or added the length plus width times 2)
• “I also want you to calculate the area, or space inside the rectangle.” (15 square centimeters)
• “How did you find your answer?” (counted the grid squares inside; multiplied length times width; or multiplied the number of rows by the number of columns)

“Some of you used the grids to help you while others used another strategy such as a formula. Let’s practice a few more examples. After graphing these, use any strategy you want to find the areas and perimeters. Use the grids on your paper to help you. We will work with the formulas for these figures in the next part of our lesson.”

Provide students with 3–5 additional sets of ordered pairs to form triangles and quadrilaterals. Give them 5–7 minutes to graph them and to find the perimeters and areas. Be sure to help students gain strategies for working with the angled sides. For example, with a parallelogram, the area can be found by “cutting and sliding” a triangular end piece to match with the angle on the opposite side in order to visualize the area as a rectangle. Triangles can be pictured as the larger rectangle or parallelogram containing them, but then get divided in half to find area. Have students check their answers with a partner after 5 minutes. Check for accuracy by walking around to monitor and by asking students to come to the board or overhead to show their responses at the end of the work time.

Reviewing Areas of Polygons Activity

At this point in the lesson, determine how much time needs to be spent learning or reviewing the area formulas for triangles and quadrilaterals. Spend a short time reviewing even if students already have a solid understanding. Spend more time explaining the formulas and why they work if students had difficulty with the Entrance Ticket questions.

“Using your Entrance Tickets, I was able to see that this class is already proficient at finding (perimeters of____, and areas of ____). [IS.4 - All Students] What we need to (work on/or review) is how to find (perimeters of _____, and areas of _____).”

“In any polygon, or figure with straight sides, we just go around the outside and add the measures of each side. What if a figure had another marking, such as an altitude or height marking inside? Do we include that in the perimeter too?” (Draw an example such as a triangle with 3 side measures and an altitude line with a measure inside.) (No, just the 3 sides are included.)

“What about for the area? What measurements are needed?” (It varies depending on the figure, but generally just the base and perpendicular height, not including the angled sides.)

Hand out the Areas Review sheet (M-7-6-2_Areas Review and KEY.docx). This sheet will be completed and used as a formula page throughout the remainder of the unit. Another option for use later in class is the Formula Sheet for Areas of Polygons (M-7-6-2_Formula Sheet for Areas of Polygons.docx).

Either work through the examples as a class, assign students pairs to work on one polygon and present it, or give students time with a partner to complete it. Be sure that all students have the opportunity to check the accuracy of their responses and correct errors. If, based on student Entrance Tickets, students are having difficulty with understanding the figures, names of their parts, or computing areas with the formulas, spend extra time here teaching these concepts. It is expected that for many students this material will be review. This would also be an excellent opportunity to go over the questions found on the Entrance ticket with the class, or have students who answered the questions correctly present their solutions. A solid understanding of these figures and formulas is necessary to proceed with the next step in the lesson and for Lesson 3.

Investigating Compound Figures Activity

Tell students, “In our last activity, we practiced calculating the perimeter and area of triangles and quadrilaterals. In this activity, I am going to challenge you by combining some of the figures you already know into compound figures you may not know. You will need to use your problem solving skills to make the calculations. Try to find the perimeter and area of this figure.” Give students 1–3 minutes, and ask for strategies and solutions.

                                                      

There will likely be a variety of strategies. Be sure that these strategies get discussed by a student or yourself:

Perimeter option 1: Use subtraction of measurements provided vertically to get the missing vertical length (12 – 3 = 9), and subtraction of horizontal lengths to get the missing horizontal value (15 – 9 = 6). Then add all sides.

Perimeter option 2: Note that all vertical segments on the right side of figure could be slid together to form a segment the same length as the left vertical side (12), so just like a rectangle you will have 12 + 12 for vertical measures. Similarly, the bottom measure (15) will be equivalent to the 9 at the top and the unknown horizontal segment, so you have the equivalent of 2 horizontal lengths of 15. Perimeter can be found by 2(12 + 15) or 2(12) + 2(15).

Area option 1: Area can be found by dividing the figure into two rectangles with a line inside. Either a vertical line can be used to get a 9 x 12 and a 3 x 6 rectangle, or a horizontal line can be used to get a 3 x 15 and a 9 x 9 rectangle. In either case find the area of each rectangle and add the two areas.

Area option 2: Think of this as a large rectangle with a piece missing from (or cut out of) one corner. Outline the larger rectangle. Calculate the large outlined rectangle (12 x 15), and the rectangular piece that is “cut out” (6 x 9). Subtract the smaller cut-out from the large rectangle to remove it from the area calculation.

Ask students to practice another example. This one is more difficult.

Answers:

   P = (5)(6) = 30 cm

 A = (3.5)(9) ÷ 2 + (9 + 6) ÷ 2 • (5.5)= 57 cm²

There may be a variety of correct strategies suggested. Be sure that dividing the pentagon into a trapezoid and a triangle is discussed. This strategy involves calculating the area of the trapezoid and triangle separately and then adding them together. The perimeter is found by multiplying 5 times 6 centimeters since the tic marks indicate congruency of all of the sides.

Continue with enough examples that students feel confident in finding strategies to work with the compound figures. Students will be working with a partner to practice additional problems such as these in the Perplexing Polygons activity, so the number of additional examples may be limited.

The figures below are additional shapes you may choose to use either before or after the trapezoid example. Insert dimensions to suit your students’ needs. Use whole numbers for students who are having difficulty with the calculations or strategies. Select decimal or fractional values for students who need a challenge.

                                          (Note: This example is two polygons.)

Note: For optional group or individual practice with compound figures, use the Compound Figures worksheet (M-7-6-2_Compund Figures and KEY.docx).

Partner Practice: Perplexing Polygons Activity

Place students in pairs if they have been working individually. Provide each student with the Perplexing Polygons sheet (M-7-6-2_Perplexing Polygons and KEY.docx). Encourage partners to discuss how to solve and label each problem and why. They may use the strategies already presented in class for the examples, or may find their own strategies. Remind the class that each pair will be selected at random to present one problem to the class. Allow approximately 15–20 minutes of work time before presentations. Provide assistance as needed during work time. As the work time is coming to a close remind students that their work should be shown and answers should be properly labeled.

At the end of work time, use a strategy such as drawing numbers or rolling a number to assign a problem to each pair to present. During presentations, encourage suggestions by audience members to assist presenters who are having difficulty explaining their work or who have made errors. Allow students time to adjust or add to their solutions after the presentations.

After the presentations, summarize the perimeter and area strategies studied in this lesson for compound figures, and correct any lingering misconceptions. Answer any student questions that arise.

Partner Quiz

With students still in pairs, give the Partner Quiz (M-7-6-2_Partner Quiz and KEY.docx). Allow students approximately 8–15 minutes to complete the quiz. Use the results to determine appropriate extension activities for each student.

Extension: [IS.5 - Struggling Learners]

• 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: coordinate graph, kite, ordered pair, parallelogram, quadrant, quadrilateral, rectangle, rhombus, square, trapezoid, triangle, x axis, y axis. Keep a supply of Vocabulary Journal pages on hand so students can add pages as needed. Bring up instances of perimeter and area examples as seen throughout the school year. Use polygon examples in studying ratio and proportional reasoning, as well as similarity and scale factor units. Ask students to bring in pictures or objects that are compound figures which require strategies from this unit. Display these examples for students to work on. Discuss the use and meaning of such examples in each particular context. Continue to distinguish the difference between labeling perimeters with standard units and areas with square units. Require students to use appropriate labeling in both verbal and written responses.
• Small Group: Revisiting the Compound Polygons:Use this activity for the entire class or for students who found the perimeters and areas of compound figures troublesome during the lesson or on the Partner Quiz. Provide students with centimeter grid paper. Prepare a variety of triangles and quadrilateral shapes ranging in size from 3 to 10 centimeters in length and width. Cut the figures out of construction paper or posterboard material.
  • Ask students to trace two of the paper cut-outs, lining up edges without gaps or overlaps, to form a compound figure as they trace. It will be beneficial for students if they also line the figures up along grid lines on their paper.
  • Have students repeat this with several different shapes until they have traced three or four compound figures.
  • Instruct students to calculate the perimeter and area of each figure. Have students measure in centimeters. They can use the grid lines from the centimeter grid paper to assist them in finding the lengths of the sides or use a centimeter ruler.
  • Encourage students to use the area formulas and strategies from the lesson to find the areas of their figures. Once they calculate the areas, they should be encouraged to check their work by counting the number of unit squares actually shown inside their compound figures on the grid paper.
  • Optional: Have students record a small sketch of the compound figure, its dimensions, and their perimeter and area calculations onto the Small Group Record sheet (M-7-6-2_Small Group Record.docx) to turn in after the activity.
  • Remind students to label their perimeter answers with centimeter labels and the areas with square centimeter labels. If time permits, ask students to carefully trace a new compound figure that uses three of the paper cut-outs, with one of the figures on the inside of the others to take some of the area away. Ask them to calculate the perimeter and area of this more complex figure.
  • Fabulous Formations: Students who have shown proficiency at calculating perimeters and areas of triangles, quadrilaterals, and compound figures can solve problems that include compound figures with circles, semi-circles, and/or more than two polygons.

Instruct students to use the approximation 3.14 for pi, and round answers to the nearest hundredth. Calculators may be used for this activity.

Provide each student with a copy of Fabulous Formations (M-7-6-2_Fabulous Formations and KEY.docx). Allow students to work with a partner or in a small group.

If time permits, ask students working on this extension activity to present one problem to the class. Instruct them to guide the class in solving it by asking for input from the class, and leading them through parts that the class is not sure about.

• Station: Create a Character: Provide students with centimeter or ¼ inch grid paper, markers, and Create a Character record sheet (M-7-6-2_Create a Character.docx).

Instruct students to draw, and label, the x- and y-axis at the center of the page. Ask students to design a cartoon style character using polygons and compound figures on their grid paper. They should take care to use the intersection points on the grid paper for vertex points of the polygons.

Next, students mark a starting point and make a list of coordinate pairs needed to create their character on their record sheet. The ordered pairs must be listed in the order that they are plotted, and when connected in order, they will form the characters’ outline. Students should indicate when a new section or detail of the figure should not be connected to the previous point by using a phrase such as “stop here” or “lift your pencil here.” If time permits, students exchange lists of coordinate pairs to create another student’s character on a new piece of grid paper.

• Technology: Locate It! If computers are available for student use, students can gain extra practice with coordinate graphing by using activities on mathematics education Web sites. Instruct students to go to one of the following sites or one you select:
  • Game in which student names the coordinate to locate aliens

http://www.mathplayground.com/locate_aliens.html

• Game in which student selects a coordinate pair from a list of four choices to catch a mole, with multiple difficulty levels

http://funbasedlearning.com/algebra/graphing/points3/